Cosmology: Part II. Asaf Pe er Asymptotic behavior of the universe. (t) 1 kr 2 +r2 (dθ 2 +sin 2 θdφ 2 )

Transcription

1 Cosmology: Prt II Asf Pe er 1 Mrch 18, 2014 This prt of the course is bsed on Refs. [1] - [4]. 1. Asymptotic behvior of the universe The Robertson-Wlker metric, [ ] dr ds 2 = dt (t) 1 kr 2 +r2 (dθ 2 +sin 2 θdφ 2 ). (1) is the most generl metric tht describes universe which is sptilly homogeneous nd isotropic, nmely its sptil term is mximlly symmetric. The curvture constnt k cn get three vlues, k = 1, 0, +1 describing n open, flt nd closed universes, respectively. The scle fctor (t) is obtined by using this metric in Einstein s eqution, The results re two equtions, nd (ȧ G µν = 8πGT µν Λg µν. (2) ä = 4πG (ρ+3p), (3) 3 ) 2 = 8πG 3 ρ k 2, (4) which re known together s Friedmnn Equtions. These equtions describe the evolution of the scle fctor, hence of the universe s whole The Big Bng While it is possible to solve the Friedmnn equtions exctly in vrious simple cses, it is often more useful to know the qulittive behvior of vrious possibilities. 1 Physics Dep., University College Cork

2 2 Consider first universe with no cosmologicl constnt: Λ = 0. Consider the behvior of universes filled with fluids of positive energy (ρ > 0) nd nonnegtive pressure (p 0). By the first of Friedmnn s equtions (Eqution 3) we must hve ä < 0. Since we know from observtions of distnt glxies tht the universe is expnding (ȧ > 0), this mens tht the universe is decelerting nmely, the expnsion rte is decresing. This is wht we should expect, since the grvittionl ttrction of the mtter in the universe works ginst the expnsion. The fct tht the universe cn only decelerte mens tht it must hve been expnding even fster in the pst; if we trce the evolution bckwrds in time, we necessrily rech singulrity t = 0. Notice tht if ä were exctly zero, (t) would be stright line, nd the ge of the universe would be H0 1. Since ä is ctully negtive, the universe must be somewht younger thn tht. This is demonstrted in Figure 1. Fig. 1. In universe with no cosmologicl constnt, we know tht it is expnding (ȧ > 0) nd decelerting (ä < 0). Thus, there must hve been point in the pst when = 0. This is the Big Bng. This singulrity t = 0 is the Big Bng. It represents the cretion of the universe from singulr stte, not explosion of mtter into pre-existing spcetime. It might be hoped tht the perfect symmetry of our FRW universes ws responsible for this singulrity, but in fct it s not true; the singulrity theorems predict tht ny universe withρ > 0ndp 0musthvebeguntsingulrity. Ofcoursetheenergydensitybecomes rbitrrily high s 0, nd we don t expect clssicl generl reltivity to be n ccurte description of nture in this regime; hopefully consistent theory of quntum grvity will be ble to fix things up.

3 Future evolution: open nd flt universes The future evolution is different for different vlues of k. For the open nd flt cses, k 0, The second of Friedmnn s equtions, Eqution 4 implies ȧ 2 = 8πG 3 ρ2 + k. (5) The right hnd side is strictly positive (since we re ssuming ρ > 0), so ȧ never psses through zero. Since we know tht tody ȧ > 0, it must be positive t ll time. Thus, the open nd flt universes expnd forever they re temporlly s well s sptilly open. (Plese keep in mind wht ssumptions go into this nmely, tht there is nonzero positive energy density, ρ > 0. Negtive energy density universes do not hve to expnd forever, even if they re open.) How fst do these universes keep expnding? Consider the quntity ρ 3 (which is constnt in mtter-dominted universes). Recll tht we wrote the energy conservtion eqution s 0 = µ T µ 0 = 0 ρ 3ȧ (ρ+p). (6) By simple lgebr, we cn write this eqution in the form The right hnd side is either zero or negtive; therefore d dt (ρ3 ) = 3 2 ȧp. (7) d dt (ρ3 ) 0. (8) Thus, in n ever-expnding universe, where Eqution 8 implies tht ρ 2 must go to zero in the limit. From Eqution 5 we thus find tht in this limit ȧ 2 k. (9) (Remember tht this is true for k 0.) Thus, for k = 1 the expnsion pproches the limiting vlue ȧ 1, while for k = 0 the universe keeps expnding, but more nd more slowly Future evolution: closed universes For the closed universes (k = +1), Eqution 4 becomes ȧ 2 = 8πG 3 ρ2 1. (10)

