Imperial College QFFF, Cosmology Lecture notes. Toby Wiseman

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1 Imperil College QFFF, 07-8 Cosmology Lecture notes Toby Wisemn

2 Toby Wisemn; Huxley 507, emil: Books This course is not bsed directly on ny one book. Approprite reding for the course is; Weinberg, Cosmology lso n older book Grvittion nd cosmology Kolb nd Turner, The erly universe (somewht dted now!) Liddle nd Lyth, Cosmologicl infltion nd lrge scle structure Pecock, Cosmologicl physics Mukhnov, Physicl foundtions of cosmology Dodelson, Modern cosmology For issues of GR, my fvourite book is obviously Wld, Generl Reltivity. Mthemtic Ihvewrittensomemthemticnotebookstoillustrtecertinclcultions. These cn be downloded from my PWP. In order to run them you will need Mthemtic. Instructions for downloding this re t;

3 Conventions Iwillusethefollowingconventions;c nd( For geometry I will use; so, c b b d r v v b + b cv c And, [r, r ]v µ R µ v so, R µ + µ µ

4 Brief history of universe 0 4 s, 0 9 GeV Plnck scle - quntum grvity - GUTs??? 0 ( 5, 40) s, GeV Infltion - quntum perturbtions creted, reheting Hot big bng begins... 0 ( 5, 8) s, GeV Bryogenesis 0 0 s, 00 GeV Electrowek scle - drk mtter nnihiltes to give relics?? 0 4 s, 0 K, 00 MeV Qurks condense to hdrons 0 s, 0 K, MeV Neutrino decoupling nd e + e nnihiltion (e + + e $ ) Observtion er begins... 0 s, 0 0 K, 0. MeV nucleosynthesis (n + $ p + e ) 0 5 yrs, K Mtter domintion (Rdition-Mtter equlity) 0 5 yrs, 0 4 K, ev Photon decoupling (p + e $ H), CMB formed 4

5 0 9 yrs yrs Structure formtion; First objects - universe no longer close to FRW Drk energy comes to dominte yrs,.7 K (e ective temperture of photons) Pln Now. FRW; symmetries,friedmnneqution,cosmologiesforperfectfluids nd sclr fields, observbles, CDM model. Hot mtter; stt. mech. descriptionofmtterinthermlequibilirium, out of equilibrium description (Boltzmnn eqution), relics (drk mtter). Therml history; hotrditioner,nucleosynthesis,recombintion 4. Infltion; cosmologiclpuzzles(fltness,horizon,monopole),sclr field cosmologies, slow roll infltion, genertion of fluctutions during infltion 5

6 FRW. Cosmologicl principle The universe is sttisticlly the sme t every loction in it, nd in every direction, ie. there is nothing specil bout where we live. On suitbly lrge scles (essentilly beyond scles which re grvittionlly bound - glxy cluster scles Mpc)ndftersuitbleverging,the universe should be homogeneous nd isotropic for some suitbly chosen set of observers. These observers define folition of spcetime into sptil slices. For ny spcetime filled with observers we my (loclly t lest) write the metric s, ds dt + h ij (t, x)dx i dx j () This is simply coordinte choice, where the time coordinte is the proper time s mesured for n observer sitting t the constnt the sptil loction x -nottheyrenotnecessrilyfreeflling(ie.x const is not geodesic). Techniclly we tke homogeneity nd isotropy to men we my write the metric of spcetime s, ds dt + d, d h ij (t, x)dx i dx j () where t ech constnt time t, thend isthemetriconspcewhichis homogeneous nd isotropic... Homogeneity nd isotropy sptil geometries Now t fixed time t then - the constnt time slice - is simply sptil geometry. Let us now suppress time in this discussion nd simply consider homogeneous isotropic sptil geometries. Consider first isotropy. Then t ny point in ll directions must pper the sme. Recll tht ny Riemnnin geometry loclly is flt. Consider point, nd then loclly bout tht point we my write, d h ij dx i dx j A( )d + B( )d () () 6

7 where d () d +sin d is the round unit -sphere. An importnt point is tht A nd B do not depend on nd s we wish to hve isotropy, nd lso there re no o digonl or terms in the metric for the sme reson. [More precisely the metric hs isometry group SO(), generted by the Killing vectors of the -sphere.] Let us chose convenient coordinte r B( ), nd then, rdr d A(r) + r d B 0 () ( ) S(r)dr + r d () (4) Now since ny spce is loclly flt, s r! 0wemusthvelim r!0 S(r). Now consider homogeneity. A necessry, but not su cient condition for homogeneity is tht the Ricci sclr is constnt - ie. it doesn t depend on x i. Now, R r S + rs0 (r) (5) S where R is the Ricci sclr of the metric on. Let us set this equl to constnt k. Thenchoosethenormliztionoftheconstntso, R 6k (6) Then one my solve the resulting eqution, nd with the condition tht the spce is loclly flt t r 0wehve, S(r) kr (7) 7

8 One then finds tht the Ricci tensor is; nd the metric is, d (R ) ij kh ij (8) dr kr + r d () (9) We see the Ricci tensor is proportionl to the metric - this is specil condition clled n Einstein spce. It is convenient to perform coordinte trnsformtion; for constnt so then, d r! r 0 r (0) dr 0 k r 0 + r0 d () () By performing this scling we my lwys write the metric s; dr d 0 k 0 r + 0 r0 d () d 0 () where now k 0 0, ±, nd is constnt. Note now, R 0 6k 0. The cses of k 0 re distinct, nd simply sets the size or scle of the geometry. Let us now drop the primes on r, k nd consider the cses k 0, ±. Cse k 0- flt Then clerly k 0 is the cse of Eucliden -d flt spce; d dr + r d () dx + dy + dz () The homogeneity nd isotropy re pprent in the crtesin nd sphericl coordintes; trnsltions nd rottions respectively. 8

