Gauge Invariance and. Frame Independence in Cosmology

Transcription

1 Guge Invrince nd Frme Independence in Cosmology

2 Cover design by Mrten vn Gent ISBN: Printed by: Proefschriftmken.nl Uitgeverij BOXPress Published by: Uitgeverij BOXPress, s-hertogenbosch

3 Guge Invrince nd Frme Independence in Cosmology Ijkinvrintie en Frme Onfhnkelijkheid in Kosmologie (met een smenvtting in het Nederlnds) Proefschrift ter verkrijging vn de grd vn doctor n de Universiteit Utrecht op gezg vn de rector mgnificus, prof. dr. G. J. vn der Zwn, ingevolge het besluit vn het college voor promoties in het openbr te verdedigen op mndg 30 september 2013 des middgs te uur door Jn Gerrd Weenink geboren op 30 december 1984 te Goud, Nederlnd

4 Promotor : Co-promotor: Prof. dr. E. Lenen Dr. T. Prokopec The reserch in this thesis ws supported by the Dutch Foundtion for Fundmenteel Onderzoek der Mterie (FOM) under the progrm Theoreticl prticle physics in the er of the LHC, progrm number FP 104.

5 Contents Publictions vii 1 Introduction A lucky cosmologist Pln for this thesis The guge problem of Generl Reltivity Generl Covrince Perturbtions nd guge dependence Perturbtions in n expnding universe Guge invrint cosmologicl perturbtions An nlogy: electrodynmics Infltionry perturbtions The infltionry universe The importnce of cosmologicl perturbtions The primordil power spectrum for sclr perturbtions The primordil power spectrum for tensor perturbtions Exmple: chotic infltion in polynomil potentil Frmes in cosmology The Einstein frme versus the Jordn frme Field redefinitions nd physicl equivlence Cse study: Higgs infltion Frme dependence of perturbtions Liner guge invrince nd frme independence Introduction v

6 Contents 5.2 Cnonicl ction in the Jordn frme Free ction for cosmologicl perturbtions Higgs infltion Summry A Digonlizing the qudrtic ction Non-liner guge invrince nd frme independence : the sclr sector Introduction Action nd perturbtions Non-liner guge dependence nd guge invrint perturbtions The guge invrint ction for cosmologicl perturbtions Uniqueness of guge invrint ction Frme independent sclr perturbtions Summry Non-liner guge invrince nd frme independence : the grviton sector Introduction Action nd perturbtions The cubic guge invrint ction in the Jordn frme Uniqueness of the sclr-grviton ction Frme independent cosmologicl perturbtions Nturlness in Higgs infltion Summry A Decoupling the constrint fields in the ction B Boundry terms Discussion nd Outlook Summry Outlook Bibliogrphy 157 Nederlndse smenvtting 169 Acknowledgements 183 Curriculum Vite 185 vi

7 Publictions The min chpters of this thesis re bsed on the following works: chpter 5: [1] J. Weenink nd T. Prokopec, Guge invrint cosmologicl perturbtions for the nonminimlly coupled inflton field, Phys. Rev. D82 (2010) chpter 6: [2] T. Prokopec nd J. Weenink, Uniqueness of the guge invrint ction for cosmologicl perturbtions, JCAP 1212 (2012) 031. chpter 7: [3] T. Prokopec nd J. Weenink, Frme independent cosmologicl perturbtions, rxiv: (2013) To pper in JCAP 13XX. Other publictions by the uthor (not included in this thesis): [4], J. Weenink nd T. Prokopec, On decoherence of cosmologicl perturbtions nd stochstic infltion, rxiv: (2011). [5], T. Prokopec, M. G. Schmidt nd J. Weenink, The Gussin entropy of fermionic systems, Annls Phys. 327 (2012) [6], T. Prokopec, M. G. Schmidt nd J. Weenink, Exct solution of the Dirc eqution with CP violtion, Phys. Rev. D87 (2013) vii

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9 Chpter 1 Introduction 1.1 A lucky cosmologist Let us begin this thesis with sttement: the current genertion of cosmologists is very fortunte. A cosmologist living t the beginning of the twentieth century did not yet know tht spce-time itself is dynmicl nd cn be described by the Einstein equtions of Generl Reltivity. A cosmologist in the roring twenties would hve been intrigued by certin solutions of these equtions which describe universe tht is expnding. Still, he or she my hve seen this s just n interesting theoreticl ide, only to be completely bffled by the discovery of the redshift-distnce reltion of distnt glxies in 1929, which gve empiricl support to the expnding universe. A pre-world-wr-2- cosmologist would try to mke up his mind between the ide of the Big Bng theory the universe originting from some infinitely hot nd dense stte, or the ide of Stedy Stte universe, universe tht hs no beginning nd no end. Perhps he or she would len more towrds the ltter ide, since Big Bng universe introduces countless difficult scientific nd philosophicl questions. But cosmologist fter the Wr would strt to be more convinced by the Big Bng model, since it cn explin the bundnces of light elements in our universe through process now known s Big Bng Nucleosynthesis (BBN). A hippie cosmologist would finlly find his pece nd hppiness in the Big Bng model by the discovery of the fterglow of the Big Bng in 1965, the fmous Cosmic Microwve Bckground rdition (CMB). But crisis would strike gin for cosmologist in the 1970s, when severl new problems strted to pper: the question of the mtter-ntimtter symmetry of the universe, the mystery of drk mtter which mkes up most of the mtter in our universe, nd problems concerning the initil homogeneity nd fltness of the universe. A cosmologist in the lst decde of the Cold Wr

10 2 CHAPTER 1. INTRODUCTION would hve been very plesed to see tht, in combined effort, Americn nd Russin scientists not only solved the ltter problems by the introduction of new infltionry phse in the universe, but could ctully mke predictions for certin structure in the CMB. Perhps he or she would be initilly sceptic bout the possibility to ctully observe such structure in the form of temperture nisotropies in the CMB in the ner future. But little over decde lter cosmologist would be mzed to her tht these nisotropies hd ctully been mesured, nd tht they contin extremely detiled informtion bout the contents of the universe. Only to be bffled once gin by the observtion of the recent ccelertion of our universe, suggesting tht some mysterious Drk Energy domintes our universe. Fst forwrd to tody. A lucky cosmologist tody would know tht the history of the universe is described remrkbly well by the so-clled ΛCDM concordnce model, or the Stndrd Model of cosmology. This model depends on only six prmeters (the energy densities of bryonic mtter, drk mtter nd drk energy, the mplitude nd spectrl index of the primordil power spectrum, nd the reioniztion opticl depth) nd is the simplest model tht cn quite ccurtely mtch the following observtions: the specific structure of the temperture nisotropies in the CMB, the bundnces of the light elements H, D, He nd Li, the wlls, filments nd voids in the distribution of glxies t lrge scles, glxy rottion curves nd grvittionl lensing of glxy clusters, the recent ccelertion of our universe. Even though it mtches well these observtions, some very big questions remin. To nme few: Why does our universe only consist of mtter nd no ntimtter? Is drk mtter some sort of new prticle, or should the lws of grvity be modified on lrge scles? Wht is Drk Energy? Wht hppens ner the initil singulrity of the universe? These re severl of the gretest open problems in physics t the moment, nd prtilly becuse of them cosmology s well s strophysics hve become incresingly populr over the lst few decdes. Another importnt reson for its populrity is tht cosmology, in prticulr the Big Bng theory, brings together mny other fields of physics. As we pproch the initil singulrity of the universe we pproch the regime of quntum grvity, of which string theory is typicl exmple. Prticle physics plys n importnt prt in the first moments of the universe, when, due to the expnsion nd cooling of the universe, prticles were creted nd phse trnsitions took plce. During these moments the bryon symmetry is thought to be creted, nd there re huge theoreticl nd experimentl efforts which

11 try to find out if the Stndrd Model, or n extension of it, cn explin the observed symmetry. Another exmple of successful combintion of prticle physics with strophysics is the study of drk mtter. If drk mtter is some new exotic prticle, it my interct with the Stndrd Model prticles, nd it my decy or nnihilte. Such nnihiltion processes cn be put to observtionl tests, which constrin the msses of the drk mtter prticle. But not only does cosmology bring together mny fields, it lso gives us the opportunity to study Nture under more extreme circumstnces thn would ever be possible on erth. For instnce, current genertion of prticle ccelertors cn rech energies in the TeV rnge, wheres the energy scle of, for exmple, infltion ws trillion times higher. Neutron strs hve n extremely high density nd pressure, nd the conditions in its interior re very interesting study object for condensed mtter physicists. Gmm-ry bursts re the most extreme electromgnetic events in the universe. Cosmic rys cn rech energies of severl millions times higher thn those in the LHC. Of course cosmology would not hve seen such flight in populrity, if it ws not for the enormous increse in experimentl dt. The lst decdes hve experienced incredible technologicl dvnces which enble us to look deeper nd deeper into the universe with unprecedented ccurcy. This is the ge of precision cosmology, where dozens of experiments re trying to figure out the smllest detils bout our universe. There re kilometer-long, hypersensitive interferometers tht try to ctch glimpse of the tiniest ripples in spce cused by grvittionl wves. There re huge detectors built on the bottom of the Mediterrnen Se nd drilled into the ice on the South Pole tht detect neutrinos originting from the highest-energy strophysicl processes. There re countless lnd-bsed telescopes, on desert plteus in Chile, on the ice of Antrctic, nd on top of mountins on Hwii nd the Cnry Islnds, to nme few plces. Other telescopes hng from blloon t extremely high ltitudes. Perhps most impressive re the spce telescopes going fter vrious sources such s strs, glxies, supernove, binry str systems nd pulsrs in vrious frequency bnds. All of these experiments re providing us with constnt flow of new dt tht either deepen our understnding of certin phenomen, or pose new questions tht need to be explined. All in ll, cosmologist tody is very lucky to work in such brod n interdisciplinry field which is constntly sourced by experimentl results. This yer cosmologist must hve felt especilly privileged, becuse Spring 2013 mrked the relese of the dt of the Plnck collbortion. The Plnck stellite, lunched by ESA in 2009, ws stellite mission with the primry objective to mesure the Cosmic Microwve Bckground rdition (CMB) in gretest detil. The CMB is essentilly the oldest light tht we cn possibly observe in the universe, nd hence the CMB temperture mp is populrly dubbed bby picture of the universe, see figure 1.1. The CMB is therefore used to extrct informtion bout the very erly universe, nd it hs given the most detiled informtion to dte bout 3

12 4 CHAPTER 1. INTRODUCTION Figure 1.1: Temperture fluctutions in the CMB. Fluctutions re of the order δt /T Courtesy of the Plnck collbortion [7]. its contents. One of the min results is tht the most importnt observtions re still in good greement with the ΛCDM concordnce model, which is quite mzing considering the simplicity of this Stndrd Model of cosmology. However, there re mny interesting hints tht need further explining. Exmples re nomlies such s the CMB cold spot, lck of power on lrge scles nd hemispheric symmetry, but lso tension with other observtions, such s the mesurement of the expnsion rte. But the devil is in the detils, lso in cosmology, nd these detils will keep the current genertion of cosmologists busy for yers to come. Finlly, coming bck to the first sttement: yes, the current genertion of cosmologists is very fortunte. And being prt of this genertion, I consider myself very lucky indeed. 1.2 Pln for this thesis Since cosmology is such brod field, one hs to nrrow down to certin field of expertise. Mny theoreticl cosmologists, including myself, re interested in the physics of the very erly universe. This thesis in prticulr focuses on the description of perturbtions in the very erly infltionry stge of our universe. The motivtion for this comes from the connection between infltion nd the structure in the CMB, s ws mentioned in the previous section. The generl ide is tht vcuum fluctutions during infltion

13 cn be relted to the temperture nisotropies in the CMB. In this thesis we go bck to the bsics of this ide. In chpter 2 we define perturbtions in n expnding universe. The definition of these perturbtions is generlly not unique, since the perturbtions re dependent on the choice of coordinte system. This is known s the guge problem of generl reltivity. However, we shll see tht it is possible to give physicl mening to perturbtions by defining guge invrint perturbtions. Chpter 3 is devoted to the connection between infltionry perturbtions nd the CMB nisotropies. We explin the bsics of n infltionry universe nd demonstrte the mechnism through which vcuum fluctutions in infltion cn leve n imprint in the CMB. In chpter 4 we introduce the concept of frmes in cosmology. The ide is generlly tht one cn write down two field theoreticl models of generl reltivity plus sclr field (two frmes) which seem to be very different, but re ctully relted vi field redefinitions nd therefore equivlent. The motivtion for studying different frmes comes from Higgs infltion, introduced in the chpter, which mkes explicit use of these frme trnsformtions. Similr to the guge problem for perturbtions, here we run into the frme problem t the perturbtive level. Perturbtions in one frme or nother do not coincide. We introduce the ide of frme independent perturbtions, perturbtions tht re invrint under frme trnsformtions. The min objectives for this thesis re () the computtion of the ction for guge invrint perturbtions t different orders in perturbtion theory, nd (b) the demonstrtion of frme equivlence t the perturbtive level. The min results re presented in chpters 5, 6 nd 7. In chpter 5 we study first order perturbtions nd compute the qudrtic ction for liner guge invrint perturbtions. We show tht the guge invrint perturbtions re frme independent t the liner level s well. In chpters 6 nd 7 the results re extended to non-liner perturbtions. Chpter 6 focuses on sclr-type perturbtions, tensoril fluctutions re included in chpter 7. We explicitly compute the cubic ctions for these perturbtions nd show tht there exist unique non-liner guge invrint vribles which re lso frme independent. We lso elborte on nonliner reltions between different guge invrint vribles nd the ssocited reltions between their ctions nd n-point functions. We conclude in chpter 8. 5

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15 Chpter 2 Reltivity The guge problem of Generl 2.1 Generl Covrince Einstein s celebrted theory of Generl Reltivity currently provides our best understnding of grvity. It describes the curvture of spce-time nd the motion of objects in such spce-time, s well s how mss nd energy influence the curvture itself. Generl Reltivity my be studied in field theoretic setting, for which the ction reds S = S EH + S M = d 4 x { } 1 g 2 M P 2 R + L M. (2.1) Here S EH is the Einstein-Hilbert ction tht describes the dynmics of the metric g µν vi the Ricci sclr R. S M nd L M re the ction nd Lgrngin density for mtter fields, respectively. The (reduced) Plnck mss is defined by M 2 P = 8πG N in units where = c = 1. The equtions of motion tht cn be derived by vrying the ction with respect to the metric re the Einstein equtions G µν = 8πG N T µν, (2.2) where G µν = R µν 1 2 Rg µν is the Einstein tensor tht is covrintly conserved, µ G µν = 0. The stress energy tensor is defined by T µν = ( 2/ g)δs M /δg µν. In the most simple scenrio the mtter content of the universe is described by single sclr field Φ with the ction S = 1 d 4 x { } g MP 2 R g µν µ Φ ν Φ 2V (Φ). (2.3) 2 The stress-energy tensor for sclr mtter reds T µν = µ Φ ν Φ + g µν ( 1 ) 2 gαβ α Φ β Φ 2V (Φ). (2.4)

16 8 CHAPTER 2. THE GAUGE PROBLEM OF GENERAL RELATIVITY The theory of Generl Reltivity is covrint it is invrint under coordinte reprmetriztions. This feture is lso clled diffeomorphism invrince. A nturl consequence of generl covrince is tht physicl observbles do not depend on our choice of coordintes. Since the line element ds 2 = g µν dx µ dx ν describes the physicl distnce between two points, generl covrince sttes tht under coordinte trnsformtion d x µ = ( x µ / x ν )dx ν the line element is invrint, ds 2 = d s 2. This implies tht the metric trnsforms s tensor g µν ( x) = xα x β x µ x ν g αβ(x). (2.5) Generl covrince lso mens tht the form of physicl lws is invrint under coordinte trnsformtions. This is most esily seen from the ction for generl reltivity, Eq. (2.3). It cn be shown tht under coordinte trnsformtion x x the mesure is invrint, d 4 x g = d 4 x g. Furthermore, Φ( x) = Φ(x). (2.6) nd likewise the Ricci sclr is invrint, R = R. Thus the ction (2.3) is mnifestly covrint, s it is composed of sclrs, tensors nd covrint derivtives. Now, when we vry the ction with respect to the metric g µν, or with respect to the metric g µν relted to the first by Eq. (2.5), we obtin the Einstein equtions (2.2). Of course, fter coordinte trnsformtion the Einstein equtions re constructed from the metric g µν nd sclr field Φ, but the form of the eqution is nevertheless the sme. Generl covrince is n extremely helpful feture in Generl Reltivity, becuse we re free to choose coordinte system which is most suitble for specific problem. For exmple, the (sptil prt of the) FLRW metric of n expnding universe (2.22) is most conveniently expressed in Crtesin coordintes. On the other hnd, polr coordintes re the vribles of choice for the sphericlly symmetric Schwrzschild solution. Moreover, the demnd of generl covrince mkes it possible to write down strightforwrdly the llowed terms in the ction. This cn be compred to the fmous SU(3) SU(2) U(1) guge symmetries nd Lorentz invrince of the Stndrd Model. In sense, the Stndrd Model is constructed by writing down ll possible terms tht re Lorentz invrint nd stisfy the guge symmetries, plus the dditionl requirement of renormlizbility. Likewise, the Stndrd Model is often extended by introducing n extr symmetry (e.g. supersymmetry) nd writing down ll possible terms tht stisfy the new symmetry. In very similr fshion one cn verify the vlidity of terms in field theory for generl reltivity by checking whether or not they re generlly covrint. New terms cn be dded which re constructed of sclrs, tensors nd covrint derivtives, nd tht re therefore covrint. So, generl covrince is in mny spects blessing for us. Unfortuntely, the freedom to choose coordinte system turns into somewht of nightmre when we strt

17 9 studying fluctutions on top of fixed bckground the perturbtions themselves become dependent on the choice of coordintes. This is problemtic, since physicl observbles should not depend on our choice of coordintes. The coordinte dependence cn be understood in the following wy. By demnding generl covrince of the theory of generl reltivity we hve ctully introduced extr, non-physicl degrees of freedom into our theory tht ensure covrince. The sitution cn be compred to tht of the photon, which is described with the vector field A µ in Lorentz invrint theory. The vector field A µ contins four degrees of freedom, but s we know the photon hs only two physicl polriztions. Thus Lorentz invrince hs somehow obscured the true physicl content of the theory. It is exctly this wht hppens in theory of Generl Reltivity tht is generlly covrint. The coordinte dependence of perturbtions shows tht some degrees of freedom re not physicl. In the next section we shll introduce the perturbtions nd explin how they become dependent on coordinte reprmetriztions. 2.2 Perturbtions nd guge dependence We now discuss perturbtions on top of n rbitrry bckground. There re two equivlent pproches to study the trnsformtions of these perturbtions under coordinte reprmetriztions the pssive pproch nd the ctive pproch nd we shll discuss both now. There exists lot of literture on the topic, see for exmple [8 14] Pssive pproch In the pssive pproch perturbtion of quntity Q (for exmple the sclr or metric) is defined s the difference between the quntity nd fixed function t the sme physicl point p. If we choose some system of coordintes x µ then the perturbtion is δq(x) = Q(x) Q(x). (2.7) The function Q is fixed in the following sense: in nother coordinte system x µ the function Q depends in exctly the sme wy on the coordintes x µ s it does on the coordintes x µ in the originl coordinte system. Thus we cn define perturbtion in the second coordinte system δ Q( x) = Q( x) Q( x). (2.8) Now, importntly, it is this seprtion between the field nd the fixed function tht induces coordinte dependence of the perturbtion by going from one coordinte system to the other. Let us give n explicit exmple of this by considering perturbtion of

18 10 CHAPTER 2. THE GAUGE PROBLEM OF GENERAL RELATIVITY the sclr field on top of homogeneous bckground, If we now tke the infinitesiml coordinte trnsformtion the field perturbtion trnsforms s Φ(x) = φ(t) + ϕ(x). (2.9) x µ x µ = x µ + ξ µ, (2.10) ϕ(x) ϕ(x) = ϕ(x) φξ 0. (2.11) Here we used the generl expressions (2.7) nd (2.8) nd the sclr trnsformtion lw (2.6), nd the fct tht the bckground φ(t) is by definition not ffected by the coordinte trnsformtion. Similrly, by considering perturbtion of the metric round homogeneous bckground g µν (x) = ḡ µν (t) + δg µν (x), (2.12) we find tht under the infinitesiml coordinte trnsformtion the metric perturbtion trnsforms s δg µν (x) δg µν (x) = δg µν (x) µ ξ ν ν ξ µ. (2.13) Here µ is the covrint derivtive nd ξ ν = g νσ ξ σ. We gin used the expressions (2.7) nd (2.8) nd we expnded the tensor trnsformtion lw (2.5) to first order in coordinte trnsformtions. Eqs. (2.11) nd (2.13) show tht the perturbtions re not invrint under chnge of coordinte system. If we compre the generl covrince in the theory of Generl Reltivity to guge symmetry in the Stndrd Model, then chnge of coordinte system cn be viewed s guge trnsformtion. The trnsformtion is chrcterized by the guge prmeter ξ µ (x) (in this cse vector). The perturbtions re therefore guge dependent. From now on we shll frequently mke use of the terms guge trnsformtion nd guge dependence, though one should keep in mind tht these do not hve the sme mening s in the Stndrd Model Active pproch Wheres the pssive pproch dels with explicit coordinte trnsformtions to demonstrte the guge dependence of perturbtions, the ctive pproch provides more intuitive, geometricl picture. In the ctive pproch we consider physicl mnifold M nd bckground mnifold M. The physicl fields Q re defined on M nd the bckground fields Q on M. The perturbtion δq describes the difference between the complete field Q nd the bckground field Q. However, the comprison between the

19 11 physicl nd bckground quntities cn only be mde meningful once they re compred in the sme spce-time. There should therefore be certin prescription to relte the points in the bckground spce-time to those in the physicl spce-time. In Generl Reltivity such prescription is clled mp, one-to-one correspondence between the bckground nd physicl spce-time, nd choice of mpping my be clled guge choice. Such mpping is not unique, since Generl Reltivity is diffeomorphism invrint, mening we hve freedom to choose mpping between different spce-times. Such freedom my be clled guge freedom. A chnge of the mp between M nd M is then clled n (ctive) guge trnsformtion. To see how the guge trnsformtion cts, let us consider quntity Q in the physicl spce-time nd define two different mppings from M to M. One choice mps Q to Q M, the other to Q M. It cn be shown [8, 9, 13, 15, 16] tht Q M nd Q M re relted by Q = e ξ Q. (2.14) Here we dropped the subscript, but it is understood tht both Q nd Q re evluted in the bckground spce-time. ξ µ is the vector field tht genertes one prmeter group of diffeomorphisms from the bckground spce-time to itself, Ψ λ : M M, nd is defined by dx µ /dλ = ξ µ, with the x µ coordintes defined on M. In the lnguge of symmetries nd guge theories, the prmeter ξ µ my be clled the guge prmeter. is the Lie derivtive defined by 1 ξ Q = lim λ 0 λ (Ψ λq Q), (2.15) where the mp Ψ λ pulls bck the vlue of Q t point Ψ λ (p) to tht t p. We cn now see how the guge trnsformtion (2.14) cts on perturbtions by seprting Q nd Q in bckground vlue nd perturbtion, Q = Q + δq nd Q = Q + δq (ll quntities re evluted in the bckground spce-time). Since the reltion (2.14) holds to ll orders, it is convenient to seprte the perturbtions order by order. Thus we use nd similrly we seprte the guge prmeter δq = λδq (1) λ2 δq (2) + O(λ 3 ), (2.16) ξ µ = λξ µ (1) λ2 ξ µ (2) + O(λ3 ). (2.17) Here the prmeter λ is ssumed to be some smll prmeter. When we spek of first order perturbtions, we re referring to terms tht re of order λ. Likewise, second order perturbtions re ll terms up to nd including λ 2. Armed with this we cn write

20 12 CHAPTER 2. THE GAUGE PROBLEM OF GENERAL RELATIVITY the guge trnsformtion (2.14) order by order, such tht Q = Q δq (1) = δq (1) + ξ (1) Q δq (2) = δq (2) + ξ (2) Q + 2 ξ (1) Q + 2 ξ (1)δQ (1). (2.18) Note tht the bckground quntities re not ffected by the guge trnsformtion. In order to see how the perturbtions of, for exmple, the sclr field nd metric chnge under guge trnsformtion, we use tht the Lie derivtive cts on sclr, vector nd tensor of rnk 2 s ξ f = ξ µ µ f ξ V µ = ξ ν ν V µ V ν ν ξ µ ξ T µν = ξ σ σ T µν + T σν µ ξ σ + T µσ ν ξ σ. (2.19) By mking use of these expressions nd metric comptibility σ g µν = 0, we immeditely find for the sclr nd metric perturbtions, see Eqs. (2.6) nd (2.5), to first order in perturbtion theory ϕ(x) = ϕ(x) + φξ 0 δg µν (x) = δg µν (x) + µ ξ ν + ν ξ µ. (2.20) All quntities re here understood to be of first order in the prmeter λ. If we compre now the first order guge trnsformtions of sclr nd metric perturbtions in the pssive pproch, Eqs. (2.11) nd (2.13), to those in the ctive pproch (2.20), we see tht they re the sme up to sign chnge of ξ. In fct, the pssive nd ctive pproch cn be shown to be equivlent [13, 15]. Essentilly the chnge in the choice of mpping induces chnge in coordinte system, which cn be written to second order s x µ = x µ λξ µ (1) 1 [ ] 2 λ2 ξ µ (2) ξµ (1) νξ(1) ν. (2.21) It is now esily seen tht, to first order, the induced coordinte trnsformtion is the sme s the infinitesiml coordinte trnsformtion used to demonstrte the guge trnsformtion in the pssive pproch. The equivlence cn lso be demonstrted to higher order, for exmple second order, by using the coordinte trnsformtion bove for the pssive pproch nd compre it to the second order guge trnsformtion in Eq. (2.18). We mention tht the pssive pproch is commonly used for first order perturbtions, s one cn esily see how coordinte chnge ffects the perturbtions. The pssive pproch ws used in, for exmple, the seminl works [10, 17, 18]. For higher

21 13 order perturbtions the ctive pproch is more populr, s it provides strightforwrd reltion for the guge trnsformtion t higher order, see Eq. (2.14). The ctive pproch hs been used in [12, 14 16]. 2.3 Perturbtions in n expnding universe So fr we hve demonstrted the guge dependence of metric perturbtions δg µν on top of generl homogeneous bckground ḡ µν (t). We re interested in perturbtions on top of n expnding universe. A homogeneous nd isotropic expnding universe cn be described by the Friedmn-Lemître-Robertson-Wlker (FLRW) metric, with line element ds 2 = g µν dx µ dx ν = dt d x d x. (2.22) Here = (t) is the scle fctor, which describes the expnsion of the universe. By convention, the scle fctor tody is equl to one. We now consider smll fluctutions on top of the FLRW metric. The metric field for four dimensionl spce-time hs in principle ten independent components. The perturbed metric should therefore contin ten degrees of freedom. A convenient wy to study these perturbtions is vi the Arnowitt-Deser-Misner (ADM) metric [19], whose line element is defined s ds 2 = N 2 dt 2 + g ij (dx i + N i dt)(dx j + N j dt). (2.23) Geometriclly, spce-time hs been sliced up in sptil hypersurfces whose geometry is described by the sptil metric g ij. The slicing nd threding of spce-time is described by the lpse function N nd shift vector N i. The lpse function nd shift vector describe four degrees of freedom, wheres nother six re contined in the sptil metric. Not ll of these degrees re freedom re dynmicl, nd not ll degrees of freedom re physicl. The ADM formlism is designed to seprte the non-dynmicl degrees of freedom from the true dynmicl degrees of freedom. This is most esily seen from figure 2.1, which shows geometricl representtion of the ADM line element. The ide is tht t ech instnt of time there is sptil hypersurfce, nd stcked together these sptil hypersurfces give the complete four-dimensionl geometry of spce-time. Now, if one envisions these hypersurfces s thin, flexible sheets of iron nd one uses (perpendiculr) connectors to stck together these sheets [20], then there re few fctors tht determine the finl four-dimensionl geometry. Tke two hypersurfces t t nd t + dt. First of ll, we should specify length for the connectors, which we cn do vi the lpse function N, such tht the length is Ndt. Secondly, we wnt to know to which point connector, connected to point x i on the hypersurfce t time t, should be connected

22 14 CHAPTER 2. THE GAUGE PROBLEM OF GENERAL RELATIVITY x i + dx i Σ t+dt x i - N i dt ds Ndt x i Σ t Figure 2.1: Geometricl representtion of the ADM line element (2.23). Sptil hypersurfces t different times t nd t + dt re seprted by proper length Ndt nd sptil coordintes in different hypersurfces re shifted with respect to one nother by N i dt. on the hypersurfce t time t + dt. These (shifted) points cn generlly be written s x i N i dt, where N i is the shift function. Of course the lpse nd shift functions re locl functions of ( x, t), becuse the length of the connectors nd position of connection points depends on the hypersurfce nd the position in the hypersurfce. Thus, this geometricl picture suggests tht N nd N i re not dynmicl fields, but rther represent prmeters which determine how hypersurfces re stcked. Indeed, s we shll see in chpter 5, when the ction (2.3) is expressed in terms of the ADM metric, the lpse function nd shift vector pper without time derivtives nd cn therefore be treted s uxiliry fields. The true dynmicl informtion is contined in the sptil metric g ij, nd in the sclr field Φ, if present. Now tht we hve discussed the prcticlities of the ADM formlism, let us define

23 15 perturbtions on top of FLRW bckground. We use the decomposition g ij = (t) 2 ( e h(x)) ij = (t)2 (δ ij + h ij (x) h ik(x)h kj (x) +...) Φ = φ(t) + ϕ(x) N = N(t) (1 + n(x)) N i = (t) 1 Nni (x). (2.24) Some nottionl remrks: ll perturbtions re defined with subscripts for the sptil indices nd ll other quntities cn be derived from the ones bove. For instnce, g ij = 2 (δ ij h ij +...) nd N i = g ij N i. Further note tht we recover the FLRW bckground with cosmic time t if we set N(t) = 1, or in conforml time τ when N(t) = (t). However, it turns out to be convenient to keep generl function N, s it becomes esy to switch between different time reprmetriztions. The perturbtion of the sptil metric cn be further decomposed using the sclr-vector-tensor decomposition 1, with h ij = 2ζδ ij + i j h 2 + (i h T j) + γ ij, (2.25) i h T i = 0, i γ ij = 0, γ ii = δ ij γ ij = 0. (2.26) Some counting revels tht the sptil metric hs been seprted into two sclr, two vector, nd two tensor degrees of freedom. The sclr-vector-tensor decomposition cn be viewed s generliztion of the Helmholtz decomposition of vector, which sttes tht vector cn be decomposed into curl-free (longitudinl) prt nd divergencefree (trnsversl) prt, V = v + A = V L + V T. This decomposition cn lso be pplied to the shift vector, with N i = (t) 1 Nni (x) = (t) 1 N(t)[(t) 1 i s(x) + n T i (x)]. (2.27) i n T i = 0. (2.28) Now tht we hve defined the perturbtions on top of n expnding bckground spcetime, we could try to find, for instnce, correltion functions for these perturbtions. This sounds simple enough. For exmple, for 2-point correltion function t tree level we just insert liner perturbtions in the ction (2.3), expnd the ction to second order, nd compute the 2-point correltion functions from this ction. However, s we rgued 1 In most literture the nottion Ψ is used insted of ζ.

24 16 CHAPTER 2. THE GAUGE PROBLEM OF GENERAL RELATIVITY before, some prts of the metric re non-dynmicl. Moreover, s we hve seen in the previous section, perturbtions on top of dynmicl bckground re guge dependent. Let us mke this more explicit by tking into ccount the generl first order guge trnsformtions of the sclr field nd metric (2.20). If we pply this to the perturbtions s defined in Eq. (2.24), we find tht under n infinitesiml coordinte trnsformtion the sclr nd sptil metric perturbtions trnsform s, ϕ ϕ + φ ξ0 N i j h 2 ζ ζ + H ξ0 N i j h 2 2 i j ξ 2 (i h T j) (i h T j) 2 (i ξ T j) γ ij γ ij. (2.29) Here we hve used ξ µ = (ξ 0, ξ i ), ξ i = 1 i ξ +ξ T i. The dotted time derivtive is defined s Ẋ = dx/( Ndt), which llows n esy trnsition to different time coordinte. For exmple, for N(t) = (t) we work in conforml time. The Hubble prmeter is defined s H = ȧ/ = d/( Ndt). Next we cn show tht the lpse function nd shift vector trnsform t first order s ( ξ 0 ) n n + N 2 s 2 (i ñ T j) 2 s 2 (i ñ T j) ( 2 ξ 2 ) ( (i ξt j) 2 2 ξ 0 N ). (2.30) As consequence of the guge dependence, perturbtions in one coordinte system re different from those in nother. Why is this problemtic? Tke the following extreme exmple. Suppose we wnt to compute 2-point function of the sclr field perturbtion, ϕ(x)ϕ(y). If we define the perturbtion in one coordinte system, then we find some (nonzero) expression for the 2-point correltor. Now, we perform specific infinitesiml coordinte trnsformtion with ξ 0 = ϕ/ φ. In the new coordinte system the sclr field perturbtion vnishes, nd so does its correltor. So wht is the physicl, observble, correltor? There re in principle two wys to del with the guge dependence of perturbtions on top of dynmicl bckground:

25 17 fix the guge, or, use guge invrint perturbtions. In the first cse the otherwise rbitrry coordinte trnsformtion is fixed, i.e. the functions ξ µ re chosen to hve specific vlues. The pproch of guge fixing mkes computtions generlly simpler, becuse it usully mens tht some of the perturbtions cn be set to zero. Not surprisingly, this pproch is most commonly used. However, guge fixing cn be very dngerous. If the guge is not completely fixed, there my still remin spurious degrees of freedom in the theory. This cn led to so-clled guge rtefcts opertors or fields tht re not physicl, but purely guge. Even when the guge is completely fixed, some difficulties cn still pper. Since there re mny wys to completely fix the guge, it cn be difficult to relte results in one guge to those in nother guge. Especilly in the intrinsiclly non-liner theory of Generl Reltivity, one needs higher order guge trnsformtions to find the reltion between results in one guge nd nother. This becomes prticulrly n issue when compring the initil stte for different vribles, or when determining cosmologicl correltion functions, s shll be discussed lter in this thesis. Alterntively, one cn express the expnded theory in terms of guge invrint perturbtions. Here, guge invrince should be interpreted s invrince under coordinte trnsformtions. A disdvntge of this pproch is tht the computtions re generlly more involved, becuse ll individul perturbtions must be tken into ccount. On the other hnd, once the theory (for instnce, the ction, or eqution of motion) is expressed in terms of guge invrint perturbtions, it is very strightforwrd to compute physicl observbles. Guge invrint vribles re by definition vribles tht do not chnge under non-physicl coordinte reprmetriztion, so their correltion functions present the ctul physicl correltion functions. Also, in certin sense (which will be mde precise lter) the guge invrint ction is unique. This mens it ought to be computed only once s strting point for e.g. perturbtion theory. For these resons we shll minly use the guge invrint pproch in this work. 2.4 Guge invrint cosmologicl perturbtions Linerized guge invrint perturbtions were first discussed in the seminl work by Brdeen [17]. Here we study guge invrint perturbtions on top of n expnding universe (2.24). We hve seen how they trnsform under first order guge trnsformtions in Eqs. (2.29) (2.30), In principle there re infinitely mny guge invrint combintions we cn mke from these perturbtions, but let us highlight some of the most importnt. First of ll, note tht the tensor perturbtion γ ij is by itself guge invrint to first order. This is the trnsverse, trceless prt of the perturbed metric nd is better known s the

26 18 CHAPTER 2. THE GAUGE PROBLEM OF GENERAL RELATIVITY grviton. Becuse of the liner guge invrince, the physicl propgtor for γ ij is found immeditely from the second order ction for γ ij. All other perturbtions of the metric re guge dependent, so guge invrint combintions must be formed. Perhps the most importnt is the guge invrint sclr perturbtion w ζ = ζ Ḣ φ ϕ. (2.31) The guge invrince to first order is most esily checked from Eqs. (2.29). The sclr perturbtion ζ is commonly clled the curvture perturbtion, since the Ricci sclr in sptil hypersurfce is to first order R (3) = 4 2 ζ/ 2, such tht ζ is directly relted to the grvittionl potentil on sptil hypersurfce. Now, w ζ is often clled the comoving curvture perturbtion, or the curvture perturbtion on uniform field hypersurfces, since it reduces to ζ if we fix guge where ϕ = 0 2. Very similrly we cn define the guge invrint combintion w ϕ = ϕ φ H ζ. (2.32) It should come s no surprise tht this vrible is clled the guge invrint field perturbtion on uniform curvture hypersurfces. At first order w ζ nd w ϕ re only relted by simple time-dependent rescling w ζ = Ḣ φ w ϕ. (2.33) This simple reltion mkes it prticulrly esy to find the reltion between (the 2-point correltion functions of) sclr perturbtions on different hypersurfces. Besides the sclr nd tensor degrees of freedom ϕ, ζ nd γ ij there re still mny other degrees of freedom in the metric tht trnsform under guge trnsformtions. We cn imgine combining lso those into guge invrint combintions. For exmple, the combintions (i n T j) s ( (i h T j) ( ) 2 h ) 2, (2.34) 2 In literture the comoving curvture perturbtion is often written s R. There is nother guge invrint perturbtion, the curvture perturbtion on uniform density hypersurfces, which is the combintion ζ Hδρ/ ρ tht reduces to w ζ in the limit. The nottionl convention is to define this quntity s ζ Ψ + Hδρ/ ρ (see lso footnote 1). In this thesis, ζ denotes the guge dependent sclr perturbtion of the metric nd w ζ is the comoving curvture perturbtion.

27 19 re guge invrint under infinitesiml coordinte trnsformtions ξi T nd ξ, respectively. However, the ltter combintion is not guge invrint under under infinitesiml time trnsltions ξ 0. These guge trnsformtions cn be blnced by dditionl contributions of, for exmple, the sclr perturbtions ζ or ϕ. As we shll see in chpter 5, once we decouple ll the perturbtions in the second order ction, we obtin n ction in terms of guge invrint vribles in precisely the combintions s in Eqs. (2.32) nd (2.34). Of course, ll combintions of perturbtions derived in this section re only guge invrint to first order in perturbtions. At higher order they re still guge dependent. In tht cse the guge invrint combintions of perturbtions become non-liner. One cn imgine tht the construction of these guge invrint vribles becomes lot more involved thn the simple computtions in this section. It is still possible though to find these guge invrint combintions in consistent mnner. Often this is done by fixing the guge trnsformtion order by order in perturbtion theory, which ws used to construct second order guge invrint perturbtions in e.g. Refs. [11 15, 21, 22]. In chpters 6 nd 7 we briefly study second order guge trnsformtions (in the long wvelength limit) nd we construct non-liner guge invrint vribles. More importntly, we show tht explicit second order guge trnsformtions re not necessry to find second order guge invrint vribles. One of our min findings is tht these vribles cn be derived utomticlly from the third order ction for cosmologicl perturbtions in procedure tht seprtes the true physicl vertices from the non-physicl ones. In generl, we shll work with the ction in our computtions. One reson is tht the ction is the object tht is mnifestly covrint. Thus, order by order in perturbtion theory it should be possible to express the ction in terms of guge invrint perturbtions. At the level of the equtions of motion this is not t ll cler. Of course, the Einstein equtions (2.2) trnsform s tensor, but when we look t perturbtions (2.24) this property is somewht obscured. Even though the ction is one order higher in perturbtion theory compred to the equtions of motion, t lest we know tht it is covrint. The equtions of motion for guge invrint perturbtions re then derived from this ction by vritionl principle. Perhps the most importnt reson for using the ction is tht it is the strting point for quntiztion. Once the ction is written in terms of guge invrint perturbtions, it is possible to define cnonicl moment for the perturbtions nd impose the cnonicl commuttor. Next we cn compute, for exmple, the tree-level 2-point correltor from the qudrtic ction, nd higher order correltors nd loops from the interction vertices.

