INFLATION. Lagrangian formulation of Field Equations

Transcription

1 Mrino Mezzei INFLATION We hve lredy menioned wo problems ffeing he Ho Big Bng model: he flness problem nd he horizon problem. To hem one n dd he mgnei monopoles problem (monopoles re zero-dimensionl opologil defes, h re produed he ime of he phse rnsiion orresponding o he breking of GUTs; heirnumber densiy, oupled wih heir very high mss, would produe vlue of lerly unepble). The prdigm of inflion, whih solves hese problems, hs been proposed by Aln Guh in 98. I ssumes h here hs been n elered expnsion phse beween he imes i nd f (wih Pl < i < f << eq ), produed by n equion of se h mimis h of osmologil onsn: i f i ( ) ( ) e i H ( ) (H ~ onsn). The sle for grows s in de Sier model (whih hs nd densiy of mer negligible), insed of growing s ()~, like n EdS model in he RD er. The exponenil growh, if suffiienly prolonged, produes growh of he prile horizon d H suffiien o solve he horizon problem; onverges owrds uniy (s in models domined by he osmologil onsn), resolving he flness problem (remember lso h he urvure of he spil seion sle s ()~, nd he exponenil growh of () fore his urvure owrds zero). The problem of monopoles is resolved hrough srong diluionof heir number densiy. If inflion ours round he ime of he breking of grnd unifiion (GUT), he bove problems re solved provided where N is nmed number of e-foldings. f ln N 6 i Lgrngin formulion of Field Equions As we hve seen, o hve phse of inflion is neessry h he universe possesses, for erin ime inervl, n equion of se of he ype p -. This n be hieved in nurl wy by mens of slr field presen in he erly sges of he erly universe ( slr field hs lso he propery of being isoropi). To undersnd he mehnism i is neessry o inrodue some oneps used in Qunum Field Theory.

2 Mrino Mezzei In Clssil Mehnis he equions of moion of dynmil sysem n be derived from Lgrngin funion L L q, q ) T( q ) V( q ) ( i i i i where q i re he generlized oordines, T is he kinei energy nd V is he poenil energy. The ion S, involved in he moion of he sysem from one onfigurion ime o noher he ime, is given by S Ld nd, ording o he priniple of les ion, he evoluion of he sysem beween he wo onfigurions is h whih orresponds o he minimum vlue of S. This ondiion leds o he Euler-Lgrnge equions: d d L L q i q i These relions desribe he moion of priles, h is, of lolized objes. A field insed oupies erin region of spe, nd he Field Theory wns o lule one (or more) funions of posiion nd ime: = (x, y, z, ) (eg.: emperure, eleri poenil, he hree omponens of he mgnei field in room). While, in he mehnis of priles, he Lgrngin L is funion of he oordines q i nd of heir derivives, Field Theory works wih Lgrngin densiy L whih is funion of he field nd of is derivives wih respe o x, y, z, nd. To keep he relivisi ovrine of physis more ppren, we use spe-ime oordines x nd x, x, x x, y, z, so h he Lgrngin is he volume inegrl of L nd he ion is L L d S d L (he for /, inessenil, serves o keep he dimensions of he ion). In relivisi field heory q i is repled by he field, nd he index i is repled by spe-ime oordines x α. Sine eh ime derivive n be ssoied o similr erm involving grdien, we use ll he ovrin derivives φ x α = α φ nd Euler-Lgrnge equions beome L x x L ( )

3 Mrino Mezzei Aully his wriing is orre in Euliden spe nd in orhogonl oordines; o ke oun of more generl hoie of oordines (e.g. o-moving spil oordines) he volume elemen d x is repled wih g d x where g is he deerminn of he meri g. So he Euler-Lgrnge equion beomes ) ( gl gl In fl, si Minkowski spe, he meri is g dig(,-,-,-). Then ;,, ; z y x x ); (,, ; z y x z y x If we use o.moving oordines (r = x) in fl, expnding spe: g dig(, -, -, - ) nd g =. We hve hen ( x is he grdien referred o he o-moving oordine x) x ; x ; x Le s onsider, for insne, he following Lgrngin (densiy): m L where is rel, single slr field. In his se, i.e. Minkowski spe, ( g =), L m L nd hene Euler-Lgrnge formul requires

