4 The dynamical FRW universe

Transcription

1 4 The dynmicl FRW universe 4.1 The Einstein equtions Einstein s equtions G µν = T µν (7) relte the expnsion rte (t) to energy distribution in the universe. On the left hnd side is the Einstein tensor which cn be determined from the metric, nd is defined s G µν = R µν 1 Rg µν. (7) where R µν nd R re the Ricci tensor nd Ricci sclr respectively. On the right side of the eqution is Newton s constnt G N nd the energy momentum tensor of mtter, T µν. The component T 00 tells us bout the energy density, while T ij is relted to the flow of mteril in the j direction with nonzero momentum pointing in the i direction. For the FRW metric one finds R 00 = ä, R ij = [ ä + ) ] + k g ij, R 0i = 0, (74) where g ij is the sptil prt of the FRW metric. Noting tht gj i = δ ij, it is esy to construct the Einstein tensor with one upper nd one lower index, [ ) ] G 0 0 = + k [ ) ] G i j = ä + + k δ ij. (75) Our isotropy nd homogeneity ssumptions for the metric re justified if the energymomentum tensor T ν µ hs similr structure: independent of sptil coordintes, nd with Tj i δ ij with Ti 0 = T0 i = 0. Throughout this course we will be treting mtter to be close to n idel fluid with energy density ρ nd pressure p, for which ρ T µ ν = p p (76) p Einstein s equtions then red: ä + ) + k = ρ, (77) ) + k = p. (78) It is useful to mnipulte the equtions to simplify the second one. By subtrcting eq. (77) from eq. (78) we get n expression fro the cosmic ccelertion: ä = 4πG N (ρ + p). (79) 0

2 It my seem little disturbing to find two differentil equtions to solve for one function, (t). However, we cn mnipulte our equtions gin to show tht one combintion of the two Einstein equtions is simply equivlent to energy conservtion. If we differentite eq. (77) with respect to time, nd then mke use of both equtions we find or ) [ ρ = ä = ) k ] ) (ρ + p), (80) ) ρ = (ρ + p). (81) Note tht this eqution hs no fctor of G N, nd so is not telling us bout grvity. In fct it cn be rewritten s d ( ρ ) = p d (8) dt dt which is the energy conservtion eqution de = p dv (8) since smll cube of co-moving spce hs proper volume dv = (r sin θ/ 1 kr )drdθdφ. For most pplictions it is convenient to use the Friedmnn eqution eq. (77) s our first eqution, nd then either the eqution for the ccelertion eq. (79) or for energy conservtion eq. (81) s our second independent eqution Summry If we define then our two equtions to solve re: ) H =, (84) H + k = ρ, Friedmnn eqution ρ = H(ρ + p), Energy conservtion (85) 4. Solutions for the eqution of stte p = wρ To proceed, we need to know the eqution of stte which reltes ρ nd p. We will ssume for now tht the mtter of the universe is single mteril which stisfies Vrious cses of prticulr interest re: p = wρ. (86) Non-reltivistic, pressureless mtter: w = 0. If the universe is dominted by dilute gs of non-reltivistic prticles, or less dilute gs of wekly intercting prticles, then its pressure is negligible, nd we hve w = 0. 1