4 4 The rgument tht ρ 2 0 s still pplies; but in tht cse, the right hnd side of Eqution 10 would become negtive, which cn t hppen. Therefore the universe does not expnd indefinitely; possesses n upper bound mx. As pproches mx, the first of Friedmnn s equtions (Eqution 3) implies ä 4πG 3 (ρ+3p) mx < 0. (11) Thusäisfinitendnegtivetthispoint, soreches mx ndstrtsdecresing, whereupon (since ä < 0) it will inevitbly continue to contrct to zero the Big Crunch. Thus, the closed universes (gin, under our ssumptions of positive ρ nd nonnegtive p) re closed in time s well s spce (see Figure 2). Fig. 2. The fte of the universe depends on its curvture. A close universe (k = 1) will end in crunch, while n open or flt universes (k = 1,0) will expnd forever. 2. The cosmologicl redshift Becuse of the time dependence of the scle fctor (t), the FRW metric is not sttic. Since (t) multiplies the sptil coordintes, ny proper distnce l(t) will chnge with time in proportion to (t).: l(t) = l 0 (t) (t). (12) In prticulr, the proper seprtion between ny two observers, locted t constnt comoving coordintes, will chnge with time. Let the coordinte seprtion between two such observers

5 5 (lets sy, locted t nerby glxies) be δr, so tht the proper seprtion is δl = (t)δr. Ech of the two observers will ttribute to the other velocity δv = d (ȧ ) dt δl = ȧδr = δl (13) This leds to severl importnt physicl consequences of rther generic nture. Consider nrrow pencil of electromgnetic rdition which crosses ny two comoving observers, seprted by proper distnce δl. The trnsit time is δt = δl/c. Let the frequency of rdition mesured by the first observer be ω. Since the first observer sees the second one receding with velocity δv, he expected the second observer to mesure Doppler shifted frequency (ω +δω) where δω ω = δv c = (ȧ ) δl c = (ȧ ) δt = δ (we hve ssumed tht the two observers re seprted by infinitesiml distnce of first order δl, therefore we could introduced loclly inertil frme encompssing both observers; the lws of specil reltivity cn be pplied in this frme). Eqution 14 cn be integrted to give (14) ω(t)(t) = constnt. (15) In other words, the frequency of of electromgnetic rdition chnges due to the expnsion of the universe ccording to the lw ω 1. Note tht we hve mde implicit use of the homogeneity of the spcetime in extending the locl result to globl context. Thus, photon emitted with frequency ω 1 will be observed t some lter time with lower frequency ω 0 s the universe expnds: ω 0 ω 1 = 1 0. (16) Cosmologists like to spek of this in terms of the redshift z between the two events, defined by the frctionl chnge in wvelength: z = λ 0 λ 1 λ 1 = NoticethtthisredshiftisnotthesmestheconventionlDopplereffect; itistheexpnsion of spce, not the reltive velocities of the observer nd emitter, which leds to the redshift. The redshift is something we cn mesure; we know the rest-frme wvelengths of vrious spectrl lines in the rdition from distnt glxies, so we cn tell how much their wvelengths (17)

6 6 hve chnged long the pth from time t 1 when they were emitted to time t 0 when they were observed. We therefore know the rtio of the scle fctors t these two times. But we don t know the times themselves; the photons re not clever enough to tell us how much coordinte time hs elpsed on their journey. We hve to work hrder to extrct this informtion. However, using the definition of the redshift in Eqution 17 with 0 (t = t 0 ) is the scle fctor tody, the redshift z is often used to replce the time coordinte t or the vlue of the scling fctor (t) t tht time. The sme rgument holds when considering the motion of mssive prticles. Consider gin two comoving observers seprted by proper distnce δl. Let mssive prticle pss the first observer with velocity v. When the prticle crossed the proper distnce δl (in time intervl δt), it pssed the second observer, whose velocity (reltive to the first one) is δu = ȧ δl = ȧ vδt = vδ The second observer will ttribute to the prticle the velocity (18) v = v δu 1 vδu = v (1 v2 )δu+o[(δu) 2 ] = v (1 v 2 )v δ, (19) where we hve used the specil-reltivistic formul for ddition of velocities, which is vlid in the infinitesiml regime round the first observer. Rewriting this eqution s δv = v(1 v 2 ) δ (20) nd integrting, we get p = v 1 v 2 = constnt Inotherwords, themgnitudeofthe3-momentumdecresess 1 duetotheexpnsion. For non-reltivistic prticles, v p nd the velocity itself decys s 1. The prticle therefore slows down with respect to the comoving coordintes s the universe expnds. This is n ctul slowing down, in the sense tht gs of prticles with initilly high reltive velocities will cool down s the universe expnds. (21) 3. Distnce mesurement in the universe Roughly speking, since photon moves t the speed of light its trvel time should simply be its distnce. But wht is the distnce of fr wy glxy in n expnding universe? The comoving distnce is not especilly useful, since it is not mesurble, nd