9 Cse k - -sphere For k then is unit rdius (round) -sphere. Let us see why. Consider 4-dimensionl Eucliden spce with coordintes x A (z,x i ). Then we my write the metric s, ds (4d) dz + dr + r d () AB dx A dx B (4) Now consider embedding unit -sphere into this spce, Hence the induced metric is, z + r ) rdr zdz (5) using, d dz dr r dr + dr + r d () dr + r d () (6) + dz + r dr z z + r (7) z r Homogeneity nd isotropy; We my see the isometries of the -sphere by noting tht the full isometry group SO(4) cts on the -sphere in the obvious wy in the Crtesin coordintes X A in the 4-d Eucliden spce. One cn trnslte this ction into the z,r,, coordintes lthough it is complicted. The ction of homogeneity re the rottions of point to nother point, nd of isotropy re the rottions bout n xis. Cse k<0 - -hyperboloid The k cse is similr to the sphere. Now insted of embedding surfce in Eucliden spce, we insted embed in 4d Minkowski spcetime, ds dz + dr + r d () dz + ij dx i dx j (8) where now the extr coordinte z is time coordinte. Now hyperboloid,, with unit rdius is embedded s, z r ) rdr zdz (9) 9

10 nd its induced metric is, now using, d dz dr dz dr +r dr + dr + r d () dr + r d () (0) r z z r z +r () Homogeneity nd isotropy; We see the embedding nd metric of the hyperboloid re invrint under Lorentz trnsformtion of the 4-d embedding Minkowski spcetime, ie. under SO(, ). Hence the isometry group of the -hyperboloid is SO(, ). The ction on the Minkowski coordintes z,x i is strightforwrd, but is complicted in the coordintes r,,. The ction of homogeneity nd isotropy re generted by combintions of the boosts nd rottions... The FRW spcetime Now let us return to the full spcetime, rther thn constnt t slice, nd consider the time dependence. We hve seen tht the geometry of constnt t slice cn be written s, d kr dr + r d () () for constnt setting the scle of the spce, nd k 0, ± settingits chrcter. However we must recll tht ws constnt of integrtion, nd my be di erent depending on the sptil slice we pick. We cn use the bove coordintes on ech time slice provided we let (t). Now the full spcetime metric tkes the form, ds dt + (t) d (k), d (k) kr dr + r d (), k 0, ± () 0

11 nd the function (t) isthe sclefctor controllinghowthesizeofthehomogeneous isotropic sptil slices chnges in time. [ Note tht one might wonder why k 6 k(t)? Such topology chnge is not possible in smooth mnner. ] We use the terminology;. k 0isflt universe. k isclosed universe (sphere sptil sections). k isnopen universe (hyperbolic sptil sections) Note tht in the flt nd open cses the geometry of the sptil slices is infinite - the sptil volume is infinite. However, for the closed cse the universe hs finite volume... Another coordinte chrt We hve seen tht we my write, d (k) kr dr + r d () (4) in isotropic coordintes. There is nother convenient chrt for computing the Christo el symbol (h)i jk, d (k) h ij dx i dx j ij + nd in this chrt one cn conveniently write, kxi x j dx i dx j, x x n x m k x nm (5) (h)i jk kx i h jk (6)

12 . Properties of FRW Writing FRW in the generl form erlier, ds dt + (t) (0,±) dt + (t) h ij (x)dx i dx j (7) then one finds the metric nd inverse metric in coordintes x µ (t, x i )is, 0 0 g µ 0, g µ (8) (t)h ij 0 (t) hij where h ij is the inverse metric to h ij. The non-vnishing Christo el components re then, t ij ȧh ij i tj ȧ ij i jk (h)i jk (9) (h)i where jk is the Christo el symbol of the -d sptil geometry h ij(x). t The components, ti i tt 0 by isotropy, nd the components t tt 0hppentovnishusingthispropertimecoordintet. Then one finds the Ricci tensor components; R tt ä R ti 0 R ij R (h) ij + ȧ + ä h ij (0) where R (h) ij is the Ricci tensor of h ij nd we recll tht R (h) ij kh ij nd hence, R ij k +ȧ + ä h ij () Agin R ti 0duetoisotropy. This yields n Einstein tensor G µ R µ g µ R s, G tt k + ȧ G ti 0 G ij k ȧ ä h ij ()

13 .. Specil geodesics of FRW By symmetry, x i constnt is timelike geodesic - with proper time t. These re the worldlines our preferred observers. These observers re clled comoving observers s they free fll with the homogeneous, isotropic frme. They hve 4-velocity v µ dx µ /d dx µ /dt (, 0, 0, 0). A simple set of geodesics re the null curves pssing through r 0is isotropic coordintes, ds dt + (t) kr dr + r d () () By symmetry, world line is, constnt for these null geodesics, so the metric on their which must vnish for null curve so, ds curve dt + (t) kr dr 0 (4) dt ± p dr (5) kr for the curve. Consider null geodesic rriving t n observer t r 0 t time t t o,hvingpreviouslypssedthroughrdiusr t time T (with t o >T). Thus r is decresing with t nd so we require the negtive sign bove. Then integrting (nd flipping the limits in R dr) weobtin, Z to T Z R (t) dt 0 8 < p dr kr :.. Generl geodesics of FRW R k 0 sin (R) k sinh (R) k f k (R) (6) Consider geodesic x µ (T ( ),R( ), ( ), ( )) with ne prmeter. In order to obtin the geodesic equtions we my vry the ction, Z S d L, L g µ dx µ d dx d T + (T ) Ṙ kr + R +sin! (7)

14 where d/d (not d/dt). Consider geodesic pssing through r 0. Thenbysymmetryginit should hve, constnt. Moregenerllysymmetryimpliesthereshould be clss of geodesics with only rdil motion, i.e. with, constnt. Check: nd equtionsre(respectively; R sin J, d d R R sin cos for constnt of integrtion J. Thesereindeedstisfiedfor, nd J 0. (8) constnt, Note: We might lso consider geodesic motion which is not rdil. For this we would hve to solve the bove complicted system. However, we know from homogeneity tht we cn consider ny point s r 0,sootherrys not going through r 0 re lwys relted to ones through the origin by homogeneity. Now we re left with vrying the line element long the curve, L T + (T ) kr Ṙ (9) We hve lredy solved this in the null cse bove. Consider now the timelike k RṘ ( kr ) (40) this yields the Euler-Lgrnge eqution, d/d d Ṙ krṙ (4) d kr ( kr ) Now dividing by Ṙ/( kr )gives,!! d Ṙ ln krṙ d kr kr d d ln kr (4) 4