28 20 CHAPTER 2. THE GAUGE PROBLEM OF GENERAL RELATIVITY 2.5 An nlogy: electrodynmics In the previous sections we hve demonstrted tht perturbtions on top of dynmicl bckground re in generl guge dependent. We hve stted tht, t the perturbtive level, the ction cn be written in mnifestly covrint wy in terms of guge invrint cosmologicl perturbtions, though we hve not shown ny explicit computtions yet. One of the min results of this work is the computtion of the guge invrint second nd third order ction for cosmologicl perturbtions. In order to illustrte our methods to rrive t perturbed ction in mnifestly guge invrint form, let us first tret the simpler cse of electrodynmics. The theory of the mssless spin 1 photon serves s gret nlogy to the field theory of linerized generl reltivity nd llows us to demonstrte mny of the fetures nd steps we shll tke in the ltter cse. The photon in the flt Minkowsky spce-time my be described by the Mxwell ction S = d 4 x { 14 } F µνf µν A µ J µ, (2.35) where the field strength is defined s F µν = µ A ν ν A µ nd J µ is conserved current, µ J µ = 0. Indices re rised nd lowered by the Minkowsky metric with sign convention(, +, +, +). The Mxwell ction is mnifestly Lorentz invrint, becuse the field strength trnsforms s tensor under Lorentz trnsformtions. Now the vector potentil A µ contins nively four degrees of freedom, but s we know from our course in electrodynmics, the photon hs only two physicl polriztions. So there re some fictitious degrees of freedom in the Mxwell ction tht ct to mke the theory Lorentz invrint, but obscure the true physicl content of the theory. The sme is true for the theory of pure Generl Reltivity, which nively contins ten degrees of freedom, lthough we hve lerned tht the only physicl modes re the two degrees of freedom of the grviton. An dditionl degree of freedom ppers when some mtter ( source) is lso present in the form of sclr field, which hppens for simple infltionry models. So let us try to see for the cse of the photon how we cn eliminte the non-physicl degrees of freedom from the ction. First of ll, note tht the zeroth component of the vector potentil A 0 ppers in the Mxwell ction without ny time derivtives. This indictes tht the field A 0 is ctully non-dynmicl. In field theory such non-dynmicl field is lso clled n uxiliry field. The eqution of motion for n uxiliry field hs single solution, nd it my be re-inserted in the ction to eliminte A 0 from the ction. As we hve rgued in section (2.3) there re lso non-dynmicl degrees of freedom in the theory of Generl Reltivity. These correspond to the lpse function N nd shift vector N i. As we shll see in chpter 5, these fields nd their perturbtions pper s uxiliry fields. Thus they cn be solved

29 21 for nd their solution inserted into the ction. Alterntively they cn be decoupled from the rest of the ction, which is wht we shll employ for the photon cse. Secondly, observe tht the Mxwell ction is invrint under the guge trnsformtion A µ A µ + µ Λ. (2.36) The guge trnsformtion indictes tht one of the components of the vector potentil is non-physicl. In fct, we cn fix the guge by setting one of the components of the vector potentil to zero. If we compre now to Generl Reltivity, the guge trnsformtion (2.36) is similr to the (liner) coordinte trnsformtion (2.20). In the ltter cse we hve four guge prmeters ξ µ, which cn fix four components of the metric. However, here we do not wnt to fix the guge, but express the ction in guge invrint wy. The previous resoning hs shown us tht, out of the four nive degrees of freedom in the vector potentil, one is ctully non-dynmicl, nd there is one guge degree of freedom. We thus expect tht only two dynmicl degrees of freedom re present in the photon ction. How cn we mke this more pprent? A simple wy to do this is to decompose the vector potentil s follows A µ = (A 0, A i ) = (A 0, i ã + A T i ), i A T i = 0. (2.37) By mking use of this decomposition it is esy to see tht the trnsverse prt of the vector potentil does not trnsform under the guge trnsformtion (2.36), A T i A T i. The longitudinl prt on the other hnd does trnsform, s does A 0. However, the specil combintion à 0 = A 0 ã (2.38) is invrint under the guge trnsformtion. Compre now to the generl reltivity cse nd note tht the sclr-vector-tensor decomposition (2.25) is similr to the decomposition for the vector potentil bove. We hve seen tht the trnsverse, trceless prt of the metric does not trnsform under liner coordinte trnsformtions: compre to A T i. The other prts of the metric trnsform under liner coordinte trnsformtions, but cn be combined into guge invrint vribles: compre to Ã0. Bck to our ction: if we now insert the decomposition (2.37) into the Mxwell ction (2.35) we find fter some prtil integrtions tht S = { 1 [ d 3 xdt ( A 2 T i ) 2 ( j A T i ) 2] A T i Ji T + 1 } 2 ( iã0) 2 + Ã0J 0. (2.39) So indeed the Mxwell ction contins only two dynmicl degrees of freedom which re the trnsverse directions of the sptil prt of the vector potentil. Their eqution of motion is the fmilir wve ction in the presence of source. Additionlly there is

30 22 CHAPTER 2. THE GAUGE PROBLEM OF GENERAL RELATIVITY non-dynmicl degree of freedom, whose eqution of motion gives 2 Ã 0 = 2 (A 0 ã) = J 0. Hd we solved for the uxiliry field A 0 from the strt the non-dynmicl field would hve dropped out. The steps we shll follow in order to compute the guge invrint ction for cosmologicl perturbtions re precisely the sme s those tken bove. First, we expnd the ction for generl reltivity plus sclr field to second order (chpter 5) or third order (chpters 6 nd 7) in perturbtions. The perturbtions in these ctions re guge dependent, nd there re four non-dynmicl degrees of freedom. We cn either decouple these non-dynmicl perturbtions from the ction, s we do in chpter 5, or we cn solve for the uxiliry fields nd insert the solution into the ction, see chpters 6 nd 7. The non-physicl degrees of freedom re bsorbed into the solutions of the uxiliry fields, nd we re in the end left with only three physicl degrees of freedom. So out of degrees of freedom in metric nd sclr field, 4 re non-dynmicl, 4 re guge degrees of freedom, nd there re only 3 propgting degrees of freedom. Finlly some words bout guge fixing. The guge freedom for the photon cn be fixed in the Coulomb guge, which sets i A i = 0. As consequence the longitudinl prt of the vector potentil does not pper in the equtions of motion. A 0 cn now be solved from its eqution of motion 2 A 0 = J 0. If the source J 0 is some chrge density ρ, then A 0 presents the instntneous Coulomb potentil, hence the nme Coulomb guge. Thus, we re left with only the two physicl, trnsverse polriztions of the photon. In the cse of grvity the guge freedom cn be fixed t liner order by setting h ii = 0 nd i h ij = 0. This fixes the four guge degrees of freedom nd effectively sets ζ = h = h T i = 0. The perturbtions of the uxiliry fields n, i s nd n T i cn now be solved from their equtions of motion, nd in the cse of linerized pure grvity in vcuum they vnish. The only remining dynmicl perturbtion is the trnsverse trceless grviton γ ij. If n extr sclr field is present in the ction, its perturbtion survives too s dynmicl degree of freedom. In tht cse the uxiliry fields re not zero by their equtions of motion, but they re determined by the field perturbtion ϕ, just like A 0 ws determined by J 0. The guge freedom cn here lso be fixed by setting ϕ = 0 (s well s h T i nd h), insted of setting ζ = 0. Tking these remrks into ccount, we re still minly interested in the guge invrint perturbtions, s they immeditely present the physicl vribles. We conclude this section by sying once more tht the photon ction provides beutiful nlogy to the ction for liner perturbtions in Generl Reltivity. We shll use the strtegy bove to compute the second order ction for guge invrint perturbtions for single sclr field in n expnding universe in chpter 5. Before we proceed with our ctul computtions of the guge invrint ction for cosmologicl perturbtions though, we should first explin why we re interested in these perturbtions in the first plce.

31 Chpter 3 Infltionry perturbtions In this chpter we discuss the infltionry prdigm, perturbtions in n inflting universe nd how these perturbtions leve n imprint in the Cosmic Microwve Bckground rdition (CMB). Some excellent books nd reviews for infltion nd primordil fluctutions cn be found in Refs. [23 27]. 3.1 The infltionry universe Let us now introduce the bsics of n infltionry universe [28 32]. Our strting point is the Einstein eqution (2.2). in the expnding FLRW universe (2.22), ds 2 = dt 2 + (t) 2 d x d x. (3.1) A homogeneous nd isotropic universe my be well described by perfect fluid with stress energy tensor T µν = dig(ρ, 2 p, 2 p, 2 p), where ρ nd p re the energy density nd pressure in the fluids rest frme. The Einstein equtions (2.2) for this stress-energy tensor nd the FLRW metric re H 2 = ρ 3MP 2 Ḣ = 1 2MP 2 (ρ + p) 0 = ρ + 3H(ρ + p). (3.2) These equtions re the celebrted Friedmnn equtions which describe homogeneous nd expnding universe. The first two equtions follow from the 00 nd ij components of the Einstein equtions, nd the lst one from the covrint conservtion of the stress energy tensor. The Hubble prmeter is defined by H = ȧ/ nd describes the rte of expnsion of the universe. If we consider n expnding universe with some

32 24 CHAPTER 3. INFLATIONARY PERTURBATIONS ω ρ (t) d phys R H mtter 0 3 t rdition 3 4 t ds 1 const. e Ht = e Ht const. Tble 3.1: Scling of energy density ρ, scle fctor, prticle horizon d phys nd Hubble rdius R H in mtter, rdition nd de Sitter er, chrcterized by the eqution of stte prmeter ω. energy density ρ, the Hubble prmeter is positive nd the scle fctor lwys increses in time. By mking use of the eqution of stte p = ωρ we find from the lst eqution tht ρ 3(1+ω). (3.3) Thus, quite generlly, the energy density gets diluted when the scle fctor increses. From the first Friedmnn eqution for H 2 we cn now derive the time dependence of the scle fctor 2 (t) t 3(1+ω) (ω 1) (t) e Ht (ω = 1). (3.4) Tble 3.1 shows the scling of the energy density nd scle fctor in different er. In rdition er (ω = 1 3 ) the energy density of the universe is dominted by reltivistic prticles nd ρ 4. In mtter er (non-reltivistic prticles, ω = 0) the energy density scles s 3. This therefore mens tht the rte of expnsion decreses for common (non)-reltivistic mtter, which cn lso be seen from the second Friedmnn eqution (3.2). In tble 3.1 two importnt distnce scles hve been introduced. The first is the prticle horizon, denoted by d phys. The prticle horizon is the distnce tht light could hve trveled from some initil time t in until time t, nd thus defines the mximum cusl distnce t time t. I.e events t time t re in cusl contct (cn influence ech other) if their distnce is d phys. Light rys propgte long null-geodesics, ds 2 = 0, nd thus the prticle horizon is defined by t dt d phys = (t) t in (t ) in = (t) d 2 H. (3.5) Another importnt distnce scle is the Hubble rdius, defined by R H = H 1. (3.6)

33 As we cn see from tble 3.1, the prticle horizon nd Hubble rdius scle eqully during rdition nd mtter er. Thus, in the conventionl hot Big Bng scenrio, where the universe first underwent hot phse dominted by rdition, then cooled down nd ws dominted by mtter, the Hubble rdius nd prticle horizon re equl up to numericl fctors. The hot Big Bng scenrio hs proven very successful in explining the current constituents of our universe. As n exmple, t time of 3 min, or energy scle of 0.1MeV the light elements re known to hve been formed from protons nd neutrons in process clled Big Bng Nucleosynthesis (BBN). The correct prediction of the bundnces of H, He nd Li is gret triumph of the hot Big Bng scenrio. Moreover, the hot Big Bng scenrio predicts tht below certin energy scle (0.3eV) the electrons nd photons combine into neutrl hydrogen, such tht the photons re no longer scttered nd they cn freely strem. At this moment of decoupling the universe becomes trnsprent, nd tody we see this rdition s the Cosmic Microwve Bckground rdition (CMB), which ws fmously discovered by Penzis nd Wilson in 1965 [33]. The CMB is the most perfect exmple of blck body rdition, corresponding to temperture of pproximtely 3000K t time of emission (z = 1090), which in turn implies the current temperture of the universe T = 2.73K. Vritions of this temperture re of the order of δt/t 10 5, nd it is these temperture fluctutions tht stellite missions such s ESA s Plnck mission is fter, becuse they contin welth of informtion bout the constituents of the history of the very erly universe. The current best picture of these temperture fluctutions is shown in figure 1.1. Even though the hot Big Bng scenrio is very successful, there re still some unresolved problems concerning initil conditions. One of the mjor problems hs to do with cuslity. The problem is the following. Let us tke some physicl scle λ phys = 2π/k, where 1/k is (constnt) comoving scle. We hve seen tht the prticle horizon (nd Hubble rdius) during rdition nd mtter er scles s 2 nd 3/2, respectively. This mens tht it is possible tht some physicl scle λ phys which is well within the horizon tody, ws well outside the horizon t some erlier time. The sitution is shown in figure 3.1. This implies tht it is in principle possible to obtin informtion bout scles now tht were out of cusl contct before. This is known s the horizon problem. As n exmple, let us consider the oldest direct observtion of the erly universe, which is the CMB. If we compute the Hubble rdius t the time of decoupling nd trnslte to how we observe it tody, it turns out tht it corresponds to ptch of bout one degree in the sky. So, the CMB mp contins mny cuslly disconnected ptches. However, the temperture of the photons coming from ll these different ptches is the sme up to tiny vritions δt/t The horizon problem my thus be rephrsed s: why is the temperture of the CMB (lmost) the sme no mtter in which direction we look, even though mny regions were out of cusl contct? Another mjor problem is the so-clled fltness problem. From observtions we know 25

34 TODAY 26 CHAPTER 3. INFLATIONARY PERTURBATIONS ~ 3/2 R H λ phys ~ ~ 2 INFLATION RADIATION MATTER ln() Figure 3.1: Plot of the Hubble rdius s function of ln(). During mtter nd rdition er the Hubble rdius grows s 3 2 nd 2, respectively. Physicl scles exit the Hubble rdius during infltion nd re-enter the Hubble rdius during mtter nd rdition er. tht the universe is flt to very high ccurcy. This is why we worked with the flt FLRW metric (3.1). However, it is possible to work with more generl metric which lso contins curvture term. If we lso include some cosmologicl constnt Λ tht cn explin the recent ccelerted expnsion of our universe, we find tht the first Friedmnn eqution becomes H 2 = ρ + Λ 3 k 2. (3.7) 3M 2 P Here the prmeter k described the curvture of the universe, k > 0 for positively curved universe nd k < 0 for negtive curvture. It is possible to rewrite this eqution in terms of the density prmeter 1 = Ω tot + Ω k, (3.8) where nd Ω m = Ω tot = Ω m + Ω rd + Ω DE, Ω k = k 2 H 2, (3.9) ρ m 3M 2 P H2, Ω rd = ρ rd 3M 2 P H2, Ω DE = Λ 3H 2. (3.10)

35 27 Tody the universe is dominted by (bryonic plus drk) mtter nd drk energy. The totl energy density is observed to be close to unity tody, such tht the energy density contined in the curvture is consistent with zero, Ω k /Ω tot However, from tble 3.1 we cn infer tht the energy density in curvture grows by nd 2 in mtter nd rdition er, respectively. This mens tht, if the curvture is smll tody, it must hve been incredibly smll in the erly universe, which requires fine-tuning. This is the fltness problem. A solution to the horizon nd fltness problems ws proposed by Guth [28]. This solution is clled cosmologicl infltion. Guth relized tht both problems cn be solved by rpid ccelerted expnsion of the erly universe. Such specil er occurs when ω < 1/3, for which ä (for > 0, p < 13 ) ρ. (3.11) In the extreme cse where ω 1, we cn see tht the energy density becomes constnt. This mens tht the energy density does not get diluted by the expnsion nd the Hubble prmeter becomes pproximtely constnt. It is not difficult to see from Eq. (3.2) tht (t) e Ht, (for p ρ). (3.12) This specil de Sitter phse is shown in tble 3.1. The crucil feture of infltion is tht the Hubble rdius remins pproximtely constnt, but the prticle horizon grows exponentilly in time. This rpidly brings into cusl contct region which is bigger thn the observble universe tody, nd thereby solves the horizon problem. Moreover, the rpid ccelerted expnsion mkes our observble universe pper lmost flt, thereby solving the fltness problem. It cn be shown tht the universe must expnd t lest by fctor of e 67 to solve these horizon nd fltness problems. We cn define quntity clled the number of e-folds ( ) tend N e ln = dt H, (3.13) end where t end denotes the end of infltion nd t is some initil time before the end of infltion. Thus the fltness nd horizon problems re solved when the totl number of e-folds of infltion N e,tot 67. So fr we hve explined the generl mechnism of cosmologicl infltion, but we hve not yet discussed ny specific model. In order to hve period of ccelerted expnsion, we need some specil type of mtter which hs negtive pressure, p ρ. At first instnt this seems strnge, but it is well-known phenomenon in modern field theories. Tke the simplest exmple of sclr field theory, for which the ction becomes Eq. (2.3). If we derive the Friedmnn equtions for the homogeneous mode of the sclr t

36 28 CHAPTER 3. INFLATIONARY PERTURBATIONS V(φ) TUNNELING REHEATING φ Figure 3.2: The potentil for Originl (Old) Infltion. The sclr field is trpped in the flse vcuum nd tunnels to the true vcuum. The ccelerted expnsion is driven by the constnt energy density of the sclr field in the flse vcuum with respect to the true vcuum. field, Φ = φ(t), we find H 2 = 1 6MP 2 ( φ 2 + 2V ) Ḣ = 1 φ 2 2MP 2 0 = φ + 3H φ + V. (3.14) Here prime denotes derivtive with respect to the field, V = dv/dφ. Now, when the potentil energy density is pproximtely constnt nd much greter thn the kinetic energy, 1 φ 2 2 V, we find tht H V/(3MP 2 ), which leds to exponentil expnsion. If the sclr field is in its true vcuum, V = 0 nd infltion is not possible. However, n ccelerted expnsion cn hppen when the sclr field is displced from its true vcuum. For exmple, the sclr field cn be trpped in metstble vcuum, nd the non-zero energy density of this vcuum with respect to the true vcuum drives infltion. The sclr field is in this cse clled the inflton field, i.e. the field responsible for the infltionry expnsion. This is the originl infltion scenrio s described by Guth 1. The potentil picture is depicted in figure 3.2. In generl we cn define so-clled slowroll conditions for the inflton field, which determine whether or not infltion will tke 1 However, note tht Strobinsky lredy found infltionry solutions in R 2 infltion before [29].

37 29 plce. The slow-roll prmeters re ɛ Ḣ H 2 = 1 φ 2 2MP 2 H 2 η ɛ ɛh = 2 φ + 2ɛ. (3.15) H φ The slow-roll conditions re then ɛ 1 nd η 1, nd they correspond (up to some constnts) to 1 2 φ 2 V nd φ H φ. In the slow-roll regime the slow-roll prmeters cn be expressed in terms of derivtives of the potentil ɛ M 2 P 2 η 2M 2 P ( ) V 2 ɛ V V V V + 4M2 P 2 ( ) V 2 2η V + 4ɛ V, (3.16) V where we hve implicitly defined the potentil slow-roll prmeters ɛ V nd η V. In smll-field models of sclr field infltion the ccelerted expnsion hppens for smll field vlues. For instnce, in Guth s originl infltion model the sclr field is in flse vcuum seprted by potentil brrier from the true vcuum (see figure 3.2). In tht cse the phse trnsition is of first order nd bubbles of the true vcuum strt forming s the temperture of the universe drops nd the field tunnels from the flse to the true vcuum. This originl scenrio suffers from the grceful exit problem: in spite of continuous bubble nucletion, the flse vcuum expnds so rpidly tht bubbles never percolte. Since the typicl size of bubbles is much smller thn our horizon it is impossible to end up with our homogeneous nd isotropic universe tody. This problem is solved in the new infltionry scenrio [31], where no first order phse trnsition tkes plce. In this scenrio the potentil is very flt round the origin, but hs true vcuum wy from the origin, such tht sclr field slowly rolls towrds the true vcuum (see figure 3.3()). It is lso possible to generte n infltionry expnsion for lrge field vlues. In the chotic infltion scenrio [34] the expecttion vlue of the sclr field is initilly φ M P nd φ 0. In such model the slow-roll conditions re stisfied nd the sclr field rolls down the potentil to its minimum (see figure 3.3(b)). Chotic infltion is very ttrctive scenrio becuse it gives successful infltion for very generl potentils. Exmples re the m 2 φ 2 potentil for mssive sclr field or the λφ 4 potentil for self-intercting sclr field, which we shll discuss in more detil in section 3.5.

38 30 CHAPTER 3. INFLATIONARY PERTURBATIONS V(φ) V(φ) SLOW-ROLL REGIME REHEATING REHEATING () New Infltion φ SLOW-ROLL REGIME φ (b) Chotic Infltion Figure 3.3: Schemtic form of the potentils for the New Infltion scenrio nd Chotic Infltion scenrio. In New Infltion the inflton field sits initilly t the origin of very flt potentil nd slowly rolls to the true vcuum. In Chotic Inflton the initil field vlue is very lrge O(MP ) nd its velocity very smll, such tht the slow-roll conditions re stisfied. When the field reches the true vcuum it first oscilltes, converting its kinetic energy into het nd prticles, process clled reheting. 3.2 The importnce of cosmologicl perturbtions In the previous section we explined the infltionry prdigm nd showed tht n infltionry expnsion solves the horizon nd fltness problems of the stndrd cosmologicl model. It is however debtble how problemtic these problems relly re. For exmple, the universe could simply hve strted out extremely flt nd homogeneous over super-horizon distnces, such tht we end up with the universe s we know it tody. The horizon nd fltness problems therefore do not necessrily men tht the stndrd cosmologicl model is inconsistent, implying tht period of infltion is not must. Still, infltion provides dynmicl mechnism for generting the initil homogeneity nd fltness of the universe. But, much more importntly, infltion mkes specific predictions for the observble universe which cn be tested. These predictions follow from studying perturbtions during infltion. The ide is tht vcuum fluctutions during infltion cn, under certin circumstnces, provide the initil seeds for the structure formtion in the universe nd explin the spectrum of temperture fluctutions in the CMB. Let us elborte bit more on this generl ide. In the previous chpter we studied generl perturbtions on top of n expnding bckground. As consequence we run into the guge problem of Generl Reltivity, but

39 31 s we hve seen in section 2.4 it is possible to define the liner guge invrint sclr perturbtion w ζ nd the tensor perturbtion γ ij. Such perturbtions hve some (comoving) wve number k, or similrly, some physicl wvelength λ phys = /k. Now, the behvior of the perturbtions very much depends on the rtio of this physicl wvelength wvelength to the Hubble scle R H defined in (3.6). Generlly speking, when the wvelength of perturbtions is smller thn the Hubble rdius (sub-hubble), the solutions of the perturbtions re plne wves in Fourier spce. However, when the wvelength of the perturbtions becomes lrger thn the Hubble rdius (super-hubble), the behvior chnges completely. It cn be shown tht the perturbtion is conserved [23, 24, 35]. on super-hubble scles, tht is, ẇ ζ 0, nd γ ij 0, for k H. (3.17) This hs been demonstrted for the (guge invrint) curvture perturbtion to ll orders in perturbtion theory by invoking generl, non-liner definition of tht prticulr vrible [36]. As consequence the mplitudes of vcuum fluctutions of fields re frozen in on super-hubble scles nd, due to extreme squeezing of the quntum stte, they pper s clssicl perturbtions, in the sense tht correltors tht involve nti-commuttors (mplitudes) re lrge compred to those tht involve commuttors of perturbtions nd moment. Clssiclity is lso often ssocited with decoherence, which describes how quntum system evolves into stte which most closely resembles clssicl stte through n interction with the environment. In the cse of cosmologicl perturbtions, one ide is tht the system of super-hubble modes decoheres through n interction with sub-hubble modes. More informtion on the quntum-toclssicl trnsition nd decoherence of cosmologicl perturbtions cn be found in, for exmple, Refs. [4, 37 50]. Now, if the universe undergoes period of infltionry expnsion, the following specil sitution occurs. See figure 3.1. During infltion the Hubble rdius is pproximtely constnt, but the physicl wvelength grows exponentilly. Tht mens more nd more modes exit the Hubble volume nd re frozen in, i.e. the modes do not evolve on super-hubble scles. When infltion ends, the universe is dominted by reltivistic prticles (rdition er) nd lter by non-reltivistic mtter (mtter er). In these periods the Hubble scle grows in time, nd in fct, it grows fster thn the physicl wvelength. Thus, fluctutions tht exited the Hubble scle during infltion re-enter the Hubble scle t lter times nd perturb the cosmic fluid. These initil perturbtions of the cosmic fluid re the seeds for the formtion of structure due to grvittionl collpse. The initil density perturbtions re evolved vi trnsfer functions from the time tht they re-enter the Hubble volume to, for instnce, the lst scttering surfce, from where we cn compute the temperture fluctutions in the CMB (figure 1.1). With modern CMB experiments such s the Plnck stellite, the gol is to mesure these tem-

40 32 CHAPTER 3. INFLATIONARY PERTURBATIONS Figure 3.4: The CMB temperture ngulr power spectrum observed by the Plnck mission [7]. perte fluctutions s ccurtely s possible. The power spectrum of these fluctutions is presented in figure 3.4. In order to predict the power spectrum for temperture fluctutions, we need some initil condition for the cosmic fluid. This initil condition for the temperture power spectrum is clled the primordil power spectrum. As explined bove, this primordil power spectrum cn be computed from the vcuum fluctutions on super-hubble scles. Of course, the primordil power spectrum depends on the chrcteristics of the prticulr infltionry model. Since the temperture fluctutions in the CMB constrin the primordil power spectrum, mesurement of the CMB thus constrins our infltionry models. This shows how we cn probe the infltionry er vi the first light of the universe. Note tht in this work we re minly interested in the description of perturbtions in n infltionry universe. How exctly these primordil perturbtions re relted to temperture fluctutions in the CMB is beyond the scope of this work, nd we refer the reder to Refs. [23, 24].

41 3.3 The primordil power spectrum for sclr perturbtions 33 In this section we shll compute the primordil power spectr for the sclr perturbtion w ζ. The sclr primordil power spectrum ws first computed in Ref. [51 55] (see lso [56]). The power spectrum cn be derived from the qudrtic ction for first order perturbtions. For w ζ this ction is S (2) [w ζ ] = 1 2 d 3 xdt { 2 3 φ N H 2 ẇζ 2 ( ) } 2 i w ζ. (3.18) The dotted derivtive is here defined s Ẋ = dx/( Ndt). This ction ws derived in Ref. [57] (see lso [10]), nd in chpter 5 we shll compute this ction in more generl setting. In order to clculte the power spectrum of vcuum fluctutions of the sclr perturbtions we usully quntize the field by defining conjugte momentum nd imposing the cnonicl commuttion reltions. In Fourier spce, this normlly mens tht the field is expnded in terms of cretion nd nnihiltion opertors, w ζ (x) = = d 3 k (2π) 3 eı k x w ζ ( k, t) d 3 k [ ] (2π) 3 w ζ,k (t)â k e ı k x + w ζ,k(t)â e ı k x. (3.19) k The nnihiltion opertor destroys the ground stte â k Ω = 0 nd the cretion nd nnihiltion opertor stisfy the commuttion reltion [â k, â k ] = (2π)3 δ 3 ( k k ). The mode functions w ζ,k re solved from the eqution of motion for w ζ, which in turn is derived from the ction (3.18). The equl-time two-point sclr correltor is then w ζ (t, x)w ζ (t, x ) = d 3 k (2π) 3 w ζ,k 2 e ı k ( x x ) [d ln k] 2 w ζ (k, t) sin(k x x ) k x x. (3.20) Here the expecttion vlue is tken with respect to the ground stte, X Ω X Ω, nd we hve mde use of the commuttion reltions. Eq. (3.20) defines the (dimensionless)

42 34 CHAPTER 3. INFLATIONARY PERTURBATIONS power spectrum 2 w ζ (k, t) 2 2 w ζ (k, t) = k3 2π 2 w ζ,k(t) 2. (3.21) The min difficulty now is to compute the mode functions w ζ,k. It is convenient to rewrite the ction (3.18) by mking use of new vrible v = zw ζ, z = φ H. (3.22) This vrible is known s the Sski-Mukhnov vrible [10, 18]. If we now lso use conforml time τ by setting N = (t), such tht dτ = dt, we find the ction S (2) [v] = 1 } d 3 xdτ {(v ) 2 ( i v) 2 + (z) 2 z v2. (3.23) Here prime denotes derivtive with respect to τ. We see tht the ction for v is tht of hrmonic oscilltor with time dependent frequency. Using our knowledge from quntum field theory, we now quntize the field v by the mode expnsion v(x) = d 3 k [ ] (2π) 3 v k (τ)â k e ı k x + vk(τ)â e ı k x, (3.24) k such tht the cnonicl commuttion reltion is stisfied, [v(τ, x), v (τ, x )] = ıδ 3 ( x x ), s long s the Wronskin for the mode functions is given by W [v k (τ), v k(τ)] v k (v k) v kv k = ı. (3.25) We thus try to find the solutions which stisfy this condition. The eqution of motion for the mode functions is ) v k + (k 2 (z) v k = 0. (3.26) z φ H By mking use of the fct tht z = = 2ɛ for minimlly coupled model, we cn write to first order in slow-roll prmeters (z) z 2 The conventionl power spectrum P wζ is defined by = 2 H 2 [ 2 ɛ η ], (3.27) w ζ (t, k)w ζ (t, k ) = (2π) 3 δ 3 ( k k )P wζ = (2π) 3 δ 3 ( k k ) w ζ,k 2, nd is thus relted to the dimensionless power spectrum by 2 w ζ (k, t) = k3 2π 2 P wζ.

43 35 where the slow-roll prmeters re given in Eq. (3.15). A quick glnce of the eqution of motion (3.26) revels tht on smll wvelengths, when k 2 (H) 2, the modes v k behve s free hrmonic oscilltors with frequency k. However, for long wvelengths k 2 (H) 2 the lst term domintes, which cn completely lter the behvior of the modes. For n infltionry bckground we find ourselves in the specil sitution where ȧ = H grows in time. This cn be demonstrted in conforml time, where we find in spce of constnt decelertion (constnt ɛ) 1 H = τ(1 ɛ). (3.28) For n infltionry solution, where H is close to constnt, the conforml time runs from to 0. For erly time τ the fctor H 0, such tht the mode functions τ behve s v k exp( ıkτ). In the lte time limit τ 0 on the other hnd it τ 0 is not difficult to see tht v k τ 1 H. The reltionship (3.22) then suggests tht w ζ,k H/ ɛ, such tht the comoving curvture perturbtion is (pproximtely) conserved on super-hubble scles in infltion, due to the lmost constncy of H. We cn mke these sttements more precise by ctully solving the eqution of motion (3.26) (3.27) with the scle fctor (3.28). We cn write ( v k + k 2 ν2 1 ) 4 τ 2 v k = 0, (3.29) with ν ɛ η. (3.30) The eqution (3.29) is well-known nd we my write the generl solution s where α nd β re Bogoliubov coefficients nd [ where H (1) ν ] (2) ( kτ) = H v k = αu k + βu k, (3.31) [ u k (τ) = exp ı π ] 4 (2ν + 1) πτ 4 H(1) ν ( kτ), (3.32) ν ( kτ) nd H ν (1,2) ( kτ) re the Hnkel functions. The fundmentl solutions u k nd u k stisfy the Wronskin condition W [u k, u k ] = ı, such tht the field v(x) is cnoniclly normlized s long s α 2 β 2 = 1 (see Eq. (3.25)). The erly time limit for the Hnkel functions is H ν (1,2) ( kτ) τ 2 [ πkτ exp ı (kτ + π )] 4 (2ν + 1), (3.33)

44 36 CHAPTER 3. INFLATIONARY PERTURBATIONS such tht the fundmentl solutions t erly times become u k = τ e ıkτ 2k (3.34) Thus, if we pick α = 1, β = 0, we find tht the v k in the erly time limit re precisely the positive frequency mode functions of hrmonic oscilltor with frequency k for which the vcuum (defined by â k 0 = 0) is the lowest energy stte. Thus, the chosen normliztion corresponds to minimum energy stte in the infinite (conforml) pst. This initil lowest energy stte is clled the Bunch-Dvies vcuum [58]. Note tht this resoning is only vlid for the dibtic modes, tht is, for those modes for which kτ s τ. For the infrred modes where k 0 one generlly encounters divergenies in expecttion vlues of fields, see e.g. Refs. [59 62]. The mode functions re now properly normlized in the infinite pst, but the solutions re vlid for ll (conforml) time. We now tke the lte time limit, τ 0, which mens tht the physicl wvelength becomes much lrger thn the Hubble scle, k H. In tht limit the Hnkel functions tke the form H ν (1) ( kτ) τ 0 ı ( kτ π Γ(ν) 2 ) ν, (3.35) where Γ(3/2) = π/2. It follows tht the power spectrum for v on super-hubble scles is ( ) 2 2 v(k, τ) = k3 H 2π 2 v k 2 = ( kτ) 3 2ν, (3.36) 2π where corrections of O(ɛ) hve been neglected. By virtue of the reltion (3.22) we cn immeditely write down the expression for the power spectrum for w ζ, ( ) 2 2 w ζ (k, τ) = k3 H 2π 2 w ζ,k 2 H 2 = 2π Often the power spectrum is written s = 1 2M 2 P ɛ 2 w ζ (k, τ) = A wζ ( k k φ 2 ( kτ) 2ɛ η ( H 2π ) ns 1 ) 2 ( kτ) 2ɛ η. (3.37), (3.38) where A wζ is the mplitude of the sclr power spectrum evluted t some pivot scle k nd the spectrl index n s is n s 1 = 2ɛ η. (3.39)

45 37 To give some more mening to the pivot scle, let us illustrte this by considering the CMB. The CMB observes correltions for the temperture nisotropies for different ngulr resolution on the sky, which cn be expressed in the rnge of multipoles 2 l 2500, see figure 3.4. This corresponds to scles 10 k MPc, or 10 4 MPc 1 k.1mpc 1. The ide is tht these correltions were creted during infltion nd tht they were conserved on super-hubble scles (see figure 3.1). Vi the infltionry mechnism we cn infer tht the correltions tht we observe in the CMB were creted roughly e-folds before the end of infltion, depending on the detils of the infltionry model nd the reheting temperture [63, 64]. So some mode exited the Hubble scle 60 e-folds before the end of infltion, becme conserved nd creted the correltions tht we see tody. Now of course, the vlue of the power spectrum chnges slightly for different modes, becuse different modes exited the Hubble rdius t different times. In the CMB we therefore pick certin pivot scle, k 0 = 0.002MP c 1, where the mplitude nd spectrl index of the power spectrum re computed. Tking this scle to be creted when perturbtion exited the Hubble scle some e-folds before the end of infltion, we thus compute the mplitude 2 w ζ (k ) A wζ = (H/(2π)) 2 /(2M 2 P ɛ) when this mode crossed the Hubble rdius. To be more precise, we should compute the spectrum few Hubble times fter Hubble crossing, since the perturbtion is only conserved when k H. However, since H nd φ re lmost conserved during infltion, we cn justify evluting the power spectrum when the mode exits the Hubble scle. Now bck to the power spectrum (3.37). The sclr mplitude mesured by the Plnck mission t the pivot scle k 0 is A wζ This mens tht V/(24π 2 M 4 P ɛ) Tking typicl slow-roll vlue ɛ = O(10 2 ), we find tht typiclly V 1/ GeV during infltion. The spectrl index is equl to one for scle invrint spectrum. But generlly speking, the slow-roll prmeters re smll, but non-zero during infltion. Infltion thus predicts tht the primordil power spectrum is nerly scle invrint. With enough ccurcy we should be ble to probe the devition from scle invrince by mesuring the primordil power spectrum. One of the gret triumphs for the infltionry prdigm of the lst decde hs been the discovery of this devition from scle invrince. The ltest results from the Plnck mission show tht the spectrl index is n s = ± , which confirms the devition by lmost 6 σ. This llows us to constrin infltionry models. In essence, the potentil nd slow-roll prmeters re different for different models, which give distinguishble predictions for e.g. the tilt of the power spectrum.

46 38 CHAPTER 3. INFLATIONARY PERTURBATIONS 3.4 The primordil power spectrum for tensor perturbtions So fr we hve only discussed the primordil power spectrum for sclr perturbtions. Besides sclr perturbtions there re lso tensoril perturbtions, chrcterized by the trnsverse, trceless grviton γ ij. The power spectrum for the grviton ws first computed in Ref. [65]. We shll derive the result here strting from the second order ction for these perturbtions S[γ 2 ] = 1 2 d 3 xdt N 3 M 2 P 4 [ γ ij γ ij ( ) ] 2 k γ ij. (3.40) This ction is derived in the min text in chpter 5. Similrly to the previous section, I would like to compute now the power spectrum of the grviton perturbtions. In order to do so, we cn use n expnsion for the grviton in terms of cretion nd nnihiltion opertors γ ij (x) = 2 M P = 2 M P p=+, p=+, d 3 k (2π) 3 ɛp ij (k)eı k x γ k (t) d 3 k [ ] (2π) 3 ɛp ij (k) γ k (t)â p, k eı k x + γ k(t)â p, k e ı k x. (3.41) ] The opertors stisfy [â p, k, â p, k = (2π) 3 δ p,p δ 3 ( k k ) nd ɛ p ij is trnsverse, trceless polriztion tensor, ɛ p ii = 0 = ki ɛ p ij. It cptures the + nd polriztions of the grviton nd stisfies ɛ p ij ɛp ij = δ p,p, ɛ p ij ɛp kl = 1 2 [P ikp jl + P il P jk P ij P kl ], (3.42) p where P ij = δ ij k i k j /k 2 is the trnsverse projector in momentum spce. The equltime two-point correltor for the grviton is then γ ij (t, x)γ kl (t, x ) = 2 4 d 3 k MP 2 (2π) 3 γ k 2 e ı k ( x x ) 1 2 [P ikp jl + P il P jk P ij P kl ] d ln k 2 γ(k, t) sin(k x x ) 1 k x x 2 [P ikp jl + P il P jk P ij P kl ], (3.43) which defines the dimensionless power spectrum for grvittionl wves 2 γ(k, t) 2 γ(k, t) = 4k3 1 π 2 MP 2 γ k (t) 2. (3.44)

47 39 Now, the difficulty is gin to compute the functions γ k (t). However, we hve chosen the normliztion of the expnsion of γ ij such tht the ction becomes S[γ 2 ] = p 1 2 d 3 kdτ [ (γ p k ) 2 ) ] (k 2 (γ p k ) 2. (3.45) We see tht the ction for γ p k is essentilly twice the ction of for the Sski-Mukhnov vrible v (3.23), due to the sum over polriztions. There is difference in the time dependent mss, which is now given by k 2 /. This gives slight chnge in the tilt of the tensoril power spectrum. We cn go through the sme steps s in the previous section, nd the result for the power spectrum is 2 γ(k) = 2 H 2 π 2 ( kτ) nt. (3.46) M 2 P The tilt of the power spectrum for grvittionl wves is n t = 2ɛ. (3.47) Furthermore we cn define the so-clled tensor-to-sclr rtio r, which is[66] r = 2 γ 2 w ζ = 16ɛ. (3.48) This provides consistency test for single field slow-roll models, r = 8n t. (3.49) If we compre the sclr field power spectrum (3.37) to tht of the tensor field (3.46), we see tht the power spectrum for tensor perturbtions is suppressed by the slowroll prmeter ɛ. Temperture fluctutions in the CMB re therefore dominted by the sclr modes. Our current best constrint for the tensor-to-sclr rtio is r < 0.12, evluted t the pivot scle k 0 = MPc 1 [7]. Fortuntely, there is wy to uniquely detect primordil grvittionl wves in the CMB. This cn be done vi CMB polriztion. The detils re beyond the scope of this work, but more informtion cn be found in Refs. [23, 67]. Generlly speking, the CMB cn be polrized in two different wys: E-mode (prity-even) polriztion nd B-mode (prity-odd) polriztion. A very importnt point is tht B-modes cn only be creted by tensor perturbtions. Thus detection of the primordil power spectrum of B-modes would be smoking gun for primordil grvittionl wves generted during infltion.

48 40 CHAPTER 3. INFLATIONARY PERTURBATIONS 3.5 Exmple: chotic infltion in polynomil potentil In order to mke contct with observtions, let us now mke theoreticl predictions for the power spectrum for single field infltion in the polynomil potentil V (Φ) = b n Φ n. (3.50) The bckground vlue of the field is Φ = φ(t). The potentil slow-roll prmeters re computed by tking derivtives of this potentil ɛ V = M 2 P 2 η V = M 2 P ( V V V V ) 2 = n2 2 M 2 P φ 2 = n(n 1)M2 P φ 2. (3.51) Infltion ends when ɛ V 1, which mens tht the field vlue t the end of infltion is φ end = n 2 M P. (3.52) We re now interested in the vlue of the slow-roll prmeters when perturbtion crosses the Hubble rdius some number of e-folds before the end of infltion. The field there hs some vlue φ i. We therefore compute this number of e-folds N e = tend t i Hdt = φend φ i H φ dφ = 1 φi V MP 2 φ end V dφ = 1 2n φ 2 i M 2 P n 4. (3.53) We cn now express the slow-roll prmeters in terms of the number of e-folds before infltion ends This gives the spectrl index n 4 ɛ V = N e + n 4 η V = MP 2 V n 1 V = 2 N e + n 4. (3.54) n s = 1 2ɛ η 1 6ɛ V + 2η V = (n + 2) N e + n, (3.55) 4 nd the tensor-to-sclr rtio r = 16ɛ 16ɛ V = 4n N + n. (3.56) 4

49 41 Also we find for the mplitude of the sclr power spectrum t Hubble crossing 2 V w ζ (k ) = A wζ 24π 2 MP 4 ɛ V = b [ ( )] n 2n Ne + n 1 2 (n+2) 4 M 4 n 12π P 2 n 2, (3.57) where ll quntities should be evluted t the time of Hubble crossing. Now let s try to compute some of these vlues by using the CMB constrints. We know tht the CMB correltions were generted e-folds before the end of infltion (depending on the energy scle of infltion nd reheting temperture). If we now consider N e = 60 nd tke the potentil V = 1 2 m2 φ 2 we find tht n s 0.97, r 0.13, A wζ 10 2 m2 MP 2, (V = 1 2 m2 φ 2 ). (3.58) For the qurtic potentil we find n s 0.95, r 0.25, A wζ 10 5 λ, (V = 1 4 λφ4 ). (3.59) The mplitude of the sclr power spectrum is observed to be A wζ 10 9, such tht the we need lrge mss m GeV or very smll qurtic coupling λ Figure 3.5 shows the Plnck CMB constrints on the spectrl index nd tensor-to-sclr rtio. From our nlysis we cn see tht the simple qurtic potentil is lredy excluded, nd tht the m 2 φ 2 potentil is disfvored by lmost 2 σ. Some finl remrks: up until now we hve discussed the simplest model of infltion described by single, minimlly coupled sclr field with cnonicl kinetic terms. We hve not yet tried to mke ny contct with our known physics of the Stndrd Model. In the Stndrd Model there is sclr field, the Higgs boson, nd we my be tempted to connect this to infltion. However, for lrge field vlues the Higgs potentil is effectively qurtic potentil, which is excluded by the CMB observtions. Thus, the Higgs boson s formulted in the Stndrd Model cnnot be the inflton field driving the ccelerted expnsion of the universe. However, s we shll see in the next chpter, the theory cn be modified by dding coupling between the Ricci sclr nd the sclr field, which llows for scenrio of Higgs infltion under certin ssumptions.

50 42 CHAPTER 3. INFLATIONARY PERTURBATIONS Figure 3.5: Constrints on the spectrl index n s nd tensor-to-sclr rtio r by the Plnck mission [7]. The contours show the 1 nd 2 σ regions.

51 Chpter 4 Frmes in cosmology 4.1 The Einstein frme versus the Jordn frme In section 3.1 we introduced the simplest infltionry model: tht of single sclr field coupled minimlly to the Ricci sclr. Such n ction is sid to be in the so-clled Einstein frme. For clrity we shll write the ction (2.3) here gin, though we indicte fields defined in the Einstein frme with subscript E, S E = 1 2 d 4 x { } g E MP 2 R E g µν E µφ E ν Φ E 2V E (Φ E ). (4.1) The Ricci sclr R E is here constructed from the Einstein frme metric g µν,e. It is lso possible to couple the sclr field to the Ricci sclr vi some function F (Φ). The ction tkes the form S J = 1 2 d 4 x { } g RF (Φ) g µν µ Φ ν Φ 2V (Φ). (4.2) When the sclr field is non-minimlly coupled to the Ricci sclr, the ction is sid to be in the Jordn frme. The Einstein frme is recovered in the limit F (Φ) = MP 2, but in this work we shll consider generl field dependent coupling. The non-miniml coupling effectively chnges the Newton constnt to some field dependent function. This cn hve some interesting consequences. For exmple, suppose we tke the specific function F (Φ) = MP 2 + ξφ2. It cn be seen tht when ξ 1 nd Φ hs some lrge bckground field vlue of O(M P ) (s in chotic infltion), the effective Newton constnt is reduced, which cn enhnce the ccelertion of the universe. We shll dive deeper into this in one of the following sections on Higgs infltion.

52 44 CHAPTER 4. FRAMES IN COSMOLOGY 4.2 Field redefinitions nd physicl equivlence The Jordn frme ction (4.2) is slight generliztion of the Brns-Dicke theory [68], which fetures sclr field qudrticlly coupled to the Ricci sclr. Computtions in these models re in generl more complicted due to the non-miniml coupling of the sclr field to grvity. For exmple, the Ricci sclr ppers in the eqution of motion for Φ s some sort of effective mss. Conversely, the grvittionl Einstein-Hilbert ction couples directly to Φ, such tht the grvittionl nd mtter Lgrngins re not seprted. Consequently, the Einstein equtions do not tke the simple form (2.2), but re rther some mix of mtter nd grvittionl degrees of freedom. Moreover, if we would compute cnonicl moment for the metric nd sclr field we would find tht they re coupled in the ction, since the metric contins some double time derivtives. Fortuntely, well-known feture of these type of models is tht they cn be brought to form where the sclr field is minimlly coupled to the Ricci sclr (see for exmple Refs. [69, 70]). This is done vi field redefinitions of the metric nd sclr field. In the cse of generl coupling F (Φ) to the Ricci sclr we cn redefine the metric s where the conforml fctor is g µν,e = Ω 2 g µν, (4.3) Ω 2 = Ω 2 (Φ) = F (Φ) M 2 P. (4.4) Such trnsformtion essentilly removes the fctor F (Φ) in front of the Ricci sclr nd replces it by MP 2, but genertes extr terms which re proportionl to derivtives of the conforml fctor Ω. Due to the field dependence of this fctor these dditionl terms re ctully proportionl to the kinetic term for Φ. This leves us complicted, non-cnonicl kinetic term, but lso this cn be written in the common cnonicl form by redefining the sclr field, ( ) 2 dφe = 1 ( 1 + 6M 2 dφ Ω 2 P Ω 2) = M P 2 ( F 2 F 2 ) F V E (Φ E ) = 1 Ω 4 V (Φ) = M P 4 V (Φ), (4.5) F 2 where the prime denotes derivtive with respect to the field, Ω = dω/dφ.. We hve lso redefined the potentil, such tht the combined field redefinitions (4.3) (4.5) bring the Jordn frme ction (4.2) to the Einstein frme form (4.1). We would like to emphsize tht these field redefinitions re different from coordinte trnsformtions discussed in chpter 2.