4 Mrino Mezzei whih is he Klein-Gordon equion, desribing (in Qunum Filed Theory) prile of spin nd mss m. In nlogy wih L = T - V, in he Lgrngin wrien bove he firs erm, ½( )( ), is nmed kinei energy erm, while he seond erm, in his se qudri in (he erm orresponding o he mss), is he poenil energy erm. For slr field we will wrie he Lgrngin in he generl form L V ( ) where V() is suible poenil (V()=/ in Klein-Gordon se). If L, wrien s bove, depends on x only hrough nd is derivives, he following quniy (energy-momenum ensor) is preserved (i.e., hs four-divergene equl o zero): T g L In he se of perfe fluid we hve seen h he energy-momenum ensor hs he form T ( p ) u u p g where P is he pressure, he Energy densiy nd u is he four-veloiy (u dx /ds); in he o-moving referene frme u =(,,,). In fl spe, by using omoving oordines, he omprison of he wo relions gives: p T T T T (see he following snned pge for he proof). V x 6 V In he se in whih he field is spilly homogeneous (from whih x ϕ = ; even if x ϕ is differen from zero he erm onining x ϕ beomes rpidly negligible due o he - for ) nd he erm / (/) is negligible ompred o he poenil V(), we hve V p V h is, n equion of se h mimis h whih orresponds o he osmologil onsn! x Aully here re smll fluuions on he sle of he Hubble rdius, whih re he "seeds" of he lrge-sle sruure of he universe

5 Mrino Mezzei 5

6 Mrino Mezzei If : is negligible wih respe o V() T p g g V ( ) g eff 8 G nd he erm of poenil energy orresponds o n effeive osmologil onsn. Phse rnsiions nd Symmery Breking In he hisory of he erly universe one or more phse rnsiions hve ourred. A high energies, ording o he unified heory of he elerowek inerion, he wek nd eleromgnei inerions were mnifesions of single fore. Then, due o he progressive ooling produed by osmi expnsion, erin ime (round riil emperure T EW 5 K, E EW GeV) he universe hs undergone phse rnsiion, fer whih he wo inerions sepred. The Grnd Unified Theories (GUTs), whih emp o unify eleromgneism nd wek nd srong inerions, in urn, require phse rnsiion in he universe riil emperure T GUT 8-9 K, E GUT ~ 5-6 GeV, bove whih here ws symmery beween he hree inerions. Le s onsider n nlogy wih he mgneizion of ferromgnei meril. Above he Curie emperure T C he mgnei momens linked o he spins of oms re rndomly oriened nd rpidly fluuing, here is roionl symmery round eh poin of he meril nd he expeion vlue of (he men vlue) of he spin is null (<S> = ). However, flling he emperure below T C, ligneme of spins beomes energeilly more fvorble, nd here is phse rnsiion o mgneized se, wih <S> in erin direion î. The originl symmery is los, broken, beuse he differen domins h begin o form, independenly of eh oher, hve spins wih differen direions. In he end, when he whole mss hs urn ino domins, defes form he borders of he differen regions. In similr wy, while bove T GUT here ws symmery beween he hree inerions, below T GUT i is broken. Going bk o he se of he ferromgnei meril, he wy in whih he roionl symmery is broken in he differen porions of he mss n be mesured by he growh of he spin S nd he orienion of he differen domins. Similrly, he wy in whih he symmery beween he hree inerions breks down n be hrerized by he quiring of non-null vlues of prmeers nmed Higgs fields; his phenomenon is lled sponneous symmery breking (SSB). The symmery is presen when he Higgs fields hve zero expeion vlue; i is sponneously broken when les one of he boson fields quires n expeion vlue oher hn zero. As in he se of ferromgnei domins, defes remin he boundries of he differen regions in whih he symmery is broken in differen wys, ssuming differen ses of vlues for he Higgs fields. These defes re lled opologil defes, nd my be wo-dimensionl (domin wlls), uni-dimensionl (osmi srings) nd zero-dimensionl (mgnei g 6