15 So then, where A is constnt of integrtion. Now using L µ 0, ± wehve, Ṙ A p kr (4) so tht, T (T ) kr Ṙ µ A µ (44) Hence the curve obeys, T p A µ (45) dr dt Ṙ T A p kr p (46) A µ Consider ry through r 0tt t o nd previously through r R t t T with t o >T. Hence during the intervl T<t<t 0 then dr/dt < 0 nd so A<0. Then (flipping the limits in R); Z to T Z A dt R (T ) p A µ (T ) 0 dr p kr 8 < : R k 0 sin (R) k sinh (R) k (47) nd so we my clculte R R(T ). This obviously grees with the cse µ 0frombefore. Aside on photons/wve Consider high frequency photon - so its wvelength is much shorter thn the locl curvture scles. Then in n LIF we my write tht, A µ e ikµxµ e µ (48) for constnt polristion e µ (obeying constrints) nd constnt wve vector k µ,obeyingk 0. The4-momentumofthephotonis, p µ ~k µ (49) 5

16 The fct tht k µ is constnt my be written covrintly s, k µ r µ k 0 (50) ie. the wve crests follow null curves. Now more generlly in curved spcetime, high frequency photon my be written s, A µ e i (t,x) e µ (t, x) (5) where the phse is rpidly vrying, nd now the field k µ gin in the short wvelength limit, obeys the geodesic eqution, k µ r µ k 0 (5) nd the 4-momentum is gin p µ ~k µ. This is wht is known s the geometric optics pproximtion... Grvittionl redshift Consider photon trvelling long null geodesic (µ 0). Then its 4- momentum is p µ ~k µ with k µ obeying the null geodesic condition k µ r µ k 0withk µ k µ 0. Hence we my write, for some p µ ~ dxµ d ne prmeter. Now, ~ T,Ṙ, 0, 0 k µ dxµ d A (T ), A p kr, 0, 0 (T ) (5) (54) Consider comoving observer with 4-velocity v µ (, 0, 0, 0). They mesure the photon to hve n energy E comove p µ v µ,so, E comove +p t A (T ) (55) Hence the energy of the photon mesured by observers in the cosmologicl frme redshifts s E /(T ). 6

17 Aunitnormsptilvector(tngenttoconstntt slices) for such n observer, in the direction of the photon, is n µ (0, / p g rr, 0, 0), so tht n µ n µ ndn µ v µ 0. Then the mgnitude of momentum p comove tht the observer mesures the photon to hve in the direction n µ is, p comove p µ n µ g rr p r n r p g rr p r A (T ) A p p kr kr (T ) (56) Thus we hve, (T ) p comove constnt. In the cse k 0,thisresult is seen s conservtion lw using Crtestin sptil coordintes due to trnsltion invrince in spce. More generlly it is conservtion lw due to the isometry of homogeneity. The fct tht E comove / is simply result of the fct tht in ny LIF photon hs E p. We hve derived the fmous grvittionl redshift result, nmely tht for photon emitted t time t e with comoving energy E e (nd momentum p e E e ), nd received t t o,comovingenergye o (nd momentum p o E o ), then, E o (t e) E e (t o ) (57) Hence photon is redshifted by n expnding universe - in sense it is tunnelling out of grvittionl potentil well. Since frequency of photon E ~, onelsohs, o (t e) e (t o ) It is conventionl to define the redshift Z s, +Z e (t o) o (t e ) (58) (59) where now t o should be tken s the time tody. ThusredshiftZ(t e ) vnishes for photons emitted tody (ie. very close by) nd is Z>0forphotonsemitted in our pst. It is importnt s it provides very physicl mesure of time, Z Z(t) by, Z(t) (t o) (t) 7 (60)

18 where we think of t<t o s the time in the pst when photon which we receive tody ws originlly emitted. Inverting this reltion, t t(z) gives n elegnt reltion between comoving proper time nd redshift which is directly observed. However, this reltion clerly depends on the detil of (t), nd hence the cosmologicl model. A more pedestrin derivtion: Consider null ry emitted t r R t time t t e nd propgting to the origin t time t t o.thenwehvepreviouslyfound, Z to t e Z dt R (t) 0 dr p kr 8 < : R k 0 sin (R) k sinh (R) k (6) Now consider second ry emitted slightly lter t t t e + t e nd received t t t o + t o.then, Z to+ t 0 t e+ t e Z dt R (t) 0 Hence, di erencing these reltions we obtin, 0 Z to+ t 0 t e+ t e t e (t e ) dt (t) t o (t o ) dr p kr Z to t e dt (t) (6) (6) Now since the reltion between the emitted nd observed comoving frequencies is o e te t o then we obtin s before; +Z e (t o) o (t e ) (64) Mssive geodesics Let us continue our discussion of geodesics nd conclude with mssive prticle (µ nd 6 t). Then it hs 4-momentum p µ mv µ, p µ m dxµ d m T,Ṙ, 0, 0 mp A + (T ), (T ) 8 ma p kr, 0, 0 (T )! (65)

19 Now comoving observer mesures E comove nd p comove s; s E comove p t m + A (T ) m (66) where is the usul Lorentz fctor / p v for observed velocity v (so v / ), nd so, Then note tht, v v A + A (67) s A p + A + A (T ) A (T ) (68) And for the momentum, p comove p µ n µ m A p p kr kr (T ) m A (T ) m v (69) with observed velocity v. ThusE comove p comove + m s usul, but with red shifting momentum, p comove /(T ). Note tht s for the photon, we hve p comove constnt, which follows from the conservtion lw due to sptil homogeneity. Recll this is esily seen computing geodesics for the flt k 0cseinCrtesincoordintes where homogeneity is mnifest. Thus more generlly the redshift is relly relted to observed nd emitted comoving -momentum; +Z (t e) (t o ) p o p e (70) for ny free flling object - null or timelike. However, for photons E p. Note in terms of observed energy for mssive prticle we hve, s +Z p o Eo m (7) p e Ee m 9