53 45 The fct tht the Jordn nd Einstein frme re relted by field redefinitions is blessing for us. The reson is tht field redefinitions do not chnge the physicl content of the theory, nd in tht sense the Jordn nd Einstein frme re considered to be physiclly equivlent [71]. We cn therefore compute Jordn frme results vi the much simpler Einstein frme. This works s follows: 1. Trnsform the Jordn frme to the Einstein frme by mking use of (4.3) (4.5). 2. Compute the object of your interest (e.g. infltionry bckground evolution, correltion functions etc.) in the Einstein frme. 3. Re-express your result in terms of Jordn frme quntities using (4.3) (4.5). Becuse mny results re well-known in the Einstein frme, it should therefore be reltively strightforwrd to find similr results in non-minimlly coupled frme. Let us now give simple exmple by considering the bckground equtions of motion in the Einstein frme, H 2 E = 1 6 [ φ2 E + 2V E ] Ḣ E = 1 2 φ 2 E 0 = φ E + 3H E φe + V E. (4.6) Here the dotted derivtive on Einstein frme quntities is defined to include the fctor N E, such tht ȧ E = d E /( N E dt) nd H E = ȧ E / E. The fctor N E llows us to esily switch to e.g. conforml time by setting N E = E. We obtined these equtions by inserting the bckground Einstein frme metric g µν,e = dig( N E 2, 2 E, 2 E, 2 E ) nd sclr field Φ E = φ E into the Einstein equtions (2.2), with stress-energy tensor (2.4). In the Jordn frme we cn do something similr to find the bckground equtions of motion. However, it is much more difficult to find the form of the stress-energy tensor in nonminimlly coupled frme. Normlly, in the computtion of the grvittionl prt of the Einstein equtions, we vry the Einstein-Hilbert ction with respect to the metric, thus we re interested in the vrition δs EH. This is not too complicted becuse the vrition of the Ricci tensor δr µν vnishes s totl derivtive. Thus we quite esily find δ( gg µν R µν ) = gg µν δg µν. Unfortuntely, when some function F (Φ) is coupled to the Ricci tensor, we must tke into ccount extr terms coming from vrition of the Ricci tensor. We then find tht the stress-energy tensor in the Jordn frme is T µν = M 2 P F ( [ µ Φ ν Φ + g µν 1 ) ] 2 gαβ α Φ β Φ 2V (Φ) + ( µ ν g µν )F (Φ). (4.7)

54 46 CHAPTER 4. FRAMES IN COSMOLOGY The box opertor is defined s = g αβ α β. Now we cn find the equtions of motion in the Jordn frme by looking t seprte components of the Einstein equtions nd by the conservtion eqution µ T µν = 0. By inserting the bckground Jordn frme metric g µν = dig( N 2, 2, 2, 2 ) nd sclr field Φ = φ into the Einstein equtions we find tht H 2 = 1 6F Ḣ = 1 2F [ φ2 + 2V 6HF ] ( φ 2 + HF F ) 0 = φ + 3H φ + V 6 ( 2H 2 + Ḣ) 1 2 F. (4.8) In the field eqution for φ we recognize the bckground Ricci sclr, R = 6(2H 2 + Ḣ). A dotted derivtive on Jordn frme quntity is defined to be ȧ/ = d/( Ndt), such tht H = ȧ/. In the derivtion we needed to evlute the stress-energy tensor (4.7), which involved covrint derivtives on the function F (φ), nd hence mde the computtion more complicted. Moreover, with ll the extr terms in the Jordn frme equtions (4.8) compred to the Einstein frme ones (4.6), it is not cler wht re the dominnt terms in the slow-roll regime. Fortuntely, we could hve found the bckground Jordn frme equtions much fster by mking use of the frme trnsformtions. If we seprte the conforml fctor in bckground vlue plus perturbtion, we find from Eqs. (4.3) (4.5) tht N E = Ω N = F 1 2 M P Ω = Ω(t) + δω(x), (4.9) N E = Ω = F 1 2 M ( P ) Ω H + Ω H E = 1 Ω φ E = 1 Ω = M P F MP 2 Ω 2 M φ = P 2 Ω 2 F ( ) H + 1 F 2 F F 2 F φ V E (φ E ) = 1 Ω4 V (φ) = M P 4 V (φ). (4.10) F 2 Here F is function of the homogeneous prt of the sclr field lone, F = F (φ(t)), F = df/dφ. Note the extr fctors of Ω in the reltions between H E nd H nd φ E nd φ. They pper becuse of the difference of the dotted derivtive cting on Einstein frme

55 47 quntities, ȧ E = d E /( N E dt), or on Jordn frme quntities, ȧ = d/( Ndt). Now, if we insert these reltions between the bckground fields into the Einstein frme Friedmnn equtions (4.6), we immeditely recover the Friedmnn equtions in the Jordn frme (4.8). This demonstrtes the usefulness of the physicl equivlence between the two frmes. In principle we do not need to do computtions in the Jordn frme directly, but we cn work in the Einstein frme, nd trnsform bck to the Jordn frme using the frme trnsformtions. For exmple, in the Einstein frme it is simple to do n nlysis of the bckground infltionry solution. Tht is, in the Einstein frme we know tht potentil energy domintes over kinetic energy, nd tht φ E /H E φe 1, such tht we cn find pproximte solutions for the Hubble prmeter nd field during infltion. The result cn be re-expressed in terms of Jordn frme quntities in the end. Of course, this should gree with those results directly found in Jordn frme. 4.3 Cse study: Higgs infltion We shll now discuss possible scenrio for which infltion cn hppen in the Stndrd Model. As we hve demonstrted, n infltionry expnsion cn occur when the universe is dominted by the energy density of sclr field tht is not in the minimum of its potentil. Of course there is sclr field in the Stndrd Model, nd this is the Higgs boson. So let us see whether or not infltion is successful in the Stndrd Model. The Higgs field H is complex sclr field doublet, which in unitry guge leves only one sclr degree of freedom with non-zero vcuum expecttion vlue. Let us cll this field Φ. Its symmetry breking potentil is V = 1 4 λ(φ2 v 2 ) 2. (4.11) Here v = 246 GeV is the vcuum expecttion vlue of Φ. The Higgs mss follows from considering smll fluctutions round the minimum of the potentil, nd is given by m 2 H = 2λv2. Keeping in mind the recent observtion of the Higgs boson nd the mesurement of its mss m H 126 GeV [72, 73], we find tht the coupling constnt is λ Now we turn to infltion. During chotic infltion the Higgs field is fr from the minimum of the potentil with n expecttion vlue M P, such tht Φ v. Thus effectively the Higgs potentil during chotic infltion is qurtic potentil. Moreover, during infltion the energy density of the Higgs field domintes the energy density of the universe, such tht we cn neglect (interctions with) guge fields nd fermions. In section 3.5 we discussed the observtionl predictions for single field infltion in qurtic potentil. We hve seen tht the qurtic coupling λ must be extremely smll (O(10 13 )) in order to gree with the observed mplitude of the power spectrum. Obviously, this excludes the Stndrd Model Higgs from being the inflton field. Aprt from

56 48 CHAPTER 4. FRAMES IN COSMOLOGY the bound on the coupling constnt, we hve seen tht infltion in qurtic potentil is generlly ruled out becuse of the limits on the spectrl index nd tensor-to-sclr rtio (see figure 3.5). Thus it seems tht infltion in the Stndrd Model is not vible nd we should stop here. Fortuntely, it is possible to relx the infltionry constrints on the qurtic potentil by minor ddition to the ction. this ddition is non-miniml coupling of Φ 2 to the Ricci sclr R, such tht the ction becomes S = d 4 x { 1 g 2 (M P 2 + ξφ 2 )R 1 2 gµν µ Φ ν Φ λ } 4 (Φ2 v 2 ) 2. (4.12) ξ is here the non-miniml coupling prmeter (not to be confused with the guge prmeter ξ µ in section 2.2) nd is tken to be positive, ξ > 0. The use of nonzero non-miniml coupling ξ my be motivted s follows: if ξ does not hve the conforml coupling vlue 1 6 (which is the cse), then ξ will run with the energy scle, see Ref. [74]. In other words, if we pick ξ to be zero t some scle, it will not be zero t nother energy scle. Moreover, if we pick ξ to be lrge t some scle, then ξ will remin lrge since the running is genericlly logrithmic. Φ is in principle generl sclr field, which cn be identified with the Higgs boson. In the ltter cse interctions between guge nd fermion fields hve been neglected, ssuming tht the energy density of the universe is dominted by the qurtic sclr potentil during infltion. Due to the non-miniml coupling there is n effective field dependent Plnck mss MP,eff 2 = M P 2 + ξφ2, whose difference from the constnt Plnck mss cn only be noticed when the expecttion vlue of the field is very lrge. This hppens in chotic infltion where Φ M P / ξ. For ξ 1 the infltionry bckground is not noticebly different from tht for miniml coupling in qurtic potentil. However, when ξ 1 the behvior of the bckground chnges drmticlly. In order to study this behvior of the bckground, one possibility is to derive the Friedmnn equtions for the ction (4.12) nd solve for the bckground field nd Hubble prmeter. This hs been done in Refs. [75, 76], but is complicted due to the non-miniml coupling. It is much esier to perform some field redefinitions such tht the ction is trnsformed to the Einstein frme. Notice tht the ction bove is the sme s the Jordn frme ction (4.2) with function F (Φ) = M 2 P + ξφ 2. (4.13) Thus we cn write the ction (4.12) in the Einstein frme (4.1) using the redefinitions

57 49 (4.3) (4.5). We explicitly find tht dφ E dφ = V E (Φ E ) = 1 + ξ(1 + 6ξ)Φ 2 /M 2 P (1 + ξφ 2 /M 2 P )2 λ 4 (Φ(Φ E) 2 v 2 ) 2 (1 + ξφ(φ E ) 2 /MP 2. (4.14) )2 Now, for smll field vlues Φ M P /ξ we find tht φ E Φ nd V E (λ/4)(φ 2 E v2 ) 2, such tht the redefined field Φ E lives in the sme potentil s the Higgs field Φ. For lrge field vlues Φ M P / ξ the behvior chnges completely. Then we find tht Φ M ( ) P ΦE exp. (4.15) ξ 6MP The integrtion constnt hs been chosen such tht Φ E 0 corresponds to Φ M P / ξ, though it cn be chnged by shifting the initil vlue of Φ E. note tht in principle the integrtion constnt cn be determined by demnding tht Φ = Φ E in the minimum of the potentil. The reltion bove gives tht for Φ M P / ξ, or equivlently Φ E 6M P, the Einstein frme potentil tkes the form V E (Φ E ) λm 4 P 4ξ 2 1 [ 1 + exp ( 2ΦE /( 6M P ) )] 2. (4.16) Thus, the potentil becomes exponentilly flt in the limit Φ E 6M P, which is demonstrted in figure 4.1. Intuitively, this ensures tht infltion is successful for theory with lrge non-miniml coupling. Hence the Einstein frme provides n intuitive picture for the infltionry behvior judging from the shpe of the potentil. Let us now compute the potentil slow-roll prmeters in the Einstein frme. We consider the homogeneous prt of the Jordn nd Einstein frme fields, Φ = φ(t) nd Φ E = φ E (t). We find tht ɛ V,E = M 2 P 2 ( dve /dφ E V E ) 2 = M 2 P 2 ( dve /dφ dφ V E dφ E ) 2 M 4 P 4 3 ξ 2 φ 4 η V,E = M 2 P d 2 V E /dφ 2 E V E = M 2 P [ d 2 V E /dφ 2 V E ( ) ] 2 dφ + dv e/dφ dφ d dφ dφ E V E dφ E dφ dφ E 4 MP 2 3 ξφ 2. (4.17)

58 50 CHAPTER 4. FRAMES IN COSMOLOGY V E V Figure 4.1: Normlized potentil in the Einstein frme s function of ξφ/m P. The normliztion V = λm 4 P /(4ξ 2 ) corresponds to the vlue of the potentil in the symptotic lrge field limit. Infltion ends when ɛ V,E 1, such tht φ end = ( ) 1/4 4 M P. (4.18) 3 ξ Note tht we hve expressed the Einstein frme slow-roll prmeters in terms of the Jordn frme field. The number of e-folds before the end of infltion cn lso be expressed in terms of the Jordn frme fields, N e,e = 1 M 2 P = 1 M 2 P 3ξ 4 φe,i φ E,end φi 1 M 2 P V E dφ E dv E /dφ E ( ) 2 V E dφe dφ dv E /dφ dφ ( φ 2 i φ 2 ) end. (4.19) φ end We cn now reexpress the slow-roll fctors in terms of the number of e-folds before the end of infltion 12 ɛ V,E = ( 4Ne,E ) 2 4 η V,E = 4N e,e (4.20) For N e,e = 60 e-folds before the end of infltion we find tht ɛ V,E nd η V,E Thus we find for the mplitude of the sclr power spectrum on super-hubble

59 51 scles A wζ,e λ ξ 2. (4.21) The observed mplitude is A wζ,e 10 9, which mens tht qurtic coupling λ of O(10 1 ) is very possible when ξ Hence lrge non-miniml coupling cretes the opportunity for the Higgs boson to be the inflton field, conclusion tht ws reched in [69] (in the context of Grnd Unified Theories). The ide of (Stndrd Model) Higgs infltion hs been considered recently [77] in light of the observed constrints on the spectrl index nd the tensor-to-sclr rtio by the WMAP mission. The vlues tht re predicted by the Higgs infltion model re n s = 1 6ɛ V,E + 2η V,E 0.97 r = 16ɛ V,E , (4.22) which both gree with the constrints given by the Plnck mission, see figure 3.5 (the R 2 model hs similr predictions s the model with non-miniml coupling). Thus we conclude tht the Higgs boson cn in fct be the inflton field when we introduce lrge non-miniml coupling of the Higgs boson to grvity. 4.4 Frme dependence of perturbtions In the nlysis in the previous section we computed the predictions for the mplitude nd spectrl index of the power spectrum for Higgs infltion. In order to simplify our nlysis we mde the trnsformtion to the Einstein frme nd computed the constrints there. It ws shown tht Higgs infltion is possible s long s there is lrge non-miniml coupling to the Ricci sclr. Now, before drwing this conclusion we hve mde one crucil ssumption: the power spectr in the Jordn nd Einstein frme coincide. Let us now rgue tht this is fr from obvious. The min point is the following: in the Einstein frme we compute the power spectrum for the sclr perturbtion w ζ,e = ζ E H E φ E ϕ E, (4.23) where ll perturbtions nd bckground fields re defined in the Einstein frme. But how does this relte to the power spectrum in the Jordn frme? The guge invrint curvture perturbtion in the Jordn frme is w ζ = ζ Ḣ φ ϕ. (4.24)

60 52 CHAPTER 4. FRAMES IN COSMOLOGY Of course, we hve shown tht the fields re relted by field redefinitions, but this does not men tht the power spectr coincide. In fct, we cn show tht in generl perturbtions in different frmes do not coincide. For exmple, the field perturbtion, curvture perturbtion nd tensor perturbtion in Jordn nd Einstein frme re relted s ϕ E = dφ E dφ ϕ + O(ϕ2 ) ζ E = ζ + 1 F 2 F ϕ + O(ϕ2 ) γ ij,e = γ ij. (4.25) These reltions re obtined by tking the reltions (4.3) (4.5) nd expnding to first order in perturbtions. Thus, generlly speking, perturbtions re frme dependent, i.e. they re not invrint under frme trnsformtions. This frme dependence refers to trnsformtions of fields, which should not be confused with guge trnsformtions, which re coordinte reprmetriztions. The tensor perturbtion in the Einstein frme coincides with tht in the Jordn frme, which suggest tht the corresponding power spectr re the sme, 2 γ E = 2 γ. The sclr perturbtions ζ E nd ϕ E on the other hnd trnsform explicitly under the frme trnsformtions. So how cn we see tht the sclr power spectr coincide in the two frmes? An esy wy to see this is to mke certin guge choice where ϕ = 0. In tht cse we find tht lso ϕ E = 0 nd tht ζ E = ζ. Intuitively it is cler why this works, becuse the function F used in the redefinitions is field dependent. Once the field fluctutions re bsent, the frme trnsformtion only ffects the bckground fields, not the sclr perturbtions. In this uniform field guge we lso find tht w ζ,e = ζ E = ζ = w ζ. Thus lso the sclr perturbtions coincide, nd so do the sclr power spectr, 2 w ζ,e = 2 w ζ. This implies tht the spectrl index nd tensor-to-sclr rtio in both frmes re the sme, nd the Einstein frme constrints tht we hve computed in the previous section re equl to those in the Jordn frme. Of course, one cn express certin observbles such s the mplitude nd spectrl index in terms of either Jordn frme quntities (H, φ, etc.) or in Einstein frme quntities (H E, φe, etc.), but the point is tht the ctul vlue of these observbles is independent of the frme. All of these rguments re bsed on certin guge fixing, but in principle this should lso hold for the completely guge invrint formultion. This is wht we shll study in the next chpter. In order to prove tht the power spectr in Einstein frme nd Jordn frme coincide, we shll derive the second order ction for guge invrint perturbtions directly in the Jordn frme. Considering just the guge invrint perturbtions is insufficient, since we need the ction in order to cnoniclly quntize the theory, which in turn is needed to determine the spectrum of perturbtions. Before we continue though, let us mention briefly tht the sitution with frme trnsformtions

61 shows lot of similrities with the guge issue in chpter 2. There we strted with n originlly covrint ction, thus n ction with some symmetry trnsformtions. However, s soon s the ction is expnded we find tht the perturbtions re guge dependent, thus the symmetry seems to be broken. Here we consider symmetry between the Jordn frme nd Einstein frme, which is mnifest s long s we do not expnd the ction. However, when the ction is expnded, the perturbtions in one frme nd the other do not coincide, thus mking it difficult to pprecite the symmetry. Now, these perturbtive problems cn be resolved for the guge issue by reformulting everything in terms of guge invrint perturbtions. Bsed on the nlogy between this guge problem nd the frme problem, we might guess tht there exist perturbtions which re frme independent. Indeed, this is precisely wht we shll demonstrte in the next chpter. 53

62

63 Chpter 5 Liner guge invrince nd frme independence 5.1 Introduction The gol of this chpter is to derive the free ction for guge invrint cosmologicl perturbtions in the Jordn frme. We use the cnonicl pproch put forwrd recently in [78], which keeps trck of ll the dynmicl nd uxiliry (constrint) fields. We do not impose ny guge fixing, nor do we use ny conforml trnsformtions in the process. As result we find the mnifestly guge invrint qudrtic ction directly in the Jordn frme, with dynmicl sclr nd tensor perturbtions, s well s decoupled, non-dynmicl perturbtions. We show how to generlize the formlism to the cse of more complicted sclr sector with n internl symmetry, such s Higgs infltion. The work in this chpter is motivted by the guge nd frme problems introduced in the previous chpters. First of ll, the guge invrint ction is crucil in order to compute physicl observbles. For exmple, the qudrtic ction derived in this chpter cn be used to compute the sclr nd tensoril power spectrum. Loop corrections to these power spectr (see Refs. [79 82]) nd higher order correltion functions (Refs. [83 86]) re computed from the higher order guge invrint ction. The guge invrint free ction is first step towrds computing such corrections. The higher order ction shll be treted in the next chpter. A second motivtion for our work is to estblish the physicl equivlence of the Jordn nd Einstein frmes t the level of the free ction. Chisholm [87] nd Kmefuchi et l. [88] lredy proved lmost 50 yers go tht, lthough the field equtions my differ in detil under point-trnsformtions of the fields (i.e. trnsformtions without time derivtives of fields), the (Euler-Lgrnge) form of these equtions nd of the stress-energy tensor remins identicl. Thereby the equivlence of two frmes relted by nonliner field trnsformtions is estblished. In this pper we would like to understnd how this equivlence works in detil for the

64 56 CHAPTER 5. LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE cse of the frme trnsformtion from the Jordn to Einstein frme. Specificlly, we re interested in the equivlence t the level of liner perturbtions. As we shll see the relevnt dynmicl sclr nd tensor perturbtions re invrint under chnge of one frme to the other. Thus the primordil power spectr for these perturbtions re independent of the frme they re computed in, nd s such these frme independent perturbtions re extremely helpful in estblishing the physicl equivlence t the level of the qudrtic ction. A brief outline of this chpter: in section 5.2 we formulte the Jordn frme ction in cnonicl form nd demonstrte the clssicl equivlence to the Einstein frme. In section 5.3 we derive the mnifestly guge invrint free ction in the Jordn frme nd estblish the frme equivlence t the level of liner perturbtions. Finlly in section 5.4 we generlize our result to the cse of Higgs infltion, nd derive the free ction for sclr field theory with locl SU(N) or O(N) symmetry. 5.2 Cnonicl ction in the Jordn frme We strt with the Jordn frme ction for sclr field Φ tht is coupled to the Ricci sclr R through some function F (Φ) (see Eq. (4.2)), S = 1 d 4 x g {RF (Φ) g µν µ Φ ν Φ 2V (Φ)}. (5.1) 2 For non-minimlly coupled inflton field F (Φ) = MP 2 + ξφ2, where ξ = 1 6 is the conforml coupling vlue nd ξ = 0 corresponds to miniml coupling. In the following we shll keep F (Φ) completely generl function. In the Lgrnge formultion the ction (5.1) is invrint under coordinte trnsformtions of the metric field g µν. It is precisely this coordinte invrince however which mkes the extrction of true dynmicl fields problemtic. Becuse we re interested in the dynmicl fields in the context of cosmologicl perturbtions, we therefore wnt to brek the generl covrince of the metric by seprting spce-time into sptil surfces of constnt time. To this end we use the ADM [19] decomposition of the metric with the line element ds 2 = N 2 dt 2 + g ij (dx i + N i dt)(dx j + N j dt), (5.2) where N nd N i re clled the lpse nd shift functions respectively. Under time chnge dt the corresponding chnge in coordinte x i is Ndt in the direction perpendiculr to the sptil surfce, nd N i dt in the direction prllel to the surfce. This geometricl interprettion shows tht the lpse nd shift functions correspond to coordinte chnges, which seems to leve the sptil metric g ij s the true dynmicl field. In fct we cn determine this precisely in the Hmiltonin formultion of grvity,

65 57 which is obtined using the ADM metric. The ADM formlism is therefore necessry for first principle quntiztion nd cn be used to check the correctness of ny other quntiztion procedure. We now insert the ADM metric (2.23) into the ction (5.1) such tht we get [19] S = 1 [ d 3 xdt g{ g ij t K ij t K + N(R (3) + K 2 K ij K ij ) 2N i j (K ij g ij K) 2 ] 2g ij j ( i N N k K ik ) F (Φ) + 1 ( t Φ N i i Φ ) } 2 Ng ij i Φ j Φ 2NV (Φ). N The terms multiplying the function F originte from the originl 4-dimensionl Ricci sclr. The extrinsic curvture is (5.3) K ij = 1 2N ( tg ij i N j j N i ) K = g ij K ij. (5.4) It is understood tht the only dynmicl metric field present in the ction is the sptil metric g ij, such tht the covrint derivtives i s well s R (3) re constructed from this metric lone. Now we wnt to find the cnonicl momentum p ij nd p Φ conjugte to g ij nd Φ respectively. In order to find p ij we hve to vry the ction with respect to t g ij, which is up to fctor 1/(2N) equl to vrying the ction with respect to K ij. Since the ction (5.3) contins derivtives of K ij (nd thus double derivtives cting on the metric) we remove these by performing prtil integrtions. The ction then tkes the simpler form S = 1 2 d 3 xdt g { N ( R (3) + K ij K ij K 2) F + 2KF ( t Φ N i i Φ ) 2Ng ij i j F + 1 N ( t Φ N i i Φ ) 2 Ng ij i Φ j Φ 2NV (Φ) }. (5.5) The cnonicl moment cn now esily be extrcted nd re p Φ δs δ t Φ = g 1 ( t Φ N i i Φ ) + gkf N p ij δs = 1 ( g Kg ij K ij) F 1 gg ij F ( t Φ N i i Φ ). (5.6) δ t g ij 2 2N

66 58 CHAPTER 5. LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE In terms of these moment the extrinsic curvture becomes K = 1 gf p F p Φ X [ K ij = 1 gf g ij 1 + F 2 F X p 2pij gij F X p Φ ], (5.7) where p p ij g ij. This cn be inserted into Eq. (5.3) to obtin the ction for nonminimlly coupled sclr field in cnonicl form, S = d 3 xdt { p ij t g ij + p Φ t Φ NH N i H i}, (5.8) where H = 1 gr (3) F gf ( ) p ij g ik g jl p kl F 2 F 2 X + 1 g 1 2X p2 Φ + g 1 2 gij i Φ j Φ + gv (Φ) 1 gf 1 X F pp Φ + gg ij i j F H i =g ij j Φp Φ 2 j p ij. (5.9) The ction (5.8) reduces to the well known cnonicl ction for minimlly coupled sclr field if we set F = M 2 p. The cnonicl ction indeed shows tht the only dynmicl field is g ij, wheres the lpse N nd shift N i functions multiply the constrints. Since p ij is densitized tensor (i.e. it is composed of tensor times the volume element g, see Eq. (5.6)) the covrint derivtive is understood s j p ij = j p ij + Γ i jl pjl, where Γ i jl only depends on sptil derivtives of the sptil metric g ij. In the cnonicl ction indices re rised nd lowered by the sptil metric g ij. Furthermore we use shorthnd nottion where F = F (Φ) nd F = df/dφ, nd we define the convenient vrible p 2 X = F 2 2 F. (5.10) As consequence of the non-miniml coupling between Φ nd R, the ltter contining double derivtives, the moment p nd p Φ re coupled in the Hmiltonin H in Eq. (5.9). Since this leds to coupled equtions when we derive the Hmilton equtions for p ij nd p Φ we would like to decouple the moment. We do this by introducing shifted momentum ˆp Φ p Φ F p. (5.11) F

67 59 Since the shift in the momentum only depends on F (Φ) nd p, the trnsformtion is cnonicl, thus the resulting Hmilton equtions of motion will be equivlent for either ˆp Φ nd p Φ. In terms of the shifted momentum ˆp Φ we find tht we cn write the ction (5.8) s S = d 3 xdt [p ij t g ij + ˆp Φ t Φ + F ] F p tφ NH N i H i, (5.12) where H = 1 gr (3) F + 2 [ p ij g ik g jl p kl 1 ] 2 gf 2 p g X 2 ˆp2 Φ + g 1 2 gij i Φ j Φ + gv (Φ) + gg ij i j F H i =g ij j Φ(ˆp Φ + F F p) 2 jp ij. (5.13) The Hmiltonin H hs drmticlly simplified becuse of the shifted momentum. On the other hnd, there re dditionl terms in the kinetic prt of the ction (5.12) nd the momentum density H i. Our gol is to perturb the ction up to second order in fluctutions round FLRW bckground. Therefore we seprte ll fields in clssicl bckground plus smll perturbtion. For the moment we choose this seprtion s p ij = P(t) 6(t) (δ ij + π ij (t, x)) ˆp Φ = ˆP φ (t) (1 + ˆπ ϕ (t, x)), (5.14) nd for the fields we tke the linerized split s in Eq. (2.24) of section g ij = (t) 2 (δ ij + h ij (t, x)) Φ = φ(t) + ϕ(t, x) N = N(t)(1 + n(t, x)) N i = (t) 1 N(t)ni (t, x). (5.15) The shift N i is pure fluctution, i.e. its bckground vlue is zero. Note tht we keep working with htted quntities ˆP φ nd ˆπ ϕ to clrify tht these re not the cnonicl moment conjugte to φ nd ϕ. Moreover, we hve defined ll perturbed quntities with lower indices in order to void confusion. The inverse metric is then defined s g ij = 2 (δ ij h ij ), nd N i = g ij N j = Nn i, both up to liner order in perturbtions Bckground equtions To recover the bckground equtions we cn perturb the ction (5.12) up to liner order in perturbtions (5.14)-(5.15) nd set the resulting expressions to vnish. In generl this

68 60 CHAPTER 5. LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE gives the Hmilton equtions of motion, which re derived from the bckground ction { S (0) = d 3 xdt P t + ˆP φ t φ + 1 F } 2 F P (0) tφ NH, (5.16) where H (0) = P2 ˆP2 12F + φ 2 X V, (5.17) where F (φ) nd X(φ) re functions of the bckground fields only. Vrying this ction with respect to P nd ˆP φ gives [ P = 6 2 F H + 1 F ] 2 F φ ˆP φ = 3 X φ, (5.18) where we hve re-introduced the dotted derivtive corresponding to ȧ N 1 d/dt nd we hve identified the Hubble prmeter s H ȧ/. Since N cn be picked rbitrry, the ction (5.16) is time reprmetriztion invrint (in the sense tht time cn be rescled by n rbitrry function of time), remnnt of the diffeomorphism invrince of the originl ction (5.1). Equtions (5.18) re the on-shell expressions for P nd ˆP φ. A vrition of the bckground ction (5.16) with respect to nd φ gives the other first order Hmilton equtions Ṗ = 1 F 2 F P φ P2 ˆP φ = 1 F (P) + 2 F ˆP 2 φ 12 2 F + 3 X V ( ) 1 P 2 ( F 12 1 X ) ˆP2 φ V, (5.19) where V = dv/dφ. Finlly we cn vry the bckground ction with respect to N to find the constrint eqution P 2 ˆP2 12F = φ 2 X V. (5.20) If we insert the cnonicl moment (5.18) in Eqs. (5.19)-(5.20) we obtin the bckground Friedmnn nd field equtions H 2 = 1 [ φ2 + 2V 6HF 6F ] Ḣ = 1 ( 2F φ 2 + HF F ) φ + 3H φ ( Ḣ) 1 6 2H F + V = 0, (5.21) which were derived previously in Eq. (4.8).

69 Clssicl equivlence of Jordn nd Einstein frmes The Einstein frme is the frme in which the inflton field is minimlly coupled to grvity. In the Einstein frme the ction in cnonicl form is [78] { S (0) = d 3 xdt P E t E + P φ,e t φ E N } E H (0) E, (5.22) where H (0) E = P2 E 12 E M 2 P nd the Einstein frme moment re + P2 φ,e 2 3 E + 3 EV E, (5.23) P E = 6 2 EM 2 P H E P φ,e = 3 E φ E. (5.24) Note tht H E = ȧ E / E nd dotted derivtives on Einstein frme quntities re defined by ȧ E = d E /( N E dt), see lso the discussion fter Eq. (4.6). This ction cn be obtined from Eq. (5.16) by setting F = MP 2. The definitions for the cnonicl moment follow from vrition of the ction with respect to P E nd P φ,e, respectively. The other Hmilton equtions re obtined by vrying the ction with respect to E nd φ E, plus there is n dditionl constrint eqution tht follows from vrition with respect to N E. Together these give P2 E P E = 12 2 E M P 2 P φ,e = 3 EV E + 3P2 φ,e 2 4 E 3 2 EV E 0 = P2 E 12 E M 2 P + P2 φ,e 2 3 E + 3 EV E, (5.25) which immeditely gives the bckground Friedmnn equtions if we insert the cnonicl moment (5.24), Ḣ E = 1 2 φ 2 E 0 = φ E + 3H E φe + V E HE 2 = 1 [ ] φ2 6 E + 2V E. (5.26) As explined in section 4.2, the Jordn frme nd Einstein frme ctions re relted by conforml trnsformtion of the metric nd redefinition of the sclr field s in Eqs. (4.3) (4.5). Thus the ctions re physiclly equivlent, nd it is esy to demonstrte this

70 62 CHAPTER 5. LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE t the level of the bckground equtions of motion. In Hmiltonin formultion, this mens we wnt to show tht the clssicl Jordn frme ction (5.12) nd Einstein frme ction (5.22), s well s their corresponding equtions of motion (5.24) nd (5.25), re relted vi field redefinitions. Indeed, it is esy to see tht the following reltions for the moment do the trick P E = M P F P 1/2 F P φ,e = ˆP XM P 2 φ. (5.27) Here we hve lso used the bckground reltions (see Eqs. (4.10)) N E = F 1 2 N, E = F 1 2, V E (φ E ) = M P 4 V (φ). (5.28) M P M P F 2 Since the bckground fields in the Jordn nd Einstein frmes re relted by timedependent resclings, we thereby estblish the physicl equivlence between the two frmes t the clssicl level. In the next section we shll demonstrte the equivlence of the Jordn nd Einstein frme ctions up to second order in (quntum) fluctutions. 5.3 Free ction for cosmologicl perturbtions In this section we will derive the free ction for guge invrint cosmologicl perturbtions for ll dynmicl nd constrint (uxiliry) fields. A common pproch is to fix guge by setting either sclr field or metric perturbtions to zero, nd then to solve for the lpse nd shift perturbtions from the linerized constrint equtions, see for exmple Ref. [83]. In this pper we do not solve ny linerized constrint equtions, nor do we use guge freedom to set some fields to zero. Insted we keep ll the fields nd insert liner fluctutions (5.14)-(5.15) into the ction (5.12). The resulting ction up to second order in perturbtions reds { S (2) = d 3 xdt N P ( ) 2π ij h ij ȧ + π ij h ij + 6 ˆP φˆπ ϕ ϕ + P [ F ( ) π ij h ij φ + πij δ ij ϕ + h ϕ + 1 ( ) F ϕ 2 φ 6 F 6 F ( ) F ( + ϕ 3 ϕ + (π ij δ ij + h) F φ ) ] } H (2) nh (1) n i H (1) i, (5.29)

71 63 where h h ij δ ij. The Hmiltonin up to first order in perturbtions is H (1) = 1 2 F [ i j h ij 2 h ] [ P2 π ij δ ij F 4 h 3 F ] 2 F ϕ + 1 ] [ ] ˆP2 φ 12 X 1 [2ˆπ 3 2 X ϕ h X ϕ hv + V ϕ + 2F 2 ϕ, (5.30) where 2 = δ ij i j. The Hmiltonin up to second order in perturbtions is H (2) = F [ 14 2 h 2 h + 12 h i j h ij 12 h ij i l h jl + 14 h ij 2 h ij + F ] F ϕ( i j h ij 2 h) [ + P2 1 18F 2 π ija ijkl π kl + π ij 2 (2h ij hδ ij ) + h ij h ij 3 ( h ijh ij + 1 ) ( ) 1 8 h2 F ϕ(π ijδ ij + h 3 F 4 h) F ( ) ] 3 1 ϕ F [ ˆP φ X ˆπ ϕ h ijh ij + 1 ( ) 8 h2 hˆπ ϕ + X ϕ(2ˆπ ϕ 1 X 1 ( ) ] X h) + ϕ X [( 2 iϕ i ϕ h ijh ij + 1 ) 8 h2 V hϕv + 1 ] 2 V ϕ 2 + i ( [ ]) 1 2 hf i ϕ F h ij j ϕ + F ϕ i ϕ, (5.31) where A ijkl = δ ik δ jl + δ il δ jk δ ij δ kl. Note tht the finl term in Eq. (5.31) is totl derivtive term nd vnishes, but we give this term explicitly for future purpose. Finlly the momentum density up to first order in perturbtions is H (1) i = iϕ ( ˆP φ + P 2 F F ) P 3 ( j π ij + j h ij 1 2 ih ). (5.32) Now tht we hve found the free ction (5.29) we wnt to mke few remrks: The ction (5.29) is quite complicted due to mny coupled fields; It is uncler wht re the dynmicl degrees of freedom in Eq. (5.29); The ction (5.29) is not explicitly guge invrint. Some clrifiction is in order. The ction (5.29) contins mny different fields (e.g. π ij, h ij, ˆπ ϕ, ϕ, n nd n i ) coupled in nontrivil wy. We cn see however tht the n nd n i re non-dynmicl fields they pper without time derivtives. This hs lredy been rgued in section 2.3. They re thus uxiliry fields tht cn be decoupled from the dynmicl fields. Furthermore, there is still guge freedom in the ction (see lso section

72 64 CHAPTER 5. LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE 2.3), mening tht there re four non-physicl degrees of freedom. The 14-dimensionl phse spce of h ij nd ϕ is therefore reduced to = 6-dimensionl physicl phse-spce. Indeed, well known result from cosmologicl perturbtion theory is tht there is only one dynmicl sclr degree of freedom nd two dynmicl tensor degrees of freedom, corresponding to 6 phse-spce dynmicl degrees of freedom. This is not t ll obvious from the ction (5.29). In order to extrct the three dynmicl degrees of freedom nd show the explicit guge invrince of the ction, we only hve to do one thing: decouple ll fields by defining shifted fields tht digonlize the ction. As it turns out, the shifted fields re ll guge invrint nd there re only three dynmicl degrees of freedom. As bonus, the ction cquires nice nd simple form. As strt it is convenient to use the sclrvector-tensor decomposition of the sptil metric nd the sclr-vector decomposition of the shift perturbtion with h ij = 2ζδ ij + i j h 2 + (i h T j) + γ ij n i = is + nt i, (5.33) i h T i = 0, i γ ij = 0 = δ ij γ ij, i n T i = 0. (5.34) The ction cn now be digonlized by defining shifted fields (see Appendix 5.A for derivtion nd definitions of introduced vribles) 2 s (i ñ T j) ˆπ ϕ = ˆ π ϕ 1 2Îϕ π ij = π ij 1 2 (I ij δ ij I) nd the comoving curvture perturbtion n = ñ + I n 2V [ ( s = h ) ] J α 2 = ( (i n T j) 1 (i h T ) j), (5.35) 2 w ζ = ζ Ḣ φ ϕ, (5.36)

73 65 After some long but strightforwrd clcultions, of which we present some intermedite results in ppendix 5.A, we obtin the free ction { [ S (2) = d 3 xdt N 3 z 2 1 2ẇ2 ζ 1 2 ˆP 2 φ ( ) ] 2 i w ζ + F [ γ ij 2 8 P2 A ijkl ( l γ ) ] 2 ij 2 6 X ˆ π ϕ F π ij 2 π kl [ ] 2 [ ] 2 s (i ñ T 2 } V ñ 2 j) + F (1 α) 2 +, (5.37) where nd (see lso (5.39)) z 2 = X φ 2 (H + 1 F = 2 F )2 α = 1 P F 3 ˆP 2 φ 3 V 2 X 6 φ (H = 1 3 ( F 2 F F F 1 )2. (5.38) ) 2 X φ. (5.39) V The ction (5.37) is our most importnt result. When we compre this new free ction to the originl free ction from Eq. (5.29) we cn mke the following remrks: All shifted fields re decoupled in the ction (5.37); The only dynmicl degrees of freedom in (5.37) re 1 sclr (w ζ ) nd 2 tensor (γ ij ) degrees of freedom; All shifted fields in Eq. (5.37) re guge-invrint up to liner order in coordinte trnsformtions. The third point cn be proven in the following wy: tke the definition of the shifted fields from ppendix 5.A nd insert the liner guge trnsformtions (2.29) nd (2.30). One cn verify tht ñ, s nd ñ T i re guge invrint. For the cnonicl moment ˆπ ϕ nd π ij it is possible to see how it trnsforms by tking the definition of the unperturbed moment (5.6) nd (5.11) nd expnding to first order in perturbtions. It cn then be shown tht the shifted moment ˆ π ϕ nd A ijkl π kl re guge invrint. This cn lso be seen by relizing tht ˆ π ϕ = 0 nd A ijkl π kl = 0 re the Hmilton equtions expnded to first order in perturbtions, nd must therefore be guge invrint. Similrly ñ = 0, s = 0 nd ñ T i = 0 re the first order solutions of the uxiliry fields N nd N i.