7 7 furthermore becuse the glxies need not be comoving in generl. Insted we cn define the luminosity distnce s d 2 L = L 4πF, (22) where L is the bsolute luminosity of the source nd F is the flux mesured by the observer (the energy per unit time per unit re of some detector). The definition comes from the fct tht in flt spce, for source t distnce d the flux over the luminosity is just one over the re of sphere centered round the source, F/L = 1/A(d) = 1/4πd 2. In n FRW universe, however, the flux will be diluted. Conservtion of photons tells us tht the totl number of photons emitted by the source will eventully pss through sphere t comoving distnce r from the emitter. Such sphere is t physicl distnce d = 0 r, where 0 is the scle fctor when the photons re observed. But the flux is diluted by two dditionl effects: the individul photons redshift by fctor (1 + z) (=their energy is decresed), nd the photons hit the sphere less frequently, since two photons emitted time δt prt will be mesured t time (1 + z)δt prt. Therefore we will hve F L = 1 4π 2 0r 2 (1+z), 2 (23) or d L = 0 r(1+z). (24) The luminosity distnce d L is something we might hope to mesure, since there re some strophysicl sources whose bsolute luminosities re known ( stndrd cndles ). But r is not observble, so we hve to remove tht from our eqution. Before we do tht, we note tht nother observble prmeter for distnt sources is the ngulr dimeter. Consider distnt object of physicl size D (sy, distnt glxy) which emits photons t comoving time t 1. These photons re observed t time t 0. Assume tht the object subtends n ngle δ to the observer, then, for smll δ, we hve D = r(t 1 )δ. The ngulr dimeter distnce, d A (z) for the source is defined vi the reltion δ = (D/d A ); so, we find tht d A (z) = r(t 1 ) = 0 r(t 1 )(1+z) 1. (25) Clerly, d L = (1+z) 2 d A. Letusreturntotheproblemofexpressingd L (ndd A )intermsofmesurblequntities, nmely tht do not contin r, which is not mesurble. From the FRW metric on null geodesic (chosen to be rdil for convenience) we hve 0 = ds 2 = dt kr 2dr2, (26)

8 8 or t0 t 1 dt r (t) = dr. (27) 0 (1 kr 2 ) 1/2 For glxies not too fr wy, we cn expnd the scle fctor in Tylor series bout its present vlue: (t 1 ) = 0 +(ȧ) 0 (t 1 t 0 )+ 1(ä) 2 0(t 1 t 0 ) [ (ȧ ) = 0 1+ (t 0 1 t 0 )+ 1 (ä 2 )0 (t 1 t 0 ) ] [ = 0 1+H0 (t 1 t 0 ) 1q 2 0H0(t 2 1 t 0 ) ], (28) where we hve used the definition of Hubble s constnt nd the decelertion prmeter, (ȧ ) ( ) ä H 0 ; q 0. (29) t=t 0 ȧ 2 t=t 0 We cn now use Tylor expnsion in both sides of Eqution 27 (Keeping only the leding term in the right hnd side), to find [ r = 1 0 (t 0 t 1 )+ 1 ] 2 H 0(t 0 t 1 ) (30) Reclling tht (1+z) = 0 /(t 1 ), Eqution 28 tkes the form 1 1+z = 1+H 0(t 1 t 0 ) 1 2 q 0H 2 0(t 1 t 0 ) (31) For smll H 0 (t 1 t 0 ) this cn be inverted to yield [ ( t 0 t 1 = H0 1 z 1+ q 0 2 Substituting this bck gin into Eqution 30 gives r = 1 0 H 0 ) ] z [ z 1 2 (1+q 0)z ]. (32). (33) Finlly, using this in Eqution 24 yields Hubble s Lw: [ d L = H0 1 z + 1 ] 2 (1 q 0)z (34) Therefore, mesurement of the luminosity distnces nd redshifts of sufficient number of glxies llows us to determine H 0 nd q 0, nd therefore tkes us long wy to deciding wht kind of FRW universe we live in.