20 . Observbles Let us now consider some importnt quntities which re relted to observbles in cosmology... Comoving distnce If we re t r 0 nd distnt object is rdil coordinte position R t time t then we sy it hs comoving distnce R. Note tht if we nd the object re comoving, then this distnce remins constnt in time... Proper distnce The folition of spcetime by the fmily of cosmologicl observers llows us to define the proper distnce between two objects t given time t. Suppose we consider n observer t time t t the origin r 0,ndnobject t tht time t rdius r R. Then the proper distnce d(t, R) isthegeodesic distnce within the sptil section. The sptil slice t constnt t is; dr d (t) kr + r d () (7) nd by symmetry the geodesic in between the observer nd the object trvels on the rdil line, constnt. Hence its proper distnce is, 8 Z R dr < R k 0 d(t, R) (t) p (t) sin (R) k (7) 0 kr : sinh (R) k Note tht this is not geodesic in the full spcetime, only within the geometry of the constnt sptil slice defined by our cosmologicl observers. Note lso tht for null geodesic emitted t time t t e from rdius r R nd reching the origin r 0ttimet t o we hve, Z to Hence, we hve for null ry, t e Z dt R (t) dr p 0 kr (74) d(t o,r) (t o ) d(t e,r) (t e ) Z to t e dt (t) (75) 0

21 .. Hubble function A very importnt quntity is the Hubble function, H(t) 0 (t) (t) (76) nd its vlue tody, ie. t t 0, is clled the Hubble constnt H 0 H(t 0 ). Consider comoving observer t r 0,ndcomovingobjecttr R. Then the Hubble function determines the rte of chnge of the proper distnce between these objects. 8 d < R k 0 dt d(t, R) 0 (t) sin (R) k : sinh (R) k H(t)d(t, R) (77) Note tht d/d is independent of R. This cn be interpreted s version of Hubble s lw, nmely tht objects in n expnding universe recede from us with velocity proportionl to their distnce. This movement of objects wy from us is clled the Hubble flow. Note however tht the velocity d is not very physicl one. Let us consider now more physicl version of Hubble s lw...4 Nerby sources Consider source nerby to us t r 0,t t o,emittingttimet e nd t rdius r R, so t t o t e is smll. Now using our previous reltion for the proper distnce trvelled by null ry, d(t o,r) (t o ) Z to t e dt (t) t (t o ) (78) then we see, d(t o,r) t (79) This simply sys the proper distnce is pproximtely the light trvel time.

22 Then we my lso expnd, Hence, (t e ) (t o )+ 0 (t o )(t e t o )+... (t o )( t H o +...) (80) +Z (t o) (t e ) + th o +... ) Z ' th o (8) Thus the Hubble constnt determines the redshift of locl sources. Putting these results together we obtin Hubble s experimentl result for nerby sources; Z ' H o d(t o,r) (8)..5 Vlue of Hubble constnt The Hubble constnt H o hs vlue, usully quoted in peculir units; H ' 70 km s Mpc (8) Wht funny units! Recll prsec; pc distnce t which AU subtends n ngle of rc second! So pc.lightyr. 0 6 m. In terms of stronomicl scles; Nerest str (Proxim Centuri) pc Milky wy (our glxy) Kpc Glxy cluster -0 Mpc Hubble horizon (pprox size of observble universe) 000 Mpc These units re useful in the sense of d Hd so tht n object Mpc wy looks s though it is receding t velocity 70 km s. For reference, note tht our peculir motion, ie. the motion reltive to the cosmologicl frme, is 400 km s (which is typicl vlue). Thus on lrge scles (> cluster size) peculir motion become negligible compred to the Hubble flow. On smll scles it is n importnt e ect, dominting the motion due to cosmologicl expnsion.

23 ..6 Angulr dimeter nd Luminosity distnce So fr we hve the reltion R(t) ndz(t) fornullryprovidedweknow the cosmologicl expnsion (t). Suppose we observe light from n object nd mesure its redshift, then we know the time of emission t nd the rdius R the light ws emitted from. Now since we lso know the proper distnce reltion d(t, R) totheobject then if we hve n independent mesure of distnce then we cn check this grees. This will be check of our cosmologicl model. Turned round, if we know the distnce d nd redshift Z of mny observed objects, we cn hope to reconstruct the function (t) nddeducethecorrect cosmologicl expnsion rte (nd then using Einstein s equtions understnd the mtter driving the expnsion). However while Z is something directly nd ccurtely observed, the proper distnce d is not something tht is esy to deduce simply looking t distnce source. Hence it is useful to hve two more prcticl distnce mesures; ngulr dimeter distnce d A nd luminosity distnce d L. Angulr dimeter distnce d A Let us be comoving observers t r 0. Considercomovingobjecttrdius r R with proper size (for exmple distnt glxy). Assume we know this size - for exmple we hve good model for glxies. Suppose it emits light t time t t e,ndweobservethisttimet t o.letusseetheobject to hve n ngulr dimeter on the sky. Then the ngulr dimeter distnce is defined to be the distnce the object ppers to be from us ie. ssuming simple Eucliden sptil geometry nd infinite speed of light. So, d A (84) Now from the sptil geometry t time t e, dr ds spce (t e ) kr + r d +sin d we hve, (85) (t e ) R (86)

24 nd thus we see, Recll tht, 8 < d(t, R) (t) : d A (t e ) R +Z (t 0) R (87) R k 0 sin (R) k sinh (R) k, d(t o,r) (t o ) d(t e,r) (t e ) (88) Hence we cn relte d A to the proper distnce tody d(t o,r). For exmple if k 0, d A (t e ) R d(t e,r) (t e) (t o ) d(t o,r) ( + Z) d(t o,r) (89) Thus if we know the size of n object we cn mesure d A nd its redshift Z directly, nd then deduce its proper distnce d from these. Luminosity distnce d L Suppose we hve similr source, but now insted of knowing the size of the source, we insted know its intrinsic luminosity L energyemitted (isotropiclly) per time. When we observe the light we see n pprent luminosity ` energy t receiver per re per time. We define the luminosity distnce which is the distnce the object ppers to be from us; ` L (90) 4 d L From the sptil geometry t time t o,therelightfromthesourceisspred over -sphere with re, The pprent luminosity is given by; A(S )4 ((t o )R) (9) ` L ( + Z) A(S ) where the two fctors of /( + Z) redueto; 4 (9)