74 66 CHAPTER 5. LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE Thus our digonliztion procedure hs pid off: we hve obtined simple, mnifestly guge invrint ction (5.37) with one propgting sclr field w ζ nd propgting tensor (spin 2) degree of freedom, the grviton γ ij. A vrition of the ction with respect to the non-dynmicl π ij nd ˆ π ϕ gives the linerized Hmilton equtions of motion. On the other hnd the vrition with respect to ñ, s nd ñ T gives the solutions of the linerized constrint equtions. Therefore the free ction (5.37) contins ll the properties of linerized infltionry perturbtions, s well s the trnsition between the Hmilton nd Lgrnge formlism. Note tht by setting F = M 2 p we obtin the well-known ctions for w ζ nd γ ij from guge invrint cosmologicl perturbtion theory for minimlly coupled sclr field (Ref. [78]). We hve used these miniml ctions in sections 3.3 nd 3.4 to compute the primordil power spectr for sclr nd tensor perturbtions. Also for the nonminimlly coupled cse here we cn in principle quntize the fields by writing the ction in terms of hrmonic oscilltor with time dependent frequency nd quntize the field. But, s we shll see shortly, the free Jordn frme ction is relted to the free Einstein frme ction vi redefinitions of the bckground quntities. Thus we cn use the previously computed results in sections 3.3 nd 3.4 for the sclr nd tensor power spectrum for minimlly coupled sclr field, nd reexpress them in terms of Jordn frme quntities. Finlly we note tht, if we wnt to tret quntum corrections to, for exmple, the inflton potentil or the power spectrum, it is necessry to know the guge invrint cubic nd qurtic vertices. The guge invrint ction up to third order in perturbtion shll be constructed in the following chpters Quntum equivlence of Jordn nd Einstein frmes In section we showed the clssicl equivlence of the Jordn nd Einstein frme ctions in Hmiltonin form. Now we wnt to demonstrte the quntum equivlence of the Jordn nd Einstein frmes t the level of the free ction. Let us first consider the dynmicl sclr w ζ in the Jordn frme free ction Eq. (5.37). The form of the ction for minimlly coupled sclr field in the Einstein frme (obtined by setting F = MP 2, see lso Ref. [78]) is S (2) R E = d 3 xdt N E 3 Ez 2 E [ 1 2ẇ2 ζ,e 1 2 ( ) ] 2 i w ζ,e ze 2 = φ 2 E HE 2, (5.40) 1 where the dotted derivtive here mens ẇ ζ,e = N E tw ζ,e nd w ζ,e is the comoving curvture perturbtion in the Einstein frme. The time dependent prefctor in the ction E

75 67 (5.40) only depends on the bckground fields nd cn therefore be trnsformed to physiclly equivlent prefctor by performing conforml trnsformtion. Indeed, we cn derive the ction for w ζ in the Jordn frme (5.37) from the ction in the Einstein frme (5.40) by redefinition of the bckground fields s in Eqs. (4.10). The free ctions re however only truly physiclly equivlent if the comoving curvture perturbtion does not chnge under conforml trnsformtion, i.e. w ζ = w ζ,e. (5.41) This cn be proved in the following wy. Tke the conforml reltion between the metrics in Einstein nd Jordn frme, g µν,e = F (Φ)g µν. By expnding both sides to first order in perturbtions we cn deduce tht ζ E = ζ + 1 F 2 F ϕ. (5.42) This ws lso shown in Eq. (4.25). Similrly we cn show tht ϕ E = F 1 ϕ 2 φ E φ, (5.43) where the extr fctor of F 1 2 ppers becuse of the difference between the dotted derivtives in the Einstein nd the Jordn frme. It is now esy to see using (4.10) tht such tht H E ϕ E = Ḣ φ E φ ϕ + 1 F 2 F ϕ, (5.44) w ζ,e = ζ E H E φ E ϕ E = ζ Ḣ φ ϕ = w ζ. (5.45) Thus the comoving curvture perturbtion is invrint under conforml trnsformtion up to liner order in perturbtions. This ws first proved by Mkino nd Sski [89] nd Fkir et l. [90]. This frme independence of the comoving curvture perturbtion mens tht it is for exmple possible to compute the ction for the comoving curvture perturbtion in the Einstein frme nd then obtin the ction in the Jordn frme by performing conforml trnsformtion of the metric, which ws done by Hwng [70]. Before, Hwng nd Noh [91] lredy found the field eqution for the comoving curvture perturbtion in the Jordn frme, by considering the linerized Einstein equtions for non-minimlly coupled sclr field in the uniform field nd curvture guges. The frme independence of the comoving curvture perturbtion cn in fct be shown in the fully nonliner pproch, see Ref. [92], but we shll come bck to this in the following chpters. Now, the point is tht, since w ζ,e = w ζ, the sclr power spectrum for w ζ,e

76 68 CHAPTER 5. LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE coincides with tht of w ζ. Of course, the power spectr in different frmes re expressed in the different quntities (H E nd φ E versus H nd φ), but the ctul numericl vlue is the sme, since the bckground fields re relted s (4.10). This cn be seen most esily from the ctions (5.37) nd (5.40). The ctions re expressed in different bckground quntities, but the form of the ction nd eqution of motion for w ζ or w ζ,e is exctly the sme. This estblishes the (quntum) equivlence of the free (qudrtic) ctions in the Jordn nd Einstein frme. Now tht we hve checked the physicl equivlence for the sclr sector, let us see how the rest of the ction (5.37) trnsforms under the conforml trnsformtion. First of ll, let us give the Einstein frme ction for the grviton nd constrint fields, S (2) = d 3 xdt N E 3 E { MP 2 [ ( ( γ ij,e ) 2 γij,e ) ] 2 8 E ˆP 2 φ,e 2 6 E V E ñ 2 E + M 2 P P 2 E π ϕ,e E M P 2 π ij,e A ijkl 2 π kl,e (1 α E ) [ 2 s E 2 E ] 2 [ ] (i ñ T 2 } j),e +, (5.46) E where α E is defined in Eq. (5.68) of Appendix 5.A nd the subscript E denotes tht these quntities depend on the Einstein frme bckground fields (F = MP 2 ). Now we perform the conforml trnsformtion of the ction (5.46) using Eqs. (4.10) nd (5.27). We find tht the Einstein frme ction trnsforms to the Jordn frme ction (5.37) if the grviton nd constrint fields trnsform s ñ E = ñ s E E = s ñ T i,e = ñ T i γ ij,e = γ ij π ϕ,e = ˆ π ϕ π ij,e = π ij. (5.47) So we must show tht these reltions re correct in order to prove tht the complete qudrtic Jordn frme ction is equivlent to the Einstein frme ction. We do this s follows. By considering the conforml trnsformtion of the metric g µν,e = F (Φ)g µν

77 69 we cn show tht to first order n E = n + 1 F 2 F ϕ, h E 2 E = h 2, h T i,e E s E E = s, nt i,e = n T i, = ht i, γ ij,e = γ ij. (5.48) If one then tkes the expressions for the shifted fields ñ, s nd ñ T i from ppendix 5.A one finds tht they trnsform in precisely the sme wy s Eq. (5.47), which ws necessry to prove the frme equivlence. We hve not yet discussed how the perturbed moment trnsform under frme trnsformtion. It is convenient to consider first how the unperturbed moment between the two frmes re relted. The ction (5.12) cn be trnsformed to the Einstein frme ction through the following reltions: N E = ΩN, N i,e = Ω 2 N i, g ij,e = Ω 2 g ij,, p ij E = 1 Ω 2 X Ω 2 pij, p Φ,E = X ˆp Φ, µ Φ E = Ω 2 µφ, V E = 1 Ω 4 V, (5.49) where s before Ω 2 = F (Φ). Of course the first order perturbtions of metric nd sclr field give precisely the reltions (5.42), (5.43) nd (5.48). For the moment we find π ϕ,e = ˆπ ϕ ( F F X ) ϕ X π ij E = πij F F ϕδij. (5.50) If we now use this in the complete definition of the shifted moment ˆ π ϕ nd π ij from Eqs. (5.35) (nd Eqs. (5.60) in the ppendix) we find tht they re invrint under frme trnsformtion. Therefore the complete free Jordn frme ction (5.37) nd Einstein frme ction (5.40) nd (5.46) re physiclly equivlent. 5.4 Higgs infltion Some time go Bezrukov nd Shposhnikov [77] investigted the ide (bsed on the erlier work [69]) tht the Higgs boson cn be the inflton field if it is non-minimlly coupled to grvity. The requirement for Higgs infltion is lrge non-miniml coupling ξ 1, ensuring the fltness of the Higgs potentil for lrge field vlues, which we hve discussed in more detil in section 4.3. Since then there hs been much debte whether or not quntum corrections destroy the fltness of the Higgs potentil, thereby spoiling Higgs infltion. One nd two loop corrections hve been clculted in both the

78 70 CHAPTER 5. LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE Einstein [93 95] nd Jordn [96 98] frmes. Although there is some debte bout the clcultionl methods, ll loop clcultions predict tht Higgs infltion is vlid if the Higgs mss lies in specific rnge, testble by the LHC. In Refs. [93 98] the vlidity of Higgs infltion ws tested in the infltionry regime. Here the Higgs boson hs lrge expecttion vlue H M P / ξ nd slowly rolls down the inflton potentil. Recently [99 103] however, the vlidity of Higgs infltion ws questioned in the smll-field limit where the Higgs expecttion vlue is H M P / ξ. Hertzberg [102] considered the generl cse of theory with multiple sclr fields. It ws found tht, for the pure grvity nd kinetic sectors, the smll field effective theory hs cut-off t n energy scle of M P if there is only one sclr field, but when more thn one sclr field is involved the cut-off is M P /ξ. In the Jordn frme the cut-off M P /ξ cn be lmost directly red off from the sclr-grviton interction term. However, when considering sclr-sclr scttering vi grviton exchnge, the lowest order digrms dd up to zero for single sclr field. Therefore the ctul cut-off scle is M P. In the Einstein frme this is even more cler. After the conforml trnsformtion the cut-off ppers in dimension 6 sclr kinetic term, but this term cn be removed vi nonliner field redefinition. In the cse of multiple sclr fields the bove resoning no longer pplies. In the Jordn frme the lowest order digrms do not vnish becuse the sclr fields re not identicl, giving the cut-off M P /ξ. In the Einstein frme the unitrity violting kinetic term cnnot be removed by field redefinition, becuse it is in generl not possible to bring the kinetic term into cnonicl form for multiple sclr fields (see Ref. [104] for more detils). The rguments bove pply to the pure grvity nd kinetic sectors of the theory, but even for the single field cse Hertzberg [102] finds tht sclr selfinterctions due to the non-polynomil potentil in the Einstein frme most likely cuse unitrity problems t the scle M P /ξ. Now we switch to the Stndrd Model. In this cse the Higgs doublet contins in principle 4 sclr fields, but the 3 Goldstone bosons re eten up by the longitudinl degrees of freedom of the W ± nd Z bosons (when the symmetry is broken). Therefore one might wonder if the cut-off shows up in the terms contining these guge bosons. Indeed, Burgess, Lee nd Trott [103] showed tht in the Stndrd Model the cut-off scle M P /ξ ppers in the Higgs-guge interctions. Now the crucil point is tht the cut-off M P /ξ of the smll-field effective theory is very close to the energy scle t the end of infltion H end λ/12m P /ξ (where λ 0.13 t the electrowek scle), which is lso the point where the smll-field limit becomes vlid. This mens tht higher order opertors, needed to solve the unitrity problems t the cut-off scle M P /ξ in the smll-field effective theory, will ffect the infltionry theory nd thereby destroy Higgs infltion. Therefore it seems tht Higgs infltion is ruled out s vlid theory.

79 71 In contrst to the previous rguments, Bezrukov et l. [105] recently showed tht the effective cut-off ctully depends on the expecttion vlue of the Higgs inflton field. An intermedite region ws identified for field vlues M p /ξ < φ < M P / ξ where the cut-off scle scles s Λ = ξφ 2 /M P. The uthors showed tht ll relevnt energy scles throughout the evolution of the universe re below the corresponding cut-off scle. Still quntum corrections could spoil the unitrity of Higgs infltion, nd systemtic wy of obtining quntum loop corrections hs been proposed. Considering the ongoing discussion bout the unitrity of Higgs infltion, we would like to mke few remrks. First of ll there re so fr no rigorous clcultions of quntum corrections to the Higgs potentil or Higgs-guge interctions in the smll field limit (φ < M P /ξ) or the intermedite region (M p /ξ < φ < M P / ξ). Secondly, the cut-off scle is found in the Jordn frme by considering Higgs-grviton interctions. As we hve shown before, the inflton perturbtion ctully combines with the sclr prt of the metric to form one guge invrint vrible. Therefore, in order to consistently clculte quntum corrections to either the Higgs potentil or Higgs-guge interctions, we need to construct the completely diffeomorphism 1 invrint Higgs ction. In the previous section we derived the free ction for single inflton field. In this section we pply this to the Stndrd Model Higgs ction with non-miniml coupling to grvity. The ction reds S = d 4 x g { (M ) 2 P 2 + ξh H R g µν (D µ H) D ν H λ ) 2 } (H H v2, (5.51) 2 where H is the complex Higgs doublet with vev H 0 = v/ 2 nd D µ H = ( µ igw µ τ i 12 ) g B µ H, (5.52) is the covrint derivtive with Wµ nd B µ the SU(2) nd U(1) guge bosons with coupling constnts g nd g, nd τ = σ /2. Now, in conventionl chotic infltionry scenrios the inflton field is rel sclr field with lrge clssicl expecttion vlue. The Higgs doublet in Eq. (5.51) contins two complex sclr fields, thus it is not cler wht the inflton field is. This becomes more obvious when we choose the following decomposition of the Higgs doublet H = Φ ( ) 0 exp (iτ α ), (5.53) 2 1 where Φ nd the α re now four rel sclr fields nd the projection vector (1, 0) T ensures tht H is doublet. In this decomposition it is esy to see tht H H = 1 2 Φ2. 1 We use the terminology diffeomorphism invrince here insted of the previously used guge invrince in order to void confusion with the well known concept of guge freedom in the Stndrd Model

80 72 CHAPTER 5. LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE Furthermore, by redefinition W µ = W µ 1 g µα iα W b µ[τ, τ b ] we cn bsorb the three would-be Goldstone bosons α into the guge bosons W µ. In fct, we cn lwys perform n SU(2) rottion on the Higgs doublet in Eq. (5.53) such tht the three wouldbe Goldstone bosons dispper, which corresponds to fixing the unitry guge H = Φ/ 2 2. If we now define W µ ± = 1 ( W 1 µ iw 2 ) µ 2 Z 0 µ = (5.54) 1 g2 + g 2 ( gw 3 µ g B µ ), (5.55) the ction (5.51) becomes { S = d 4 x 1 g 2 (M P 2 ξφ 2 )R 1 2 gµν µ Φ ν Φ 1 4 λ(φ2 v 2 ) 2 m2 W v 2 gµν W µ + Wν Φ m 2 Z 2 v 2 gµν ZµZ 0 νφ 0 2 }, (5.56) with m 2 W = 1 4 g2 v 2 nd m 2 Z = 1 4 (g2 + g 2 )v 2. We see tht the first prt of the ction (5.56) is equl to the ction (5.1) with the identifiction F (Φ) = MP 2 +ξφ2 nd V (Φ) = 1 4 λ(φ2 v 2 ) 2 (see lso (4.12)). The second prt contins the Higgs-guge interction terms. If we now wnt to clculte the free guge invrint ction for the Higgs sector of the SM we cn simply do the expnsions (5.14)-(5.15) with Φ = φ(t) + ϕ(t, x), which results in (5.37). The field ϕ is in this cse identified with the Higgs boson nd φ is the clssicl bckground field with vev v, which drives the infltionry expnsion of the universe when φ is fr wy from its minimum. Since the guge bosons W µ ± nd Zµ 0 re pure fluctutions, the free ction (5.37) contins dditionl terms S (2) dd = d 3 xdt 3 { m2 W v 2 ḡµν W + µ W ν φ m 2 Z v 2 ḡµν Z 0 µz 0 νφ 2 }, (5.57) where ḡ µν = dig( N 2, 2 δ ij ) is the bckground ADM metric. So in the end we hve shown tht the Stndrd Model with non-minimlly coupled Higgs boson, which hs locl SU(2) symmetry, cn be written in mnifestly diffeomorphism wy t the level of qudrtic perturbtions. The qudrtic ction cn be used to extrct the diffeomorphism invrint propgtors for the Higgs inflton field nd guge bosons. If we wnt to clculte quntum corrections to the free propgtors nd the Higgs potentil in n 2 However, note Refs. [106, 107], which point out the delicte sitution concerning unitry guge t the quntum level. It ws found tht the Goldstone bosons give non-zero contribution to the Colemn-Weinberg potentil when the Higgs field is displced from its minimum.

81 73 invrint mnner, we need to find the guge invrint ction up to third nd fourth order in perturbtions. We shll come bck to this in the next chpters. The nlysis in this section shows tht, when the bckrection from the W ± nd Z bosons is neglected, the single sclr field nd the SU(2) Higgs doublet led to identicl qudrtic ctions for cosmologicl perturbtions in non-minimlly coupled models. The pproch tht we used in this section cn be pplied in much more generl setting thn the Stndrd Model. In fct, ny theory with locl SU(N) or SO(N) symmetry (for exmple GUT theories) cn be written in terms of one dynmicl sclr nd number of mssive guge bosons if we use the suitble generliztion of (5.53). Therefore it is lwys possible in these theories to hve single light inflton field, thus opening the wy for n infltionry er. 5.5 Summry In this chpter we derived the qudrtic ction for guge invrint perturbtions in the Jordn frme. The min result is Eq. (5.37), which shows tht the dynmicl degrees of freedom re the guge invrint curvture perturbtion w ζ nd the grviton γ ij. We showed tht the Jordn frme ction cn be derived from the Einstein frme ction by performing conforml trnsformtion of the metric, thereby estblishing the physicl equivlence of the two frmes t the level of qudrtic fluctutions. An importnt spect is the frme independence of the comoving curvture perturbtion nd the grviton. Once the ction is expressed in terms of these perturbtions, the equivlence is esily estblished. Moreover, the primordil power spectrum for sclr nd tensor perturbtions cn be computed in either frme nd the result is identicl. We lso extended the guge invrint pproch to non-minimlly coupled sclr field theories with n internl symmetry, such s the Higgs sector of the Stndrd Model with n SU(2) guge symmetry. By suitble representtion of the sclr sector in such theories we cn identify one sclr field with non-zero expecttion vlue, whose perturbtion combines with sclr perturbtion of the metric to form guge invrint combintion. The other sclr degrees of freedom decouple t the qudrtic level nd re guge invrint by themselves. Thus lso such theories cn be written in mnifestly guge invrint wy t the qudrtic level. 5.A Digonlizing the qudrtic ction In this ppendix we show how to obtin our min result for the free ction (5.37) from the ction (5.29). The min ide is tht the physicl degrees of freedom re decoupled vi digonliztion procedure.

82 74 CHAPTER 5. LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE Moment sector We first consider the sector of the ction (5.29) contining only the moment π ij nd ˆπ ϕ. We cn write this s S π (2) = d 3 xdt N ˆP { 2 ] [ ] 3 φ [ˆπ } 2 6 X ϕ 2 + Îϕˆπ ϕ P F 2 π ija ijkl π kl + I ij π ij, (5.58) where A ijkl = δ ik δ jl + δ il δ jk δ ij δ kl nd ( ) Î ϕ = h 23 X ϕ + 2n + 2 ˆP X ϕ φ 1 X [ ( )] I ij = 32 F h ij F H + 1 F P P 2 F δ ij 2 h δ ijn + 62 F (i n j) [ ( P 3 2 ( ) φ) F F ϕ + F ϕ + F P F h ij ( ) 1 ϕ] δ ij. (5.59) F Note tht on-shell, i.e. using Eq. (5.18), the term multiplying h ij in Eq. (5.59) simplifies to 2h ij. If we now define shifted moment ˆ π ϕ =ˆπ ϕ + 1 2Îϕ π ij =π ij (I ij δ ij I), (5.60) where I δ ij I ij, we find tht we cn write the momentum ction in digonlized form, S π (2) = d 3 xdt N ˆP { 2 [ 3 φ 2 6 X ˆ π ϕ 2 1 ] [ 4Î2 ϕ P2 A ijkl 18 4 π ij F 2 π kl 1 ( Iij I ij I 2)]}. 4 (5.61) The extr terms contining Î2 ϕ nd ( I ij I ij I 2) now give qudrtic contributions of n 2 nd (i n j) 2, which llows us to digonlize lso the terms contining these fctors. Lpse sector From the digonlized momentum sector of the ction, see (5.61), we collect ll the terms contining the lpse function n. We collect these terms from the rest of the ction (5.29) s well, which re only in the nh (1) term. We find tht we cn write S n (2) = d 3 xdt N 3 { V n 2 + I n n }, (5.62)

83 75 where I n = P 6 2 ḣ F [ i j h ij 2 h ] ( ) ˆPφ 3 + PF 2 2 ϕ F 2 ϕ F 2 + C ϕ ϕ C h h 2 + P s 2 (5.63) where s is the longitudinl prt of n i nd C ϕ = V + P2 F ˆP F F φ Ω 2 6 X Ω P ( ) F φ 2 2 F on-shell φ + (1 + 2F )3H φ 6F (H 2 + [ ( )] Ḣ) C h = P F H + 1 F + F P 2 F ˆP 2 φ 2 6 X + V on-shell 0. (5.64) Note tht on-shell, these expressions simplify gretly. We now shift the lpse, ñ = n 1 2V I n, (5.65) such tht we cn write S n (2) = { d 3 xdt N 3 V ñ V I2 n }. (5.66) Thus, we hve digonlized the lpse sector. Shift sector For the shift n i we do the sme. We collect ll the terms contining n i coming from Eqs. (5.61) nd (5.66) nd the n i H (1) i term in Eq. (5.29). We cn now write { ( [ (i ] S n (2) i = d 3 xdt N F n 2 ( ) )} 2 j) i n i (i n j) α + J ij, (5.67) where [ (i n j) ] 2 = (i n j) (i n j). Furthermore, α = 1 P F 3 ˆP 2 φ 3 V 2 X 6 on-shell 1 3 ( 1 ) 2 X φ, (5.68) V

84 76 CHAPTER 5. LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE nd J ij = ḣij + J hij h ij ( + δ ij αḣ + P 6 4 V [ k l h kl 2 h ] P 3 4 V ) F F 2 ϕ + J ϕ ϕ + J ϕ ϕ + J h h, (5.69) where [ ] J hij = P 3 2 F 2 H + 1 F on-shell 0 2 F J ϕ =2 F F P ( 1 ˆP 3 2 F 3 φ + PF ) = on-shell X 2H φ V 2F V ( ) F ˆPφ J ϕ =2 φ + F 3 F + P F (3 2 F 2 + P C ϕ 3 2 F V [ ] J h = P 3 2 F + 2 H + 1 F P C h 2 F 12 2 V on-shell 2 XH 2 V ( ) φ H on-shell 0, (5.70) where C ϕ nd C h were defined in Eqs. (5.64). In order to digonlize the shift sector (5.67), we seprte the longitudinl nd trnsversl degrees of the shift vector, n i = 1 i s + n T i, with i n T i = 0. (5.71) Then, up to boundry terms, the shift ction (5.67) becomes S n (2) i = d 3 xdt N [ F (1 2 ] 2 [ s (i n T j) α) 2 + ] 2 i j s + J ij 2 (i n T j) h ij. (5.72) Note tht we hve been ble to seprte the longitudinl nd trnsversl degrees of the shift vector. Now we substitute the sclr-vector-tensor decomposition of h ij, see (5.33). With this decomposition, we cn show tht [ ( i j s J ij 2 = 1 2 h ) ] J ( [ ] h (i n T j) (i h T ) j) (i n T j) ij J hij h ij =, 2 s 2

85 77 where J = δ ij J ij nd we did not write down ny totl derivtive terms. We cn now digonlize the ction (5.72) by introducing nd the ction becomes S n (2) i = 2 s 2 (i ñ T j) d 3 xdt N 3 { 1 2 F [ ( = 2 s h ) ] J α 2 = ( (i n T j) 1 (i h T ) j), (5.73) 2 [ ] 2 [ ] 2 s (i ñ T 2 [( ) ] j) (1 α) 2 + F (i h T 2 j) 8 [ ( F 1 2 h ) ] 2 } J α 2. (5.74) Guge invrint sclr ction So fr we hve digonlized the perturbed moment π ij nd ˆπ φ nd the perturbed lpse n nd shift N i functions. Wht remins from the ction re the field nd metric perturbtions ϕ nd h ij. For the interested reder who wnts to check the derivtion we present n intermedite result where we collect ll the terms contining these fields. Since there re mny terms contining these fields we will do this step by step. First of ll we collect the leftover terms in the ction (5.29) S (2) h,ϕ = { [ d 3 xdt N P F F h ϕ F [ 2 2 ( ) F ( F ϕ 2 φ + F F ) ( ϕ 3 ϕ + h φ ) ] 1 4 h 2 h h i j h ij 1 2 h ij i l h jl h ij 2 h ij + F ] [ F ϕ( i j h ij 2 h) 1 P 2 ( ) 1 P 2 ( ) ] φ F 2 1 X 6 + V hϕ 2 [ ] [ ] P2 P F + φ 2 6 X + V h 2 8 P 2 5 P F 3 + φ 2 X V h ij h ij 6 4 [ ( ) P2 1 P 2 ( ) ] φ F 2 1 X 6 + V ϕ } 2 2 iϕ i ϕ. (5.75) Remember tht we digonlized the moment by shifting them. By doing this we mnged to write ll the momentum terms from the ction in terms of qudrtic shifted

86 78 CHAPTER 5. LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE moment, see Eq. (5.61). As consequence, we obtined dditionl terms tht do not contin the moment, but do contin h nd ϕ nd combintions. We find ˆP φ 2 1 P 2 X 4Î2 φ 2 h 2 ϕ = 6 X X ϕ P ( ) φ 2 2 X 2 X ϕ 2 + P φ h ϕ + P2 φ X 8 X 2 hϕ + P φ X ϕϕ. (5.76) 6 3 X Now we consider the other prt of Eq. (5.61) with the term I ij I ij I 2. For this term we will put the cnonicl momentum P on-shell in the term multiplying h ij in Eq. (5.59). This will drmticlly simplify our clcultions. We find { P 2 1 ( Iij 18 4 I ij I 2) = F h ij h ij F 4 8 ḣ2 4P [ 3F 2 h ij h ij 1 ] [ ] 2 hḣ + P F 2 3 h ijh ij h 2 [ ( ) F 2 ( ) 2 ( ) 6 φ2 + 2P2 1 4P φ F ( ) ] 1 F F 2 ϕ 2 F F [ ( 2P F 6F 2 ( ) F ( ) )] 1 φ + 2F 2 F 2 ϕ ϕ 6 F 2 P F F F 2 ϕ2 [ 4 F ( ) F ( ) ] F ḣ ϕ 4P 1 4 φ + F 3 2 ḣϕ + 2P F F 3 2 F F h ϕ [ ( ) 2P F 2P + φ 2 ( ) ] } F F 9 4 hϕ. (5.77) F F The completion of the squre for n lso gives extr terms, { In 2 4V = F 8V P F ḣ2 2 ( P φ + PF ) 2 ϕ 2 + 4C ϕ F 3 ( P φ + PF F 6 2F 2P ( P 3 5 φ + PF ) ḣ ϕ + 2P F 2F 3 2 F Cϕḣϕ 2 [ i 2 j h ij 2 h ] ( P ( 6 2 ḣ 3 P φ + PF ) 2F + 4F ( F 2 2 ϕ P ( 6 2 ḣ 3 P φ + PF ) ) ϕ + C ϕ ϕ 2F 1 ( 2 F [ i j h ij 2 h ] ) 2 } 2F 2 ϕ 2F 2 2F ) ϕ + C ϕ ϕ ) ϕ ϕ 2 F C2 ϕϕ 2. (5.78)

87 79 As for the finl contributions coming from completing the squre in the shift sector, we use the following useful reltions: which gives F 8 V P F 1 1 α = V P F 3α 1 = 1 V [ ( F 1 J α [ ( 2 P φ 2 V 2 X ḣ 2 2 h ) 2 on-shell P 2 φ 6 X 2 h ) ] 2 2 = P 2 4 V V ( 2F H ) 2 F F 2 on-shell X φ V, (5.79) [ i j h ij 2 h ] P 4 V ] 2 F F 2 ϕ + 3J ϕ ϕ + 3J ϕ ϕ. (5.80) Now tht we hve obtined ll the terms contining h nd ϕ, we wnt to express the sclr prt of Eqs. (5.75), (5.76), (5.77), (5.78) nd (5.80) in terms of the guge invrint vrible w ζ = ζ (H/ φ)ϕ. This is tedious procedure which we will not explicitly write down in this Appendix. An importnt step in the derivtion is prtil integrtion of ll the terms contining only one time derivtive. The finl digonlized free ction we obtin is S (2) = { [ d 3 xdt N 3 z 2 1 2ẇ2 ζ 1 2 ( ) ] 2 i w ζ + F [ γ ij 2 8 ( γij ) 2 ] ˆP 2 φ P2 A ijkl 2 6 X ˆ π ϕ F π ij 2 π kl [ ] 2 [ ] 2 s (i ñ T 2 } V ñ 2 j) + F (1 α) 2 +, (5.81) where z 2 = X φ 2 (H + 1 F = 2 F )2 φ (H F 2 F F F )2. (5.82)

88 80 CHAPTER 5. LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE The free ction is presented in the min text in Eq. (5.37). For future reference we give here the on-shell expressions for the shifted fields constrint fields 2 s 2 (i ñ T j) ñ =n ζ H + = 2 s = (i n T j) F 2 F Ḣ 1 H φ ϕ + H + 1 F 2 F ( 2 h ) ( (i h T j) ) H + 1 F 2 F ( ẇ F ζ H H V F F ) 2 s 2 ( ζ + 1 F ) 2 F ϕ 1 2F X φ 2 ( H F F ) 2 ẇ ζ, (5.83) which re the first order solutions of the uxiliry fields N nd N i. Finlly we note tht we cn, in similr wy s in section 3.3 write the sclr prt of the ction (5.81) s time dependent hrmonic oscilltor. First pick conforml time τ by setting N(t) = (t), nd then define the Mukhnov-Sski vrible which chnges the sclr prt of the ction to S v (2) = d 3 xdτ 1 2 v = zw ζ, (5.84) ] [v 2 ( i v) 2 + (z) z v2. (5.85) Note tht in the Jordn frme the fctor z is different from the Einstein frme fctor z E = φ E /H E, but they re relted vi conforml trnsformtion.

89 Chpter 6 Non-liner guge invrince nd frme independence : the sclr sector 6.1 Introduction One of the min gols of this chpter is to compute the cubic guge invrint ction for sclr cosmologicl perturbtions. A motivtion for this is tht knowledge of the cubic ction is first step towrds computing non-gussinities from infltion. In the previous chpter we computed the qudrtic guge invrint ction (5.37) for cosmologicl perturbtions. As we hve seen in the section 3.3, from this qudrtic ction we cn compute the primordil power spectrum, which is closely relted to the 2-point function for super-hubble perturbtions. However, the sclr primordil power spectrum only presents two observble prmeters in the CMB, nmely its mplitude nd spectrl tilt. This lredy constrins nd excludes mny populr models, but it is not enough to nil down precisely the infltionry potentil. Fortuntely we cn do better. In principle we could mesure higher order temperture correltions in the CMB, for exmple 3-point correltor. Since such higher order correltion is lso believed to originte from 3-point functions of super-hubble perturbtions during infltion (which is closely relted to the primordil bispectrum), this cn better constrin our infltionry models. A wy to see this is by simply noticing tht there cn be mny shpes for 3-point function in the CMB mp. Certin infltionry models predict lrge contribution to prticulr shpes, nd thus (non-)detection of certin shpes cn further constrin the model spce of infltion. One of the simplest infltionry models is tht of single sclr field in slow-roll scenrio. In his seminl work [83] Mldcen found the third order ction for infltionry perturbtions nd showed tht locl non-gussinity, chrcterized by the prmeter f NL, re too smll to be observed, f NL 1. Since then there hs been n increse in infltionry models tht predict lrge (locl, or other type of) non-gussinity tht

90 82 CHAPTER 6. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE SCALAR SECTOR my be observed. For exmple, in more generl sclr theories in e.g. [84] nd [85] the locl non-gussinity cn be lrge nd observble, f NL 1. Of course there exist mny other models, often involving multiple fields, with lrge non-gussinity. Since the im of this work is not to discuss non-gussinities nd scn the model spce, I shll refer the interested reder to some reviews [12, 108, 109] nd references therein. Recently the ESA Plnck mission [110] hs given the strongest constrints on primordil non-gussinity. For exmple, non-gussinity of the locl type is constrined by f NL = 2.7 ± 5.8. On one hnd this hs excluded mny models, which is good, but on the other hnd it does not point in the direction of some specific infltionry models. In this chpter we return to the strting point for computing the primordil bispectrum, which is the cubic ction. Importntly, the cubic ction nd 3-point functions hve been derived for ζ, the curvture perturbtion (see for exmple [83]). As we hve discussed in chpter 2, ζ is in principle guge dependent quntity. However, it cn hve guge invrint mening once it is defined on prticulr hypersurfce. For exmple, the curvture perturbtion on uniform field hypersurfces, w ζ is guge invrint perturbtion. There exist however mny other guge invrint perturbtions. At the liner level they re only relted by resclings, but t second order different guge invrint vribles re relted by non-liner trnsformtions. In this chpter we set out to find the cubic ction for second order guge invrint cosmologicl perturbtions. In generl, the guge invrint cubic ctions for different vribles re different, but we show tht the bulk prt of the guge invrint ction coincides for different vribles nd it is the bulk ction tht determines the evolution of non-gussinity. In tht sense the evolution of non-gussinity is unique. The difference between the ctions lies in boundry terms, which generte dditionl, disconnected prts of the bispectrum. A second gol of this chpter is to compute the cubic guge invrint ction for single sclr field non-minimlly coupled to grvity. As we hve seen in chpter 4 such n ction is in the Jordn frme, which is relted to the Einstein frme by field redefinitions. This should lso hold t the perturbtive level, but is not obvious since perturbtions re generlly frme dependent: perturbtions in one frme do not coincide with those in nother. However, s we hve seen in the previous chpter, it is possible to express the perturbed qudrtic ction in terms of liner frme independent cosmologicl perturbtions. In the second prt of this chpter we demonstrte tht prticulr second order guge invrint vrible is lso frme independent. When the cubic ction is expressed in terms of this vrible, it is very esy to trnsform bck nd forth between Jordn nd Einstein frme. This is exctly wht we shll use to compute the cubic ction for sclr perturbtions in the Jordn frme. A short outlook for this chpter: in section 6.2 we define the Einstein frme ction nd sclr perturbtions. In section 6.3 we construct different guge invrint vribles t second order nd show how they re relted. In section 6.4 we construct the cubic

91 83 ction for different guge invrint sclr perturbtions nd show how these ctions re relted. In the second prt, section 6.6, we discuss different frmes nd perturbtions in those frmes, nd compute the cubic ction for sclr perturbtions in the Jordn frme by using frme independent vribles. An importnt remrk for this chpter: since computtions re performed in the Einstein frme for the lrgest prt of this chpter, we shll define Einstein frme quntities nd perturbtions without subscripts. In section 6.6 we shll relte our results to those in the Jordn frme, where ll Jordn frme quntities nd perturbtions re defined with subscript J. Note tht this is different thn wht is used in the rest of this thesis, see for exmple chpter 4, chpter 5 nd chpter Action nd perturbtions In the first prt of this chpter we shll work with the Einstein frme ction for sclr field minimlly coupled to grvity, see Eq. (2.3). As we hve seen in the previous chpter, it is convenient to use the ADM metric (2.23) which seprtes dynmicl degrees of freedom from uxiliry ones. The ction then becomes S = 1 { d 3 xdtn g MP (R 2 (3) + N 2 ( E ij E ij E 2)) 2 } where + N 2 ( 0 Φ N i i Φ ) 2 g ij i Φ j Φ 2V (Φ) E ij = 1 2 ( 0g ij i N j j N i ), (6.1) E = g ij E ij, (6.2) nd R (3) is the sptil sclr curvture computed from (sptil derivtives of) g ij lone. This ction is most esily obtined from Eq. (5.5) by setting F = MP 2 nd identifying E ij = NK ij. We re interested in the third order ction for guge invrint perturbtions, which due to the non-linerity of generl reltivity mens tht the perturbtions re second order (see Eq. (2.16) nd below for the mening of certin order in perturbtion theory). We thus insert the following perturbtions Φ = φ(t) + ϕ(x) g ij = (t) 2 e 2ζ(x) δ ij N = N(t) (1 + n(x)) N i = (t) 1 N(t) [ (t) 1 i s(x) + n T i (x) ]. (6.3)

92 84 CHAPTER 6. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE SCALAR SECTOR These perturbtions re simplified version of those in Eqs. (2.24) (2.25). Let us explin why we choose this simplifiction. First of ll, we re in this chpter interested in the dynmicl sclr perturbtion t second order. In the previous chpter we hve seen tht the dynmicl, liner sclr w ζ is combintion of ζ nd ϕ. We thus consider these perturbtions, but leve the tensor perturbtion γ ij for the next chpter. Wht bout other perturbtions of the metric (2.25), nmely the other sclr h nd the vector h T i? The bsence of h nd h T i cn be understood in the following wy. As cn be seen from Eq. (2.29) the metric perturbtions h nd h T i trnsform under guge trnsformtion only with the guge prmeters ξ nd ξi T, respectively. In other words, they trnsform under sptil guge trnsformtions lone. It is possible to fix the sptil guge freedom by setting h = 0 nd h T i = 0. In tht cse we only look t temporl guge trnsformtions, nd the guge invrint perturbtions we consider must be invrint under t t + ξ 0. A different wy to understnd the bsence of h nd h T i is by looking t the lpse nd shift functions. From Eq. (6.1) it is cler tht N nd N i re non-dynmicl, uxiliry fields. In the previous chpter we hve seen tht the non-dynmicl perturbtions of these uxiliry fields cn be decoupled from the dynmicl perturbtions by digonliztion procedure. The n, s nd n T i combine with the other perturbtions into guge invrint, non-dynmicl perturbtions ñ, s nd ñ T i. In fct, the metric perturbtions h nd h T i re completely bsorbed into the shifted constrint fields s nd ñ T i, s cn be seen from their definitions (5.83). If we tke the on-shell expressions s = 0 nd ñ T i = 0, the metric perturbtions h nd h T i completely drop out of the ction. Insted of decoupling the uxiliry fields from the rest of the ction, it is simpler to solve for the fields order by order in perturbtion theory. Their solution cn then be re-inserted in the ction. Compring to the previous sector, this corresponds to setting the shifted perturbtions (5.35) to zero, i.e. ñ = 0, s = 0 nd ñ T i = 0. To see tht these pproches re equivlent, let us derive the equtions of motion for the lpse nd shift fields 0 = 1 { } g [R N 2 (E ij E ij E 2 )]MP 2 N 2 ( t Φ N i i Φ) 2 g ij i Φ j Φ 2V (Φ) 2 0 = j [ N 1 (E jk g ik Eδ ij )M 2 P ] N 1 ( t Φ N i i Φ) i Φ. (6.4) If we insert the perturbtions (6.3) nd expnd to liner order, we find the solutions n = ζ H + n T i =0 2 s 2 = 2 2 φϕ 2HM 2 P [ ] ζ H + χ, (6.5)

93 85 where 2 χ 2 = φ 2 2H 2 MP 2 ẇ ζ, (6.6) nd w ζ = ζ Hϕ/ φ. These indeed gree with the conditions ñ = 0, s = 0 nd ñ T i Eq. (5.83) in the limit where F = MP 2. = 0 in 6.3 Non-liner guge dependence nd guge invrint perturbtions Second order guge trnsformtions In section 2.14 we studied the guge dependence of perturbtions in dynmicl spcetime. Eq. (2.14) shows the guge trnsformtion to ll orders in perturbtion. We now study more precisely how the sclr field perturbtion ϕ nd sclr metric perturbtions ζ in FLRW bckground (6.3) trnsform under second order guge trnsformtions. From these trnsformtions we re then ble to deduce wht re the second order guge invrint vribles. We re not interested in second order perturbtions of the non-dynmicl constrint fields n, s nd n T i, becuse it turns out we only need the first order solutions of the uxiliry fields in order to derive the cubic ction for the dynmicl sclr perturbtion. Furthermore, in order to simplify the second order guge trnsformtions for ϕ nd ζ we neglect terms with sptil derivtives, since these re unimportnt in the long wvelength limit. Moreover, since we hve fixed the sptil guge freedom by setting h = 0 = h T i, we only hve to consider temporl guge trnsformtions t t + ξ 0. For the sclr field Φ the guge trnsformtions (2.18) ct on the perturbtion s ϕ ϕ + φ Nξ [ φ Nξ 0 + φ( Nξ ] 0 ) + 2 ϕ Nξ 0 + O(ξ i, i ξ 0, i ϕ). (6.7) As before the dotted derivtive denotes φ = dφ/( Ndt). Note tht the terms O(ξ i, i ξ 0, i ϕ) re of second order in perturbtions. Similrly, the metric tensor trnsforms under (2.18). If we consider the curvture perturbtion ζ then ζ ζ + H Nξ [Ḣ Nξ 0 + H( Nξ 0 ) + 2 ζ ] Nξ 0 + O(ξ i, i ξ 0, i ζ). (6.8) Here H ȧ/. Also here we hve not explicitly written other higher order terms tht include ξ i nd/or sptil derivtives of ζ or ξ 0. The precise trnsformtions of ζ nd ϕ cn be found, for exmple, in Refs. [11, 14 16, 22].

94 86 CHAPTER 6. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE SCALAR SECTOR Second order guge invrint vribles From Eqs. (6.7) nd (6.8) combintion cn be formed which is guge invrint under temporl guge trnsformtions to first order. This combintion is the comoving curvture perturbtion w ζ introduced before in Eq. (2.31) nd whose (Jordn nd Einstein frme) ction ws derived in the previous chpter, see Eq. (5.37). For completeness, the comoving curvture perturbtion is w ζ = ζ Ḣ φ ϕ, (6.9) nd is clled the comoving curvture perturbtion. This guge invrint combintion of sclr metric nd field perturbtions ppers not to be unique, in the sense tht it cn be rescled by ny function depending on the bckground fields ( φ, H). These bckground quntities re by construction fixed nd do not induce dditionl guge trnsformtions. Thus there re in principle rbitrrily mny guge invrint combintions t first order relted by resclings. However the comoving curvture perturbtion is the unique guge invrint combintion tht reduces to ζ in the guge ϕ = 0. It is therefore lso known s the curvture perturbtion on uniform field hypersurfces. Alterntively, we cn define nother guge invrint vrible, the field perturbtion on uniform curvture hypersurfces, by w ϕ = ϕ φ H ζ. (6.10) Although guge invrint to first order in perturbtions to second order w ζ chnges under the guge trnsformtion ( ) w ζ w ζ 1 φ 2 H φ Ḣ H 2 [H 2 ( Nξ 0 ) Ḣ φ ϕh Nξ 0 ] + ẇ ζ Nξ 0 + O(ξ i, i ξ 0, i w ζ ) w ζ + (2) ξ w ζ, (6.11) where the second order guge trnsformtion of w ζ is indicted s (2) ξ w ζ, s in Ref. [111], with n dditionl subscript. The comoving curvture perturbtion w ζ cn be mde guge invrint to second order by dding qudrtic perturbtions in ϕ to its definition. For exmple, we know tht to first order ϕ chnges under guge trnsformtion s ϕ ϕ + φ Nξ 0. Therefore we cn define W ζ = w ζ [( φ H φ Ḣ H 2 ) H 2 φ 2 ϕ2 2 1 φẇ ζ ϕ w ζ + F ϕ [w ζ, ϕ], (6.12) ]

95 87 which is guge invrint to second order 1. This guge invrint vrible is the second order generliztion of the comoving curvture perturbtion, or the curvture perturbtion on uniform curvture hypersurfces, since it reduces to ζ in the guge ϕ = 0. In Refs. [14, 21] this vrible is constructed by fixing the vector field ξ µ t ech order such tht ϕ = 0. Here, we hve constructed the guge invrint vrible by demnding tht the guge trnsformtion t ech order is countered by pproprite terms. It is lso possible to derive second order guge invrint generliztion of the field perturbtion on uniform curvture hypersurfces, whose first order definition is given by w ϕ in Eq. (6.10). Since this vrible is relted to w ζ t first order s w ϕ = φϕ/h, it trnsforms in similr wy s Eq. (6.11) under second order guge trnsformtion ( ) w ϕ w ϕ + 1 H φ 2 φ H φ Ḣ H 2 [ φ 2 ( Nξ 0 ) φ H ζ φ Nξ 0 ] + ẇ ϕ Nξ 0 + O(ξ i, i ξ 0, i w ζ ) w ϕ + (2) ξ w ϕ, (6.13) In this cse we blnce the temporl guge dependence by qudrtic terms in ζ which trnsform s ζ ζ + H Nξ 0. Now we cn define nother vrible W ϕ = w ϕ 1 2 [ ( ) ] H φ φ H φ Ḣ φ2 H 2 H 2 ζ H ẇϕζ w ϕ + F ζ [w ϕ, ζ], (6.14) which is guge invrint to second order s well (see footnote 1). Agin, this vrible reduces to ϕ in the guge ζ = 0, nd is thus the correct second order generliztion of the field perturbtion on uniform curvture hypersurfces. In Refs. [14, 21] this vrible is constructed by fixing the guge such tht ζ = 0 t ech order. In principle we could hve picked ny combintion of qudrtic perturbtions in ζ nd ϕ to blnce the second order guge dependence of w ζ, or of w ϕ. The dvntge of the bove vribles is tht they reduce to the liner perturbtions if one of the sclr perturbtions is set to zero. For exmple, in the cse where ϕ = 0, W ζ ζ, or when ζ = 0, the other guge invrint vrible W ϕ ϕ. As it turns out, this mkes it very useful to find the guge invrint ction t third order in terms of these vribles. We will discuss this in the next section. It is obvious tht the two guge invrint vribles W ζ nd W ϕ re not equl t second order. Their difference cn be expressed in terms of guge invrint second 1 Here we hve not written terms with vectors, tensors nd sptil derivtives of w ζ which re unimportnt on super-hubble scles. The complete second order guge invrint vribles cn be found in, for exmple, [14].