3 Highly reltivistic prticles: w = 1/. When the universe is dominted by prticles whose energy is so much greter thn their mss tht they cn be treted s mssless, then one finds w = 1/. Note tht for this specil vlue, the energy momentum tensor eq. (76) is trceless, T µ µ = 0. There is reson for this: if ll your prticles re mssless, then they move t the speed of light nd there is no specil frme (unlike for mssive prticles, where there is rest frme). But T µ µ hs dimensions of mss, nd is invrint under Lorentz trnsformtions. The only dimensionful quntity round is the energy of the prticles, but tht is not Lorentz invrint, so T µ µ must vnish. Cosmologicl constnt (vcuum energy): w = 1. In non-grvittionl physics one cn lwys dd constnt to the Lgrngin without consequence. It shifts the vlue of the energy, but since only energy differences re mesured, there is no innte definition of zero energy. However, grvity couples to energy, nd shifting the Lgrngin by constnt shifts T µ ν. The constnt term is clled the cosmologicl constnt; it is divergent quntity in quntum field theory, nd so nively one would think it should be very big, but s we will see, observtions of our cosmos tell us it is very smll. Since dding constnt to the Lgrngin is Lorentz invrint, it follows tht the chnge in T µ ν must be proportionl to g µ ν δ µν. Thus it must be tht such form of mtter must hve p = ρ, or w = 1. Note tht when w < 0, the pressure is negtive, which implies tension...the mteril would wnt to contrct in the bsence of grvity. Mtter with w = 1/. Although there is no reson to expect mtter with w = 1/, it is interesting to note from eq. (79) tht for w < 1/, the expnsion of the universe is ccelerting (ä > 0) while for w > 1/, it is decelerting (ä < 0). From the eqution for energy conservtion eq. (85) we find ρ = H(1 + w)ρ or dρ ρ = (1 + w)d (87) with solution ρ (1+w). (88) Note tht non-reltivistic mtter hs ρ, which ccounts for dilution of the prticles dues to the expnsion of the volume they re in; for reltivistic mtter, ρ 4, where the extr fctor of 1 ccounts for the redshift of the energy; nd cosmologicl constnt hs n energy density tht remins constnt s the universe expnds, where the increse of energy comes from the universe expnding ginst the negtive pressure. Next we look t the Friedmnn eqution, eq. (85). A sttic solution is possible with H = 0 provided tht k 0 nd k = ρ. (89) This solution ppeled to Einstein on philosophicl grounds before he lerned of Hubble s discovery of cosmologicl expnsion. It is not stble configurtion smll perturbtions bout this solution cuse it to collpse or expnd. If H 0 then we cn rewrite the Friedmnn eqution s k = H (Ω 1), (90)

4 where Ω(t) = ρ(t) ρ crit (t), ρ crit(t) H (t) (91) ρ c ρ crit (t 0 ) = H 0 = 1.88h 10 9 gm cm = h GeV cm. In principle then we see how to tell wht k is: we mesure the totl energy density ρ of the universe, we determine ρ c by mesuring the Hubble constnt nd then by constructing the rtio we find Ω. If Ω > 1 we hve k = 1, nd if Ω < 1 then k = 1, while if Ω = 1, k = 0. Of course, if 1, the right hnd side my be close to zero, even if k 0; this is the ide behind the theory of infltion. As we will see lter, we hve evidence tht Ω is very close or equl to one. Tht mens tht the curvture term in our universe is negligible. Now suppose the universe contins ll three types of energy: reltivistic (ρ R ), nonreltivistic (ρ M, M is for mtter ), nd cosmologicl constnt (ρ Λ ); we hve seen tht these scle s 4,, nd 0 respectively. Then the totl energy density, s function of the scle fctor, is given by ρ = ρ R + ρ M + ρ Λ ( ( 0 ) 4 ( 0 ) ) = ρ c Ω R + ΩM + ΩΛ. (9) We see tht the erly universe (smll ) will be rdition dominted, while the lte universe is dominted by the cosmologicl constnt, if nonzero, or mtter if it is. Everything with the subscript 0 on ρ nd H refers to tody s vlue; the terms Ω M, Ω R nd Ω Λ lwys refer to tody s vlues. Using this formul we cn find useful formul for the evolution of Ω. From eq. (89) we hve We cn rewrite H H0 Plugging the bove result into the previous one, we get Ω 1 = (Ω 0 1) H0. (9) 0H = ρ crit(t) = ρ crit(t) ρ ρ c ρ ρ c = 1 ( ( 0 ) 4 ( 0 ) ) Ω R + ΩM + ΩΛ. (94) Ω Ω 0 1 Ω 1 = ( 1 Ω 0 + Ω 0 ) ( R + 0 ) ( ). (95) ΩM + ΩΛ 0 We see tht in the erly universe (Ω 1) (Ω 0 1)(/ 0 ), so tht if Ω 0 1 then Ω must hve stisfied Ω = 1 to high ccurcy in the erly universe, nd will evolve to Ω = 0 in the fr future if Ω Λ = 0. In contrst, if Ω 0 > 1, then we must lso hve hd Ω = 1 to high ccurcy in the erly universe, but in the future Ω will grow nd eventully diverge. If we set k = 0, it is simple to solve for for the evolution of the universe for simple eqution of stte p = wρ. By tking the time derivtive of the Friedmnn eqution we get HḢ = ρ = [ H(1 + w)ρ] = H (1 + w). (96)

5 Thus with solution dh + w) = (1 dt, (97) H H = (1 + w)t. (98) I hve chosen the integrtion constnt such tht H = t t = 0. Writing H = ȧ/ nd integrting gin, we find { 0 e ±λt w = 1, (t) = 0 (t/t 0 ) /(1+w) (99) w 1 4