25 The energy of ech photon is redshifted by o / e /( + Z) The rte of receiving photons is reduced reltive to emission frequency by the sme fctor. Recll our previous clcultion of grvittionl redshift precisely showed in section.. (see eqution 6) tht t o ( + Z) e. Hence we see, tht; ` L 4 d L L ( + Z) A(S ) Reclling tht d A (t e ) R, wesee, ) d L (+Z)(t o ) R (9) d L (+Z) d A (94) Agin we cn relte d L to the proper distnce tody d(t o,r). For exmple if k 0, d L (+Z)(t o ) R (+Z)d(t o,r) (95) Experiment We mesure distnt source, nd determine its redshift Z. Either knowing its size, or its luminosity L we then determine its distnce d A or d L. Recll, d A (t e ) R ( + Z) (t o) R, d L (+Z)(t o ) R (96) nd so our mesurements give us directly Z nd (t o )R. Suppose we understnd (t) ndwishtotestourcosmologiclmodel.then our mesurement for source gives both Z nd R. But the photon reltions, 8 Z to < +Z (t o) (t e ), t e dt (t) : R k 0 sin (R) k sinh (R) k (97) reltes these s we know R R(t e )ndz Z(t e )soweknowr R(Z). Thus we cn check the greement. In prctice we use lots of sources to build up the reltion between Z nd R, nd then constrin prmeters in our model to fit this. 5

26 .4 Horizons nd the big bng The inverse of the Hubble function defines distnce scle, denoted the Hubble scle. This is sometimes loosely referred to s the horizon size. Generlly it doesn t refer to ny horizon, but is n importnt physicl scle tht essentilly determines the scle beyond which the spcetime doesn t look flt for cusl physics. However in certin situtions there re two types of horizon - prticle nd event horizons. Suppose we hve n expnding universe which strted t t t BB t Big Bng, so tht (t BB ) 0. Using the null reltion for ry reching r 0t time t strting t rdius r R H t the Big Bng; Z t t BB dt 0 (t 0 ) 8 < : R H k 0 sin (R H ) k sinh (R H ) k (98) Thus this gives reltion R H R H (t). The Prticle horizon size d H (t) t time t is the proper rdius of the comoving volume with this rdius R H (t). Recll the proper distnce, nd hence we see, 8 < d(t, R) (t) : R k 0 sin (R) k sinh (R) k Z t dt 0 d H (t) d(t, R H (t)) (t) t BB (t 0 ) (99) (00) Existence of prticle horizon Note tht this integrl my or my not converge depending on the behviour of (t) tthebigbng. Weshlldiscussthislter. Forusulmtter nd rdition this does converge (with (t t BB ) / for hot rdition). However, for exotic inflton mtter, this does not. Event horizon If the integrl Z t dt 0 (t 0 ) 8 < : R E k 0 sin (R E ) k sinh (R E ) k (0) 6

27 converges, it mens tht n event t r 0ndtimetcn only influence comoving observers within rdius r pple R E (t). An observer outside the rdius R E will never see the event, however long they wit. This comoving size is known s the event horizon for the event t r 0,timet, ndits proper size is, Z dt 0 d E (t) d(t, R E (t)) (t) (0) t (t 0 ) Note tht if the universe recollpses then t my hve mximum vlue t mx. In this cse the event horizon is defined in the obvious wy s, 8 Z tmx < t dt 0 (t 0 ) : R E k 0 sin (R E ) k sinh (R E ) k (0) Comment: The existence of drk energy leds to n event horizon, where d E H0. Hence s the universe expnds, objects which re not grvittionlly bound to us move further wy ( d Hd). Once they move beyond properdistnced E they cn never influence us gin. Note tht while the terminology horizon is used, these re not proper cusl horizons in the sense of blck holes. Their definition is dependent on the choice of n observer. The region ssocited to one observers horizon will not coincide with tht of nother..5 Perfect fluids Consider the stress tensor for single perfect fluid; T µ ( + P ) u µ u + Pg µ (04) where, P re the energy density nd pressure, nd u µ is the locl fluid 4- velocity (so u ). The reltion P P ( ) istheeqution of stte of the fluid. Recll tht for such fluid the dynmics is simply determined by the conservtion of stress-energy, which yields reltivistic hydrodynmics. r µ T µ 0 (05) 7

28 Importnt: We my hve severl such fluids in spcetime nd provided they don t exchnge energy-momentum with other mtter, ech one will individully obey the bove equtions. The equtions of hydro re best described by projecting into the u µ nd orthogonl directions. Firstly project the conservtion eqution into the direction u µ ; u µ r T µ 0 ) +( + P ) r u 0 (06) reclling u. This gives n evolution eqution for the density. Let us now project onto n orthogonl direction n µ to the motion u µ,sotht u µ n µ 0. Note tht n µ must be spce like (since v µ is timelike). Then, n µ r T µ 0 ) n µ (( + P ) u r u µ + g P )0 (07) nd this is equivlent to, ( + P ) u r u µ + µ + u µ P 0 (08) gin reclling tht u. The quntity? µ µ + u µ u is projector into the directions orthogonl to u µ.ithsthepropertytht, for ny n µ orthogonl to u µ.then, gives n evolution eqution for the velocity.? µu µ 0,? µn µ n (09) ( + P ) u r u µ? µ@ P (0) We will derive this lter, but tking P w for constnt w, then, Dust / cold mtter: w 0,soP ' 0 Rdition / hot mtter: w,sop ' Vcuum energy: w 8

29 Note in the cse of vcuum energy it isn t relly fluid; T µ g µ () However, the condition r µ T µ 0thenimplies@ µ 0,ndhencethe energy density is not dynmicl, but constnt in time nd spce. It is nturl to define the cosmologicl constnt so we my write, T µ 8 G N () 8 G N g µ () nd then we move this term to the LHS of the Einstein equtions, R µ g µ R + g µ 8 G N Tµ (4) where T µ is the stress tensor due to other mtter - not cosmologicl term. Now is n inverse length squred, where the length gives the rdius of curvture ssocited to the cosmologicl constnt..6 The cosmologicl stress tensor Now consider cosmologicl mtter in FRW, ds g µ dx µ dx dt + (t) h ij (x)dx i dx j (5) so tht the stress tensor is homogeneous nd isotropic. Then since there is no preferred direction we must hve T ti 0,ndT ij / h ij.duetohomogeneity we must hve T tt is function of time only, nd T ij f(t)h ij for function of time f. Thenwedefine the totl density nd totl pressure of cosmologicl mtter in FRW to be, T tt tot (t), T ij (t)p tot (t)h ij (x) (6) where T µ is the sum of ll mtter components. Why? Consider perfect fluid which is homogeneous nd isotropic. Then u µ (, 0, 0, 0) nd (t), P P (t). Thus, T tt ( + P ) u t u t + Pg tt T ti 0 T ij Pg ij Ph ij (7) 9