9 9 4. Evolution of the scle fctor (t) We cn solve Friedmnn s Eqution (Eqution 4) nd obtin the evolution of (t) in the vrious scenrios. Since the universe is composed of mtter, rdition nd vcuum energy, the evolution depends on nswering two questions: (i) whether the universe is flt, open or closed; nd (ii) wht is the dominnt energy content of the universe. In Cosmology prt I, we showed tht for ll three ingredients (mtter, rdition nd vcuum energy), we cn write the eqution of stte in the form p = ωρ, from which the conservtion of energy becomes ρ m 3 [ω = 0] p = ωρ ρ 3(1+ω) ρ r 4 [ω = 1/3] ρ Λ 0 [ω = 1] Where ρ r is the energy density in both rdition nd reltivistic mtter, ρ m is the energy density in non-reltivistic mtter nd ρ Λ is the vcuum energy density. Thus, ρ m (t) = ρ m (t 0 ) (35) ( 0 ) 3 = ρc Ω m (1+z) 3 (36) where Ω m ρ m,0 /ρ c ; ρ c = ρ crit,0 = 3H 2 0/8πG is the criticl density; nd the subscript 0 represent present time vlues. nd Similrly, ρ r (t) = ρ r (t 0 ) ( 0 ) 4 = ρc Ω r (1+z) 4, (37) ρ Λ (t) = ρ Λ (t 0 ). (38) Observtions suggest tht t present epoch Ω totl Ω 1; Ω m 0.3 Ω r (39) Thus, t present, mtter domintes over rdition (nd vcuum over both). But when looking into equtions 36, 37 nd 38, clerly when looking t the pst (z increses), rdition energy density grows fster thn mtter (nd vcuum) energy densities s we go to erlier phses of the universe. At some time, t = t eq in the pst (corresponding to vlue = eq nd redshift z = z eq ) the rdition nd mtter hve hd equl energy densities. From Equtions 36, 37 nd 39 we get (1+z eq ) = 0 eq = Ω m Ω r (40)

10 10 Since the temperture of the rdition grows s 1 (recll Stefn-Boltzmnn s lw, T 4 ρ r (1+z) 4 ), the temperture of the universe t this epoch ws T eq = T 0 (1+z eq ) = K = 0.9 ev (41) where T 0 = 2.7 K is the temperture of the CMB rdition. We cn now solve Friedmnn Eqution (Eqution 4) for vrious geometries nd contents of the universe. Let us focus on the flt universe (k = 0). For mtter dominted universe, we find ( ) 1/3 9 8πG (t) = 4 3 Ω mρ c 3 0 t 2/3, (42) while for rdition-dominted flt universe ( (t) = 4 8πG ) 1/4 3 Ω rρ c 4 0 t 1/2. (43) Clerly, t very erly times, t t eq, the energy density of the universe is dominted by rdition, while t lter times, t t eq it is dominted by mtter. Given tht we cn consider present dy epoch s being mtter-dominted, the Equilibrium time, t eq is esily obtined using Equtions 40 nd 42. For universe which is dominted by positive vcuum energy density, ρ Λ = Λ/8πG, the solution (for closed, flt nd open universes) is (Λ ) ] 1/2t sinh[ (k = 1) 3 [ (t) exp ± ( ) ] Λ 1/2t (k = 0) (44) 3 (Λ ) ] 1/2t cosh[ (k = +1) 3 All these solutions represent, in fct, the sme spce-time, just in different coordintes. This spcetime is known s de Sitter spce, which is mximlly symmetric. Solution lso exists with Λ < 0, which is known s nti de Sitter spce. 5. Therml history of the Universe Going bck in time, when the redshift ws higher thn z eq , the universe ws dominted by rdition. In the rdition dominted phse of the universe, its temperture ws T > T eq 1 ev, nd it chnged with time ccording to T 1 (1+z). At these times, the content of the universe ws different thn tody. Atomic nd nucler structure hve binding energies of the order of few tens ev nd MeV, respectively. Thus,