96 88 CHAPTER 6. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE SCALAR SECTOR order prt. For exmple, substituting the equlity ζ = w ζ + Ḣ φ ϕ in the definition of W ϕ (6.14), we find W ϕ = φ [ ( ) ] W ζ 1 φ H 2 φ Wζ 2 1 H H W ζẇζ = φ H [W ζ + Q(W ζ, W ζ )]. (6.15) So, the two guge invrint vribles re relted by bckground fctor φ/h nd term qudrtic in W ζ nd its derivtives, which is therefore guge invrint by itself. Note tht, since t first order W ζ = HW ϕ / φ, Eq. (6.15) cn lso be esily inverted. Both guge invrint vribles W ζ nd W ϕ cn be clled physicl degrees of freedom, in the sense tht they do not depend on the non-physicl guge degrees of freedom. The question is if the vribles describe the sme physics. One could imgine tht, since the vribles re not equl, the 2-point nd 3-point functions re lso different nd this my give different results. Of course, in order to clculte the 2-point nd 3-point functions nd describe the dynmics of the guge invrint vribles, one should study the ction for them, which is wht we do next. 6.4 The guge invrint ction for cosmologicl perturbtions Guge invrince t zeroth order We now insert perturbtions on top of fixed homogeneous, isotropic nd expnding bckground into the ction (6.1). If we consider the bckground lone, the ction is S (0) = d 3 xdt { 3M N 3 P 2 H } 2 φ 2 V (φ). (6.16) This ction is trivilly covrint, in the sense tht the bckground quntities trnsform t zeroth order under coordinte trnsformtions, such tht the bckground fields re fixed functions of the coordintes. In other words, if = (t) in one coordinte system, thn = ( t) in nother coordinte system.

97 89 The hmiltonin constrint, momentum eqution nd field eqution for the bckground re found by vrying the ction with respect to the N, nd φ, respectively, 3M 2 P H 2 = 1 2 φ 2 + V (φ) 2M 2 P Ḣ = φ 2 0 = φ + 3H φ + V (φ). (6.17) It turns out tht it is useful to define vrible z s z φ H, (6.18) such tht the vrious slow-roll prmeters cn be written s, z 2 ɛ Ḣ H 2 = 1 2 MP 2 η ɛ ɛh = 2 ż zh. (6.19) Here the sme definitions hve been used s in [84]. These slow-roll prmeters re very useful for finding the dominnt contributions to n point functions from the ction Guge invrince of qudrtic ction The ction to liner order in perturbtions vnishes due to the clssicl bckground equtions of motion. The first non-trivil ction of perturbtions is the second order ction in perturbtions. This ction ws derived for generl non-minimlly coupled theory in the previous chpter, see Eq. (5.37). For simple minimlly coupled theory we set F = MP 2, such tht the ction for the comoving curvture perturbtion is S (2) [w ζ ] = d 3 xdt N 3 z 2 { 1 2ẇ2 ζ 1 2 ( ) } 2 i w ζ, (6.20) up to totl derivtive terms. z ws defined in Eq. (6.18). We cn lso derive the qudrtic ction for w ϕ, which is t the liner level relted to w ζ s w ϕ = ( φ/h)w ζ. The ction my be written s S (2) [w ϕ ] = d 3 xdt N 3 { 1 Using the equtions of motion we find tht 2ẇ2 ϕ 1 2 ( i w ϕ ) } [ 3 ż] 2 3 z w2 ϕ, (6.21) [ 3 ż] 3 z = φ 2 MP 4 V 2 φ H2 MP 2 H V V. (6.22)

98 90 CHAPTER 6. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE SCALAR SECTOR Guge invrince of cubic ction In order to study non-gussinities in the Cosmic Microwve Bckground rdition (CMB), n-point functions such s the primordil bispectrum or trispectrum must be derived. The 3-point functions re found from the third order ction in cosmologicl perturbtions. Of course, the physicl bispectrum is only found when the perturbtions in the (tree-level) cubic ction re physicl, i.e guge invrint. Schemticlly, the third order ction in ζ nd ϕ tkes the form S (3) [ζ, ϕ] = d 3 xdt N 3 { AO(ζ 3 ) + BO(ζ 2 ϕ) + CO(ζϕ 2 ) + DO(ϕ 3 ) }. (6.23) One cn find the explicit form of the ction from Eq. (6.1) by inserting the perturbtions (6.3) nd substituting the solution of the constrint equtions (6.4). We only hve to do this to first order in perturbtions, see Eq. (6.5). The third order solutions multiply the bckground equtions of motion, nd the second order solutions multiply the Hmiltonin nd momentum constrint evluted t first order. The explicit form of the AO(ζ 3 ) terms is S (3) ζ = d 3 xdt N 3 M 2 P ( 3ζ ζ H { ( ζ 2 + 2ζ ζ H ) [ i j s ζ i j s ζ 2 2 ) ( 2 ζ 2 ζ + ζ H ( 2 ) 2 ] s ζ 2 ) ( i ) 2 ζ } 2 is ζ i ζ 2 s ζ 2 z 2 M 2 P ζ 2 ( 3ζ ζ ) H, (6.24) up to sptil derivtives nd boundry terms, where s ζ is the first order solution for s in the guge ϕ = 0 (see Eq. (6.5)), 2 s ζ 2 = 2 2 ζ H z 2 M 2 P ζ. (6.25) We lso present the explicit form of the DO(ϕ 3 ) terms: { ( S (3) ϕ = d 3 xdt N 3 1 ( ) ) 2 z 3 4 MP 2 ϕ ϕ 2 i ϕ + ϕ iχ ϕ [ ( ) 2 ( ) ( ) ] 4 zϕ χ ϕ i j χ ϕ i j χ ϕ 2 [ 1 z MP 6 V 1 4 z M 2 P 2 2 i ϕ V 1 ] } 6 V ϕ H z3 MP 4 ϕ 2 ϕ H z2 MP 2 ϕ 2 2 χ ϕ 2, (6.26) where 2 χ ϕ 2 = 1 2 z 2 M 2 P ( ϕ z ). (6.27)

99 91 The perturbed ctions (6.24) nd (6.26) were first derived by Mldcen [83]. The AO(ζ 3 ) terms hve been derived for generlized sclr theories in Ref. [84], see lso [85]. Mnifest guge invrince: cubic ction for W ϕ The ction (6.23) does not pper to be covrint due to the guge dependence of ζ nd ϕ, see Eqs. (6.7) nd (6.8). However, we know tht the complete ction is covrint, nd should be covrint to this order s well. To mke this more mnifest, we cn try to express the cubic ction in terms of the liner guge invrint vribles w ζ or w ϕ, since the third order terms in (6.23) only trnsform under liner guge trnsformtions. In doing so we hve severl possibilities. The first option is to eliminte ll ϕ dependence in the ction by replcing ϕ = w ϕ + zζ, (6.28) where gin z = φ/h such tht S (3) = d 3 xdt N 3 { DO ( wϕ 3 ) + EO(ζ 3 ) + F O(ζ 2 w ϕ ) + GO(ζwϕ) 2 } = S (3) GI [w ϕ] + S (3) GD [w ϕ, ζ]. (6.29) The first prt S (3) GI (w ϕ) now only depends on O(wϕ) 3 terms, nd is therefore explicitly guge invrint. The second prt S (3) GD [ζ, w ϕ] is on the other hnd guge dependent, becuse it contins cubic terms ζ 3, ζ 2 w ϕ nd ζwϕ 2 tht cnnot blnce ech other s guge trnsformtions. However, the originl ction (2.3) is diffeomorphism invrint, nd this guge dependence should somehow cncel. One should not forget tht lso the second order ction chnges under guge trnsformtion. Although (6.21) is guge invrint to first order, to second order in guge trnsformtions it is not. The guge trnsformtion (6.13) thus genertes third order terms from the second order ction (6.21). Following Rigopoulos [111], under second order guge trnsformtion of w ϕ, S (2) [w ϕ ] S (2) [w ϕ ] + { d 3 xdt N 1 3 δs (2) [w ϕ ] 3 (2) ξ δw ϕ w ϕ }, (6.30) where (2) ξ w ϕ ws defined in (6.13). In order to keep the generl covrince t ech order in perturbtion theory, this second order guge trnsformtion of S (2) [w ϕ ] must be blnced by pproprite liner guge trnsformtions of S (3). This mens tht the guge dependent terms in Eq. (6.29) must be proportionl to the first order equtions of motion, such tht S (3) GD [ζ] = { d 3 xdt N 1 3 δs (2) } [w ϕ ] 3 F ζ [w ϕ, ζ], (6.31) δw ϕ

100 92 CHAPTER 6. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE SCALAR SECTOR where F ζ [w, ζ] is defined in Eq. (6.14) 2. Of course, the guge dependent prt S (3) GD [w ϕ, ζ] is precisely such tht it counters the second order guge trnsformtion of S (2) [w ϕ ] in (6.30). Now it is strightforwrd to express the ction in terms of the second order guge invrint vrible W ϕ, S (2) + S (3) = d 3 xdt N 3 { 1 2Ẇ 2 ϕ 1 2 ( i W ϕ ) } [ 3 ż] 2 3 z W ϕ 2 + S (3) GI [W ϕ]. (6.32) Thus, by this procedure one obtins the third order ction which is mnifestly guge invrint up to second order in guge trnsformtions. The dynmicl sclr degree of freedom in this ction is W ϕ, which is the field perturbtion on uniform curvture hypersurfces. We hven t mentioned explicitly wht is S (3) GI [W ϕ], tht is, wht re the guge invrint cubic vertices for W ϕ needed for the clcultions of non-gussinities. The esiest wy to find this is to set ζ = 0 from the strt. In tht cse W ϕ coincides with ϕ nd S (3) GI [W ϕ] ζ 0 d 3 xdt N 3 { DO ( ϕ 3)}. (6.33) Hence, S (3) GI [W ϕ] cn be found immeditely fter the replcement ϕ W ϕ in the DO(ϕ 3 ) terms of (6.23) [111]. Using these terms from Eq. (6.26) the result is S (3) GI [W ϕ] = where d 3 xdt N 3 { zw ϕ 1 4 z M 2 P W ϕ [Ẇ 2 ϕ + [ ( 2 ) 2 χ Wϕ i j χ Wϕ 2 [ 1 z MP 6 V 1 4 z M 2 P 2 ( ) ] 2 i W ϕ i χ Wϕ Ẇϕ ] i j χ Wϕ 2 V 1 ] 6 V Wϕ χ Wϕ 2 = 1 2 z 2 M 2 P z 3 M 4 P i W ϕ } HWϕẆϕ H z2 MP 2 Wϕ 2 2 χ Wϕ 2, (6.34) ( W ) ϕ. (6.35) z This demonstrtes the usefulness of the field perturbtion on uniform curvture hypersurfces W ϕ in combintion with the uniform curvture guge ζ = 0. For resons tht shll become cler in section 6.5, we shll now rewrite the cubic ction (6.34) in terms of W ϕ /z. After performing some prtil integrtions nd using the bckground equtions 2 This implies tht the terms of O(ζ 3 ) should drop out in the third order ction (6.23) fter the replcement of ϕ (6.28).

101 93 of motion (6.17), we find S (3) GI [W ϕ] = 1 2 d 3 xdt N 3 M 2 P z 4 M 4 P [ 1 2 z 6 M 6 P { ( ) 2 Wϕ W ϕ W ϕ z z z 1 2 [ (Wϕ z ) 2 ( i W ϕ z ( i j 2 ( Wϕ ) 2 W ϕ z + z ) ) 2 ] ( ) Wϕ i ( Wϕ z 2 z z 2 ( Wϕ MP 2 z 1 2 ) W 2 ϕ z 2 ) i W ϕ z [ ż zh ] ] } (6.36) In Eq. (6.36) different orders in slow-roll cn esily be distinguished. Using the slow-roll prmeters (6.19) it is cler tht the first line contins terms of order ɛ 2 (for the vrible W ϕ /z), wheres the second line is subleding in slow-roll. In the derivtion of the guge invrint ction for W ϕ we hve set ζ = 0. Wht hppens if we would hve tken ϕ = 0 from the strt? In tht cse the third order ction (6.23) contins only terms AO(ζ 3 ), nd W ϕ is nonliner expression in ζ. Therefore the third order ction must contin terms proportionl to the first order eqution of motion for ζ, which cn be bsorbed in the second order ction by identifying W ϕ (ϕ = 0). These terms proportionl to the eqution of motion were identified first in Ref. [83]. It ws shown tht fter the field redefinition ζ (H/ φ)w ϕ (ϕ = 0), the cubic ction for ϕ is obtined by replcing W ϕ ϕ. In our lnguge of guge invrince, we redefine ζ to the guge invrint vrible W ϕ (in the guge ϕ = 0), then restore the dependence on ϕ in W ϕ, nd finlly set ζ = 0 to get the ction for O(ϕ 3 ) terms.. Mnifest guge invrince: cubic ction for W ζ In the previous section we found the mnifestly guge invrint cubic ction in terms of W ϕ, the field perturbtion on uniform curvture hypersurfces. Just s well we could hve constructed the guge invrint ction for W ζ, the curvture perturbtion on uniform field hypersurfces. Strting point is gin the schemtic third order ction (6.23) nd we follow the sme steps s in the previous prt. Insted of eliminting ϕ in terms of w ϕ nd ζ s in Eq. (6.28), we eliminte ζ ζ = w ζ + 1 z ϕ. (6.37) As before, the third order ction S (3) cn then be seprted in guge invrint prt depending only on O(wζ 3 ) terms, nd guge non-invrint prt with cubic terms depending on combintions of ϕ nd w ζ. Of course, this guge dependent prt must gin

102 94 CHAPTER 6. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE SCALAR SECTOR blnce the guge trnsformtion of S (2) (6.30). Therefore S (3) = d 3 xdt N { ( 3 AO (w ζ ) 3) } + E O(ϕ 3 ) + F O(ϕ 2 w) + G O(ϕwζ) 2 = = S (3) S (3) GI [w ζ] + GD [w ζ, ϕ] { (3) S GI [w ζ] d 3 xdt N 1 3 δs (2) } [w ζ ] 3 F ϕ [w ζ, ϕ]. (6.38) δw ζ Agin we cn do redefinition of w to the second order guge invrint W ζ using (6.12) such tht S (2) + S (3) = d 3 xdt N 3 z 2 { 1 2Ẇ 2 ζ 1 2 ( ) } 2 i W ζ (3) + S GI (W ζ). (6.39) Thus, following this procedure one obtins the mnifestly guge invrint ction t third order expressed in terms of the guge invrint vrible W ζ. The simplest wy to (3) find the guge invrint vertices in S GI [W ζ] is now to set the field perturbtion ϕ = 0. Then, the guge invrint vrible becomes W ζ (ϕ = 0) = ζ, nd S (3) GI [W ζ] ϕ 0 d 3 xdt N 3 { AO ( ζ 3)}. (6.40) So, if we set ϕ = 0, the third order ction for ζ immeditely gives the guge invrint ction in terms of the curvture perturbtion on uniform field hypersurfces fter the replcement ζ W ζ. Using the AO(ζ 3 ) terms derived in (6.24) (see lso [83]), S (3) GI [W ζ] = where d 3 xdt N 3 M 2 P ( 3W ζ Ẇζ H { 1 2 z 2 M 2 P ) [ i j s Wζ 2 s Wζ 2 2 W ζ i W ζ i j s Wζ 2 i W ζ = 2 W ζ 2 H z 2 MP 2 Ẇζ ( 2 ) 2 ] s Wζ z 2 M 2 P 2 ( 3W ζ Ẇζ H 2 is Wζ ) i W ζ 2 s Wζ 2 (6.41) Ẇ ζ. (6.42) This demonstrtes the convenience of working with the guge invrint vrible W ζ in combintion with the guge ϕ = 0 In the derivtion of (6.41) we hve performed severl prtil integrtions with respect to spce nd time. The specific boundry terms do not contribute to the bispectrum. However, there re certin temporl boundry terms tht cn contribute to the bispectrum. This is discussed in the next section. },

103 Uniqueness of guge invrint ction In the previous section two third order ctions for cosmologicl perturbtions were derived which were mnifestly guge invrint up to second order in guge trnsformtions. The generl trick is tht the guge dependent prts of the third order ction could be bsorbed in the second order ction, which defined guge invrint vrible. In Eq. (6.32) the guge invrint cubic ction ws expressed in terms of W ϕ, in Eq. (6.39) in terms of W ζ. Compring the guge invrint ctions (6.32) nd (6.39), we cn see tht the prts qudrtic in the guge invrint vribles re relted by rescling W ϕ = zw ζ. Of course, this must be true since t liner order the different guge invrint vribles re relted by rescling, w ϕ = zw ζ. Thus the tree-level propgtor for W ϕ /z is the sme s the one for W ζ. The guge invrint prts of the ction which re cubic in the guge invrint vribles, Eqs. (6.36) nd (6.41), pper not to be relted by rescling, S (3) GI [ W (3) ϕ/z] S GI [W ζ]. (6.43) This implies tht the guge invrint vertices for W ϕ /z differ from those for W ζ. On the other hnd, the guge invrint cubic vertices originte from the sme ction. This presents n opportunity to find out exctly how the guge invrint ctions differ Non-liner trnsformtions To illustrte wht is the difference between guge invrint ctions for non-linerly relted vribles, let us tke generl ction for second order guge invrint vrible W X, S[W X ] = d 3 xdt N 3 z 2 { 1 2Ẇ 2 X 1 2 ( ) } 2 i W X + S (3) GI [W X]. (6.44) This vrible is non-linerly relted to nother second order guge invrint vrible W Y, W X = W Y + Q(W Y, W Y ), (6.45) where the Q(W Y, W Y ) is completely generl function bi-liner in W Y, which cn include temporl nd/or sptil derivtives of W Y. For simplicity we consider here two vribles which coincide t the liner level, but re different non-linerly. The ction for

104 96 CHAPTER 6. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE SCALAR SECTOR W Y then becomes S[W Y ] = + + { d 3 xdt N 3 z 2 1 Y 2 1 ( i W Y 2Ẇ 2 { d 3 xdt N 3 Q 1 δs (2) } [W Y ] N 3 δw Y { d 3 xdt N 1 [ z 2 ] i QẆY ) 2 } + S (3) GI [W Y ] [ Q ]} iw Y, (6.46) where δs (2) /δw Y represents the eqution of motion following from the qudrtic ction, i.e. 1 δs (2) (W Y ) = 1 [ 14 ] N 3 δw Y 3 3 z 2 Ẇ Y W Y. (6.47) These terms re relted to the vrition of the complete ction s 1 δs(w Y ) 3 = 1 δs (2) (W Y ) δw Y 3 + O(WY 2 ) = 0, (6.48) δw Y which vnishes by the vritionl principle. In the third order ction the terms QδS (2) /δw Y therefore re zero t the cubic level upon inserting the solutions of the equtions of motion (6.48), nd consequently these terms do not contribute to the 3- point function t tree-level. Inspecting the tree-level cubic guge invrint ction for W Y (6.46) we see tht the bulk ction S (3) GI [W Y ] coincides with the guge invrint cubic ction for W X (6.44). In this sense the bulk guge invrint cubic ction cn be clled unique. Thus, the evolution of the 3-point function is independent of the choice vribles W X or ny nonlinerly relted vrible, chrcterized by W Y. Of course there re lso boundry terms in the ction for W Y (6.46). In the in-in or Schwinger-Keldysh formlism [ ] they cn contribute to the 3-point function. In this formlism n equl-time expecttion vlue of n opertor O(W ) my be defined s Ω, t in O(W (t)) Ω, t in = [DW + DW ]O(W (t))ρ[w + (t in ), W (t in )] ( t ) exp i dt (L[W + (t )] L[W (t )]) δ [W + (t) W (t)], t in (6.49) where ρ[w + (t in ), W (t in )] is the density mtrix t initil time t in, which for pure initil stte equls Ψ [W (t in )]Ψ[W + (t in )]. W ± is here second order guge invrint vrible on the + or prt of the complex in-in contour, which could be, for instnce, W X or W Y.

105 97 In generl, the opertor O(W (t)) depends on both W + nd W fields. In the simple cse of n equl-time 3-point function, opertor ordering is not importnt, one cn drop the ± subscripts from W s nd O(W (t)) = W ( x 1, t)w ( x 2, t)w ( x 3, t). Coming bck to the boundry terms, the sptil ones do not contribute to the 3-point function. On the other hnd, the temporl boundry terms give in generl nonzero contribution. For cosmologicl correltion functions (6.49) the initil time is often tken t t in =. Strictly speking, this is not the correct procedure. As t the quntum field theory of grvity becomes strongly coupled (tht is, in tht limit the physicl moment k/ ) nd perturbtion theory fils. In prctice one cn define n in-in expecttion vlue by strting from some finite initil time t in t which perturbtion theory is well defined. Thus boundry terms t t in cn contribute. One cn define the initil stte for, for instnce, the guge invrint vrible W X to be Gussin. As consequence the initil stte for ny other non-linerly relted vrible W Y is explicitly non-gussin. This initil non-gussinity is then evolved through the bulk ction S (3) GI (W Y ). It is importnt to distinguish how much non-gussinity is dynmiclly generted from some Gussin initil stte, nd how much comes from potentilly non-gussin initil stte. For exmple, if one observes non-gussinity for the vrible W Y, but one defines the initil stte for W X to be Gussin, then some of the finl non-gussinity finds its origin in the evolution of the initil non-gussinity in W Y. Let us now discuss the contributions to the 3-point function coming from the temporl boundry terms t time t. There cn be vrious types of boundry terms. Boundry terms of the type W 3 (t) cnnot contribute to the 3-point function, becuse the δ-function in Eq. (6.49) forces the + nd vertices to be equl t time t. In fct, these boundry terms do not nturlly pper fter the trnsformtion to different guge invrint vrible, s cn be seen in (6.46). Nonetheless these terms cn pper fter dditionl prtil integrtions of terms W 3 (t) nd W 2 (t)ẇ (t) in the bulk ction. This ws exctly done for the ction of W ϕ in going from Eq. (6.34) to (6.36), which justifies the use of (6.36) s the guge invrint cubic ction for W ϕ. Similr boundry terms were found in obtining the cubic ction for W ζ, Eq. (6.41). One hs to be creful with other boundry terms such s αw 2 Ẇ or βw Ẇ 2, becuse they give in generl non-negligible contributions to the bispectrum. The reson is tht these terms contin the cnonicl momentum Π W, which hs non-vnishing commuttion reltion with W. In fct these type of boundry terms generte disconnected prts of the bispectrum nd thus contribute to locl f NL. In the generl exmple bove, it cn be seen tht the temporl boundry terms in the ction for W Y re precisely of the form S [W Y ] = d 3 xπ WY (t)q(t). Consider now n exmple where Q(W Y, W Y ) = α(t)w 2 Y + β(t)w Y Ẇ Y. (6.50) Then using the expecttion vlue s defined in Eq. (6.49) one cn compute to lowest

106 98 CHAPTER 6. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE SCALAR SECTOR (tree-level) order tht W X (x 1 )W X (x 2 )W X (x 3 ) = W Y (x 1 )W Y (x 2 )W Y (x 3 ) + 2α(t) ( W Y (x 1 )W Y (x 2 ) W Y (x 1 )W Y (x 3 ) + sym) ( + β(t) ẆY (x 1 )W Y (x 2 ) W Y (x 1 )W Y (x 3 ) ) + W Y (x 1 )W Y (x 2 ) ẆY (x 1 )W Y (x 3 ) + sym, (6.51) where sym stnds for other cyclic contributions nd (x i ) = ( x i, t) re t equl time 3. In words: the 3-point function for the vrible W X is computed using the guge invrint cubic vertices in S (3) GI [W X], the result expressed on the left-hnd side, but it is lso directly relted to the 3-point function for W Y, computed using guge invrint vertices for W Y, plus dditionl disconnected prts coming from the boundry terms, which dd up to the right-hnd side of (6.51). Note tht the reltion between the 3-point functions in (6.51) cn be immeditely derived by inserting the non-liner reltion between W X nd W Y (6.45) into the left-hnd side of (6.51) nd using Wick s theorem. Thus in order to compute the 3-point function of one guge invrint vrible, one cn use the guge invrint ction for nother vrible (which my hve more convenient form) nd dd disconnected pieces ccording to the non-liner reltion. Since time t in (6.51) is rbitrry, reltion (6.51) holds lso for t = t in, telling us how re initil non-gussinities in the vribles W X nd W Y relted. For exmple, if the initil stte on Σ WY is Gussin, then the initil non-gussinity on Σ WY will be given by the terms multiplying α nd β in (6.51) evluted t t = t in Prcticl exmple: different guge invrint vribles The generl discussion in this section demonstrtes tht the cubic guge invrint ctions ( ) for different (i.e. non-linerly relted) guge invrint vribles re relted, in the sense tht they both hve the sme, unique, bulk ction, but they differ by boundry terms nd terms proportionl to the eqution of motion. This is not lwys obvious. For exmple, the terms proportionl to the eqution of motion cn 3 An lterntive wy to compute the 3-point correltor (6.51) is by mking use of the interction picture [83, 118], where n expecttion vlue to lowest order in perturbtion theory is given by t O(W (t)) = i dt [ O(W (t)), H int (t ) ], ( = 1). (6.52) t 0 If for the interction Hmiltonin H int one considers the prt with the boundry terms in (6.46) only, which re of the form Π W (t)q(t), then it is strightforwrd to find the disconnected pieces in (6.51) using the cnonicl commuttion reltions.

107 99 be seprted, prtilly integrted, nd the remining terms cn be written such tht it is not cler tht they re totl derivtive terms. Therefore guge invrint ctions for different, non-linerly relted vribles cn pper very different. To illustrte this, let us now consider prcticl exmple. The guge invrint vrible W ζ is non-linerly relted to W ϕ /z s in (6.15), which is n exmple of (6.45). According to the bove discussion, their 3-point functions should therefore be relted 1 z 3 W ϕ(x 1 )W ϕ (x 2 )W ϕ (x 3 ) = W ζ (x 1 )W ζ (x 2 )W ζ (x 3 ) (6.53) 1 ż 2 zh ( W ζ(x 1 )W ζ (x 2 ) W ζ (x 1 )W ζ (x 3 ) + sym) +..., where the terms of higher order in slow-roll hve been neglected. Note tht this reltion cn lso be inverted to give the 3-point function of W ζ in terms W ϕ. This is purely bsed on the non-liner reltion between the guge invrint vribles, but it should follow from the ctions s well. The guge invrint ctions for W ϕ /z nd W ζ coincide t the qudrtic level, but the cubic ctions Eq. (6.36) nd (6.41) look very different t first sight. For instnce, the ction for W ϕ /z (6.36) is of second order in slow-roll (ɛ 2 = 1 4 z4 /M 4 P ), wheres the ction for W ζ (6.41) seems to be of first order. Mldcen [83] showed tht it is possible to relte the cubic ction for ϕ in the ζ = 0 guge with the cubic ction for ζ in the ϕ = 0 guge. The two ctions differ by terms proportionl to the liner eqution of motion (which do not contribute to the tree-level ction) nd by some boundry terms. When trnslted to our lnguge of guge invrint vribles, the cubic ction for W ϕ cn be relted to tht for W ζ, up to boundry terms. After mny prtil integrtions of (6.41), the result is: S (3) GI [W ζ] = { d 3 xdt N 3 MP z 6 [ Ẇζ 2 W ζ 1 16 M 6 P + Q(W ζ, W ζ ) M 2 P = S (3) GI [W ζ] + z 4 M 4 P [ 1 ζ 2 W ζ + 1 2Ẇ 2 ( i j 2 Ẇ ζ 1 N 3 δs (2) [W ζ ] δw ζ d 3 xdt N 3 {Q(W ζ, W ζ ) ( ) 2 ( i W ζ i W ζ Ẇζ ) ( ) ] i j 2 Ẇ ζ W ζ + 1 z 2 2 MP 2 Ẇ ζ Wζ 2 } 2 Ẇζ ) i W ζ ] [ ] ż zh } 1 δs (2) [W ζ ]. (6.54) N 3 δw ζ The function Q(W ζ, W ζ ) is defined in the long-wvelength limit in Eq. (6.15). This is precisely the form of the ction predicted fter insertion of the non-liner reltion W ϕ /z = W ζ + Q(W ζ, W ζ ), s in (6.46), up to boundry terms. The guge invrint vertices in the bulk ction S (3) GI for W ζ coincide with those for W ϕ /z. Thus, the bulk

108 100 CHAPTER 6. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE SCALAR SECTOR ctions coincide nd therefore the evolution of the 3-point function is unique. The cubic guge invrint ction for W ϕ my be clled the unique ction which seprtes the different levels of slow-roll. A guge invrint ction in terms of ny other guge invrint vrible, e.g. W ζ cn be brought to this unique form fter mny prtil integrtions nd extrcting terms proportionl to the eqution of motion. The bispectrum for W ζ cn now be computed by mking use of the prtilly integrted bulk ction, including possible contributions coming from boundry terms which cn give rise to disconnected contributions to 3-point functions, see Eq. (6.51) (6.53). Alterntively, one cn redefine the field W ζ to the non-linerly relted W ϕ nd use the ction for tht guge invrint vrible. The 3-point function for W ζ is then computed from Eq. (6.53). The non-liner reltion between different guge invrint vribles prescribes wht this field redefinition should be Boundry terms, hypersurfces nd observtions So fr we hve not shown the boundry terms in (6.54). They were explicitly computed for ζ (or W ζ in ϕ = 0 guge) in Refs. [119] nd [120] for sclr field Lgrngins which re generl function Φ nd its kinetic term. Both rech similr conclusions: boundry terms with time derivtives of W ζ contribute to the bispectrum, nd the dominnt terms in the slow-roll pproximtion give exctly the sme contribution s in Eq. (6.53), which is wht one finds fter field redefinition to non-liner vrible in W ζ. 4 In our lnguge this is nothing more thn switching between different guge invrint vribles. One remrk here is tht the boundry terms in Refs. [119] nd [120] do not dispper completely fter redefining W ζ to new non-liner vrible. On the other hnd, Eq. (6.46) suggests tht ll boundry terms re incorported fter switching to non-liner vrible. This must be so becuse under non-liner trnsformtion 3-point functions of different vribles re relted s (6.51), irrespective of specific ction. When describing now different guge invrint vribles with their corresponding ctions, the sme reltion of the bispectr should follow from the ction, for both the bulk nd the boundry prt. The origin of these dditionl boundry terms not removed by the field redefinition my reside in dditionl boundry terms of the qudrtic ction. The form of the qudrtic ction in (6.20) is only reched fter severl prtil integrtions, which 4 In fct, in Ref. [120] the procedure is slightly different thn stted bove. After prtil integrtions Burrge et l. do not keep ny boundry terms proportionl Wζ 2Ẇζ. The reson is precisely tht these would generte disconnected prts of the 3-point function. The prtil integrtions performed re only the llowed ones: those tht do not contribute to the bispectrum t ll, or those tht re slow-roll suppressed contributions. The field redefinition is slightly different in their work, nd it does not coincide with the vrible W ϕ/z (note: ll computtions re performed in the comoving, ϕ = 0, guge.)

109 101 generte dditionl second order boundry terms. Moreover, the originl, unperturbed ADM ction (6.1) lso contins severl sptil nd temporl boundry terms tht in principle contribute t every order. The ltter boundry terms hve been described in prt in Ref. [10]. Together these boundry terms must dd up to guge invrint second order boundry term, expressed in the liner w ζ, since fter ll the originl strting point is the covrint ction (2.3). Now, under non-liner field trnsformtion these boundry terms will generte lso cubic boundry terms, which my blnce the extr boundry terms mentioned before. A different wy to see this is to come bck to the procedure of finding the guge invrint cubic ction, outlined in Sec Here the non-liner guge trnsform of the second order bulk ction ws blnced by guge dependent terms in the third order ction, which re proportionl to the liner eqution of motion. This in turn defined second order guge invrint vrible (W ϕ or W ζ depending on the procedure). Similrly, lso the second order boundry terms trnsform under non-liner guge trnsformtions. They cn be written in guge invrint wy fter incorporting the guge dependent third order boundry terms. In spite of these remrks the conclusion of [119] nd [120] still stnds: the dominnt contribution to the bispectrum coming from the boundry terms for W ζ is tken into ccount by switching to different guge invrint vrible W ϕ /z nd using (6.53). Note tht one hs to be creful concerning the initil stte. We hve seen tht different guge invrint vribles re non-linerly relted, nd therefore their n-point functions re relted vi disconnected pieces. As consequence, the initil stte for one vrible my pper non-gussin, wheres they re in fct Gussin for nother guge invrint vrible. It is importnt to distinguish the mount of non-gussinity tht is generted by the cosmologicl evolution, nd non- Gussinity tht origintes from the choice of guge invrint vribles, either initilly or t time of observtion. Let us now mke connection to observtions. Ultimtely we re interested in describing correltion functions of W ζ, the second order guge invrint comoving curvture perturbtion. The reson is tht W ζ is closely relted to the grvittionl potentil in the cosmic fluid, which in turn gives rise to the formtion of structure vi the fluid equtions. Fluctutions in the grvittionl potentil t the time of photon decoupling cn be observed using the CMB. The ide is the following: fter photons decouple (t the zero density hypersurfce) they climb out of the potentil wells creted by (lrge scle) fluctutions in the grvittionl potentil W ζ nd trvel towrds us 5. We observe these photons s the CMB vi stellites such s WMAP or Plnck. In the rest frme of the CMB the grvittionl potentil W ζ hs some vlue t the position of the observer, 5 One should consider here W ζ s some generlized guge invrint curvture perturbtion, which is constructed of ζ nd components of the cosmic fluid.

110 102 CHAPTER 6. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE SCALAR SECTOR which cn be set to zero. We re then interested in the chnge of W ζ from the lst scttering surfce to tody. This chnge cn be seen s differences in redshift for photons coming from different prts of the sky, which cn be directly relted to temperture fluctutions. This process ws first described by Schs nd Wolfe [121], who derived very simple reltion between temperture fluctutions nd fluctutions of the grvittionl potentil t the lst scttering surfce, δt/t = 1 3 ζ LS. In our guge invrint lnguge this would be generlized to (δt/t ) GI = 1 3 W ζ LS, where (δt/t ) GI is relted to the guge invrint definition of density perturbtions (δρ/ρ) GI. This cn then be relted to W ζ on super-hubble scles during infltion s (δt/t ) GI = 1 5 W ζ INF. 6. So, we re interested in these fluctutions t Hubble crossing, becuse they re cn be directly relted to the observed temperture fluctutions in the CMB. The guge invrint cubic ction for W ζ is given in Eq. (6.41), but this ction does not clerly seprte the dominnt contributions in the slow-roll pproximtion. However, the prtilly integrted form of the ction (6.54) does show the different orders in slow-roll, which mkes this form convenient for studying the dominnt contributions to the 3-point function for W ζ. Finlly some notes on the notion of hypersurfces. Hypersurfces re commonly ssocited with choice of guge, for exmple the uniform field ϕ = 0 or the uniform curvture ζ = 0 guge. A trnsformtion from one hypersurfce to the other is chieved by guge trnsformtion of the form x µ x µ + ξ µ + O(ξ 2 ) (see Eq. (2.21)). In our guge invrint lnguge there is no rel notion of notion of hypersurfces, but insted there re different guge invrint vribles. When we spek of hypersurfce in guge invrint setting, we men certin choice of guge invrint vribles, which reduces to some specific perturbtion in certin guge. For instnce, the curvture perturbtion on uniform field hypersurfces W ζ reduces to ζ in the guge ϕ = 0. Another hypersurfce is then described by nother guge invrint vrible, which is non-linerly relted to the first. 6.6 Frme independent sclr perturbtions So fr we hve discussed perturbtions of the Einstein frme ction (2.3). It is lso possible to study perturbtions in the Jordn frme, where non-miniml coupling between the sclr field nd the Ricci sclr is present. The Jordn frme ction is (see lso chpter 4) S J = d 4 x { 1 g J 2 R JF (Φ J ) 1 } 2 gµν J µφ J ν Φ J V J (Φ J ). (6.55) 6 The super-hubble fluctutions do not evolve on super-hubble scles during infltion. However, there is some evolution for super-hubble fluctutions in, for exmple, mtter dominted er, but it only leds to rescling, such tht W ζ LS = 3 5 W ζ INF [10][26].

111 103 In order to indicte the difference with the Einstein frme quntities we hve used in the first prt of this chpter, we shll use subscripts J for the quntities expressed in the Jordn frme. The function F (Φ J ) presents the generl coupling between the Ricci sclr nd the sclr field Φ J. Setting F (Φ J ) = MP 2 tkes us bck to the minimlly coupled cse. The Einstein frme nd the Jordn frme re relted by combined conforml trnsformtion of the metric nd redefinition of the sclr field (see section 4.2) with g µν = Ω 2 g µν,j ( ) 2 dφ = 1 dφ J Ω 2 + 6Ω 2 Ω 2 V (Φ) = 1 Ω 4 V J(Φ J ), (6.56) Ω 2 = Ω 2 (Φ J ) = F (Φ J) MP 2. (6.57) Since these re just field redefinitions of the metric nd sclr field, no physicl informtion is expected to be lost in the frme trnsformtion. This is wht we refer to s physicl equivlence of Jordn nd Einstein frme. The physicl equivlence is very useful, becuse it mens we could obtin ny results, such s the power spectrum, in the Jordn frme by trnsforming bck the well-known Einstein frme results using the bove reltions (6.56). Insted of deling with the difficult non-miniml coupling, we merely hve to del with modified potentil (6.56). In section 4.2 we hve explicitly demonstrted the equivlence t the level of the bckground equtions of motion, nd in section we hve shown this for the bckground ction (in Hmiltonin formultion). This follows immeditely from the bckground reltions between quntities in the Einstein nd Jordn frme, N = Ω N J = Ω J ( H = 1 Ω H J + φ = 1 Ω dφ φj 1 Ω dφ J ) Ω Ω 1 Ω Ω 2 Ω φ 2 J V = V J Ω 4, (6.58) with Ω 2 Ω 2 (φ J ) = F (φ J )/M 2 P. A dotted derivtive on Jordn frme quntity implies tht it is derivtive with respect to N J dt, wheres dotted derivtive on n Einstein

112 104 CHAPTER 6. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE SCALAR SECTOR frme quntity implies it is with respect to Ndt. The bckground reltions (6.58) re derived by inserting perturbtive expnsion for the fields on both sides of Eqs. (6.56) nd tke the lowest order terms. In the Einstein frme the expnsions re given in Eqs. (6.3), nd for the Jordn frme we insert similr perturbtions g ij,j = J (t) 2 e 2ζ J (x) δ ij Φ J = φ J (t) + ϕ J (x) N J = N J (t) (1 + n J (x)) NJ i = 1 J (t) N J (t)( 1 J (t) is J (x) + n T i,j(x)). (6.59) With this expnsion we cn lso study the equivlence t the level of perturbtions. However, we hve shown in section 4.4 tht perturbtions in the Jordn frme generlly do not coincide with those in the Einstein frme, mking it difficult to relte perturbtive results between different frmes. However, in section it ws shown tht w ζ, the linerly guge invrint curvture perturbtion on uniform field hypersurfces, is lso frme independent. Tht is, w ζ = ζ Ḣ φ ϕ = ζ J H J φ J ϕ J w ζ,j, (6.60) such tht the guge invrint w ζ in Einstein frme coincides with the guge invrint w J in the Jordn frme to first order. Using this one cn immeditely write down the second order Jordn frme ction vi frme trnsformtion from the second order Einstein frme ction (6.20), which ws done in, for instnce, Ref. [70]. By mking use of the reltion z 2 φ 2 MP 2 MP 2 = 1 φ 2 J + 6 Ω2 H2 Ω 2 MP 2 (H J + Ω Ω) z2 J 2 F, (6.61) where z J is defined by we cn write S (2) J [w J] = d 3 xdt N J 3 JF zj 2 = φ 2 J + 6 Ω2 (H J + Ω Ω), (6.62) 2 { [ zj 2 1 2ẇ2 ζ,j 1 F 2 ( ) ]} 2 i w ζ,j. (6.63) In chpter 5 (see Ref. [1]) we hve derived the sme ction directly in the Jordn frme, leding to the min result (5.37). Note tht, lthough the Jordn frme nd Einstein frme ctions (6.55) nd (2.3) look originlly quite different (non-miniml or miniml coupling), t the level of perturbtions the ctions hve unique form when expressed in terms of frme independent J

113 105 perturbtions. Of course we cn express the prefctor 3 z 2 in terms of Einstein frme quntities H, φ etc., or Jordn frme quntities H J, φ J etc., but on the solutions of the bckground equtions of motion it is the sme time dependent function. Therefore, if we only consider the sclr sector there is no rel notion of preferred frme. Still, from the clcultionl point of view we my prefer one frme or the other. For the sclr field lone, it is simpler to work in the Einstein frme, becuse insted of the non-miniml coupling there is merely modified potentil. However, if one includes other mtter fields in the ction, it my be convenient to work with metric for which their kinetic terms re cnonicl 7, such tht their motion is described in the stndrd wy. For this metric the sclr field my be non-minimlly coupled to grvity, thus described in the Jordn frme. Let us now consider the frme equivlence t higher order in perturbtions. Even though w ζ nd w ζ,j re frme independent t first order, they do not coincide t second order. The reson is tht the frme trnsformtion becomes of second order. Let us perturb the conforml fctor Ω 2 = Ω 2 (Φ J ) to second order, Ω(Φ J ) = Ω + Ω ϕ J Ω ϕ 2 J, (6.64) where Ω d Ω/dφ J. Using this nd the generl reltions between Jordn nd Einstein frme (6.56), we cn write second order reltions for ϕ nd ζ ϕ = dφ ϕ J + 1 d 2 φ dφ J 2 dφ 2 ϕ 2 J J ζ = ζ J + Ω Ω ϕ J + 1 Ω 2 Ω ϕ2 J 1 2 ( ) 2 Ω ϕ Ω 2 J. (6.65) If we now tke the liner guge nd frme independent vrible w ζ, we see tht it is relted to w ζ,j s [ w ζ = w ζ,j 1 (HJ + Ω ) d 2 φ/dφ 2 ( ) 2 ] J Ω Ω 2 φ J Ω dφ/dφ J Ω + ϕ Ω 2 J. (6.66) Thus the first order guge invrint vrible w ζ is not convenient vrible for compring Jordn frme results with Einstein frme results t third order, since the vrible itself is frme dependent. Motivted by the results in the previous section, we could 7 It is in principle possible tht the metric differs for different mtter fields, such tht the kinetic term for some mtter field which is cnonicl with respect to one metric, is non-cnonicl for nother metric. The Strong Equivlence Principle (SEP) tells us tht the grvittionl motion for some test body does not depend on its constitution, which implies tht there is one metric nd it couples similrly to ll mtter. So fr no devitions hve been found from the SEP, but violtions of the SEP t very high energies or distnce scles re only wekly constrined (or not t ll).