30 nd thus the density nd pressure gree. Note however, tht while ny mtter with cosmologicl symmetry hs stresstensorwiththeformbove,itdoesnot men it behves like single perfect fluid with simple locl eqution of stte P tot (t) F ( tot (t)). In generl mtter with hve complicted non-locl (in time) eqution of stte. Let us consider the conservtion eqution r µ T µ 0forstresstensor shring the cosmologicl symmetry. Recll tht this pplies to the totl stress tensor (s result of the Binchi identities), but lso pplies seprtely to the stress tensor of ny mtter component tht doesn t interct with other mtter. The time component of the conservtion eqution implies; 0 r µ T t T tt µ µ T t ȧ ( i it T t t )+ȧ j ti T i j j i P i j µ t T µ +ȧ ( + P ) (8) This pplies to both the totl stress tensor, but lso seprtely to ny component tht doesn t interct with other components. The only ssumption is tht the mtter hs the cosmologicl symmetry..7 Cosmologicl perfect fluids Consider now the behviour of single perfect fluid with cosmologicl symmetry tht doesn t interct with other mtter. Let it hve eqution of stte P w for constnt w. Thenconservtionimplies, +ȧ ( + w) 0 ) k (+w) (9) for constnt of integrtion k. We usully write this constnt in terms of the scle fctor tody, o,ndthevlueofthefluiddensitytody, o,s, 0 (+w) 0 (0) Consider the importnt cses; Dust / cold mtter: w 0;,sothemttersimplydilutes. 0

31 Rdition / hot mtter: w ; dilutes nd its constituents redshift. 4,sotherditionboth Vcuum energy: w, s we sw before we hve constnt..8 Dynmics nd the Friedmnn eqution We hve previously sttes the components of the Einstein tensor. Recll, ds dt + (t) h ij (x)dx i dx j, R (h) ij kh ij G tt k + ȧ, G ij k ȧ ä h ij () Then the Einstein equtions re G µ 8 G N T µ where we emphsise tht T µ is the totl stress tensor, nd hence is conserved nd in our FRW pproximtion must hve cosmologicl symmetry. Using equtions (7) we then rrive t two importnt reltions. Firstly the tt component directly gives; ȧ + k 8 G N tot ) H 8 G N tot k () This is the Friedmnn eqution nd directly determines the Hubble function H ȧ/. Note tht we lso hve the mtter conservtion eqution; +ȧ ( tot + P tot ) 0 () Suppose we hve single fluid for mtter. Recll the fluid equtions of motion re simply given by stress energy conservtion. Then given the universe t time t i with scle fctor (t i )nddensity (t i ), with known eqution of stte P P ( ) thenthefriedmnnndconservtionequtions llow us to integrte forwrd in time to determine (t) nd (t). For combintion of perfect fluids (with cosmologicl symmetry), then the totl energy nd pressure will be the sum of the vrious components tot X i i, P tot X i P i (4) nd ech will be conserved seprtely; i +ȧ ( i + P i ) 0 (5)

32 (Obviously this implies the totl stress tensor is conserved). Agin the Friedmnn eqution, the individul conservtion equtions nd equtions of stte P i P i ( i ), llow one to integrte (t) nd i (t) forwrdintime. More generlly, for ny mtter with cosmologicl symmetry, the Friedmnn eqution together with the mtter equtions of motion llow integrtion forwrd in time. For exmple, for mssive sclr field the eqution of motion is, r r µ r µ V 0 ( ) (6) nd the stress tensor is, T g µ (@ ) + V ( ) (7) nd it is esy to show the mtter eqution of motion implies the stress tensor is conserved. Then for cosmologicl symmetry the sclr eqution becomes; + ȧ V 0 ( ) (8) nd so knowing, nd t time t i then the Friedmnn eqution nd sclr eqution llow integrtion forwrd in time. In this cse, tot + V ( ), P tot V ( ) (9) Note tht the cse m 0isP (so w - sti equtionofstte). However, for generl m this is not of simple P P ( ) form. Wht bout the sptil components of the Einstein equtions? The sptil ij-components my be combined with the tt component to eliminte terms involving H to give nother eqution which is useful; ä 4 G N ( +P ) (0) Note tht this eqution is independent of k. An importnt point is tht (due to the Binchi identities) this eqution is equivlent to the Friedmnn eqution nd totl mtter conservtion eqution. Thus it contins no new informtion, but nicely summrises how ä behves.

33 Accelertion nd expnsion Cosmologicl expnsion is chrcterised by ȧ>0. However, ccelertion is chrcterised by ä>0. We see for this to hold we require; +P<0 () where we note this is the totl energy density nd pressure. Suppose the stress tensor is dominted by perfect fluid with eqution of stte P w. In this cse ccelertion implies, w< () Hence we see cold or hot mtter (w 0, )givedecelertedexpnsion. However drk energy gives ccelertion. Note tht if >0thenifȧ>0tsometime,thentheuniverseremins expnding unless k (the closed cse). In the cse k 0, thisis decelerted or ccelerted expnsion depending on the sign of ä. In the closed cse k,ifwehveä<0thenifthesclefctorrechesrdius such tht ȧ 0so, 8 G () then fter this point ȧ<0, nd the universe will recollpse to big crunch..9 Simple cosmologicl solutions Consider the flt cse k 0withsingleperfectfluidP w for constnt w with w >. Then from conservtion we lredy sw; o (+w) o (4) Putting this into the Friedmnn eqution; ȧ 8 G N o o (+w) H o Hence (tking the positive root for n expnding universe); +w o (+w) (5) ȧ H o (+w) o (6)