7 Mrino Mezzei monopoles). During he phse rnsiion h leds o he breking of he symmery period of exponenil expnsion my lso our: he inflion. Le's see how. For simpliiy we onsider single Higgs field, he slr field. We onsider gin Lgrngin L V ( ) The equion of moion, generlizion of Klein-Gordon (KG) equion, beomes V Free prile ses re he soluion of his equion wih only qudri erm in in he poenil V(), like in KG se; he oeffiien of his erm speifies he mss m of he prile: V()=/, =m /ħ. The vuum se, whih by definiion is he se in whih here re no priles, ours when V/; in he KG se his ours. Higher-order erms in V() orrespond o he inerions beween hese priles. The equion wrien bove dmis he soluion onsn ny vlue of for whih V/. The vuum (no priles) se will herefore be one of hose in whih he expeion vlue of ssumes one of hese onsn vlues. There re severl possibiliies: I my be h he V/ hs only one soluion. In order for he energy o be bounded below, his should orrespond o minimum of V() nd lso orresponds o he unique vuum of he heory On he oher hnd here my be muliple soluions of V/. The mxim of he poenil re unsble, bu ll he minim re possible vu of he heory. If here is more hn minimum, he lowes would be he ulime vuum, he "rue vuum" of he universe. However, he universe my be, erin momen, in lol minimum wih higher vlue of he poenil; i would be in "flse vuum", wih he possibiliy, for insne by unnel effe, o move o he rue vuum. Poenil shpe for old inflion (see below). The Inflion phse, for his poenil, orresponds o he rpping of he field in he well ϕ=. 7

8 Mrino Mezzei In some ses here my be severl suh minim h hve he sme vlue of he poenil, nd he vuum is degenere. This Figure orresponds o poenil of he form V ( ) V () wih μ nd λ rel onsns (λ > if he poenil is bounded from below). The firs erm on he r.h.s. looks like mss erm nd he seond like n inerion, bu he sign of he mss erm is wrong, he mss should be imginry! However, ϕ= is mximum for he poenil, nd we hve wo, degenere, minim orresponding o possible vu or groud ses for v Peurbion heory involves n expnsion of L in ϕ round minimum of he poenil. We rbirrily hoose one of he wo minim, for insne +v, nd define new field η φ v. We wrie he poenil s funion of he new field η nd now he Lgrngin is L = ( αη) α η μ η + erms ubi nd higher in η + ons. whih possess he righ sign of he mss erm nd omplied inerions. If we hose he oher minimum he mss erm remins he sme (only he η erm hnges his sign). This is n exmple wih only wo possible vlues for he rue vuum, bu more generl poenils n led o n infinie number of possible vlues in whih he rue vuum my end. Here we see wodimensionl (omplex) se for he poenil [we subsiue φφ o φ nd (φφ ) o φ, where φ is he omplex onjuge of φ]. True possible vu 8

9 Mrino Mezzei orrespond o poins belonging o he irle of he minimum vlues of V(). The rndom hoie of one of he minim generes sponneous symmery breking (SSB), similr o he formion of domin wih priulr orienion of he spins of is oms in porion of ferromgnei meril h ools below T C. The poenil wrien bove hs his form, wih negive oeffiien for, emperure T =. Bu in he erly universe, when he emperure is very high, o ke ino oun his effe, orreions o V() produe n effeive poenil wih ddiionl erms proporionl o T. In his wy he oeffiien of is posiive emperure high enough, he minimum of he poenil is =, nd he symmery is unbroken. During he ooling of n expnding universe, ording o deils h depend on he priulr shpe of V (), he sponneous symmery breking will ke ple: Through phse rnsiion of firs order, in whih he field, iniilly in =, rosses, by unnel effe, poenil brrier wihin whih i remins rpped for erin ime; inflion, wih /, ours during his rpping phse. This is he model iniilly proposed by Guh, nmed old inflion, whih presens, however, some problems. In f, phse rnsiion of firs order ours hrough he formion of bubbles of he new phse in he middle of he old phse; hese bubbles expnd, ollide nd olese unil he new phse ompleely reples he old one. Bu in he model of Guh, o hve phse inflionry suffiienly long, he probbiliy of forming bubbles is low nd, sine he flse vuum expnds exponenilly, he bubbles n no olese nd he rnsiion o he rue vuum does no our. Through phse rnsiion of he seond order, in whih he field evolves smoohly from one phse o noher. This is he model of new inflion, proposed by Linde, Albreh nd Seinhrd in 98, in whih he field evolves very slowly (slow-roll) from he ondiion of flse vuum = o he rue vuum. Agin, if he evoluion from = kes ple slowly nd V () V () for long enough ime before flling ino he rue vuum, we hve phse of inflion (we will see, ler, wh re he ondiions for his o hppen). 9