114 106 CHAPTER 6. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE SCALAR SECTOR check if second order guge invrint vrible is frme independent, in the sense tht it hs exctly the sme form in the Jordn nd Einstein frmes. Indeed, it cn be shown tht the curvture perturbtion on uniform field hypersurfces, W ζ from Eq. (6.12), coincides with the sme guge invrint vrible in the Jordn frme, [( W ζ = w ζ + 1 φ 2 H φ = w ζ,j ) Ḣ H 2 H 2 ) [( φj H J φj ḢJ H 2 J φ 2 ϕ2 2 1 φẇ ζ ϕ ] ] HJ 2 ϕ φ 2 2 J 2 1 φj ẇ ζ,j ϕ J J W ζ,j, (6.67) where we did not write the terms with vectors, tensors or sptil derivtives. The frme independence of W ζ ws shown explicitly in [122], but other, more generl, proofs exist s well [92, 123, 124]. This mens tht we cn directly find the third order ction in the Jordn frme from the third order Einstein frme ction expressed in terms of W ζ, which cn be used, for exmple, to find f NL for non-minimlly coupled theory [122, 123, 125]. We only hve to replce the bckground Einstein frme quntities by corresponding Jordn frme quntities using (6.58). Thus from Eqs. (6.41) nd (6.61) we strightforwrdly find S (3) GI [W ζ,j] = where { d 3 xdt N J 3 JF 1 z 2 ( J 2 F W i W ζ,j ζ,j J ( ) [ + 1 Ẇ ζ,j i j s 3W ζ,j Wζ,J 2 H J + 1 F 2 2 F J + 2 } is Wζ,J i W ζ,j J J 2 s Wζ,J 2 J 2 s Wζ,J 2 J = 2 2 J ) i j s Wζ,J 2 J zj 2 F Ẇ ζ,j 2 ( 3W ζ,j ( 2 s Wζ,J 2 J Ẇ ζ,j H J + 1 F 2 F ) 2 ], (6.68) W ζ,j H J + 1 F 2 F + 1 zj 2 2 F Ẇζ J. (6.69) The 3-point function for non-minimlly coupled theory is found in similr wy. Thus, one first trnsforms to the Einstein frme where one cn use previously computed results for the 3-point function for W ζ, nd this result cn be expressed in Jordn frme quntities by going bck to the Jordn frme. Of course we cn lso trnsform the prtilly integrted Einstein frme ction for W ζ (6.54), tht shows the seprtion between different orders in slow-roll, to the Jordn )

115 107 frme. After the frme trnsformtions we find { ( S (3) GI,J = d 3 xdt N J 3 1 z 4 ( ) ) 2 J JF [W 4 F 2 ζ,j Ẇζ,J 2 i W ζ,j + where 1 zj 6 16 F 3 W ζ,j + 1 N J 3 J [ ( ) ( Ẇζ,J 2 i j 2 Ẇ i j ζ,j 2 } δs (2) J (w J) δw J Q J (W ζ,j, W ζ,j ) J Ẇ ζ,j )] zj 2 F Ẇζ,JWζ,J 2 H J + 1 F 2 F ) i W ζ,j ] ( i 2Ẇζ,J 2 Ẇζ,J [ żj zj 1 ] F 2 F, (6.70) ż J zj 1 F 2 F Q J (W ζ,j, W ζ,j ) = 1 2 H J + 1 F Wζ,J 2 Ẇζ,JW ζ,j. (6.71) 2 F H J + 1 F 2 F In the lst expression we hve only written the terms without sptil derivtives. As explined in Sec. 6.5, the boundry terms re ccounted for by performing field redefinition to new guge invrint vrible Ω W ϕ,j = W ζ,j 1 z J 2 ż J z J 1 F 2 F H J + 1 F 2 F Wζ,J 2 Ẇζ,JW ζ,j. (6.72) H J + 1 F 2 F The combintion Ω W ϕ,j /z J is of course the frme trnsformed Einstein frme perturbtion W ϕ /z = HW ϕ / φ, which is seen most esily fter trnsforming both sides of the non-liner reltion between guge invrint vribles (6.15). The 3-point function for W ζ,j is then clculted from the 3-point function for W ϕ,j plus disconnected prts Ω 3 W ϕ,j (x 1 ) W ϕ,j (x 2 ) W ϕ,j (x 3 ) = W ζ,j (x 1 )W ζ,j (x 2 )W ζ,j (x 3 ) (6.73) z 3 J 1 2 z J zj 1 2 F F H J + 1 F 2 F ( W ζ,j (x 1 ) W ζ,j (x 2 ) W ζ,j (x 1 ) W ) ζ,j (x 3 ) + sym +..., where z J is defined in Eq. (6.62) nd terms of higher order in slow-roll hve been neglected. Although Ω W ϕ,j /z J is the frme trnsformed field perturbtion on uniform curvture hypersurfces in the Einstein frme, Wϕ,J is not the field perturbtion on uniform curvture hypersurfces in the Jordn frme: it does not reduce to ϕ J in the guge ζ J = 0. The reson is tht the field trnsformtion genertes extr terms in ϕ J from the trnsformtion of ζ, see Eq. (6.65). The correct definition of the field perturbtion on uniform curvture hypersurfces in the Jordn frme is ) ] φ2 J W ϕ,j w ϕ,j 1 2 [ H J φ J ( φj H J φj ḢJ H 2 J HJ 2 ζj ẇ ϕ,j ζ J H J. (6.74)

116 108 CHAPTER 6. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE SCALAR SECTOR The question is how to compute the 3-point function for the field perturbtion on uniform curvture hypersurfces in the Jordn frme: wht is the ction for W ϕ,j? It cn be shown tht W ϕ,j is non-linerly relted to the field perturbtion on uniform curvture hypersurfces in the Einstein frme s HW ϕ φ = H JW ϕ,j φ J [ + H J ( 1 φj φ J H J 1 F 2 F F F ) ( HJ W ϕ,j φ J z J z J 1 2 F F H J + 1 F 2 F ] H 2 J φ 2 J W 2 ϕ,j ) W ϕ,j φ J, (6.75) where sptil derivtives hve been neglected. In the previous section 6.5 we hve shown tht the cubic ctions for non-linerly relted vribles differ only by boundry terms, which cn give disconnected contributions to the bispectrum. Thus the bispectrum for the field perturbtion on uniform curvture hypersurfces in the Jordn frme contins connected prt coming from the Einstein frme ction (6.36), plus disconnected prt from the non-liner reltion (6.75). Alterntively, one could use the direct reltion ( ) H J W ϕ,j = W ζ,j 1 1 φj Wζ,J φ J 2 φ 2 1 W ζ,j Ẇ ζ,j, (6.76) J H J H J by replcing ll quntities by Jordn frme quntities in (6.15). Then one finds the connected prt of the bispectrum for W ϕ,j from (6.68), nd disconnected pieces from (6.76). Both methods should give the sme result for the 3-point function for W ϕ,j, even though we did not compute the explicit ction for W ϕ,j. Now some words bout the specil sitution when ϕ J = 0. In tht cse the conforml fctor Ω 2 only hs bckground vlue nd W ζ,j (ϕ J = 0) = ζ J = ζ = W ζ (ϕ = 0). Thus the cubic terms in ζ J not only directly provide the guge invrint vertices, but cn lso be trnsformed to Einstein frme vertices, nd vice vers. The third order Jordn frme ction for ζ J ws derived in [126], nd it ws shown tht it cn be found from the Einstein frme ction [83 85] in Ref. [127]. Moreover, one could imgine tht t higher order in perturbtions one cn construct guge invrint vrible which reduces to ζ J in the guge ϕ J = 0, just s we did before for W ζ to second order. When ϕ J = 0 it now becomes lmost trivil to show tht the curvture perturbtion is invrint under frme trnsformtions. Thus, the curvture perturbtion on uniform field hypersurfces is frme independent to ll orders [124]. Finlly remrk bout previous results found in Higgs infltion. Often computtions re done in both frmes, exmples being quntum corrections of the Higgs potentil [93 97], or the computtions of the cut-off scle for which the theory becomes non-perturbtive [ , 105, 128]. The Einstein frme results re then compred to

117 109 direct Jordn frme computtions by trnsforming them to the Jordn frme. More often thn not, trnsformed Einstein frme results do not exctly gree with wht is found in the Jordn frme. A recent exmple is clcultion of the field dependent cut-off in [128], which ppers different in one frme or the other. However, the result of this section is tht the Jordn frme ction cn be found directly from the Einstein frme vi field trnsformtion. The most cler wy to see this is tht everything cn be expressed in frme independent vribles. Thus the cut-off scle should be the sme whether you compute it directly in the Jordn frme, or vi trnsformed Einstein frme results. The reson for the confusion nd difference between results obtined in different frmes in the references bove is due to non-covrint formlism, where the vribles become frme dependent. For exmple, in Einstein frme computtions often the non-perturbtive cut-off scle is found from expnding the non-polynomil potentil in powers of ϕ nd neglecting metric fluctutions. ϕ is then frme dependent vrible. There is no confusion when using frme independent vribles: quntum corrections or cut-off s computed directly in the Jordn frme re exctly the sme when first computed in the Einstein frme nd trnsformed bck to the Jordn frme. The cutoff scle for sclr perturbtions cn therefore in principle be computed directly from (6.68), nd from higher order generliztion of tht ction. The third order ction including tensoril contributions shll be computed in the next chpter. 6.7 Summry In the first prt of this chpter we hve focused on sclr perturbtions on top of n expnding bckground in the Einstein frme. These perturbtions re generlly guge dependent. It is possible to construct guge invrint vribles by tking certin combintions of these perturbtions. At second order in coordinte trnsformtions there re in principle infinitely mny guge invrint vribles, two specific exmples being the curvture perturbtion on uniform field hypersurfces (6.12) nd the field perturbtion on uniform curvture hypersurfces (6.14). These vribles re relted in non-liner wy (6.15). We hve outlined the procedure for finding the guge invrint ctions for these vribles t third order, nd we hve shown explicitly the guge invrint cubic ctions (6.41) nd (6.36). Next we hve demonstrted tht, due to the non-liner reltion between these vribles, the cubic ctions pper different, but ctully their bulk ctions re the sme (6.54). They differ by boundry terms, which genericlly give disconnected contributions to the bispectrum. This brings us to the spect of uniqueness. Once you pick certin initil hypersurfce, or equivlently choose specific guge invrint vrible, for exmple the co-

118 110 CHAPTER 6. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE SCALAR SECTOR moving curvture perturbtion in Gussin stte, then there is unique ction for this vrible tht evolves the initil (Gussin) stte nd cretes non-gussinity through the evolution. If the finl hypersurfce or guge invrint vrible, is different from the initil, boundry terms must be tken into ccount due to the non-liner reltion between different vribles. We lso commented on initil non-gussinity: lthough for one guge invrint vrible the initil stte is Gussin, it is genericlly non-gussin for nother vrible. Some of the finl non-gussinity for certin vrible therefore origintes from some initil non-gussinity. In the second prt of this chpter we discussed different conformlly relted frmes. It ws shown tht the cubic ction for guge invrint perturbtions in the Jordn frme cn be obtined directly from the Einstein frme ction (6.68). The trick is to identify the vrible tht hs exctly the sme form in either frme. We hve presented proof tht the second order curvture perturbtion on uniform field hypersurfces W ζ is such frme independent cosmologicl perturbtion (6.67). Thus the bispectrum for the comoving curvture perturbtion cn be found vi the Einstein frme bispectrum by trnsforming bck nd forth between the two frmes. The bispectrum for nother guge invrint vrible in the Jordn frme, such s the field perturbtion on uniform curvture hypersurfces, is then found from the non-liner reltion between this vrible nd the comoving curvture perturbtion (6.76).

119 Chpter 7 Non-liner guge invrince nd frme independence : the grviton sector 7.1 Introduction In the previous chpter we computed the mnifestly guge invrint cubic ction for sclr cosmologicl perturbtions. One of the min gols of this chpter is to include the grviton in our clcultions nd derive the cubic guge invrint ction for pure grviton interctions nd sclr-grviton interctions. Knowledge of such interctions is needed to compute, for instnce, one-loop corrections to the sclr power spectrum. Such loops cn consist of only two sclrs, for which we need the guge invrint sclr 3- nd 4-point vertices, but lso of two grvitons or combintion of the sclr nd grviton perturbtions. The sclr 3-point vertices hve been computed in the previous chpter, nd in this chpter we compute the guge invrint cubic vertices for sclrgrviton interctions. Even though the grviton, or trnsverse-trceless prt of the sptil metric, is guge invrint to liner order in guge trnsformtions (see Eq. (2.29)), it trnsforms under second order guge trnsformtions. Thus, in order to find the mnifestly guge invrint ction t third order, we need to find the second order guge invrint grviton. Moreover, the sclr-grviton vertices give dditionl contributions to the second order guge invrint sclr perturbtion. As we shll see in this chpter, these second order vribles utomticlly follow from the procedure through which we construct the cubic guge invrint ction. Also, in this chpter we come bck to the question of uniqueness. In the previous chpter we hve shown tht there re infinitely mny guge invrint perturbtions t second order, which re ll relted by non-liner trnsformtions. At the level of the ction, there re lso seemingly different guge invrint ctions, but it cn be shown tht they only differ by boundry terms nd terms proportionl to the eqution of mo-

120 112 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR tion, which vnish on-shell. Thus, the evolution of non-gussinity is independent of the choice of guge invrint vribles, nd in tht sense there is unique ction tht describes this evolution. However, 3-point functions for different vribles generlly differ by disconnected pieces due to the boundry terms t initil nd finl time. In this chpter we show tht exctly the sme resoning cn be mde once we include the second order guge invrint grviton. Another importnt gol of this chpter is to demonstrte the reltion between sclrgrviton interctions in the Jordn frme nd those in the Einstein frme. As we hve seen in the previous chpter, t the perturbtive level it is most esily seen tht the frmes re equivlent once the ction is expressed in terms of frme independent sclr perturbtions. In this chpter we show tht lso prticulr choice of second order guge invrint grviton is frme independent, s re dditionl contributions to the guge invrint sclr perturbtion. We shll do ll our computtions directly in the Jordn frme, where quntities re defined without subscript, nd indicte ll Einstein frme quntities with the subscript E. A brief outline for this chpter. In section 7.2 we introduce the Jordn frme ction nd perturbtions. In section 7.3 we compute the guge invrint cubic vertices for the grviton nd for sclr-grviton interctions on different hypersurfces in the Jordn frme. Finlly, in section 7.5 we show how the guge invrint cubic ction in the Jordn frme is relted to tht in the Einstein frme. 7.2 Action nd perturbtions We consider single sclr field in the Jordn frme, i.e. coupled non-minimlly to the Ricci sclr, S = 1 d 4 x g {RF (Φ) g µν µ Φ ν Φ 2V (Φ)}. (7.1) 2 For minimlly coupled theory F (Φ) = M 2 P, where M P is the reduced Plnck mss. This ction hs been shown before in chpter 4, Eq. (4.2). We re interested in finding the ction (7.1) up to third order in perturbtions. Since the theory of generl reltivity is covrint, some of the degrees of freedom in the ction re not physicl. In order to eliminte non-physicl degrees of freedom from the ction, nd to seprte dynmicl from constrint degrees of freedom, it is most convenient use the ADM metric [19], defined in Eq. (2.23). For completeness: ds 2 = N 2 dt 2 + g ij (dx i + N i dt)(dx j + N j dt). (7.2) Here g ij is the sptil metric nd N nd N i re the lpse nd shift functions, respectively. In terms of the ADM metric the ction (7.1) cn be written (up to boundry terms) s

121 113 (see lso Eq. (5.5)) S = 1 { d 3 xdt g NRF (Φ) + 1 ( E ij E ij E 2) F (Φ) 2 2 N N EF (Φ) ( t Φ N i i Φ ) + 2g ij i N j F (Φ) + 1 ( t Φ N i i Φ ) } 2 Ng ij i Φ j Φ 2NV (Φ), (7.3) N where F (Φ) = df (Φ)/dΦ nd the mesure g, the Ricci sclr R nd covrint derivtives i re composed of the sptil prt of the metric g ij lone. The quntities E ij nd E re relted to the extrinsic curvture K ij s E ij = NK ij, with E ij = 1 2 ( tg ij i N j j N i ) (7.4) E = g ij E ij. (7.5) We now consider perturbtions on top of the homogeneous FLRW bckground, similr to Eq. (6.3) in the previous chpter, with g ij = 2 e 2ζ (e γ ) ij Φ = φ + ϕ N = N (1 + n) N i = 1 N(t)( 1 i s + n T i ), (7.6) i n T i = 0, i γ ij = 0, γ ii = δ ij γ ij = 0. (7.7) The perturbtion γ ij is thus the trnsverse trceless tensoril perturbtion, nd is often referred to s the grviton. The bckground vlues of (7.6) cn be inserted in the ction (7.3). The bckground Friedmnn equtions nd field equtions re obtined by vrition of this ction with respect to N, nd φ. This gives, respectively, H 2 = 1 6F Ḣ = 1 2F [ φ2 + 2V 6HF ] ( φ 2 + HF F ) 0 = φ + 3H φ + V 6 ( 2H 2 + Ḣ) 1 2 F. (7.8) Here we hve defined the dotted derivtive s ȧ d/( Ndt), nd the Hubble prmeter is H = ȧ/. These equtions hve been shown previously, see Eq. (4.8), but they hve been repeted here due to their frequent use in this chpter.

122 114 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR In this chpter we re minly interested in the sclr perturbtions ζ nd ϕ, nd in the trnsverse trceless grviton γ ij. The bsence of other metric perturbtions h nd h T i in Eq. (7.6) compred to Eq. (2.25) cn be viewed s prtil sptil guge fixing, or s consequence of the decoupling from the dynmicl degrees of freedom. This ws discussed in detil in section 6.2. As one observes from Eq. (7.3), there re no kineticlike terms for N nd N i. These fields therefore ct s constrint or uxiliry fields. Thus, they cn be solved for nd their solution cn be inserted bck into the ction [83]. For the second order ction, it is only necessry to find the solution for N nd N i to first order in perturbtions. Any second order term for N multiplies the bckground eqution for H 2 (7.8), s this eqution is precisely derived by vrying the ction with respect to N. A second order perturbtion of N i ppers s totl sptil derivtive, nd such term vnishes likewise. For the third order ction, it is lso sufficient to know the solutions for N nd N i to first order in perturbtions. The third order terms vnish due to the bckground equtions of motion, nd the second order perturbtions of N nd N i multiply the first order solutions of the constrint equtions. The first order solutions for N nd N i re found from the equtions of motion for these vribles. Strting from the ction (7.3) we find 0 = 1 { g [R N 2 (E ij E ij E 2 )]F (Φ) + 2N 2 EF (Φ)( t Φ N i i Φ) 2 } 2g ij i j F (Φ) N 2 ( t Φ N i i Φ) 2 g ij i Φ j Φ 2V (Φ) 0 = j [ N 1 (E jk g ik Eδ ij )F (Φ) δ ij N 1 F (Φ)( t Φ N i i Φ) ] N 1 ( EF (Φ) + t Φ N i i Φ) i Φ. (7.9) The second eqution gives, to first order in perturbtions 0 = i [2(2HF + F )n 4F ζ 2 φϕ ] 2F ϕ + 2HF ϕ 2F φϕ nit, (7.10) which present the solutions 1 n = [2F 2HF + F ζ + φϕ ] + F ϕ HF ϕ + F φϕ n T i =0. (7.11) The other constrint in Eq. (7.9) cn lso be expnded to first order in perturbtions, nd this presents the solution for the sclr shift perturbtion 2 s. By using the equtions

123 115 of motion (7.8) nd the first order solution for n we find 2 s 2 = 1 H χ 2 1 2F z2 2 F 2 F ( ζ + 1 F 2 F ϕ where for simplicity we hve defined (s in Eq. (5.38)) ) + χ (ζ Ḣφ ) ϕ, (7.12) z 2 φ ( H F 2 F F F ) 2. (7.13) The solutions (7.11) nd (7.12) gree with the expressions (5.83) which were obtined fter decoupling procedure. The solutions my be inserted into the ction (7.3) perturbed to second nd third order. The remining terms in the ction only depend on ϕ, ζ nd γ ij. Before we strt to compute the guge invrint ction for the dynmicl perturbtions, let us note tht there re lterntive wys to del with the non-dynmicl degrees of freedom N nd N i. As we hve seen in chpter 5 (see lso Refs. [78][1]), the perturbtions of N nd N i cn be decoupled from the rest of the qudrtic ction. As consequence extr terms pper in the second order ction for the remining dynmicl vribles, which, not surprisingly, re precisely those tht one would get fter replcing n nd s by their first order solutions. The decoupled perturbtions re guge invrint nd their eqution of motion is the first order solution of the constrint eqution. At third order the perturbtions for N nd N i cn lso be decoupled, with the decoupled fields giving the second order solutions for the constrint equtions, nd extr terms in the third order ction corresponding to the substitution of the first order solutions of the constrint fields in the ction. The procedure is explined in ppendix 7.A. 7.3 The cubic guge invrint ction in the Jordn frme We now wish to find the guge invrint ction t third order for the dynmicl sclr nd tensor. Of course, the ction (7.3) is guge invrint by definition, since it origintes from the mnifestly covrint ction (7.1). However, the perturbtions (7.6) re not guge invrint (see section 2.3). Thus, we set out to find the mnifestly guge invrint ction t the perturbtive level.

124 116 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR Review of the qudrtic ction In chpter 5 we computed the qudrtic ction for liner guge invrint perturbtions in the Jordn frme. Let us briefly review the min results, see Eq. (5.37). After expnding the ction (7.3) up to second order in perturbtions (7.6) nd inserting the first order solutions of the constrint equtions (7.11) (7.12), we find the guge invrint ction for the grviton: S (2) γ = d 3 xdt N 3 F 4 [ γ ij γ ij ( ) ] 2 l γ ij. (7.14) The mnifestly guge invrint ction for the curvture perturbtion on uniform field hypersurfces is S (2) = 1 ( ) ] 2 d 3 xdt wζ 2 2 N 3 z [ẇ 2 ζ 2 i w ζ, (7.15) where z 2 is defined in (7.13) nd w ζ is defined s w ζ = ζ Ḣ φ ϕ. (7.16) Alterntively, we cn express the ction in terms of nother guge invrint vrible, the field perturbtion on uniform curvture hypersurfces, The qudrtic ction for this vrible is S (2) w = 1 [ (Hwϕ d 3 xdt ϕ 2 2 N 3 z 2 φ w ϕ = ϕ φ H ζ. (7.17) ) 2 ( i ) ] 2 Hw ϕ. (7.18) φ The first order equtions for motion for the grviton nd the guge invrint curvture perturbtion re δs (2) ( =0 = 3 F 1 ( 3 ) 2 ) δγ ij 4 3 F γ ij + F 2 γ ij δs (2) δw ζ =0 = ( 3 z 2 ẇ ζ ) + 3 z w ζ. (7.19) The first order eqution of motion for the guge invrint field perturbtion is relted to tht of the curvture perturbtion s δs (2) = 0 = Ḣ δs (2). (7.20) δw ϕ φ δw ζ wζ = Hwϕ φ

125 117 This is of course true since t first order the guge invrint curvture perturbtion nd field perturbtion re relted vi rescling by bckground quntities w ζ = Ḣ φ w ϕ. (7.21) The fct tht the liner guge invrint vribles re relted by time dependent resclings, plus the fct tht the grviton decouples nd is guge invrint by itself, mkes it reltively esy to relte sclr nd grviton 2-point functions for different guge invrint vribles. At higher order this is no longer the cse, nd we shll discuss this next Sclr-sclr-grviton vertices on uniform field hypersurfces We continue by finding the guge invrint vertices for sclr nd grviton interctions in the Jordn frme. Due to the non-liner nture of generl reltivity the second order perturbtions (7.6) trnsform non-linerly under guge trnsformtions. As such, the guge invrint sclr nd tensor perturbtions receive qudrtic contributions. This implies tht some of the vertices for sclr nd grviton interctions tht nively pper in the third order ction fter inserting the perturbtions (7.6) in (??), re ctully nonphysicl. When the nive third order ction is expressed in terms of second order guge invrint vribles, some of the cubic vertices re bsorbed into the qudrtic ction for these vribles. This gretly complictes the construction of the third order ction for sclr nd grviton interctions. However, there is systemtic wy to isolte the physicl cubic vertices nd t the sme time find the second order guge invrint sclr nd tensor perturbtion. The procedure ws outlined in the previous chpter (see Ref. [2]) where the cubic vertices for sclr interctions were found. Here we pply the procedure to the cubic ction for sclr-grviton interctions, strting from the interction vertices for two sclrs nd grviton. The nive sclr-sclr-grviton vertices re found by collecting those third order terms from the ction (7.3) expnded up to third order in perturbtions (7.6). This gives S ssg = 1 2 d 3 xdt N 3 { 4 [F ζ + F ϕ + F n] γ ij i j ζ 2 + 2F γ ij i ζ j ζ + [ 3F ζ F i j s ϕ + nf ] γ ij + F i j s i n 2γ ij F jϕ + γ ij i ϕ 2 j ϕ } 2 k γ ij k s. (7.22) Note tht s nd n re the first order solutions of the constrint fields, nd re hence liner in sclr perturbtions ζ nd ϕ ccording to (7.11) nd (7.12). Guge invrince

126 118 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR is not mnifest in the ction (7.22), becuse the sclr perturbtions ζ nd ϕ re not linerly guge invrint. The curvture perturbtion on uniform field hypersurfces, w ζ, on the other hnd is guge invrint to first order. So let us find the guge invrint cubic vertices for sclr-sclr-grviton interctions by inserting into the ction bove ζ = w ζ + Ḣ φ ϕ. (7.23) This substitution seprtes the ction (7.22) into mnifestly guge invrint prt with w 2 ζ γ vertices, nd guge dependent prt with ϕ2 γ nd w ζ ϕγ vertices. Schemticlly The guge invrint vertices re S GI wζ 2γ = 1 { d 3 xdt 2 N 3 F 4 [ ] i j w ζ w ζ + n wζ γij 2 S ssg = S GI w 2 ζ γ + SGD ϕ 2 γ + SGD w ζ ϕγ. (7.24) [ 3w ζ n wζ ] γij i j s wζ 2 + 2γ ij i w ζ + i j s wζ 2 j w ζ k γ ij k s wζ }, (7.25) where for convenience we hve seprted the first order solutions of the constrint equtions in guge invrint nd guge dependent prt, ( ) ( ) 1 n = ẇ ζ + n H + 1 wζ + F 2 F ϕ φ ϕ φ ( ) 2 s 2 = w ζ + χ 2 ϕ H + 1 F 2 φ 2 2 s w ζ 2 ϕ 2 φ. (7.26) 2 F Note from Eq. (7.12) tht the χ prt of the first order solution is guge invrint by itself. Although we did not write the guge dependent prt of the third order ction S GD S GD ϕ 2 γ + w ζ ϕγ explicitly, it is esy to see tht it is nonzero. This seems to pose mjor problem, since it suggests tht guge invrince is broken t the perturbtive level. However, we should remind ourselves tht in the third order ction we del with second order perturbtions. Likewise, these perturbtions trnsform to second order under guge trnsformtions. As consequence, the first order guge invrint curvture perturbtion w ζ nd grviton γ ij re no longer guge invrint. Schemticlly we cn write these second order guge trnsformtions s (see Refs. [111] [2] nd Eq. (6.11)) w ζ w ζ + (2) ξ w ζ γ ij γ ij + ( (2) ξ γ) ij. (7.27)

127 119 Under such guge trnsformtions the second order ctions for w ζ nd γ ij trnsform s S (2) γ S (2) 2 γ S (2) w 2 ζ S (2) + 1 wζ 2 2 { d 3 xdt N 2 3 δs (2) 3 ( (2) ξ δγ ij { d 3 xdt N 2 3 δs (2) 3 (2) ξ δw ζ γ) ij w ζ } }, (7.28) up to totl derivtives. This second order guge trnsformtion of the qudrtic ction cn only be blnced by guge dependent terms in the third order ction which re proportionl to the eqution of motion. So in fct, in order for the third order ction to be guge invrint under second order trnsformtions, we expect to hve guge dependent prt in the third order ction fter the substitution (7.23) into (7.3). This guge dependent prt must be proportionl to the liner equtions of motion. Indeed, fter mny prtil integrtions we find tht Sϕ GD 2 γ + SGD w ζ ϕγ = 1 2 { d 3 xdt N 3 2 δs (2) [ i ϕ 3 δγ ij φ δs (2) δw ζ j ϕ φ [ i j 2 ( i ϕ j s wζ φ + )] jϕ i s wζ φ )] }. (7.29) ( 1 ϕ 4 φ γ ij The complete ction to third order in perturbtions cn now be written in mnifestly guge invrint wy by defining second order guge invrint vribles, γ ζ,ij =γ ij + iϕ j ϕ φ φ W ζ =w ζ + i j 2 ( 1 4 ϕ φ γ ij ( i ϕ j s wζ + jϕ φ ) i s wζ + O(γϕ) (7.30) ) φ + O(ϕ 2, w ζ ϕ). (7.31) The first vrible is the second order guge invrint tensor perturbtion, nd my be clled the grviton on uniform field hypersurfces, since it reduces to γ ij in the guge ϕ = 0. Similrly, W ζ is the second order curvture perturbtion on uniform field hypersurfces. The mnifestly guge invrint qudrtic ctions for these perturbtions hve precisely the sme form s Eqs. (7.14) nd (7.15), i.e. S (2) γ 2 ζ S (2) W 2 ζ = 1 [ d 3 xdt 2 N 3 F ( ) ] 2 γζ,ij γ ζ,ij γ ζ,ij 4 = 1 ( ) ] 2 d 3 xdt 2 N 3 z [Ẇ 2 ζ 2 Wζ. (7.32)

128 120 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR The vertices for sclr-sclr-grviton interctions cn be found immeditely from Eq. (7.25) by replcing the first order guge invrint vribles by their second order generliztions (which does not chnge nything for the cubic vertices). Thus we get S (3) Wζ 2 γ = 1 ζ 2 d 3 xdt N 3 F { 4 [ W ζ + n Wζ ] γζ,ij i j W ζ 2 [ 3W ζ n Wζ ] γ ζ,ij i j s wζ γ ζ,ij i W ζ + i j s Wζ 2 j W ζ k γ ζ,ij k s Wζ }. (7.33) Note tht the guge invrint vribles cn still contin other terms t second order, s is suggested in Eqs. ( ). The O(γϕ) terms in the definition for γ ζ cn only originte from the sclr-grviton-grviton ction t third order, which is precisely wht we shll discuss in the next prt. The O(ϕ 2, w ζ ϕ) terms in the definition for W ζ originte from the cubic sclr interctions. They were discussed in the previous chpter nd were given in the long wvelength limit in Eq. (6.12) (see Refs. [111] nd [2]). In conclusion, we hve presented nd pplied systemtic method to find the mnifestly guge invrint ction t third order. The trick is to simply insert the first order guge invrint vrible (7.23) into the nive cubic vertices. Not only does this method give the physicl cubic vertices, but it lso gives the correct form of the second order guge invrint vribles. Usully the second order guge invrint vribles re found by fixing the second order guge trnsformtion of the perturbtions (see e.g. [14, 21]). Here no explicit second order guge trnsformtions were necessry 1. The only ingredient tht we hve used is the mnifest generl covrince of the complete ction (7.1), nd the notion tht mnifestly guge invrint ctio cn be written down order by order in perturbtion theory. The method cn lso be generlized to higher order perturbtion theory. Tke the mnifestly guge invrint ction for second order perturbtions, Eqs. (7.32) nd (7.33). The ction (7.3) expnded to fourth order in perturbtion theory contins gin ζ, ϕ nd γ ij. Now replce ζ = W ζ + Hϕ/ φ nd γ ij = γ ζ,ij. The fourth order ction seprtes in guge invrint prt with second order guge invrint vribles W ζ nd γ ζ,ij, nd guge dependent prt. The guge dependent prt must be proportionl to the liner equtions of motion, or to δs (3) /δw ζ or δs (3) /δγ ij. The first prt must be there to blnce the third order guge trnsformtion of the qudrtic ction, nd cn be bsorbed into the definition of third order guge invrint vribles to the qudrtic ction guge invrint to cubic order. The second prt must be there to blnce the second order 1 Of course, one could check tht the second order guge invrint vribles which follow from the ction procedure gree with those constructed vi the guge trnsformtion method s in [14].

129 121 guge trnsformtion of the cubic ction, nd cn be bsorbed into the definition of second order guge invrint vribles in the cubic ction. Note tht we replced w ζ W ζ in the cubic vertices, which is only llowed when we consider the ction up to third order in perturbtions. For the qurtic ction we must tke into ccount contributions to the cubic ction tht originte from the qurtic ction. In the end, the form of the cubic prt of the ction is the sme s in Eq. (7.33). Note tht the cubic nd qurtic ctions re needed to compute one-loop corrections to the power spectrum. One could go to even higher order in perturbtion theory, where in generl the guge dependent prt of the nth order ction cn be bsorbed into lower order ctions by defining guge invrint vribles t (n 1)th order Sclr-grviton-grviton vertices on uniform field hypersurfces So fr we hve only discussed the sclr-sclr-grviton vertices. The guge invrint sclr-grviton-grviton nd pure grviton vertices cn be computed using the sme method s described in the previous section. We strt with the nive sclr-grvitongrviton vertices from the expnsion of (7.3) S sgg = 1 { d 3 xdt 2 N 3 i j ζ i γ jk j ζ 2F γ ik γ kj 2 2F γ ik + (3F ζ + F ϕ) 1 4 γ kl γ kl (F ζ + F ϕ) 1 i γ kl 4 nf ( 1 4 γ kl γ kl + 1 i γ kl 4 ) i γ kl 1 2 F γ i γ kl kl i γ kl i s }. (7.34) Agin we seprte the cubic terms in mnifestly guge invrint vertices composed of w ζ nd γ ij, plus guge non-invrint terms, by using (7.23). The guge invrint vertices re S GI w ζ γ 2 = 1 2 d 3 xdt N 3 F { 2γ ik γ kj i j w ζ 2 2γ ik i γ jk j w ζ 1 2 γ i γ kl i s wζ kl } + (3w ζ n wζ ) 1 4 γ kl γ kl (w ζ + n wζ ) 1 i γ kl i γ kl 4, (7.35) nd the guge dependent prt of the ction (7.34) (with ϕγ 2 terms) becomes fter severl prtil integrtions Sϕγ GD = 1 { } d 3 xdt 2 2 N 3 ϕ φ 2 δs (2) γ ij 3, (7.36) δγ ij

130 122 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR up to boundry terms. Hence, this guge dependent prt of the sclr-grviton-grviton ction my be bsorbed by redefinition of γ ij. Considering the previously defined second order grviton on uniform field hypersurfces γ ζ,ij, this precisely describes the O(ϕγ) contribution in Eq. (7.30). Thus, the complete guge invrint grviton to second order is γ ζ,ij =γ ij + iϕ j ϕ φ φ ( i ϕ φ j s wζ + jϕ φ ) i s wζ ϕ φ γ ij. (7.37) The mnifestly guge invrint cubic vertices for sclr-grviton-grviton interctions re then S (3) W ζ = 1 ( d 3 xdt γ ζ 2 2 N 3 i j W ζ F { 2 γ ζ,ik γ ζ,kj 2 ) i γ ζ,jk j W ζ 1 2 γ i γ ζ,kl i s Wζ ζ,kl + (3W ζ n Wζ ) 1 4 γ ζ,kl γ ζ,kl (W ζ + n Wζ ) 1 i γ ζ,kl i γ ζ,kl Pure grviton vertices on uniform field hypersurfces }. (7.38) Since the second order tensor perturbtion γ ij is guge invrint to first order, the pure grviton vertices re utomticlly guge invrint. Thus the O(γ 3 ) found fter n expnsion of (7.3) re the physicl vertices. In the cubic vertices we cn replce γ ij by its second order guge invrint generliztion, γ ζ,ij, which gives the mnifestly guge invrint vertices S (3) γ 3 ζ = 1 2 d 3 xdt N 3 F 4 { γ ζ,ij i γ ζ,kl j γ ζ,kl + γ ζ,kl i γ ζ,kj j γ ζ,il γ ζ,ik i γ ζ,jl j γ ζ,kl } (7.39) Sclr-grviton interctions on uniform curvture hypersurfces In the previous sections we constructed in systemtic mnner the cubic guge invrint ction for the second order grviton nd curvture perturbtion on uniform field hypersurfces. Similrly, we cn find the guge invrint ction for sclr field nd tensor perturbtions on uniform curvture hypersurfces. The strting point is gin the ction (7.3), which expnded to third order for sclr-grviton interctions gives Eqs. (7.22) nd (7.34). Insted of replcing ζ by its linerly guge invrint vrible w ζ, we now mke use of Eq. (7.17) to replce ϕ = w ϕ + φ H ζ. (7.40).

131 123 This seprtes the cubic vertices into mnifestly guge invrint prt involving linerly guge invrint sclr perturbtion w ϕ nd tensor γ ij, plus guge dependent vertices tht involve the guge dependent vrible ζ. Anlogous to the previous sections, the guge dependent prt of the cubic ction cn be brought to form which is proportionl to the liner equtions of motion for w ϕ nd γ ij. These terms my be bsorbed into the qudrtic ction by defining second order vribles γ ϕ,ij =γ ij ζ H γ ij + ( iζ j ζ H H i ζ j s wϕ + ) jζ i s wϕ H H ( ) W ϕ =w ϕ i j 1 φζ 2 4 H 2 γ ij + O(ζ 2 ), (7.41) where we hve seprted the first order constrint solutions (7.11) nd (7.12) into guge invrint nd guge dependent prt n = Ḣ 1 F ( ) ( ) ( ) H φ w 2 F Hwϕ ζ ζ ϕ + + n H + 1 wϕ + F φ H H 2 F ( ) 2 s 2 = 2 1 F 2 F w 2 + χ H + 1 F 2 F ϕ φ 2 ζ 2 H 2 2 s w ϕ 2 ζ 2 H. (7.42) In the guge ζ = 0 the perturbtions γ ϕ,ij nd W ϕ reduce to the grviton γ ij nd field perturbtion ϕ. Therefore, they re clled the second order guge invrint grviton on uniform curvture hypersurfces nd field perturbtion on uniform curvture hypersurfces, respectively. The O(ζ 2 ) terms in the second order perturbtion W ϕ follow from the pure sclr interctions, which hve been discussed (in prt) in the previous chpter, see Eq. (6.14) nd Refs. [111] nd [2]. The qudrtic ction for the perturbtions (7.41) is S (2) γ = 1 [ d 3 xdt ϕ 2 2 N 3 F ( ) ] 2 γϕ,ij γ ϕ,ij γ ϕ,ij 4 S (2) [Wϕ] 2 = 1 [ (HWϕ ) 2 ( ) ] 2 d 3 xdt 2 N 3 z 2 HW ϕ, (7.43) φ φ where gin z 2 ws defined in Eq. (7.13). The interction vertices re found by replcing w ϕ nd γ ij in the guge invrint prt of the third order ction by their second order generliztions. This gives the sclr-sclr-grviton vertices S (3) W 2 ϕ γϕ = 1 2 { ( d 3 xdt N 3 n Wϕ F F W ) ϕ φ + γ ϕ,ij i W ϕ j W ϕ γ ϕ,ij i j s Wϕ 2 + F i j s Wϕ 2 2F i n Wϕ γ ϕ,ij k γ ϕ,ij k s Wϕ } j W ϕ φ, (7.44)

132 124 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR nd the sclr-grviton-grviton vertices S (3) W ϕ γ = 1 ϕ 2 2 { d 3 xdt N 3 F W ϕ φ n Wϕ F ( 1 4 γ ϕ,kl γ ϕ,kl 1 i γ ϕ,kl 4 ( 1 4 γ ϕ,kl γ ϕ,kl + 1 i γ ϕ,kl 4 nd finlly the pure grviton vertices S (3) γ 3 ϕ = 1 2 d 3 xdt N 3 F 4 { γ ϕ,ij i γ ϕ,kl j γ ϕ,kl i γ ϕ,kl + γ ϕ,kl i γ ϕ,kj ) i γ ϕ,kl ) 1 2 γ ϕ,kl i γ ϕ,kl j γ ϕ,il i s Wϕ } γ ϕ,ik i γ ϕ,jl, (7.45) j γ ϕ,kl (7.46) In these interction ctions the n Wϕ nd s Wϕ re obtined by replcing w ϕ W ϕ in Eqs. (7.42). }. 7.4 Uniqueness of the sclr-grviton ction We hve now computed two guge invrint ctions for cubic sclr nd grviton interctions. Eqs. (7.33), (7.38) nd (7.39) describe the interctions for the curvture perturbtion nd grviton on uniform field hypersurfces, W ζ nd γ ζ,ij. Eqs. (7.44), (7.45) nd (7.46) describe the interctions for the field perturbtion nd grviton on uniform curvture hypersurfces, W ϕ nd γ ϕ,ij. If we compre the guge invrint vertices on different hypersurfces, we see tht they re not the sme. They re however relted in reltively simple wy. We cn show this by writing the cubic ctions on different hypersurfces in the sme form. For exmple, the sclr-sclr-grviton interctions Eqs. (7.33) nd (7.44) my be prtilly integrted in order to find, up to boundry terms, S (3) Wζ 2 γ = 1 ζ 2 { d 3 xdt N 3 1 i j χ 2 z2 W ζ γ ζ,ij 2 + z 2 γ i W ζ ζ,ij [ ] + F i j χ k γ ζ,ij k χ 2 + i j γ ζ,ij W ζ 2 4 H + 1 F 2 F [ ( 1 1 i W ζ j W ζ i W ζ j χ + H + 1 F 2 F H + 1 F 2 F j W ζ 2 δs (2) 3 δw ζ + jw ζ ) ] } i χ 2 δs (2) 3, δ γ ζ,ij (7.47)

133 125 nd S (3) W = 1 ϕ γϕ 2 2 { d 3 xdt N 3 1 HW 2 z2 ϕ i j χ γ ϕ,ij φ 2 [ γ ϕ,ij + F i j χ k γ ϕ,ij 2 [ 1 F 2 F F F + H H k χ 1 F 2 F F F + i j 2 i W ϕ φ j W ϕ φ + z 2 γ i HW ϕ j HW ϕ ϕ,ij φ φ ] ( 1 F W ϕ 2 F 2 δs (2) 4 φ H + 1 F 3 φ ) δw ϕ H 2 F ( i W ϕ j χ φ + jw ϕ i χ φ ) ] 2 3 δs (2) δ γ ϕ,ij } (7.48) Similrly, the sclr-grviton-grviton ctions (7.38) nd (7.45) become fter some prtil integrtions, S (3) W ζ = 1 γ ζ 2 2 nd S (3) W ϕ γ = 1 ϕ 2 2 d 3 xdt N 3 { z 2 d 3 xdt N 3 { z W ζ γ ζ,ij HW ϕ φ ( γ ζ,ij γ ζ,ij + γ ζ,ij W ζ H + 1 F 2 F γ 1 F 2 F ϕ,ij H δs (2) 3 δ γ ζ,ij } ( γ ϕ,ij γ ϕ,ij + γ ϕ,ij F F 1 H ( HWϕ φ γ ζ,ij ) F 2 γ k γ ζ,ij k χ ζ,ij, (7.49) ) γ ϕ,ij F 2 γ k γ ϕ,ij ϕ,ij } ) 2 3 δs (2) δ γ ϕ,ij k χ. (7.50) We see tht the ctions for W ζ nd γ ζ,ij on the one hnd, re lmost of the sme form of those for W ϕ nd γ ϕ,ij on the other hnd. They only differ by terms proportionl to the eqution of motion nd boundry terms (which cn be found in ppendix 7.B). In fct, we cn see tht the ctions (7.47) nd (7.48) become of the sme form once we identify W ζ γ ϕ,ij = γ ζ,ij γ ζ,ij H iw ζ j W ζ H H HW ϕ φ ( i W ζ H j s Wζ + jw ζ H i s Wζ = W ζ + i j γ ζ,ij W ζ H + O(W 2 ζ ), (7.51) The O(W 2 ζ ) terms hve been discussed in the previous chpter, see Eq. (6.15). The ction for the curvture nd grviton perturbtion on uniform field hypersurfces is thus relted to the ction for the field nd grviton perturbtion on uniform curvture hypersurfces vi non-liner field redefinitions. Of course, Eqs. (7.51) re nothing more thn ).