34 nd integrting, ( + w) (+w) H o (+w) o (t k) (7) for constnt of integrtion k 0. Wenowseewhywerestrictedtow>. We conventionlly fix this by tking the big bng 0 to be t time t 0, so, ( + w) (+w) o H o t (8) Then the density goes s, o (+w)! (+w) ( + w) (+w) H o t o (9) ( + w) H o t Note then tht for ll vlues of w we hve t. Hubble function The Hubble function is; H ȧ ( + w) t Null geodesics Null geodesics re governed by the eqution (recll k 0); (40) where, R Z to t e dt (t) Z to t e dt t Z ( + w) (+w) t o H o dt t o t e (+w) (4) ( + w) i (+w) ht +w to (+w) (4) +w t e Lower limit: For the lower limit to converge for t e! 0werequire; 0 < +w ( + w) ) w> 4 or w< (4)

35 Thus for non-ccelerting mtter <wprecisely gives finite lower limit. For ccelerting mtter in the rnge pple w< it does not. (The rnge w< isunlikelytobephysicllyrelevnt). Upper limit: For the upper limit to converge for t o!we require the opposite; +w ( + w) < 0 ) <w< (44) Thus this requires ccelerted expnsion. Prticle horizon In such model the prticle horizon size (recll k 0)ttimet t o is; Z to d H (t o ) (t o ) 0 dt Z ( + w) (t) (+w) t o H o dt t 0 (+w) (45) Hence it is finite for non-ccelerting mtter with <wsuch s dust w 0 nd rdition w /. Then the lower limit gives zero, nd so, d H (t o ) ( + w) (+w) ( + w) H o +w (t o) +w ( + w) ( + w) H o +w +w (+w) (46) H o reclling tht H o (+w) t o. Event horizon In this model the event horizon size (recll k 0)ttimet t o is; Z dt Z ( + w) d E (t) (t o ) t o (t) (+w) H o dt t t o (+w) (47) 5

36 Thus we see for <w< tht the upper limit gives zero nd so up to sign we get the sme s for the prticle horizon bove, d E (t) ( + w) (+w) ( + w) i H o ht +w (+w) +w t o ( + w) ( + w) H o +w +w where the quntity is positive due to the rnge of w. (48) H o Cold mtter nd k 0 Consider our specil cse of cold mtter. Note tht this cse is clled the Einstein-de Sitter model (lthough it hs nothing to do with de Sitter spcetime). Then, o o (49) Then we hve decelerted expnsion with; o H ot, H t (50) Recll (s for ll w) then t.theprticlehorizonsizeis; but there is no event horizon. d H (t o ) H o (5) Hot mtter/rdition w / nd k 0 For rdition we hve, o 4 o (5) Agin this yields t. As for dust we hve decelerted expnsion; (H o t), H o t 6 (5)

37 nd the prticle horizon is; d H (t o ) H o (54) nd gin there is no event horizon without ccelertion. Specil cse: Cosmologicl constnt w nd k 0 This is n exmple of de Sitter spcetime. Then s we sw erlier the energy density is constnt, o 8 G N (55) with constnt inverse length squred. Let us ssume tht > 0- positive cosmologicl constnt, sothttheenergydensity P>0. This negtive pressure leds to ccelerted expnsion of the spcetime. Then the Friedmnn eqution gives; H ȧ 8 G N o (56) so (tking the positive root); 0 ep (t to), H H o r (57) where the Hubble function is constnt. An importnt point is tht now! 0st!. So the singulrity or big bng is not t t 0,butis n infinite proper time in the pst. Forthisresonthereisnoprticle horizon; Z to Z dt to (t) dte p (t to) q h e p (t to) i to! (58) Another importnt feture is tht we see exponentil expnsion. So the 7

38 rte of ccelertion is very quick. This ccelertion leds to n event horizon; d E (t) (t o ) q r Z t o h e with the upper limit now giving zero. Z dt (t) dt e t o p i (t to) t o p (t to) (59).0 Some other cosmologicl solutions More generlly the de Sitter solutions re simple to write down for k ±, 0 nd > 0. They solve the eqution, Let us define; R µ g µ R + g µ 0 ) R µ g µ (60) H o r (6) s bove. Flt slicing of de Sitter, k 0: we hve lredy seen this; ds dt + e Hot ijdx i dx j (6) where! 0st!. Globl de Sitter, k +: ds dt + H o cosh H o td k+ (6) Note tht in this cse we hve no 0 bigbng,ndfort<0wehve contrcting universe! 8

39 Hyperbolic slicing of de Sitter, k : ds dt + H o sinh H o td k (64) where! 0st! 0. In fct ll these re coordintes on the sme spcetime, with the globl k +csecoveringthewhole of de Sitter, nd the others only covering portions of it. Thus in fct for k 0ndk, then! 0isn trelly the big bng t ll but rther just coordinte singulrity. Mtter plus rdition (k 0): Consider now gin the cse k 0with both mtter nd rdition perfect fluid. Now, ȧ 8 G N m + r (65) nd thus, Then, so, ȧ A + B, A 8 G N m o, p d dt ) t t k A + B p A (t t k ) r m o 6 G B 8 G N B A r 4 o (66) r + B A (67) (t) r r 0 (t)+ r 0 (68) m m Note we my set the big bng to t 0bychoosingnppropritet k ; p 6 Gt p m o (t) r r 0 (t)+ r 0 + / r o (69) m m m Note tht this is not strightforwrd to invert to obtin (t). Note tht setting r 0werecovert / (i.e.. t / ). Note tht tking m! 0 is possible - divergence must be bsorbed into t k -ndthenoneisleftwith 9

40 t (ie. t / ). Mtter plus (k 0): One cn lso solve for mtter plus cosmologicl constnt. Then, ȧ 8 G N 0 + m (70) nd so, 8 G N dt r d + m o h p ln + p p i + m o (7). Generl cosmologies Let us use the nottion tht t o is the time tody, with H o nd o being the vlues of H(t o )nd tot (t o )tody.wethendefinethecriticl density; crit 8 G N H o (7) This is the mtter density flt (k 0)wouldhveinordertoreproduce the expnsion rte H o.experimentllywefind, where we recll h 0.7. crit ' h kg m proton/m (7) We my write the Friedmnn eqution tody s; Hence we see; crit o 8 G N k (74) By construction for flt universe o crit Acloseduniverseisover dense: o > crit An open universe is under dense: o < crit 40