10 Mrino Mezzei V() inflion V() slow-roll osillions nd reheing Orders of mgniude for Inflion The expnsion in he Erly Universe, if we negle he urvure (whih, however, ends rpidly owrds zero due o he enormous growh of he sle for), will be given by he equion 8G nd he energy densiy is given by R wih g* ( T) kt R V( ) Where V() orresponds o he energy densiy of he field h, high emperure, hs is minimum in =. Unil ρ is domined by ρ R, he universe behves s in he model of EdS domined by rdiion, wih () ~ /. Bu while ρ R sle s /, ρ Λ remins onsn. A erin ime i, ρ R ~ ρ Λ, nd, from h momen, he expnsion beomes domined by n "effeive" osmologil onsn Λ eff, s in he exponenilly expnding de Sier model:

11 Mrino Mezzei ( ) exp H os i 8 G V ( ) 8 G V ( ) where i is he sle for i. A he equliy ime, T=T, R.. i eff g* ( T ) kt kt erg/m GeV kt erg/m where kt 5 is he energy sle in unis of 5 GeV. To his energy densiy orresponds n "effeive" osmologil onsn 5 GUT 8G. 8 ( kt 5. ( kt 5) m If we ompre his vlue (GUT) for kt5=, wih h of he osmologil onsn ody ( -56 m - ) we ge huge rio (fine-uning?): ) GeV m GUT 8 The Hubble onsn, during he phse in whih he sysem is rpped in he flse vuum nd he expnsion ours exponenilly, is H 6.6 ( kt ) GeV.6 If we ke kt5=, nd we wn h H f 6 o solve he horizon nd flness problems, hen we hve h 6 ( kt 5 ) s 5 f 6 s H s he epoh of he end of inflion, while he sr, using model of EdS domined by rdiion, is given by

12 Mrino Mezzei s H i 7 These re order of mgniude esimes nd depend on he vlue of kt doped. Dynmis of he Inflon Le us derive he evoluion equion of inflon, i.e. he slr field, sring from he Lgrngin densiy: x d g S L For n expnding universe, spilly fl, in orhogonl oordines, g = nd he Euler-Lgrnge equions re pplied o he quniy L: ) ( V L If he inflon is solely dependen on ime, nd no on he spil oordines, only will be differen from zero nd ) ( V L nd he Euler-Lgrnge equions give: L L L d dv L Puing ogeher nd simplifying we ge ( H / )

13 Mrino Mezzei H dv d whih represens he evoluion of he inflon. This equion, if we refer o he ypil poenil of new inflion, hs wo differen regimes, one lled "slow roll" nd one during whih rpid osillions round he minimum develop. Le's look hem in more deil. ) Slow-roll regime: is his phse here is slow "rolling" of he non-elered field whih orresponds o he phse of inflion. In his regime he erm φ is negligible nd he equion of moion redues o H dv d h is, he friion due o he expnsion is dynmilly blned by he elerion due o he slope of he poenil. By using he derivive of he bove relion, relling h H is essenilly onsn during inflion, nd nming V he d V/d, he ondiion φ Hφ gives V" H H 9H V" G 9 8G V V" V GV Anoher ruil ondiion o hve p = ρ is h φ / V(φ), whih leds o V ' H V V ' V 9H V ' 8G V 8GV V 9 The wo onsrins on he poenil, η, ε re he slow-roll ondiions. b) Fs osillions: A he end of he inflion phse, he poenil "flls" in he rue vuum nd he inflon osilles rpidly round he minimum. If nohing more hppened, we would hve osillions undergoing redshif s ime goes on, in universe h hs lredy ooled drmilly during he inflionry dibi expnsion. In order h he herml hisory of he universe evolves s suggesed by he evidene (e.g. BBN) i is required h he energy of he flse vuum is onvered ino mer nd rdiion wih erin effiieny. This proess is lled reheing. We hve lredy noed h inflion rpidly dilued mgnei monopoles beuse he energy densiy of he slr field remins onsn, while he densiy of monopoles