134 126 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR non-liner reltions between the different guge invrint vribles. It cn be checked strightforwrdly tht the guge invrint vribles (7.31) nd (7.41) stisfy the reltions (7.51). The exercise in this section shows tht the non-liner reltion between different guge invrint vribles lso follows from the ction. Note tht we did not consider the boundry terms here. They re discussed in the next section. Some finl words on uniqueness of the sclr-grviton interctions. We hve seen tht the ctions on different hypersurfces differ by terms proportionl to the liner equtions of motion nd boundry terms. This mens tht the complete non-liner equtions of motion for W ζ nd γ ζ,ij re ctully equl to those for HW ϕ / φ nd γ ϕ,ij, since the boundry terms do not contribute nd the terms proportionl to the eqution of motion vnish t liner order. Thus, the evolution of the fields, nd of n-point functions, is the sme whether one works on one hypersurfce or nother. In tht sense, there is unique bulk ction tht describes the evolution of non-gussinity, s ws lso discussed in section 6.5 (see lso figure 7.1). Still, n-point functions cn most certinly receive contributions from boundry terms in the ction, which we shll discuss next point functions nd boundry terms Due to the non-liner reltion between different vribles, their 3-point functions re relted vi disconnected pieces. For exmple, in the cse of pure sclr interctions, the disconnected pieces re proportionl to the squre of the sclr power spectrum, see for exmple [83]. In the cse of sclr-grviton interctions, the 3-point function for one set of guge invrint vribles differs from tht for nother set by product of the sclr nd grvittionl power spectrum. For exmple, if we consider only the leding contributions in the non-liner reltions (7.51), we find tht the sclr-grviton-grviton 3-point function on different hypersurfces re relted s Ḣ φ W ϕ(x 1 ) γ ϕ (x 2 ) γ ϕ (x 3 ) = W ζ (x 1 ) γ ζ (x 2 ) γ ζ (x 3 ) 1 H ( Wζ (x 1 )W ζ (x 2 ) γ ζ (x 2 ) γ ζ (x 3 ) + W ζ (x 1 )W ζ (x 3 ) γ ζ (x 2 ) γ ζ (x 3 ) ) (7.52) The spce-time coordintes re evluted t equl time. Here we used tht γ ϕ,ij = γ ζ,ij γ ζ,ij W ζ /H +..., nd the dots denote terms tht involve sptil derivtives. Furthermore, we hve mde use of the Wick contrction nd the fct tht W ζ γ ζ = 0. Considering the reltion (7.52), it hs been shows tht in the so-clled squeezed limit where one of the moment is much smller thn the other two the sclr-grvitongrviton reduces to product of the sclr nd grviton 2-point correltors multiplied by the spectrl index for tensor fluctutions, n T, see Ref. [83]. For the cse of the sclr-

135 127 sclr-grviton correltor, it cn be seen from the non-liner reltions (7.51) tht the correltor on different hypersurfces differs by non-locl terms. Eq. (7.51) is just n exmple of two guge invrint vribles tht re non-linerly relted, but in principle it cn be used for ny non-linerly relted vribles. This is very useful when computing 3-point functions for certin vrible from n ction, since one cn choose to work with non-linerly relted vrible for which the ction tkes form which is most suitble in specific situtions. For exmple, during the slowroll infltionry expnsion it is useful to use the ction for W ϕ nd γ ϕ,ij, since the orders in slow-roll re seprted nd it is esy to see wht terms re dominnt. This is wht ws used in Ref. [83] to compute the bispectrum for minimlly coupled sclr field. However, note here tht the initil stte for W ϕ nd γ ϕ,ij is tken to be Gussin, which then implies tht the initil stte for other vribles is non-gussin. We shll come bck to this lter. Now some words on the boundry terms. We hve derived Eq. (7.52) bsed purely on the non-liner reltions (7.51). Eq. (7.52) should lso follow from the ction, but this is not obvious when we consider the rewritten ctions on the uniform field hypersurfce (7.47) nd (7.49) versus the ctions on uniform curvture hypersurfces (7.48) nd (7.50). The bulk ctions re of the sme form, nd therefore led to similr 3-point functions. Terms proportionl to the equtions of motion do not contribute to bispectrum, s they vnish when evluted on the solutions of the first order eqution of motion. This suggests tht the disconnected contributions should follow from the boundry terms tht re obtined fter prtil integrtions. Indeed, in the in-in formlism temporl boundry terms contining time derivtives of fields cn contribute to the bispectrum. These boundry terms were discussed for the pure sclr cse in Refs. [120][119] [111][2]. There it ws found tht the temporl boundry terms precisely provide the dominnt contributions (during slow-roll infltion) to the disconnected pieces of the bispectrum. Thus, in the slow-roll regime the 3-point function for W ζ cn be computed vi the 3- point function for W ϕ. Figure 7.1 illustrtes the sitution. We wnt to compute guge invrint n-point correltors t time t by evolving some initil stte t t in. For this we use the guge invrint ction for perturbtions, which we hve computed up to third order for sclr-grviton interctions in this work. At both the initil time t in nd the finl time t we cn choose different hypersurfce. In figure 7.1 we schemticlly show the guge invrint uniform field nd uniform curvture hypersurfces 2. As we hve seen in the previous section, 2 As explined t the end of section 6.5.3, our notion of hypersurfce differs from tht commonly used in literture. Here hypersurfce is defined by choice of guge invrint perturbtions nd the guge invrint wve function is function of these vribles. Another hypersurfce is defined by other guge invrint perturbtions, which re relted to the first by non-liner trnsformtions.

136 128 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR Figure 7.1: Schemtic representtion of the initil hypersurfces t t = t in nd finl hypersurfces t time t. The depicted hypersurfces re the guge invrint uniform field hypersurfce (solid line) defined by guge invrint perturbtions W ζ nd γ ζ, nd the guge invrint uniform curvture hypersurfce (dshed line) defined by guge invrint perturbtions W ϕ nd γ ϕ. Since the bulk ction for the perturbtions on different hypersurfces is the sme, the evolution of their correltors from t in to t is unique. It is understood tht our notion of hypersurfce is different from the common one, which is ssocited with guge trnsformtions. Here the difference in hypersurfces is ment to illustrte the difference in guge invrint vribles, nd the ssocited difference in wve functions Ψ[W ζ, γ ζ ] nd Ψ[W ζ, γ ζ ]. the ctions on different hypersurfces re relted by boundry terms nd terms proportionl to the eqution of motion. This implies tht the evolution of non-gussinity is unique, nd there exists some bulk ction which describes this evolution. Now, depending on the choice of initil nd finl hypersurfce, nd on the initil non-gussinity, we find some non-gussinity t the finl time t. Tke the following exmple: t time t we re interested in correltors of W ζ nd γ ζ, since the CMB photons decouple on the uniform density hypersurfce (which coincides with the uniform field hypersurfce on super-hubble scles). Let us tke the sclr-grviton-grviton correltor W ζ γ 2 ζ. Suppose we wnt to know how much non-gussinity is creted by the pure evolution, nd how much origintes from the originl stte. If the initil stte is Gussin for vribles on the uniform field hypersurfce, mening tht the only non-zero correltors t initil time re 2-point functions of W ζ nd γ ζ, then the non-zero correltor W ζ γ 2 ζ t time t is creted only by the evolution. However, this chnges if the uniform curvture

137 129 hypersurfce, defined by vribles W ϕ nd γ ϕ, is in n initil Gussin stte. Then, due to the non-liner reltions between vribles on different hypersurfces, the initil stte for W ζ nd γ ζ is non-gussin. This initil non-gussinity is of the locl type (see Eq. (7.52)) nd origintes from the boundry terms in the ction. Some of the finl non- Gussinity, described by the correltor W ζ γ ζ 2, is therefore generted by the difference between initil nd finl hypersurfce. In order to mke this more rigorous, let us consider the generting functionl in the in-in formlism (see e.g. Ref. [118] for brief overview of the formlism): W [J W+, J W ] = [DW + DW D γ + D γ ] ρ(t in )δ(w + (t) W (t))δ( γ + (t) γ (t)) [ t ] ( ) exp ı (S + S ) + ı JW+ W + J W W + J γ+ γ + J γ γ, t in (7.53) where S ± = S[W ±, γ ± ]. The form of the generting functionl is the sme for both the uniform field nd uniform curvture hypersurfce, hence we did not write subscripts ϕ or ζ for the W nd γ fields. The ctions re functionl of W nd γ nd re integrted from initil time t in to the finl time t. The initil density mtrix for pure initil stte cn be written s ρ(t in ) = Ψ [W (t in ), γ (t in )]Ψ[W + (t in ), γ + (t in )]. (7.54) When the initil stte is not pure, then ρ in will not be fctorizble functionl of (W +, γ + ) nd (W, γ ), but insted it will depend on non-fctorizble correltors such s W + W, W + γ etc. Equl time expecttion vlues re computed by tking functionl derivtives of the generting functionl (7.53) with respect to the sources J i, (i = W ±, γ ± ). Now, let us consider the uniform field hypersurfce nd choose the initil stte to be Gussin, i.e. ρ(w ζ (t in ), γ ζ (t in )) = Ψ G Ψ G. Next, we mke the trnsition to the uniform curvture hypersurfce. We hve demonstrted tht the cubic ctions for non-linerly relted vribles re the sme up to terms proportionl to the eqution of motion nd boundry terms. Schemticlly S (3) [W ζ, γ ζ ] = S (3) bulk [W ϕ, γ ϕ ] + S (3) bd [W ϕ, γ ϕ ]. The boundry terms t initil time t in cn be bsorbed into the density mtrix, which then becomes non-

138 130 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR Gussin [ Ψ G [W ζ+, γ ζ +] e ıs(3) bd,+ ΨG φ ] H W ϕ+ Q(W ϕ+, γ ϕ+ ), γ ϕ+,ij Q ij (W ϕ+, γ ϕ+ ) Ψ ng [W ϕ+, γ ϕ+ ] [ Ψ G[W ζ, γ ζ ] e ıs(3) bd, Ψ G φ ] H W ϕ Q(W ϕ, γ ϕ ), γ ϕ,ij Q ij (W ϕ, γ ϕ ) Ψ ng[w ϕ, γ ϕ ]. (7.55) Here we hve written the non-liner trnsformtions (7.51) schemticlly s γ ϕ,ij = γ ζ,ij +Q ij (W ζ, γ ζ ) nd HW ϕ / φ = W ζ +Q(W ζ, γ ζ ) (which cn be inverted by expressing Q nd Q ij in terms of W ϕ nd γ ϕ. In ddition, the source terms trnsform non-linerly under the chnge of hypersurfce such tht, for exmple, W ζ J ζ W ϕ J ϕ. As result, we obtin the generting functionl for W ϕ nd γ ϕ with non-gussin initil stte. Thus, n initil stte which is Gussin on one hypersurfce is in generl non-gussin on nother hypersurfce. Let us discuss one more spect of the boundry terms. In the pure sclr cse it ws shown tht the non-liner field redefinition (7.51) tkes into ccount the dominnt contributions coming from the boundry terms in the ction for W ζ. However, in the previous chpter we hve rgued tht the field redefinition should completely tke cre of ll boundry terms (by which the ction on uniform field hypersurfces differs from tht on uniform curvture hypersurfces). The rgument is strightforwrd: the third order ction for W ζ nd γ ζ,ij nd the ction for W ϕ nd γ ϕ,ij re both mnifestly guge invrint ctions tht originte from the sme covrint ction (7.1). The vribles re relted vi specific non-liner reltion (7.51). Thus, lso the mnifestly guge invrint ctions should be relted by this non-liner reltion. The qudrtic ctions for γ ϕ,ij nd W ϕ (7.43) chnge schemticlly under the non-liner trnsformtion (7.51) s S (2) γ 2 ϕ = S (2) γ 2 ζ d 3 xdt N 3 { Q ij (W ζ, γ ζ ) 2 3 δs (2) + 1 [ 3 ] F 3 2 γ ζ,ij Q ij (W ζ, γ ζ ) [ i F 2 δ γ ζ,ij i γ ζ,kl Q kl (W ζ, γ ζ )] }, (7.56)

139 131 nd S (2) W = S (2) + 1 ϕ 2 Wζ 2 2 d 3 xdt N 3 { [2 3 z 2 Ẇ ζ Q(W ζ, γ ζ ) Q(W ζ, γ ζ ) 2 3 δs (2) ] i [ δw ζ 2z 2 iw ζ Q(W ζ, γ ζ ) ]}. (7.57) We hve seen tht the third order ction for W ζ nd γ ζ,ij cn be written s the third order ction for HW ϕ / φ nd γ ϕ,ij, plus terms proportionl to the eqution of motion nd boundry terms, see Eqs. (7.47) nd (7.49). The terms by which the ctions differ re precisely the eqution of motion terms s expected by the non-liner trnsformtions (7.56) (7.57). The boundry terms re however not precisely the expected ones. In ppendix 7.B we give explicit expressions for the boundry terms nd show tht the non-liner redefinitions (7.51) do not completely tke cre of the boundry terms. Bsed on guge invrince of the ction, ll boundry terms must be tken into ccount by the non-liner field redefinition. We suspect tht the discrepncy is due to two fctors tht we did not consider. First, boundry terms re lredy present in the second order ction (7.18). Under non-liner redefinition (7.51) these second order boundry terms generte third order boundry terms. Second, boundry terms re lso present in the ADM formultion of the ction, i.e. by going from Eq. (7.1) to Eq. (7.3). In totl, we expect tht complete tretment of the boundry terms ensures tht the ction on one hypersurfce is relted to tht on nother, t the bulk s well s t the boundry level. Whtever these boundry terms re, t t in they cn be bsorbed in the definition of the initil stte. 7.5 Frme independent cosmologicl perturbtions So fr we hve worked exclusively in the Jordn frme nd derived the physicl vertices for the mnifestly guge invrint vribles. In chpter 4 however we hve shown tht the Jordn frme ction with non-miniml coupling (7.1) cn be brought to the Einstein frme ction with miniml coupling S = 1 2 d 4 x g E {R E g µν E µφ E ν Φ E 2V E (Φ E )}. (7.58) by combined redefinition of the metric nd sclr field, see Eqs. (4.3) (4.5). No physicl content is lost by field redefinitions, nd thus, in tht sense, the Jordn nd Einstein frme re physiclly equivlent. Generlly speking, it is complicted to do computtions directly in the Jordn frme due to the non-miniml coupling. Fortuntely we cn exploit the physicl equivlence

140 132 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR of Jordn nd Einstein frme. It is possible to first trnsform to the Einstein frme, then perform the relevnt computtions in tht simpler frme, nd finlly trnsform the results bck to the Jordn frme. For exmple, in section 4.2 we used the reltions between bckground quntities in different frmes (4.10) in order to show tht the bckground equtions of motion in the Jordn frme (7.8) cn be derived directly from the Einstein frme equtions (4.6). In section the sme ws done in cnonicl formultion, i.e. we showed the equivlence between the bckground Hmilton equtions in Jordn nd Einstein frme. In section 4.4 we pointed out tht the equivlence is not tht esily seen t the perturbtive level. The reson is tht perturbtions in one frme do not coincide with those in nother, mking it complicted to relte, for instnce, correltion functions in different frmes. However, in section we hve given proof tht the guge invrint sclr perturbtion w ζ, defined in Eq. (7.16) is lso frme independent, w ζ,e = w ζ. Also it cn be seen tht the trnsverse, trceless prt of the metric γ ij is frme independent, simply becuse the conforml trnsformtion g µν,e = Ω 2 (Φ)g µν cnnot ffect the tensoril prt of the metric. Anlogous to the discussion of guge invrince, where mnifestly guge invrint form of the ction is reched when using guge invrint vribles, here the ction is mnifestly equivlent in terms of frme independent perturbtions. Wht we men is tht, when expressed in terms of w ζ nd γ ij, the ctions in Jordn nd Einstein frme re directly relted vi (trivil) frme trnsformtions of the bckground lone. For exmple, by mking use of the bckground reltions (4.10) it cn be seen tht ze 2 MP 2 = 1 M 2 P φ 2 E H 2 E = 1 Ω 2 M 2 P φ ( H F 2 F F F ) 2 z2 F, (7.59) where z E = φ E /H E nd z 2 ws defined in Eq. (7.13). By mking use of this reltion nd the frme independence of w ζ nd γ ij we cn immeditely derive their qudrtic ctions in the Jordn frme (7.15) nd (7.14) from the well-known results in the Einstein frme (5.40) nd (5.46). Thus, if one computes certin results in the Einstein frme using these frme independent perturbtions, such s the sclr or tensoril power spectrum, the corresponding Jordn frme result is immeditely obtined by trnsforming the bckground quntities. This ws used in, for exmple [69, 89, 90, 129]. Note tht the power spectrum for the comoving curvture perturbtion is expressed in terms of H nd φ in the Jordn frme, but in terms of H E nd φ E in the Einstein frme. However, the ctul vlue of the mplitude nd spectrl index is independent of the frme, once the equtions of motion for H nd φ, or H E nd φ E re solved nd their vlues computed t Hubble crossing. This is true since the comoving curvture perturbtion is frme independent.

141 133 As we hve seen in section 7.3, t higher order the guge trnsformtions become nonliner, such tht the mnifestly guge invrint ction t third order is expressed in terms of second order guge invrint vribles. In nlogy to this, we expect tht mnifestly equivlent ction cn be expressed in terms of second order frme independent perturbtions. Bsed on the previous section, we re led to believe tht the second order curvture perturbtion nd grviton on uniform field hypersurfces re lso frme independent. This ws verified in chpter 6 for the sclr perturbtion W ζ, see Eq. (6.67) nd Refs. [2, 122], lthough only the sclr guge invrint prt importnt on long wvelengths hs been considered. Here we computed the tensoril contributions to the second order guge invrint vrible W ζ, nd for those contributions we cn show using Eqs. (4.10) nd (4.25) tht W ζ,e = w ζ,e + ( ) i j 1 ϕ E γ ij,e + O(ϕ φ 2 E, w ζ,e ϕ E ) E 2 = w ζ + i j 2 4 ( 1 ϕ 4 φ γ ij ) + O(ϕ 2, w ζ, ϕ) = W ζ. (7.60) Agin, we hve shown this for the O(ϕ 2, w ζ ϕ) terms in Eq. (6.67). Next we consider the grviton. The trnsverse trceless prt of the metric is not ffected by the redefinition trnsformtion g µν,e = Ω 2 g µν. However, the trnsverse trceless prt γ ij is only guge invrint t liner order. We hve seen in Eq. (7.37) tht the second order guge invrint grviton on uniform field hypersurfces ctully receives contributions proportionl to ϕ. Thus t the non-liner level the frme trnsformtions ffect the guge invrint grviton. Nonetheless, we cn show for the grviton on uniform field hypersurfces tht γ ζ,ij,e =γ ij,e + iϕ E E φe j ϕ E =γ ij + iϕ φ j ϕ φ ( i ϕ E E φe ( i ϕ φ j s wζ,e E φe E j s wζ + jϕ φ i s wζ ) ϕ E φ E γ ij,e + jϕ E i s wζ,e E φe E ) ϕ φ γ ij = γ ζ,ij. (7.61) Here we used tht 1 E s w ζ,e = 1 s wζ, where s wζ is defined in (7.26). Thus W ζ,e = W ζ nd γ ζ,ij,e = γ ζ,ij, such tht the guge invrint sclr nd tensor perturbtions on uniform field hypersurfce re frme independent to second order 3. Of course, in the guge ϕ = 0 it becomes prticulrly cler tht W ζ nd γ ζ,ij do not trnsform under frme trnsformtion, which cn be generlized to rbitrry order to show tht the curvture perturbtion is frme independent [124]. By similr resoning, lso the grviton on uniform field hypersurfces is frme independent to ll orders. 3 The frme independence of the guge invrint curvture perturbtion cn be demonstrted to ll orders [92, 130] by using the fully non-liner generliztion of the curvture perturbtion [36, 131, 132].

142 134 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR Now tht we hve proven tht W ζ nd γ ζ,ij re frme independent vribles, it is trivil to show tht the cubic Jordn nd Einstein frme ctions, expressed in those vribles, re physiclly equivlent. Tke the cubic vertices for sclr-grviton interctions in the Jordn frme, Eqs. (7.33) nd (7.38). The sme vertices re obtined by tking the corresponding Einstein frme ction (setting F = MP 2 ) nd trnsforming the bckground quntities ccording to Eq. (4.10). Thus one cn use the Einstein frme ction to compute, for exmple, the 3-point function for W ζ in terms of H E nd φ E, nd then re-express everything in terms of H nd φ to find the 3-point function in the Jordn frme. Agin, once these bckground quntities re solved for through their equtions of motion nd their vlues inserted in the 3-point function, we would find exctly the sme number. For exmple, f NL computed in the Einstein frme coincides with tht computed in Jordn frme, nd is thus invrint under (non-liner) field redefinitions [122]. Generlly speking, ny n-point function of frme independent vribles does not depend on the frme in which it is computed. Although the cubic ction for W ζ nd γ ζ,ij is mnifestly equivlent, the perturbtive physicl equivlence of Jordn nd Einstein frme is not so obvious when the ction is expressed in terms of the guge invrint field nd grviton perturbtions on uniform curvture hypersurfces, W ϕ nd γ ϕ,ij. The cubic interction vertices for these perturbtions in the Einstein frme re (set F = MP 2 in Eqs. (7.44) nd (7.45)) S (3) Wϕ,E 2 γ = 1 { d 3 xdt ϕ,e 2 N E 3 EMP 2 where z E = φ E /H E nd S (3) W ϕ,e = 1 γ ϕ,e 2 2 Here χ E is defined by z 2 E 2M 2 P H E W ϕ,e φ E γ ϕ,ij,e i j χ E 2 E + z2 E i H E W ϕ,e j H E W ϕ,e MP 2 γ ϕ,ij,e + M 2 i j χ E k γ ϕ,ij,e k χ E P E φe E φe 2, E E E (7.62) d 3 xdt N E 3 EM 2 P 2 χ E 2 E { z 2 E 8M 2 P ( H E W ϕ,e γ ϕ,ij,e γ ϕ,ij,e + γ ϕ,ij,e φ E 1 2 γ ϕ,ij,e k γ ϕ,ij,e E k χ E E } E } ) γ ϕ,ij,e E. (7.63) ( ) = z2 E HE W ϕ,e 2MP 2. (7.64) φ E Now, if we compre with the Jordn frme ctions (7.44) nd (7.45), we see tht the ctions in the two frmes re not simply relted by re-expressing the bckground fields s

143 135 Eq. (4.10). The reson is tht the second order field perturbtion on uniform curvture hypersurfces re not frme independent, W ϕ,e W ϕ nd γ ϕ,ij,e γ ϕ,ij. Even so, it is not cler how exctly the different frmes re relted t the perturbtive level when working on the uniform curvture hypersurfce. Fortuntely, we cn mke this more explicit by severl prtil integrtions of the Jordn frme ction, which mke it look more like the trnsformed Einstein frme ction. In fct, we hve lredy performed these prtil integrtions in section 7.4, nd the resulting ctions re Eqs. (7.48) nd (7.50). These ctions re of the sme form s the Einstein frme ctions (7.62) (7.63) fter frme trnsformtion of the bckground, up to terms proportionl to the liner eqution of motion nd boundry terms. In nlogy to the reltion between the ctions on different hypersurfces in section 7.4, we cn now identify 1 F 2 F F F 1 F 2 F F F γ ϕ,ij,e = γ ϕ,ij + H H E W ϕ,e φ E = HW ϕ φ H [ W ϕ φ i j 2 1 F ] j) χ 2 F i W ϕ j W ϕ γ ϕ,ij + H + 1 F φ φ 2 (i W ϕ φ 2 F ( ) 1 W ϕ γ ϕ,ij + O(W 4 ϕ) φ 2. (7.65) The O(W 2 ϕ) terms hve been described in Eq. (6.75) of the previous chpter. Eqs. (7.65) show tht the Jordn nd Einstein frme ctions on uniform curvture hypersurfces re relted by combined trnsformtions of the bckground (4.10) nd non-liner trnsformtions of the perturbtions (7.65) 4. This demonstrtes the physicl equivlence for the third order ction expressed in guge invrint vribles on the uniform curvture hypersurfce, lthough the ction is not mnifestly equivlent. We mention tht the vribles W ϕ,e nd γ ϕ,ij,e on one side, nd W ϕ nd γ ϕ,ij on the other, re defined on the sme uniform curvture hypersurfce. The frme trnsformtion merely seems to mix up the sclr nd tensoril degrees of freedom t the non-liner level, s cn be seen from (7.65). This is not so strnge, considering the conforml trnsformtion (4.3) mixes grvittionl nd sclr degrees of freedom. In the end, the guge invrint perturbtions on the sme hypersurfce but in different frme re relted by non-liner trnsformtions, just like guge invrint perturbtions on different hypersurfces but in the sme frme re non-linerly relted. The non-liner reltion (7.65) between the guge invrint vribles in different frmes implies tht the n-point functions in different frmes re relted vi resclings nd disconnected terms. For exmple, in the previous chpter we hve seen tht the 3- point function for W ϕ,e differs from the 3-point function for W ϕ by rescling in terms of H, H E, φ, φe nd by squres of the 2-point function. From the non-liner frme 4 The non-liner reltions (7.65) cn lso be derived from the definitions of the guge invrint perturbtions in the Einstein frme (7.41) nd replcing (4.10) nd (4.25).

144 136 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR reltions in this work we deduce tht the sclr-grviton-grviton correltion function in Einstein nd Jordn frme re relted s H E W ϕ,e (x 1 ) γ ϕ,e (x 2 ) γ ϕ,e (x 3 ) = Ḣ ( W ϕ (x 1 ) γ ϕ (x 2 ) γ ϕ (x 3 ) φ E φ 1 F 2 F H F F ( Wϕ (x 1 )W ϕ (x 2 ) γ ϕ (x 2 ) γ ϕ (x 3 ) + W ϕ (x 1 )W ϕ (x 3 ) γ ϕ (x 2 ) γ ϕ (x 3 ) ) ) +..., (7.66) where the dots denote sptil derivtive terms. Thus, cre must be tken when computing quntities in the (defining) Jordn frme vi the Einstein frme, if one works on the uniform curvture hypersurfce. An exmple of delicte sitution is the computtion of the nive cut-off in Higgs infltion, which we shll discuss next. 7.6 Nturlness in Higgs infltion The mnifestly guge invrint nd frme independent formultion of the perturbtive ction is n idel strting point for studying the nturlness problem in Higgs infltion [ , 105, 128]. The nturlness problem hs lredy been discussed in section 5.4, but we shll go into more detil here. First we briefly review recent computtions of the cut-off following Ref. [105], then we reddress the question of nturlness in the frme independent frmework The cut-off in Higgs infltion revisited A usul requirement for renormlizble theories is tht the unitrity bound is not violted in perturbtion theory, i.e. scttering mplitudes should not become bigger thn unity. In prticulr there is the requirement of tree unitrity [133], which sttes tht N prticle tree mplitudes A N should not grow more rpidly thn E 4 N, where E is the center of mss energy. If the mplitude grows fster, perturbtion theory fils t some cut-off scle Λ. This usully mens tht some new physics should enter t this energy scle. When the relevnt energy scles of the theory under considertion re well below the cut-off scle, the theory cn be considered nturl, in the sense tht perturbtion theory is vlid nd there is no need for new physics. Conversely, there is nturlness problem if typicl energy scles re higher thn the cut-off scle. The cut-off is most esily computed when the perturbtive ction is written in cnonicl form, such tht the propgtor goes s 1/k 2, where k is the 4-momentum. Next, the cut-off cn be red off from the vertices of dimension higher then 4, which should be suppressed s Λ 4 D. This hs been done for Higgs infltion in Ref. [128] in both the

145 137 Einstein nd Jordn frme, nd we outline the derivtion here. In the Jordn frme the strting point is the ction (7.1) with F (Φ) = M 2 P + ξφ2, which is the ction for Higgs infltion in unitry guge nd guge interctions re neglected. Next one inserts perturbtions of the metric nd sclr field round homogeneous bckground g µν (x) = ḡ µν (t) + δg µν (x) Φ(x) = φ(t) + ϕ(x). (7.67) ḡ µν is in principle generl bckground metric, but is of course the FLRW metric in n expnding universe. The kinetic terms for sclr nd metric fluctutions in the qudrtic ction re not digonl (due to the coupling ξφ 2 R) nd not cnoniclly normlized. We cn define new perturbtions δg ˆ µν nd ˆϕ for which these kinetic terms become cnoniclly normlized, such tht their propgtors scle t high energies s 1/E 2, where E = k 5 0. Next we cn compute the interction vertices for these redefined perturbtions. We now identify the dominnt term with dimension higher thn 4. In the Jordn frme, this vertex is ξϕ 2 δg, where δg = ḡ µν δg µν, which in terms of the cnoniclly normlized fields becomes ξ M 2 P + ξφ2 M 2 P + ξφ2 + 6ξ 2 φ 2 ˆϕ2 ˆδg. (7.68) We cn see tht t high energies this vertex scles s E 2. If we consider 2 2 scttering process of ˆϕ vi exchnge of grvittionl sclr δg, ˆ we see tht the totl mplitude scles s E 2 /Λ 2 t high energy. The cut-off scle is precisely the inverse of the opertor bove. The cut-off in the Jordn frme is thus Λ = M 2 P + ξφ2 + 6ξ 2 φ 2 ξ M 2 P + ξφ2. (7.69) In Ref. [105] the cut-off ws lso computed vi the Einstein frme. In tht frme the non-miniml coupling term is bsent nd the grvittionl nd field kinetic terms re cnonicl. Still, the cut-off scle reppers in the non-polynomil potentil nd shows roughly the sme behvior s bove (though it is not exctly the sme!). From the cutoff s shown bove it cn be seen tht Λ ξφ in the regime where φ M P / ξ, Λ ξφ 2 /M P for M P / ξ φ M P /ξ nd Λ M P /ξ when φ M P /ξ. The uthors of Ref. [128] then rgue tht ll relevnt energy scles in these regimes re lower thn the cut-off scle, such tht the perturbtive expnsion is vlid nd Higgs infltion is nturl. In spite of these results there re mny problems with the previous computtions. Let us discuss the min ones here. 5 The propgtor scles s 1/k 2 = 1/( k k 2 ), so depending on whether k 0 k or k 0 k it scles s 1/E 2 or 1/ k 2. In the center of mss frme k is zero.

146 138 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR 1: The computtion of the cut-off is guge dependent As we hve seen in chpter 2 both the metric fluctutions δg µν nd sclr field perturbtion ϕ re guge dependent. The cut-off determines the scle of vlidity of perturbtion theory, but we should consider this for physicl, i.e. guge invrint perturbtions. A vertex such s ϕ 2 δg is explicitly guge dependent, nd my be non-physicl. We hve shown in chpters 6 nd 7 tht some vertices re ctully bsorbed into the definition of guge invrint vribles. Moreover, we hve lso seen tht some degrees of freedom in the metric re non-dynmicl. They ct s uxiliry fields nd should be solved for, which generte dditionl vertices tht my cncel the problemtic vertex. In fct, this lredy hppens t the level of the qudrtic ction. Nively, the field perturbtion hs n effective mss term m 2 eff ϕ2 = ( ξ R + V )ϕ 2 +.., where R = 6(2H 2 + Ḣ) is the bckground Ricci sclr. Such mss term is huge, in the sense tht ξ R/H 2 1. However, only light inflton field, with m 2 eff /H2 1 cn generte nerly scle invrint power spectrum. Thus, such mss term is disstrous for the model nd nively rules out Higgs infltion. But, when contributions from the uxiliry fields re tken into ccount, the ξr contribution is cnceled, leving only light effective mss for the inflton field. A similr cncelltion hppens t higher order in guge invrint formultion. 2: The cnonicl redefinition mixes up frmes A crucil step in the computtion of the cut-off ws the definition of new perturbtions ˆϕ nd ˆδg which cnoniclly normlize the kinetic terms. However, this redefinition is in fct nothing more thn trnsformtion from the Jordn to Einstein frme t the level of perturbtions 6! On the other hnd, the cut-off is computed from the term ξφ 2 R in the Jordn frme ction. So somehow one computes cut-off from Jordn frme vertex using Einstein frme perturbtions, which is very bizrre. Moreover, we would like to emphsize tht, lthough the Einstein frme is often referred to s the frme in which both the grvittionl ction nd sclr field ction re written in cnonicl form, the Einstein frme is not cnonicl. Cnonicl formultion mens tht the kinetic sectors for different fields re decoupled (nd normlized), such tht one cn strightforwrdly extrct the cnonicl momentum nd quntize the theory. However, in the cnonicl Einstein frme the grvittionl field still couples to the sclr sector s g E g µν E µφ E ν Φ E. Conversely, the sclr field still couples to the kinetic term for the metric perturbtions in the g E R E term vi the uxiliry fields in the metric (which contin ϕ E in its first order solution). Hence, the Einstein frme is formulted in non-cnonicl wy, just like the Jordn frme. The true cnonicl formultion is only reched once one inserts 6 This cn be explicitly checked by compring Eqs. (2.9) nd (2.10) in Ref. [105] to the trnsformtions (4.25) nd the expnsion of g µν,e = Ω 2 g µν to first order in perturbtions.

147 139 perturbtions nd decouples physicl nd non-physicl degrees of freedom, which we hve done in chpter 5. The (Jordn frme) result is Eq. (5.37), but s we hve shown it hs exctly the sme form in Einstein frme. Thus in cnonicl formultion there is no rel notion of frme nymore. 3: Inequivlence of Jordn nd trnsformed Einstein cut-off As we hve mentioned, the cut-offs computed directly in the Jordn frme, or vi the Einstein frme, re similr, but not exctly the sme. However, they should coincide, becuse no physicl content is lost in the frme trnsformtion. Of course the origin of this problem is relted to the lredy mentioned problems, nmely non-covrint formultion nd mixing-up of different frmes Frme independent computtion of cut-off The problems bove re ll voided once the theory is written in mnifestly guge invrint nd frme independent wy. First of ll, when one mkes use of guge invrint vribles, ll vertices re physicl vertices. Moreover, the guge invrint perturbtions decouple in the qudrtic ction, which mkes it very esy to write the theory in cnoniclly normlized form. And obviously, when frme independent perturbtions re used, results in Jordn nd Einstein frme re equivlent. So let us compute the cut-off now in our frme independent formultion, following the sme steps s bove. First we perform our clcultions in the Jordn frme, for which the qudrtic ctions for the physicl perturbtions W ζ nd γ ζ re given in Eqs. (7.32). We cn define cnonicl vribles v =zw ζ = 2ɛF W ζ v γ,ij = 1 2 F γ ζ,ij, (7.70) where we hve mde of useful reltion to relte z (defined in Eq. (7.59)) to the slow-roll prmeter ɛ in the Einstein frme, ɛ = ḢE H 2 E = z2 E 2M 2 P = z2 2F. (7.71) In terms of the cnonicl fields the second order ctions in conforml time τ ( N(t) = (t)) become S (2) v = 1 d 3 xdτ [ v 2 ( 2 i v) ] 2 S (2) = 1 d 3 xdτ [ v vγ,ij 2 γ,ij 2 ( l v γ,ij ) ], (7.72) 2

148 140 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR where prime denotes derivtive with respect to conforml time. We did not write the time dependent mss terms, becuse we re only interested in the behvior t high energy scles, for which both the sclr nd grviton propgtors re proportionl to 1/E 2. Next we wnt to see wht re the vertices for these fields. The pure sclr vertices re given in Eq. (6.70), the grviton-sclr-sclr in Eq. (7.47), the grviton-grvitonsclr in Eq. (7.49) nd the pure grviton vertices in Eq. (7.39). We re-express these in terms of the cnonicl fields (7.70) nd collect the dominnt vertices from the ction, which gives (up to numericl fctors): v 3 : v 2 v γ : vvγ 2 : vγ 3 : ɛ F v[v 2 + ( i v) 2 ] 1 F v γ,ij i v j v ɛ F v[v 2 γ,ij + ( l v γ,ij ) 2 ] 1 F v γ,ij i v γ,kl j v γ,kl. (7.73) We cn see tht ll of these vertices re of dimension 5, nd thus must be suppressed by cut-off scle 7. This cn be seen when we consider 2 2 scttering process, where the ingoing nd outgoing prticles cn be ny combintion of sclrs nd grvitons, medited by either sclr or grviton propgtor. In the high energy limit E = k 0 k these mplitudes scle s (E/Λ) 2, where Λ is the (comoving) cut-off scle. In the high energy limit the pure sclr or sclr-grviton-grviton vertices give the biggest contribution, leding to the cut-off scle Λ F M 2 = = P + ξφ 2. (7.74) ɛ ɛ Here Λ is the comoving cut-off scle, nd Λ/ the physicl cut-off. So, for lrge (infltionry) field vlues φ M P / ξ, Λ/ ξφ/ ɛ nd ɛ 1, such tht the cut-off scle is well bove the Plnck scle. When infltion ends, ɛ = O(1), the cut-off scle coincides with the Plnck scle Λ/ = M P. Thus the cut-off is lwys t or bove the Plnck scle. We cn now repet the sme clcultion in the Einstein frme, which strightforwrdly 7 It comes s no surprise tht the cnoniclly normlized sclr-sclr-grviton nd pure grviton vertices re not suppressed by fctor of ɛ. The reson is tht in the de Sitter limit ɛ 0 the curvture perturbtion ζ becomes pure guge mode, nd is completely bsorbed by the guge invrint lpse nd shift perturbtions. The only remining dynmicl perturbtions re the sclr ϕ nd the grviton γ ij. The term g µν µφ νφ in the originl ction then gives the interction term γ ij i ϕ j ϕ, which is not ɛ suppressed. Likewise, the pure grviton vertices re lwys present nd re not suppressed by powers of ɛ.