11 11 when the temperture of the universe ws higher thn these vlues, toms nd nucleons could not hve existed s bound objects. Going even further bck in time, when the temperture of the photons exceeded the rest mss of the electrons (m e 0.5 MeV, the photons energies ws high enough to produce lrge numbers of electrons nd positrons. These prticles hd the sme typicl temperture (T), mking them ultr-reltivistic. Thus, s function of temperture, T (or time, t), the universe would be populted by different types of elementry prticles. To work out the physicl processes t some time t, we need to know the distribution functions f A ( x, p,t) f A ( p,t) of these prticles. Here, A lbels the different species of prticles. The dependence of f A on x is excluded, becuse of the homogeneity of the universe. We cn get the distribution function f A ( p,t) by noting tht the the different species interct constntly through the vrious processes, scttering ech other nd exchnging energy nd momentum. As long s the rte of this rections Γ(t) is much higher thn the rte of expnsion of the universe, H(t) = ȧ/, these interctions produce nd mintin thermodynmic equilibrium, with ll intercting prticles hving the sme temperture, T(t). We cn thus ssume tht the prticles my be treted s idel Bose or Fermi gs, for which the distribution function is given by f A ( p,t)d 3 p = g A 1 (2π) 3 E p µ A e T A (t) ±1 d 3 p (45) where g A is the spin degenercy fctor of the species, µ A (T) is the chemicl potentil, E p = ( p 2 + m 2 ) 1/2 nd T A (t) is the temperture of the species t time t. The upper sign (+1) corresponds to Fermions, while the lower sign (-1) to Bosons. At ny instnt in time, the universe lso contins blck body distribution of photons, with temperture T γ (t). If prticulr species of prticles is coupled to the photons (nmely, Γ Aγ H), then these prticles will hve the sme temperture s the photons, T A = T γ. Since this is usully the cse, the photon temperture is often referred to s the temperture of the universe. As the universe expnds, its temperture decreses; furthermore, the number density of prticles decreses, nd so grdully prticles decouple from ech other. Once species A is completely decoupled, ll the prticles of tht species no longer interct (efficiently) with other prticles, nd they simply move long geodesics. After decoupling, the distribution functionfreezes; thus,denotingthedecouplingtimebyt = t D,where = D,thedistribution function t t > t D is given by ( f dec (p,t) = f,equi p (t) ) (46) D

12 12 where f equi is the distribution function just before decoupling, nd we considered the fct tht ll prticles with momentum p t time t must hve hd momentum p[(t)/ D ] t t D. Given the distribution function, the number density, n, energy density, ρ nd pressure, p re given by (omitting the subscript A nd the time dependence for clrity) nd n = ρ = p = f( k)d 3 k = g (2π 2 ) m Ef( k)d 3 k = g (2π 2 ) m 1 k 2 3 E f( k)d 3 k = g (6π 2 ) m (E 2 m 2 ) 1/2 EdE e E µ T ±1 (E 2 m 2 ) 1/2 E 2 de e E µ T ±1 (E 2 m 2 ) 1/2 de e E µ T ±1 The bove expressions for n, ρ nd p simplify considerbly in some limiting cses. When the prticles re highly reltivistic (T m) nd non-degenerte (T µ), we get: ρ g { E 3 de (2π 2 ) m e E T ±1 = gb (π 2 /30)T 4 (Bosons) 7 g 8 F(π 2 /30)T 4 (Fermions) We cn therefore express the totl energy density contributed by ll reltivistic species s ρ totl = ( ) π 2 g i Ti 4 + ( ) ( ) 7 π 2 π 30 8 g i Ti 4 2 = g totl T 4 (51) where bosons g g totl = bosons fermions ( ) 4 Ti g i + 7 T 8 fermions (47) (48) (49) (50) ( ) 4 Ti g i. (52) T Note tht we hve considered the possibility tht not ll the species hve the sme temperture. If ll the species hve the sme temperture, we hve g totl = g B g F. The pressure due to the reltivistic species is p ρ/3 = g(π 2 /90)T 4. The number density cn be found in similr wy: n g { E 2 de (ζ(3)/π 2 (2π 2 ) m e E T ±1 = )g B T 3 (Bosons) 3 g 4 F(ζ(3)/π 2 )g F T 3 (Fermions) (53) where ζ(3) is the Riemnn zet function of order 3. Combining equtions 50 nd 53, we find tht the men energy per prticle, E = ρ/n is 2.70T for Bosons, nd 3.15T for Fermions.