41 Now consider n FRW universe with mtter given by cosmologicl constnt, cold mtter M nd hot mtter/rdition R,llofwhichrenot intercting with the other components. Then we write, 8 G N 0 M M,o 0 4 R R,o (75) for mtter nd rdition densities M,o, R,o tody. Then we hve, H + k 8 G N M,o + R,o Now let us use the criticl density to rewrite this expression by defining; (76) nd we lso define, i i,o crit, 8 G N crit H o (77) 8 G N crit k H o k k o (78) These i for i,m,r,k give the frction of the energy density t t t o in component compred to the energy density crit. Note tht for k 0 then i give the frction in component compred to the totl energy density. For k 6 0weshouldthinkof crit s the e ective totl energy density including thinking of the sptil curvture s mtter component. Then we my write, H 8 G N 0 crit + k + M R (79) nd so the Friedmnn eqution (note we hve lso used ll the fluid equtions to get here) tkes the simple form; H H o + k 0 + M R (80)

42 nd this reltion completely determines the dynmics of the scle fctor. Note tht for t t o we obtin the importnt reltion; X i i + k + M + R (8) An importnt comment; clerly for very smll, rdition (if present) lwys domintes. At lte times if!then (if present) domintes. In tht cse then; H! p H 0 (8) Note tht for our universe we believe 0.7. Hence H, whichtodyis H o,isveryclosetoitsfinlvlue,soh! 0.8H 0. To obtin the dynmics we tke the positive root (for n expnding universe) nd integrte; t Z (t) d q H o R (8) where we ssume tht we hve big bng 0tt 0. Amorephysicl prmeteriztion is in terms of redshift. Define; x(t) (t) o +Z(t) (84) Note then tht; t 0,0 ) x 0 t t o, o ) x (85) Then, t H o Z +Z(t) nd the ge of the universe tody is; 0 t o H o Z 0 dx x p + k x + M x + R x 4 (86) dx x p + k x + M x + R x 4 (87) 4

43 Consider null geodesic received t t t o nd r 0,emittedttimet nd rdius R. Recllourusulreltion; 8 Z to < Now, t dt 0 (t 0 ) F k(r), F k (R) F k (R) Z o (t) o H o dt d d Z +Z Anicewytowritethisis; " p k Z o R(Z) p sinh H o k where we recll k k/( oh o ). Z o (t) : R k 0 sin (R) k sinh (R) k d H() (88) dx x p + k x + M x + R x 4 (89) +Z dx x p + k x + M x + R x 4 # (90). The CDM model We now consider the CDM universe which is our stndrd model of cosmology. It is universe tht is strted by infltion, goes through period of hot big bng, nd t lte times is dominted by Cold Drk Mtter (CDM) together with conventionl mtter, nd cosmologicl constnt. At lte times in the CDM universe we ssume tht the universe hs expnded to the point where rdition is irrelevnt, so r. In fct s we discuss lter for our universe r 0 5.Thusweneglectit. We lso ssume tht k. Note tht this does not strictly require k 0, but just tht the size of the sptil sections re su ciently lrge tht ny sptil curvture is irrelevnt. This is justified by the theory of infltion we will discuss lter. 4

44 Now reclling tht we hve the constrint (setting k r 0); + m (9) then t lte times in the universe (ie. now) there re just two prmeters to mesure. Firstly H o,ndsecondlysy m,thefrctionofenergydensityin mtter reltive to the cosmologicl constnt. Recll the ngulr dimeter nd luminosity distnces for n object t redshift Z re; d A ( + Z) o R(Z), d L (+Z) o R(Z) (9) The most elegnt determintion of the CDM prmeters is from supernove. Their redshift is directly mesured. Certin (Type IIA) supernov re believed to be stndrd cndles. Once their light curves hve been mesured, we believe we cn infer their totl luminosity L. Then we my determine their luminosity distnce from observtion of pprent luminosity `, s` L/(4 d L ). Then, d L (Z) (+Z) o R(Z) ( + Z) H o Z +Z dx x p + M x (9) give the reltion between mesured luminosity distnce nd redshift. Note tht this integrl cn be computed in closed form - it is given in terms of n elliptic integrl. However the nlytic expression is not terribly useful. After observing lots of supernove, one fits the dt to this reltion to determine the best fit of the prmeters H o nd m.onefmouslyfinds, h ' 0.7, H o 00hkms Mpc m ' 0. ) ' 0.7 (94) Hence there is positive cosmologicl constnt (drk energy). 44

45 Current stte of the rt mesurements re from Plnck stellite from full CMB nlysis As you cn see these gree nicely! h ' 0.67 ± 0.05 ' ± 0.0 (95) It is gret mystery why the cosmologicl constnt is so smll (ccording to ny QFT clcultion it should be enormous!), nd yet is of order the energy density in mtter tody. No one knows the nswers to these questions. The ide tht drk energy my be dynmic (so clled quintessence) rther thn cosmologicl constnt my try to explin this, but the rel issues I believe should be ddressed by quntum grvity. There re two importnt epochs; Accelertion: Recll tht, ä 4 G ( +P ) 4 G + m,0 ( + Z) (96) Hence the redshift Z cc t which ccelertion begn in the lte universe ws, m ( + Z cc ) ) Z cc 0.67 (97) which is round 6. Gyr go. -mtter equlity: Defined s the time or redshift Z m when the energy densities in the two components (which drive the Friedmnn eqution) re equl. Hence. m ( + Z meq ) ) Z meq 0. (98) which is round.6 Gyr go. 45

46 . Drk mtter The m 0. mttercomponentisthetotlcoldmtter. Thisistheusul bryonic mtter - strs nd gs - but lso drk mtter. There re vrious techniques in stronomy to mesure the mount of bryonic mtter in the universe. However there re lso methods to mesure the totl mss tht grvittes in system - for exmple wek lensing. The mss of glxies ssessed by directly ccounting for the bryonic mtter, or looking t the totl grvitting mss is considerbly di erent. Likewise the glctic dynmics tht would follow only form the observed bryonic mtter doesn t reproduce observed dynmics t ll. The impliction is tht cold mtter is ctully composed not only of bryons but lso non-luminous drk mtter. If we split the cold mtter into drk mtter nd bryonic component, so, then di erent methods gree tht, m DM + B (99) B m 0.5 (00) ie. round only 5% of cold mtter is bryonic. The rest is drk mtter. Since m 0. (Plnck) then, B 0.05, DM 0.7 (0) 46