149 141 gives Λ E E = M P. (7.75) ɛ If we trnsform bck to the Jordn frme, we obtin the sme result s (7.74) for the physicl cut-off Λ/, which is wht we lso wnted to show. In fct, the comoving cutoffs coincide, Λ = Λ E. The difference between the physicl cut-offs in Eqs. (7.74) nd (7.75) comes from the different definitions of nd E, nd hence physicl moment re frme dependent. Wht hppens if we hd tken the high momentum limit, k k 0? In tht cse lso the pure grviton vertices nd sclr-sclr-grviton vertices re contributing to 2 2 scttering process. These vertices re not suppressed by fctor ɛ/ F, but only by 1/ F, nd hence the cut-off is lower. It is esy to see tht the Jordn frme cut-off is nd the cut-off in the Einstein frme Λ = F = MP 2 + ξφ2, (7.76) Λ E E = M P, (7.77) which gin is relted to the Jordn frme result by redefinition of the bckground. The Jordn frme cut-off is shown in figure 7.2. The importnt point is tht lso here the cut-off is never smller thn the Plnck scle. This mens tht the perturbtive expnsion is vlid t lest up to energy scles below the Plnck scle for theory with some non-miniml coupling to grvity, such s Higgs infltion. Thus, the bove nlysis indictes, t lest for the clss of 2 2 scttering processes, tht there is no nturlness problem in Higgs infltion. There re some cvets to this sttement. First of ll we hve only considered the sclr-grvity sector of Higgs infltion, but neglected interctions with e.g. guge fields. In Ref. [103] it ws stted tht the (low) cut-off scle of M P /ξ lso ppers in Higgsguge interctions. However, these vertices re guge dependent s well, so the problem my be bsent in guge invrint formultion including the guge fields. Secondly, we hve mde use of the prtilly integrted ctions (6.70), Eq. (7.47) nd Eq. (7.49) for the computtion of the cut-off. Hd we not used these prtilly integrted ctions, we would hve found disstrous terms in the ction. For exmple, in the pure sclr ction for W ζ (in Jordn frme (6.68) or Einstein frme (6.41)) the dominnt terms seem to be of order ɛ, such tht the vertices in cnoniclly normlized vribles become, for exmple 1/( ɛf )vv 2. This suggests cut-off scle Λ/ = ɛf, or in Einstein frme Λ/ E = M P ɛ. As consequence there my be nturlness problem for Higgs infltion, becuse ɛ 1, which mens tht the cut-off cn be well below the Plnck

150 142 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR INFLATION Figure 7.2: Physicl cut-off Λ/ = M 2 P + ξφ2 in the Jordn frme. scle, nd my become comprble to the energy scle of infltion H. The sme cut-off scle ppers in the sclr-grviton-grviton vertices before prtil integrtions (7.38). Worse still, the sclr-sclr-grviton vertices before prtil integrtions (7.33) re in terms of cnoniclly normlized vribles proportionl to 1/(ɛ F ), which gives n even lower cut-off Λ/ = F ɛ. However, we hve seen tht, in ll these cses, the ction only ppers to be of certin order in slow-roll, but fter some prtil integrtions the dominnt contributions cncel nd the ction is ctully suppressed by higher order slow-roll prmeters. So, the pure sclr vertices re ctully of order ɛ 2, nd the sclrsclr-grviton nd sclr-grviton-grviton vertices re of order ɛ. This ensures tht Higgs infltion is nturl, in the sense tht the perturbtive expnsion is vlid ll the wy up to the Plnck scle. Of course, we hve mde some estimtes of the cut-off scle, but more rigorous computtion is required. Also, we hve not discussed the boundry terms nd whether or not they contribute in some wy to the cut-off scle. These questions re currently under investigtion [134]. 7.7 Summry The im of this chpter ws to compute the guge invrint ction t third order for single sclr field in the Jordn frme, with emphsis on the grviton nd its interctions

151 143 with the sclr perturbtions. We hve demonstrted method to find the mnifestly guge invrint ction order by order in perturbtion theory. The procedure relies on seprting the higher order ction in guge invrint plus guge dependent prt nd bsorbing the ltter into the definition of new non-liner guge invrint vrible. By doing so one not only obtins the physicl vertices, but t the sme time finds the correct higher order guge invrint vribles. The method thus provides n lterntive wy to find guge invrint vribles directly from the ction, without hving to resort to non-liner guge trnsformtions. In section 7.3 we computed the cubic guge invrint ction for the second order curvture perturbtion nd grviton on uniform field hypersurfces, nd for the second order field perturbtion nd grviton on uniform curvture hypersurfces, both directly in the Jordn frme. We demonstrted tht the different ction re relted vi nonliner trnsformtions of the guge invrint vribles, which re precisely the nonliner reltions between the vribles on different hypersurfces (7.51). The ctions on different hypersurfces only differ by terms proportionl to the eqution of motion nd boundry terms, such tht the evolution of n n-point function is the sme for one set of guge invrint vribles or nother. In this sense the guge invrint ction is unique. Still, the n-point functions, for exmple the bispectrum, for one set of vribles differs from tht for nother by disconnected pieces due to the non-liner reltion (7.51). In section 7.5 we discussed the equivlence to the Einstein frme. We hve demonstrted tht the perturbed ction cn be written in mnifestly equivlent form by expressing it in terms of frme independent perturbtions. We hve shown tht the guge invrint curvture perturbtion nd grviton on uniform field hypersurfces in the Jordn frme coincide with the corresponding perturbtions in the Einstein frme, nd re thus frme independent (7.60) (7.61). In terms of these vribles, the perturbed ction in the Jordn frme cn be obtined from the Einstein frme ction by frme trnsformtion of the bckground fields lone. Moreover, we hve shown tht the field perturbtion nd grviton on uniform curvture hypersurfces in the Einstein frme re non-linerly relted to their counterprts in the Jordn frme (7.65). This cn be derived from the perturbtions themselves, but lso follows from the ction for these vribles. As consequence, n-point functions for perturbtions on uniform curvture hypersurfces differ by disconnected pieces between different frmes. In conclusion, we hve shown tht, whether one tkes the ction for guge invrint perturbtions on different hypersurfces in the sme frme, or the ction for the sme guge invrint perturbtions in different frmes, they re ll relted vi nonliner trnsformtions, which mkes it very convenient to find n-point functions on specific hypersurfce/in specific frme vi the n-point functions on different hypersurfce/in different frme. As finl ppliction of our frme independent nd guge invrint formultion of

152 144 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR perturbtions we hve reddressed the question of nturlness in Higgs infltion. By considering only sclr nd grvittionl perturbtion we hve found tht the lowest bound for the Jordn frme cut-off is Λ/ = M 2 P + ξφ2, nd the cut-off is Λ/ E = M P in the Einstein frme. Thus, the perturbtive expnsion in Higgs infltion is vlid t lest up until the Plnck scle. 7.A Decoupling the constrint fields in the ction We now present n lterntive method to del with the uxiliry fields in the ction. Insted of solving for them to first order in perturbtions, s ws done in the min text nd in [83], we cn decouple them from the dynmicl fields in the ction. This ws used in chpter 5, see lso Refs. [1, 78]. There, cnonicl formultion of the ction ws used s strting point. Here we strt with the Lgrngin formultion of the ction nd decouple the constrint fields s first step. After inserting the perturbtions (7.6) in the ction (7.3) nd expnding to second order, we cn collect only those terms tht contin perturbtions of the lpse nd shift field. After performing some prtil integrtions nd mking use of the bckground equtions of motion the ction tkes the form where nd S (2) [n, s] = 1 2 d 3 xdt N ( { 2V 3 n 2 + I n n + 2 2HF + F ) n sol 2 s 2 I n =6(2HF + F ) ζ + (6HF φ) ϕ + (6H 2 F + 6HF 2V )ϕ }, (7.78) 2(2HF + F ) 2 s (2F ζ + F ϕ), (7.79) n sol = 1 [2F 2HF + F ζ + φϕ ] + F ϕ HF ϕ + F φϕ. (7.80) In terms of n sol we my re-express I n s ( I n = 4V n sol 2(2HF + F 2 ) s ) 2 2 s sol 2, (7.81) where 2 s sol 2 = 1 H F F 2 2 ( ζ + 1 F ) 2 F ϕ + 1 2F φ ( H F 2 F F F ( ) 2 ζ Ḣ ) φ ϕ. (7.82)

153 145 Now we define new vribles s nd ñ s (compre to Eqs. (5.65), (5.73) nd (5.83)) 2 s 2 2 s 2 2 s sol 2 ñ n 1 4V I n = n n sol 2(2HF + F ) 2 s 2. (7.83) In terms of these vribles, the ction (7.78) cn be rewritten s (compre to Eqs. (5.66) nd (5.74)) S (2) [n, s] = 1 { d 3 xdt 2 N 3 2V ñ 2 + (2HF + F ) 2 ( ) 2 2 s V 2 ( + 2V n 2 sol + 2 2HF + F ) n sol 2 s sol 2 }. (7.84) Now, the first line in the ction re the completely decoupled qudrtic ctions for ñ nd s. The equtions of motions for these vribles re simply ñ = 0 nd 2 s/2 = 0, which give the first order solutions of the constrint equtions, n = n sol nd 2 s/ 2 = 2 s sol / 2. The terms on the second line in the rewritten qudrtic ction re precisely the terms tht we would hve obtined if we would hve solved the constrint equtions to first order in perturbtions nd inserted the solution in the ction. This is wht we wnted to proof, lthough we hve so fr only shown it for the second order ction. The extr terms in Eq. (7.84) re needed to construct the mnifestly guge invrint ction for w ζ. This implies tht the vribles ñ nd s re guge invrint by themselves, which cn be shown explicitly [1]. The guge invrince cn be exploited to decouple non-dynmicl degrees of freedom from dynmicl ones t higher order. The resoning is very similr to tht in section Let us consider the schemtic form of the cubic ction for the lpse perturbtion lone S (3) [n] = 1 { } d 3 xdt 2 N 3 2V n 3 + n 2 Q (1) + nq (2). (7.85) Here the Q (1) nd Q (2) re liner nd qudrtic functions of the dynmicl perturbtions ζ, ϕ nd γ ij (nd of s, but for simplicity we neglect it). The explicit form of these functions cn be derived when expnding Eq. (7.3) up to third order in perturbtions. We wnt to decouple the non-dynmicl degrees of freedom from the dynmicl ones, but this nively does not seem possible. Here is where we exploit the guge invrince of the ction. Although ñ is guge invrint to first order, it trnsforms under second order guge trnsformtions s ñ ñ + (2) ξ ñ. (7.86)

154 146 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR This induces chnge in the qudrtic ction (7.84) S (2) [ñ] S (2) [ñ] + 1 { } d 3 xdt 2 N 3 2 δs (2) 3 δñ (2) ξ ñ =S (2) [ñ] + 1 { } d 3 xdt 2 N 3 4V ñ (2) ξ ñ. (7.87) Such second order guge trnsformtion ws lso shown for the dynmicl perturbtions (7.28), nd similr guge trnsformtion cn be derived for the ction for s. Now, the only wy in which such second order guge trnsformtion of the qudrtic ction cn be blnced, is by terms proportionl to the equtions of motion for ñ in the third order ction. Since the eqution of motion for non-dynmicl field is simply proportionl to the field itself, we should find ll the terms in the third order ction proportionl to ñ. We cn do this systemticlly for the ction (7.85) by replcing n by ñ using Eq. (7.83). The result is S (3) [ñ] = 1 2 [ d 3 xdt {2V N 3 ñ 3 4V ñ 3 2ñn sol 3 ] 2 n2 sol Q(1) ñ 4V Q(1) n sol Q(2) 2V 4V } + 2V n 3 sol + n 2 solq (1) + n sol Q (2). (7.88) The terms on the first line re guge invrint vertex for the non-dynmicl ñ, nd terms proportionl to the liner eqution of motion. The ltter cn be bsorbed into the qudrtic ction by defining new second order guge invrint vrible Ñ = ñ 3 2ñn sol 3 2 n2 sol Q(1) ñ 4V Q(1) n sol 2V Q(2) 4V, (7.89) such tht the cubic ction (7.90) becomes S (3) [Ñ] = 1 { } d 3 xdt 2 N 3 2V Ñ 3 + 2V n 3 sol + n 2 solq (1) + n sol Q (2). (7.90) Thus we hve found decoupled cubic vertex for Ñ, plus remining terms which re the sme s those obtined by replcing n in Eq. (7.85) by its first order solution. In Eq. (7.90) we hve neglected the s terms, but in very similr wy they cn be bsorbed into the definition of second order guge invrint shift perturbtion S. So, finlly we re left with decoupled, mnifestly guge invrint cubic ction for the non-dynmicl fields Ñ nd S, nd extr terms tht we would hve obtined when the constrints would hve been replced by their first order solution. This is precisely wht

155 147 we wnted to prove. The method outlined here is nice nd systemtic wy to decouple the non-dynmicl sector from the dynmicl one, while t the sme time finding the second order guge invrint constrint fields (which re the second order solutions of the constrint equtions). The procedure here cn be extended to higher order, for exmple fourth order. First, we found the second order guge invrint ction in terms of the linerly guge invrint ñ nd s. Next, we insert these quntities in the third nd fourth order ction. From the third order ction we now find the definition for second order perturbtions Ñ nd S. We cn now replce in the third nd fourth order ction ñ = Ñ + n(2) sol +... nd s = S + s (2) sol +..., where n(2) sol nd s(2) sol re the second order prts of the solution of the constrint eqution. This gives guge invrint cubic vertices for Ñ nd S. Finlly, the terms proportionl to Ñ nd S in the fourth order ction cn be bsorbed in the qudrtic ction for Ñ nd S by defining third order guge invrint vrible. The remining terms in the ction re those where n nd s re replced by their second order solution n (2) sol nd s(2) sol. This ws lso found in Ref. [85], nd pplied in Refs. [80, 86] to find the fourth order ction for perturbtions in the uniform curvture guge. 7.B Boundry terms Here we discuss some of the boundry terms tht pper due to the prtil integrtions section 7.4. We only show the temporl boundry terms, s these re terms tht cn possibly contribute to the bispectrum. We strt with the boundry terms in the sclr-grviton-grviton ction. By going from Eq. (7.38) to Eq. (7.49), we find the following temporl boundry terms S W ζ γ 2 ζ = 1 2 d 3 x { 3 F H + 1 F 2 F ( W ζ 1 4 γ ζ,ij γ ζ,ij 1 γ ζ,ij 4 Likewise, going from Eq. (7.45) to Eq. (7.50) we obtin S W ϕ γ 2 ϕ = 1 2 d 3 x { 3 F H + 1 F 2 F F 2F 1 H ( HWϕ φ ) ( 1 4 γ ϕ,ij γ ϕ,ij 1 γ ϕ,ij 4 ) } γ ζ,ij. (7.91) ) } γ ϕ,ij. (7.92) In section it ws rgued tht by the non-liner reltion (7.51) the complete ction for W ϕ nd γ ϕ should trnsform into tht for W ζ nd γ ζ, both the bulk nd the boundry prt. If we consider the sclr-grviton-grviton ction lone, nd see wht boundry

156 148 CHAPTER 7. NON-LINEAR GAUGE INVARIANCE AND FRAME INDEPENDENCE : THE GRAVITON SECTOR terms for W ζ nd γ ζ we get under the trnsformtion (7.51), we find { 1 d 3 3 F x ( F W ζ 1 2 H + 1 F 2F H 4 γ ζ,ij γ ζ,ij 1 ) } γ ζ,ij γ ζ,ij 3 F 4 2 γ W ζ ζ,ij H γ ζ,ij 2 F = S W ζ γ + 1 { d 3 x 3 F W ( ζ 1 ζ 2 2 H 4 γ ζ,ij γ ζ,ij + 1 ) } γ ζ,ij γ ζ,ij. (7.93) 4 The first line contins terms coming from the boundry terms in Eq. (7.92), plus terms tht re generted from the qudrtic ction through the trnsformtion, see Eq. (7.56). Eq. (7.93) suggests tht the sclr-grviton-grviton boundry terms re not relted vi the non-liner trnsformtion (7.51). In the min text we rgue why the boundry terms must be relted s well, nd we discuss wht my cuse the discrepncy. Similrly, we cn compute the boundry terms for sclr-sclr-grviton interctions. By going from Eq. (7.33) to Eq. (7.47), we find the following temporl boundry terms S W 2 ζ γ ζ = 1 2 { [ d 3 3 F x H + 1 F 2 F ( 1 i W ζ j χ γ ζ,ij 2 + jw ζ ]} + 4 γ ζ,ij i W ζ j W ζ i χ 1 H F F i W ζ j W ζ ). (7.94) Likewise, fter mny prtil integrtions to get Eq. (7.48) from Eq. (7.44), we find the following boundry terms, S W 2 ϕ γ ζ = 1 2 d 3 x { 3 1 F 2 F F F H [ F 2 γ ϕ,ij ( 1 F 2 F H F F F i W ϕ j W ϕ γ ϕ,ij φ φ i W ϕ φ ]} j W ϕ φ iw ϕ φ j χ φ jw ϕ φ ) i χ φ. (7.95) Also here we find tht, fter the redefinition of W ϕ nd γ ϕ using Eq. (7.51), the totl boundry terms for sclr-sclr-grviton interctions do not gree with those in (7.94).

157 Chpter 8 Discussion nd Outlook 8.1 Summry In this thesis we hve studied perturbtions in n infltionry universe. The motivtion origintes from the ide tht vcuum fluctutions during infltion cn form the initil seeds of structure formtion in the universe nd leve n imprint in the CMB. In this work we hve gone somewht bck to the bsis by studying these vcuum fluctutions in very simple model of infltion described by the ction for grvity nd single sclr field. In chpter 2 we hve introduced mjor problem tht rises when studying perturbtions of the metric field nd sclr field on top of dynmicl bckground. The theory of Generl Reltivity is covrint, i.e it is invrint under coordinte reprmetriztions. This fundmentl, useful property of Generl Reltivity turns into problem when studying perturbtions. Perturbtions become dependent on the choice of coordinte system, which mkes it difficult to see wht re the truly physicl degrees of freedom. Generl covrince is symmetry of Generl Reltivity nd resembles the guge symmetries in the Stndrd Model. Therefore, the coordinte reprmetriztion dependence of perturbtions is often clled guge dependence, nd we refer to the ssocited complictions s the guge problem. As in guge theories, it is possible to define guge invrint combintions which cn be scribed physicl mening. In cosmologicl setting these re referred to s guge invrint cosmologicl perturbtions. Since the metric formultion of Generl Reltivity contins guge degrees of freedom, s well s non-dynmicl, constrint degrees of freedom, the theory reminds us of the formultion of electrodynmics in terms of the vector potentil. In section 2.5 we hve exploited this nlogy by demonstrting how to extrct the dynmicl nd physicl degrees of freedom in electrodynmics, which gve preview for the computtion of

158 150 CHAPTER 8. DISCUSSION AND OUTLOOK the mnifestly guge invrint ction for cosmologicl perturbtions. The finding of this guge invrint ction t qudrtic nd third order in perturbtions is one of the min results of this work. In chpter 3 we hve reviewed the bsics of infltionry cosmology. We hve explined how super-hubble perturbtions during infltion form some initil condition for the cosmic fluid during the lter rdition nd mtter ers. Specific computtions of the primordil power spectr for sclr nd tensor perturbtions were performed, strting from the mnifestly guge invrint ction for these perturbtions. These computtions resulted in the fmous nerly scle invrint power spectr for cosmologicl perturbtions. As demonstrtion, we computed specific vlues for the mplitude nd spectrl index in single field sclr field models of infltion in polynomil potentil. In chpter 4 we hve introduced nother problem considering perturbtions in n expnding universe: the frme problem. The ction for grvity nd sclr field mtter cn be formulted in different wys in field-theoretic setting: either the sclr field is not, or minimlly, coupled to the Ricci sclr the Einstein frme, or the sclr field is nonminimlly coupled to the Ricci sclr the Jordn frme. Although the ctions seem to be completely different, it turns out they cn be relted to ech other by redefinitions of the metric nd sclr field. This is very useful property, becuse it mens we cn compute ny Jordn frme result vi the much simpler nd well-known Einstein frme results. We hve exploited this property in order to compute the primordil power spectrum in Higgs infltion, which fetures the Stndrd Model Higgs boson non-minimlly coupled to the Ricci sclr. However, lthough the frme equivlence is cler for the unperturbed theory, it is not so obvious t the perturbtive level, becuse perturbtions in one frme generlly do not coincide with those in nother. Thus it is not cler how to relte, for exmple, n-point functions in one frme with those in nother. Even though frme trnsformtions re something else thn coordinte reprmetriztions, the frme problem shows lot of similrities to the guge problem. In both cses there is some sort of symmetry frme equivlence nd covrince, respectively nd in both cses this symmetry is not mnifestly present t the perturbtive level. So, similr to the concept of guge invrint perturbtions, we hve lso introduced the concept of frme independent cosmologicl perturbtions perturbtions tht hve the sme form in either frme. The equivlence between two frmes is very esily seen when writing, for exmple, n n-point function or ction in terms of these vribles, nd such n ction my be clled mnifestly equivlent. The demonstrtion of the equivlence t the level of the qudrtic nd third order ction is nother min result of this thesis. In chpter 5 we presented the originl work published in Ref. [1]. We computed the

159 151 mnifestly guge invrint ction t the qudrtic level, directly in the Jordn frme. At this order it is possible to completely decouple the dynmicl degrees of freedom, both from ech other nd from the non-dynmicl degrees of freedom vi strightforwrd digonliztion procedure. The min result (5.37) contins one dynmicl sclr perturbtion, the curvture perturbtion on uniform field hypersurfces w ζ, nd one dynmicl tensor perturbtion, the trnsverse trceless grviton γ ij. Additionlly there re four decoupled non-dynmicl perturbtions, whose equtions of motion represent the liner solutions of the constrint equtions. We lso gve proof tht both the guge invrint curvture perturbtion nd the grviton re frme independent to first order. The free ction in the Jordn frme is thus relted to tht in the Einstein frme vi redefinitions of the bckground fields lone, in is thus mnifestly equivlent. Moreover, the power spectrum for w ζ is invrint under frme trnsformtions, in the sense tht it cn be expressed in terms of Jordn frme or Einstein frme bckground quntities, but it is the sme function of time. Chpters 6 nd 7 contin the originl work published in Refs. [2, 3]. Here we extended our computtions to the third order ction, with focus on pure sclr interctions in chpter 6, nd on grviton nd sclr-grviton interctions in chpter 7. In chpter 6 we gve explicit second order guge trnsformtions for the sclr perturbtions nd defined second order guge invrint vribles. In prticulr, we defined two specil second order perturbtions: the curvture perturbtion on uniform field hypersurfces W ζ (6.12), which reduces to the curvture perturbtion ζ in the uniform field guge where ϕ = 0, nd the field perturbtion on uniform curvture hypersurfces W ϕ (6.14), which reduces to the field perturbtion ϕ in the uniform curvture guge ζ = 0. At the liner level these guge invrint perturbtions re merely relted by time dependent rescling, nd hence it is strightforwrd to relte their qudrtic ctions nd 2-point functions. However, t second order W ζ nd W ϕ re non-linerly relted. We outlined procedure for computing the mnifestly guge invrint ction for these perturbtions. The procedure strts from the nive third order ction for (guge dependent) sclr perturbtions, obtined fter eliminting the uxiliry or constrint fields from the ction by solving for them to first order in perturbtions. Next, this nive third order ction is seprted in mnifestly guge invrint prt, expressed in terms of liner guge invrint perturbtions, nd guge dependent prt. Since the originl, unperturbed ction is covrint, i.e. guge invrint, the guge dependent terms must be proportionl to the liner eqution of motion. They cn then be bsorbed into the qudrtic ction by defining new, second order vribles, which re precisely the second order guge invrint vribles mentioned before. In specific guge these cubic mnifestly guge invrint ctions for W ζ or W ϕ then reduce to the cubic ctions for ζ nd ϕ, respectively. This ws used in chpter 6 s shortcut to find the mnifestly guge

160 152 CHAPTER 8. DISCUSSION AND OUTLOOK invrint ction for W ϕ (6.36) nd W ζ (6.41), both in the Einstein frme. In chpter 7 we pplied the procedure rigorously, without guge fixing, in order to compute the guge invrint vertices for sclr-grviton interctions directly in the Jordn frme. Not only did this give the guge invrint vertices for different sets of guge invrint vribles (see Eqs. (7.33), (7.38) nd (7.39) or Eqs. (7.44), (7.45) nd (7.46)), but we lso obtined explicit expressions for the second order guge invrint vribles on uniform field hypersurfces, W ζ (7.31) nd γ ζ (7.37), nd for those on uniform curvture hypersurfces, W ϕ nd γ ϕ (7.41). These sets of vribles re relted by non-liner field redefinitions. Chpters 6 nd 7 lso contined n extensive discussion on uniqueness of the guge invrint ction. Although the mnifestly guge invrint cubic ctions for different sets of guge invrint perturbtions pper very different, it cn be shown tht they only differ by boundry terms nd terms proportionl to the liner equtions of motion. These terms do not contribute to the second order eqution of motion, nd therefore the evolution of non-gussinity my be clled unique. However, n-point functions of different sets of guge invrint vribles re generlly different becuse of contributions coming from the boundry terms t initil nd finl time. We hve shown tht n initil stte which is Gussin for one set of guge invrint vribles, is generlly non-gussin for nother set of vribles. Moreover we hve shown tht the boundry terms t finl time cn generte disconnected contributions to the bispectrum, see Eqs. (6.53) nd (7.52). These disconnected pieces lso follow directly from the non-liner reltions between different sets of guge invrint vribles. Furthermore, connection ws mde to physicl observbles. The super-hubble curvture perturbtion cretes grvittionl potentil wells for the CMB photons. These photons trvel towrds us from the surfce of lst scttering nd lose some energy when they climb out of these potentil wells. We observe the fluctutions in the grvittionl potentil t the time of decoupling s temperture fluctutions in the CMB (the Schs- Wolfe effect). Temperture fluctutions nd the (guge invrint) curvture perturbtion t the time of Hubble crossing re relted in very simple wy. Finlly we hve demonstrted the frme equivlence for the cubic guge invrint ction. This is most clerly seen when the ction is expressed in terms of the frme independent perturbtions W ζ nd γ ζ. We hve presented explicit proofs of the frme independence t second order for these vribles. The ction for these vribles is then mnifestly equivlent. We hve lso shown tht other guge invrint vribles W ϕ nd γ ϕ re non-linerly relted between different frmes. Agin, this shows lot of similrities with the story bout guge invrince, where different sets of guge invrint vribles re relted by non-liner redefinitions. In the frme story the sme guge invrint vribles in different frmes re non-linerly relted. This lso mens tht, for exmple, n-point functions for these guge invrint vribles in the Jordn frme

161 153 cn be computed vi Einstein frme n-point functions by resclings nd the dding of disconnected pieces. Let us conclude with the following: generl covrince is fundmentl symmetry of Generl Reltivity. It is possible to write covrint ction in mnifestly guge invrint wy t every order in perturbtion theory by mking use of guge invrint perturbtions. In principle there re infinitely mny guge invrint vribles, but they re ll relted by resclings nd non-liner redefinitions. Likewise, ll ctions for guge invrint perturbtions re relted by boundry terms nd terms proportionl to the equtions of motion for perturbtions. If we lso consider frme trnsformtions nd wish to write the ction in mnifestly equivlent form t the perturbtive level, only one (set of) guge invrint vrible(s) is singled out from the infinitely mny others. These specil vribles re the guge invrint curvture perturbtion nd grviton on uniform field hypersurfces. It is not surprising tht these guge invrint vribles re invrint under the frme trnsformtions. On the uniform field hypersurfce the frme trnsformtion only ffects the bckground fields, but not the sclr nd tensor fluctutions of the metric. Therefore, the guge invrint generliztions of these perturbtions re lso not ffected by the frme trnsformtion. Thus, there is unique set of guge invrint vribles for which the ction is both guge invrint nd mnifestly equivlent. As finl ppliction we hve used the guge invrint nd frme independent formultion of cosmologicl perturbtions in order to re-ddress the nturlness problem in Higgs infltion in section 7.6. In this frmework we find tht the Jordn frme cutoff is Λ/ = MP 2 + ξφ2. The Einstein frme cut-off is Λ E / E = M P, nd is hence relted to the Jordn frme cut-off by redefinition of the bckground. These simple clcultions of the cut-off for certin clss of processes indicte tht the perturbtive expnsion in Higgs infltion is vlid t lest up until the Plnck scle M P, such tht there is no nturlness problem in Higgs infltion. This originl work is currently being prepred for publiction[134]. 8.2 Outlook The work in this thesis is in mny spects stepping stone for rnge of possible new projects. First of ll, in this work we hve only computed the tree-level guge invrint cubic ction. We hve not computed explicit 3-point functions, though they hve been computed in fixed guge (see for exmple the erly works [83 85]). Often in these computtions the initil stte is tken to be Gussin for fluctutions of the sclr field

162 154 CHAPTER 8. DISCUSSION AND OUTLOOK (in the guge invrint lnguge, the initil stte for the guge invrint field perturbtion on uniform curvture hypersurfces is Gussin), which implies tht there is some initil non-gussinity for the (guge invrint) curvture perturbtion. It is n interesting question to sk how much of the finl non-gussinity is purely generted by the evolution, nd how much origintes from the initil stte. Another very interesting topic is the question of loops in infltion. Generlly speking, cosmologicl correltion functions receive corrections from loop contributions, the simplest exmple being one-loop correction to the power spectrum [80, 81, 118, ]. Quite generlly, these corrections come in the form of logrithms (see Ref. [61] for n overview of the types of logrithms nd literture), for instnce, logrithm of the scle fctor [118]. This suggests tht, lthough the (guge invrint) curvture perturbtion is conserved on super-hubble scles t tree-level, it is time dependent when quntum corrections re tken into ccount. However, recently there hs been lot of debte in literture bout the time (in)dependence of the curvture perturbtion [62, ]. It is of gret interest to see how these (different) results compre to results obtined in guge invrint frmework. The guge invrint tree level ction in this work provides n excellent strting point for computing loop correction to the power spectrum. It is possible to quntize the ction by defining cnonicl moment, finding the Hmiltonin nd evolve the sclr 2-point correltor using the in-in formlism, tking into ccount one-loop effects. For these effects we must lso compute the fourth order guge invrint ction, nd the steps to do so hve been outlined in this work. Of course, if we mke use of frme independent cosmologicl perturbtions in order to compute loop corrections, it is strightforwrd to obtin results for theory with non-miniml coupling. Indeed, they re identicl to those for the minimlly coupled Einstein frme with modified potentil.. With respect to the different frmes, we would lso like to focus our ttention on the issue of nturlness in Higgs infltion [99 103, 105]. In section 7.6 we hve performed first clcultion of the cut-off in the guge invrint nd frme independent frmework for cosmologicl perturbtions. These clcultions suggest tht the cut-off is not M P /ξ, s suggested in Refs. [99, 100, 102, 103], or more generl field dependent cut-off which in some regimes reduces to M P /ξ [105], but the cut-off is insted Λ/ = M 2 P + ξφ2 M P. In the derivtion we hve only considered sclr nd grviton perturbtions, but no other fields. It needs to be checked whether the cut-off scle is still M P when we include guge fields, whose interctions with the Higgs boson nd grvity should lso be formulted in guge invrint wy. Moreover, we hve yet to study the boundry terms in the ctions for frme independent perturbtions nd see whether or not they cn contribute to cut-off scle. Aprt from these directions there re mny extensions to our work. For exmple, one could consider single sclr field theory with non-stndrd kinetic terms (see, for

163 155 exmple, Refs. [84, 85]) nd formulte it in mnifestly guge invrint wy t the perturbtive level. The sme could be done for F (R) grvity[29, 144], which in fct shows lot of similrities to the Jordn frme, since it cn be written in Einstein frme form by redefinitions of the metric. But we could lso consider n inflton potentil with multiple sclr fields nd tke guge invrint pproch to perturbtions. These multiple field scenrios hve gined lot of ttention, becuse, for instnce, it is possible tht some new, hevy physics my leve some (oscilltory) fetures in the power spectrum [ ]. In this respect perhps nother interesting new direction is to compre our results to the effective field theory pproch to infltion [154]. Another interesting extension is the description of guge invrint perturbtions in spce-times with other types of fields thn sclr fields, for exmple vector infltion [155], or guge invrint perturbtions for generl cosmologicl fluid. So, the study of guge invrince nd frme independence opens mny doors in new directions, s well s connects to mny reserch res t the fundmentl level. The explortion of these res provide deeper understnding of the theory of cosmologicl perturbtions nd my give some new insights to the next genertion of lucky cosmologists.

164

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177 Nederlndse smenvtting De titel vn mijn proefeschrift kn vrij vertld worden nr Ijkinvrintie en Frme Onfhnkelijkheid in de Kosmologie. Voor vrijwel iedereen, inclusief de meeste onderzoekers in de kosmologie, klinkt dit niet bepld bekend in de oren. Vndr deze smenvtting die in principe voor iedereen te volgen zou moeten zijn. Angezien kosmologie een enorm uitgebreid vkgebied is, is het een grote uitdging om op een begrijpelijke en beknopte mnier uit te leggen wr mijn onderzoek over gt. En dt is eigenlijk ook wel weer heel leuk. Als onderzoeker krijg je nmelijk niet heel vk de mogelijkheid om n een breder publiek duidelijk te vertellen wr je je dgelijks mee bezig houdt. Dus lees verder en hopelijk heb je n het einde vn dit verhl een beter beeld vn het wel en wee vn een theoretisch kosmoloog. Kosmologie Wnneer ik iemnd vertel dt ik promovendus in de kosmologie ben, krijg ik vk de vrg: O kosmologie, is dt niet iets met sterren en plneten en dergelijke? Hels moet ik dn ntwoorden dt niet kosmologen, mr sterrenkundigen zich drmee bezig houden. Kosmologen bekommeren zich lleen om zken die zich op veel grotere schl fspelen. Zij proberen het heell ls geheel te beschrijven. En j, plneten, sterren en sterrenstelsels zijn een onderdeel vn ons heell, mr voor een kosmoloog zijn dit slechts detils. Een kosmoloog ziet het heell ls een enorme pn soep met een beplde tempertuur en een beplde dichtheid. Misschien dt er hier en dr een gehktblletje of een stukje groente in rond zwemt, mr dt verndert weinig wnneer je de pn soep ls geheel beschouwt. Een belngrijk bsisprincipe in de kosmologie is dn ook het kosmologische principe: op grote schl ziet ons heell er in lle richtingen hetzelfde uit (isotroop) en op elke punt bezit het dezelfde eigenschppen (homogeen). Uiterrd is dit op kleine fstnden zeker niet het gevl (de rde ziet er heel nders uit dn de

178 170 Bibliogrphy zon bijvoorbeeld), mr gemiddeld genomen over zeer grote fstnden ziet het heell op één punt precies hetzelfde uit ls op ieder nder punt. Dn rest nu de vrg: hoe beschrijven we het heell ls geheel? Ntuurkundigen beschrijven beplde verschijnselen met ntuurwetten. Zo kn je de ntrekking vn mgneten en positieve en negtive ldingen beschrijven met de wetten der elekrodynmic. Processen in de kernen vn tomen worden gekenmerkt door de sterke en zwkke wisselwerking. Dit is echter lleml irrelevnt op grotere fstnden. In ons zonnestelsel bijvoorbeeld is het de zwrtekrcht die de plneten in hun bnen houdt. Ook sterrenstelsels, zols onze eigen Melkweg, worden bij elkr gehouden door de zwrtekrcht. Mr kijken we op nog grotere schl, dn merken we eigenlijk weinig vn deze zwrtekrcht. En hier zit dn ook het probleem: het heell bestt voor het grootste gedeelte uit lege ruimte. Zols gezegd zijn er her en der wt sterrenstelsels en -clusters die bij elkr worden gehouden door zwrtekrcht, mr dit heeft geen effect op het heell ls geheel. Hoe beschrijven wij dus lege ruimte? Gelukkig is dit mogelijk gemkt door Einstein toen hij in 1915 zijn Algemene Reltiviteitstheorie publiceerde. In deze theorie wordt ruimte zelf ls een dynmisch object beschouwd. Ruimte wordt bijvoorbeeld gekromd in de omgeving vn een beplde mss. Je kn dit visuliseren door een trmpoline te nemen wrbij het elstische doek de ruimte in twee dimensies voorstelt. Neem nu een pr ronde stenen met een verschillende mss: dit zijn de sterren. Als je de stenen op de trmpoline legt zie je dt het elstische doek, de ruimte dus, gekromd is rondom de mss. Als je nu een pingpongblletje (een licht object in de ruimte) met beplde snelheid op de trmpoline fschiet dn zie je dt deze wordt fgebogen door de mss s op de trmpoline. Dit verklrt de werking vn de zwrtekrcht door middel vn kromming vn de ruimte. Al vrij snel pste Einstein zijn Algemene Reltiviteitstheorie toe op een sttisch heell. De smenstelling vn zo n heell verndert niet in de tijd, en het heeft dus geen begin en geen eind. Echter, het bleek dt een sttisch heell met een beplde mssdichtheid binnen fzienbre tijd in elkr stort onder zijn eigen zwrtekrcht. Angezien lle wrnemingen wezen op een sttisch heell introduceerde Einstein zijn kosmologische constnte, die zo werkte dt hij de zwrtekrcht blnceerde. Niet lng drn bleek dit echter niet te kloppen. Een uitdijend heell In het begin vn de jren 20 vn de vorige eeuw ontwikkelden de Russische wiskundige Alexnder Friedmnn en de Belgische priester Georges Lemître theoretische modellen die een uitdijend heell beschreven. Deze modellen werden gezien ls een rdige theoretische mogelijkheid vn de Algemene Reltiviteitstheorie, mr werden niet heel se-

179 171 rieus genomen ngezien er geen wrnemingen wren die lieten zien dt het heell evolueert. Dit vernderde in 1929 toen Edwin Hubble ontdekte dt ndere sterrenstelsels vn ons f bewegen. Hoe verder weg een sterrenstelsel stt, hoe sneller het vn ons f beweegt. Dit zijn precies de eigenschppen vn een uitdijend heell. Een leuke mnier om je zo n uitdijend heell voor te stellen is door brooddeeg te nemen met drin krenten vermengd. Deze krenten zijn de sterrenstelsels, en ons Melkwegstelsel is dr één vn. Het deeg zelf is de ruimte tussen de sterrenstelsels. Als het brooddeeg rijst (het heell zet uit) dn zie je vnuit het oogpunt vn één krent lle ndere krenten vn je f bewegen. De krenten zelf zetten niet uit: de zwrtekrcht vn sterren onderling overheerst het effect vn de uitzetting vn het heell en houdt het sterrenstelsel bij elkr. Een uitdijend heell heeft een ntl vergnde gevolgen. Allereerst kun je je fvrgen hoe het verder gt met ons heell. Remt de uitzetting vn het heell f en zl het uiteindelijk weer in elkr storten? Of blijft het voor ltijd uitzetten en drijft lles uit elkr? Je kn ook ndersom denken: ls het heell uitzet, dn betekent dt dt het vroeger veel kleiner ws. Sterker nog, ls je ver genoeg terug gt in de tijd dn komt lles weer bij elkr. Dit suggereert dt het heell een begin heeft en dus ook mr een beplde tijd bestt: volgens de huidige wrnemingen is ons universum 13.8 miljrd jr oud. Dit heeft geleid tot het idee vn de Big Bng. Deze theorie zegt dt ons heell is begonnen in een toestnd met een extreem hoge tempertuur en dichtheid, en drn door uitdijing is fgekoeld en uitgedund. Over het echte begin kunnen we eigenlijk niet veel zeggen. Hier gn de fstnden nr nul en de dichtheid en tempertuur nr oneindig. Dit noemen we een singulriteit, en hier zijn onze ntuurwetten niet meer vn toepssing. In de prktijk mken kosmologen zich dr niet heel druk om. Zij kiezen ls strtpunt vn het heell simpelweg een punt dt een frctie n de singulriteit ligt. Hier heeft het universum l een kleine mr eindige grootte, en zijn de tempertuur en dichtheid extreem hoog, mr niet zo hoog dt ze niet beschreven kunnen worden met de wetten der ntuur. De Big Bng theorie doet een ntl belngrijke voorspellingen die bevestigd zijn door verschillende wrnemingen. Alvorens dr op in te gn zl ik eerst wt moeten verduidelijken. Sterren en plneten zijn opgebouwd uit verschillende elementen, zols wterstof (H), helium (He) en lithium (Li). Deze elementen zijn op hun beurt weer opgebouwd uit tomen. Atomen bestn uit een kern vn protonen en neutronen met dromheen een wolk vn elektronen. Protonen en neutronen bestn dn weer uit qurks, de kleinste bouwstenen vn mterie. Zols wellicht bekend worden l deze kleinste deeltjes veelvuldig gecreëerd in deeltjesversnellers, zols de Lrge Hdron Collider in Zwitserlnd. Hier botsen protonen met enorme snelheid op elkr, zodt een extreem hoge tempertuur en dichtheid bereikt wordt en kleinere deeltjes worden gecreëerd. Deze omstndigheden zijn precies ook nwezig tijdens de Big Bng: er

180 172 Bibliogrphy is een zogenmde oersoep met een zeer hoge tempertuur wrin llerlei kleine elementire deeltjes zols qurks, elektronen en fotonen (beter bekend ls strling of licht) rondzwemmen. Als het heell uitzet koelt de soep f en klonteren de verschillende deeltjes smen. Qurks vormen smen protonen en neutronen, die vervolgens weer smenklonteren tot toomkernen en elementen. Met behulp vn de Big Bng theorie kn berekend worden in welke concentrties de lichte elementen zols wterstof en helium in ons heell voorkomen. Dit blijkt uitstekend overeen te komen met verschillende wrnemingen en is een belngrijke bevestiging vn de theorie. De Big Bng theorie voorspelt ook dt mterie op grote schl specifieke structuren vormt onder invloed vn de zwrtekrcht, en dit is eveneens wrgenomen. De belngrijkste voorspelling vn de Big Bng theorie is de zogenmde kosmische chtergrondstrling. Dit is strling die vrijkomt wnneer een elektron en een proton combineren tot een wterstoftoom. Boven een tempertuur vn enkele duizende grden zijn protonen en elektronen niet n elkr gekoppeld: het heell is een plsm. In zo n plsm kunnen fotonen (elektromgnetische strling) zich niet vrij bewegen, omdt ze voortdurend verstrooid worden door de gelden deeltjes. Het heell is dus ondoorzichtig. Ps ls het heell voldoende fkoelt kunnen de protonen en elektronen combineren tot een neutrl wterstoftoom. In ons heell vond deze gebeurtenis plts jr n de oerknl. Vnf dt moment werd het heell trnsprnt en verspreidde de strling zich in lle richtingen. Als de Big Bng theorie correct is dn zouden wij deze strling dus vnuit lle richtingen moeten kunnen wrnemen (vndr de nm kosmische chtergrondstrling). En inderdd, deze strling is in 1967 voor het eerst gedetecteerd door ltere Nobelprijswinnrs Penzis en Wilson. Deze ontdekking wordt gezien ls de grootste bevestiging vn de Big Bng theorie. De chtergrondstrling is een extreem belngrijke bron vn informtie voor kosmologen. De reden is dt deze strling het eerste licht is dt we nu wr kunnen nemen en ons dus het oudste directe beeld vn het heell oplevert. De strling blijkt extreem uniform te zijn: uit lle richtingen heeft de strling vrijwel exct dezelde tempertuur (strling kn gessocieerd worden met tempertuur) tot een nuwkeurigheid vn 1 op Dit komt overeen met het eerder genoemde kosmologische principe, dt zegt dt het heell overl en in lle richtingen dezelfde eigenschppen gezien. Een mp vn de kleine tempertuurfluctties vn de chtergrondstrling gezien over het hele heell is te zien in figuur 1.1. Dit is de meest recente meting vn de chtergrondstrling, gedn door de Plnck steliet (een project vn ESA). De kleine fluctuties in tempertuur lijken compleet willekeurig, mr zijn dt in feite niet. Hier komen we lter op terug.

181 173 Kosmologische infltie Een korte smenvtting vn wt er tot nu toe verteld is. Kosmologen proberen de de globle structuur en evolutie vn het heell te beschrijven. Dit kn middels de Algemene Reltiviteitstheorie, wrbinnen een wiskundige beschrijving vn een uitdijend heell een mogelijkheid is. Dit heeft tot geleid tot de ontwikkeling vn de Big Bng theorie. Binnen deze theorie begint het heell in een dichte, hete fse wrin deeltjes rzendsnel bewegen (dit wordt het rditie tijdperk genoemd) en wordt drn door uitzetting en fkoeling gevolgd door een fse wrin deeltjes lngzm bewegen (het mterie tijdperk). Dit model kn met behulp vn wiskunde worden beschreven en kn een verklring geven voor enkele vn de belngrijkste gebeurtenissen in het heell, wronder de nwezigheid vn de kosmische chtergrondstrling. So fr so good. Mr hels houdt het hier niet op. Er zijn nmelijk nog enkele problemen met de Big Bng theorie die een ndere uitleg vereisen. Eén vn die problemen is het zogenmde horizon probleem. Zols gezegd heeft de kosmische chtergrondstrling ngenoeg dezelfde tempertuur in elke richting. Echter, dit is erg onwrschijnlijk in het stndrd Big Bng model zols hierboven beschreven. De reden is ls volgt: ls we uitrekenen welke fstnd een signl, dt zich voortplnt met de lichtsnelheid, heeft kunnen fleggen vnf de oerknl (het begin) tot n het tijdstip dt de chtergrondstrling werd uitgezonden ( jr), dn is deze fstnd veel kleiner dn de grootte vn het heell. Dit betekent dt informtie zich niet over het hele heell heeft kunnen verspreiden: verschillende gebieden in het heell hebben met geen mogelijkheid met elkr in cusl contct kunnen stn. Toch zien we dt het hele heell overl vrijwel dezelfde tempertuur heeft. Hoe is dit mogelijk? Een oplossing voor dit horizon probleem en enkele ndere problemen is het principe vn kosmologische infltie, bedcht door Aln Guth en nderen in het begin vn de jren 80. Het idee is dt het heell in de eerste frctie vn een seconde n de oerknl rzendsnel uitzet. Zo snel dt het heell in slechts seconde miniml een fctor in volume uitzet. Hierdoor wordt het heell veel groter dn het wrneembre heell vndg de dg en stn lle gebieden met elkr in cusl contct. N deze periode is het heell extreem leeg en extreem koud. Door een proces wt heropwrming wordt genoemd wordt de nwezige energie in het heell omgezet in hitte en worden de bekende elementire deeltjes gecreëerd. Drn ontwikkelt het heell zich verder volgens de stndrd Big Bng theorie. Dit is te zien in figuur 8.1, wrmee je een rdige indruk kn krijgen vn de evolutie en uitzetting vn het heell vnf de oerknl tot n nu.

182 174 Bibliogrphy Figure 8.1: Evolutie vn het universum vnf de oerknl tot n nu. De breedte vn de trechter geeft een indictie vn de grootte vn het heell. Tijd loopt vn links nr rechts. Het heell begint met een periode vn kosmologische infltie wrin het heell rzendsnel uitzet. Deze periode zorgt ervoor dt kwntumfluctuties een ptroon chterlten in de kosmische chtergrondstrling, die getoond is n jr. Vervolgens zet het heell uit volgens de stndrd Big Bng theorie en worden sterren, sterrenstelsels en plneten gevormd. Met dnk n het NASA/WMAP Science Tem. Om zo n periode vn infltie te bewerkstelligen is er een specil soort mterie nodig. De uitzetting vn het heell wordt over het lgemeen gedreven door een beplde energiedichtheid. Bij gewone mterie wordt deze dichtheid norml gesproken verdund door de uitzetting vn het heell. Hierdoor neemt de mte vn uitzetting f nrmte het heell groeit. Tijdens een periode vn infltie neemt de uitzetting vn de ruimte juist exponentieel toe. Hiervoor is mterie nodig met een bijn constnte energiedichtheid en negtieve druk. Theoretisch gezien is dit mogelijk. Ik geef hier nu een korte, redelijk technische toelichting op voor de ge ınteresseerde lezer. In theoretische deeltjesfysic worden deeltjes beschreven door middel vn de zogenmde kwntumveldentheorie. Een veld is hier een grootheid die op ieder tijdstip en op elk punt in de ruimte een beplde grootte heeft. Excitties vn zo n veld worden gezien ls deeltjes. Er zijn verschillende soorten velden, zols een sclir veld, vector veld of tensor veld. In het Stndrd Model vn de deeltjesfysic is slechts e e n sclir