14 Mrino Mezzei dereses s / (his does no men h hey disppered ompleely, one dy hey will reurn wihin he horizon). However, in order no o be rereed by reheing, i is neessry h his does no bring gin he emperure of he universe o vlues ble o remke hem. The number of e-foldings: I is immedie o lule he number N of e-foldings. We sr from f d d H H d d ln( ) d i i i f i H d f f f H 8G V N ln d i H V ' f 8 V N d V '. G i Oher models of inflion In he model of inflion h we hve desribed, sponneous breking of he symmery ours, bu i is possible o hieve inflion even wihou SSB, s in he se of he so-lled hoi inflion proposed by Linde (98), in whih he poenil V() is simply V nd he poenil hs minimum =. The phse of inflion kes ple if, wihin he horizon, he field, due o qunum fluuions, ssumes vlue differen from zero in region of he universe nd hen reurns owrd he minimum. This is more likely o hppen he end of Plnk er, rher hn he ime of he breking of gre unifiion. To solve he problems of he sndrd model i is no neessry sge wih exponenil expnsion of he sle for; i is suffiien h p p (power-lw inflion) The required poenil hs he shpe V e In he bove disussions we hve ssumed h spe is fl, homogeneous nd isoropi. Wh hppens if i is no he se? I n be seen (see for exmple hp. 8, prgrph 6, in "The Erly Universe" Kolb nd Turner) h, unless he iniil spe urvure is so high o fore he universe o reollpse before inflion, his phse produes, for wide lss of models, huge regions uniform nd fl, whih exeed in d

15 Mrino Mezzei size he urren Hubble rdius, nd hen solve he problems of he sndrd model. Inhomogeneiy nd/or nisoropy re, however, only delyed nd will evenully repper. Cosmologil onsn nd Drk Energy How n we inerpre ody's osmologil onsn? In Einsein's equions (nd hose of Friedmnn), if we remove ll mer nd rdiion, he osmologil onsn is he only soure of he field: Λ orresponds o he densiy of he vuum. Bu, ording o Qunum Field Theory, he vuum is no he nohingness of mephysis, bu he ground se of minimum energy, wih no priles, of he field iself. We hve seen h he osmologil onsn behves s perfe fluid wih ρ Λ =ε Λ =Λ /8πG nd p Λ =-ε Λ =-ρ Λ, nd he energy-momenum ensor is digonl T p p p Moreover, i mus be expeed h he vlues of ε Λ nd p Λ, h define he se of vuum, re he sme in ny, no elered, referene frme, so hey hve o be relivisi invrins. If, for exmple, we mke Lorenz rnsformion wih veloiy v=β [γ =/(-β )] long he xis x, T αβ hnges ording o he rule where T' nd T γδ is digonl, s bove. We ge (see he following snned pge for he proof): p T ' ' T ' T ' T ' T ' p p p T T ' 5

16 Mrino Mezzei 6

17 Mrino Mezzei nd ll he oher erms of T' αβ re null. In order h ε' Λ = ε Λ, p' Λ = p Λ nd T' αβ is digonl p Λ =-ε Λ =-ρ Λ is required. Thus we see h he "flse" vuum of inflion, nd he "rue" void shre similr equion of se. We n lso onsider he vuum s "subsne" wih given ε Λ nd p Λ, in he sense h he relion dl = -dv = -pdv (sine dq = ) is sisfied. Indeed du = d (ε Λ V) = ε Λ dv = - p Λ dv if p Λ = -ε Λ. Bu n we esime he expeed vlue for ε Λ? (Noe h in he following pges =). 7