13 13 In the opposite limit, T m, the exponentil in eqution 45 is 1. In this limit, for both Bosons nd Fermions one gets ) 3/2 e m µ T (54) n g 2π 2 In this limit, ρ nm, nd p = nt ρ. 0 p 2 dpe m µ T e p2 2mT = g ( mt 2π Comprison of equtions 53 nd 54 shows tht the number (nd energy) density of nonreltivistic prticles re exponentilly dmped by fctor exp( m/t) with respect to tht of reltivistic prticles. Thus, t very erly times, t t < t eq., in the rdition dominted phse, we my ignore the contribution of non-reltivistic prticles to ρ. During the rdition dominte phse, (t) t 1/2, nd thus (ȧ ) 2 = H 2 (t) = 1 4t 2 = 8πG 3 ρ = 8πG 3 g ( π 2 30 ) T 4 (55) In terms of Plnck energy, m Pl = G 1/2 = GeV, ( ) T H(t) = 1.66g 1/2 2, (56) nd t 0.3g 1/2 ( mpl T 2 ) 1 sec m Pl ( ) 2 T g 1/2. (57) 1 MeV Remember tht the fctor g is these expressions counts the degrees of freedom of those prticles which re still reltivistic t the given temperture, T. As the temperture decreses, more nd more prticles become non reltivistic, nd g decreses; thus, g = g(t) is slowly vrying function of T: For T MeV, the only reltivistic prticles re the three neutrino species, nd the photons. Thus we hve 2 degrees of freedoms for the bosons - photons [polriztions], nd 6 degrees of freedom for the Fermions = neutrinos (3 neutrino + nti-neutrino species) ; since T ν = (4/11) 1/3 T γ, we hve g(t MeV) = 2+ 7 ( ) 4/ = 3.36 (58) 11 For MeV T 100MeV, the electron nd positron dd dditionl reltivistic degrees of freedom, (nd ech one with 2 spins, so totl ddition of 4 Fermionic degrees of freedom), nd one obtins g(mev T 100 MeV) = (59)

14 14 For T 300 GeV, ll species of the stndrd model re reltivistic, nd thus g(t 300 GeV) = (60) The key to understnd the therml history of the universe is the comprison of prticle interction rtes to the expnsion rte. So long s the interctions necessry for prticle distribution functions to djust to the chnging temperture re rpid compred to the expnsion rte, the universe will, to good pproximtion, evolve through succession of nerly therml sttes with temperture decresing like 1. We cn now describe the therml history of the erly universe. Note tht we cnnot go bckwrd beyond the Plnck epoch, t s, nd T GeV - the point t which quntum corrections to GR should render it invlid. At the erliest time, the universe ws plsm of reltivistic prticles, including qurks, leptons guge bosons nd Higgs bosons. If current ides re correct, number of spontneous symmetry breking (SSB) phse trnsition took plce during the course of the erly history of the universe. They perhps include the GUT phse trnsition t T GeV, nd the electrowek phse trnsition t T 300 GeV. During these SSB phse trnsitions, some guge bosons nd mtter prticles cquire mss through the Higgs mechnism nd the full symmetry is broken to lower symmetry. At temperture of bout T MeV (t 10 5 s), the universe undergoes trnsition ssocited with chirl symmetry breking nd color confinement, fter which strongly-intercting prticles confine into color singlets combintions - nmely, bryons nd mesons. The epoch of nucleosynthesis follows when t s (T MeV). At present 1, nucleosynthesis is the erliest test of stndrd cosmology. At time of bout s (T 1 ev), the energy density in mtter becomes equl to tht in rdition, nd the universe becomes mtter dominted. This further mrks the beginning of structure formtions. Finlly, t time of s, (T 0.1 ev), ions nd electrons combine to form toms: this is known s recombintion. When it hppens, mtter nd rdition decouple, ending the long epoch of therml equilibrium tht existed in the erly universe. The surfce of lst scttering for the microwve bckground rdition (CMB) is the universe itself t decoupling, which occurred t z dec 1180 (T dec 3220 K, or 0.28 ev), which occurred when the universe ws 378,000 yers old. 1 In Mrch 2014, BICEP2 showed the first evidence for Infltion

15 15 REFERENCES [1] T. Pdmnbhn, Structure Formtion in the Universe (Cmbridge), chpters 2 nd 3. [2] S. Crroll, Lecture Notes on Generl Reltivity, prt 8. Cosmology ( [3] S. Weinberg, Grvittion nd Cosmology: Principles nd Applictions of the Generl Theory of Reltivity (John Wiley & Sons), chpter 15. [4] J. Hrtle, Grvity: An Introduction to Einstein s Generl Reltivity (Addison-Wesley), chpters 17 nd 18.