18 Mrino Mezzei This is he so-lled osmologil onsn problem from he poin of view of Field Theory. There is seond problem linked o he oinidene beween mer-energy densiy nd osmologil onsn: why do hey re omprble ody? Mny emps hve been done o find soluion o hese problems. The disovery of SUSY led o he hope h, sine bosons nd fermions (of idenil mss) onribue eqully bu wih opposie sign o he vuum expeion vlue, he osmologil onsn should be zero. Bu SUSY is ody broken, so Λ ould be zero only in he erly universe. Aemps hs been done o produe (lmos) vnishing osmologil onsn lso wih broken SUSY. Anhropi explnions hve been proposed. In severl osmologil heories he observed big bng is jus one member of n ensemble. The ensemble my onsis of differen expnding regions differen imes nd loions in he sme speime, or of differen erms in he wve funion of he universe. If he vuum energy densiy ρ V vries mong he differen members of his ensemble, hen he vlue observed by ny speies of sronomers will be ondiioned by he neessiy h his vlue of ρ V should be suible for he evoluion of inelligen life. The nhropi bound on posiive vuum energy densiy is se by he requiremen h ρ V should no be so lrge s o preven he formion of glxies (he elered expnsion sops he growing of he mpliude of densiy fluuions). A negive vlue for he osmologil onsn, s we hve seen, s s n ddiionl selfgrviy nd fores he reollpse of he universe; if his reollpse hppens oo erly, no inelligen life n develop. 8

19 Mrino Mezzei Dynmil models of Drk Energy Mny ides hve been proposed o solve he problem of Drk Energy (if you re ineresed in his subje you n refer o he book DARK ENERGY, Theory nd Observions by Lu Amendol nd Shinji Tsujikw,, Cmbridge Universiy Press). There re bsilly wo pprohes for he onsruion of drk energy models. The firs pproh is bsed on modified mer models in whih he energy-momenum ensor T µν on he r.h.s. of he Einsein equions onins n exoi mer soure wih negive pressure. The seond pproh is bsed on modified grviy models in whih he Einsein ensor G µν on he l.h.s. of he Einsein equions is modified. Here we menion he so-lled quinessene model s one of he represenive modified mer models. Quinessene is nonil slr, uniform field Q wih poenil V(Q) responsible for he le-ime osmi elerion. Unlike he osmologil onsn, he equion of se of quinessene dynmilly hnges wih ime: p Q = w Q ρ Q wih ρ Q = Q + V Q p Q = Q V Q Q w Q = V Q Q + V Q where w Q n be in he rnge from - o +. Here we n use relions similr o hose used when working on inflion. We ssume fl, k=, universe. The evoluion of he field nd he dynmis of he universe re given by he lredy known relions Peebles nd Rr proposed poenil like Aording o nien Greek siene, he quinessene (from he Lin fifh elemen ) denoes fifh osmi elemen fer erh, fire, wer, nd ir. 9

20 Mrino Mezzei nd ssumed h high redshif he densiy of he field is subdominn wih respe o h of mer/rdiion, in order o preserve he BBN. We ssume h he sle for grows s () q The equion of he field is he soluion is doesn depend on q!!!

21 Mrino Mezzei Sine, in EdS, ρ /, boh for mer nd rdiion α=, V(Q)=ons., orresponds o he osmologil onsn, ρ Q. For α> he slr field beomes dominn wih respe o mer, even if i ws negligible high redshif. In ny se, he firs hing o undersnd is if he vuum energy densiy is onsn or vries over ime. To do h, people uses ll he vilble ses of osmologil observions o fi, for insne, liner dependene on sle of he equion of se w()=w + w ( / ) The resuls re no onlusive, nd osmologil onsn is sill onsisen wih he d (plo ken from Plnk sellie 5 resuls).

22 Mrino Mezzei SHORT COSMIC HISTORY Er (se) E T (K) Evens Plnk - 9 GeV Qunum Grviy GUT -8 GeV -6 5 GeV 9 8 GUT s SSB Inflion? Bryogenesis? Elerowek - GeV 5 Elerowek SSB Adroni - MeV Qurk-drons rnsiion Leponi.7 MeV Deoupling of e 5.5 MeV 5 9 Annihilion e + e - BBN - min. MeV 9 BBN: He, He, D, 7 Li Rdiion-Mer Equliy 6 yr ev ( ) Mer-domined er begins Reombinion 5 yr. ev The universo beomes neurl nd rnspren Void Gyr - ev.6 Void-domined er begins Tody.7 Gyr - ev.7 yr siderel yer (9) se ly ligh yer m.u. sronomil uni m p prse m H Hubble onsn. -8 h se - /H Hubble ime.86 7 h - se M solr mss.989 g R solr rdius m L solr luminosiy.9 erg se M Erh mss g R equoril Erh rdius km We use : T K ~ E GeV