4. Cosmic Dynmics: The Friedmnn Eqution Reding: Chpter 4 Newtonin Derivtion of the Friedmnn Eqution Consider n isolted sphere of rdius R s nd mss M s, in uniform, isotropic expnsion (Hubble flow). The eqution of motion for R s (t) cn be obtined from the grvittionl ccelertion t the outer edge of the sphere: d 2 R s dt 2 = GM s Rs(t) 2. Multiplying both sides by dr s /dt nd integrting converts this ccelertion eqution to n energy eqution : ( ) 2 1 drs = GM s 2 dt R s (t) +U. Mthemticlly, U is just constnt of integrtion, but physiclly it corresponds to the totl energy per unit mss t the surfce of the expnding sphere, i.e., the sum of the kinetic energy per unit mss nd the grvittionl potentil energy per unit mss. If U > 0, then the expnding sphere hs positive totl energy nd will expnd forever (the r.h.s. will lwys be positive). If U < 0, then the sphere hs negtive totl energy nd will eventully recollpse (the r.h.s. will eventully become zero). Write the rdius in the form R s (t) = (t)r s, where r s is the comoving rdius of the sphere, equl to the physicl rdius t the epoch when (t) = 1. With M s = 4π 3 ρ(t)r3 s(t), the energy eqution becomes Dividing ech side by r 2 s 2 /2 yields 1 2 r2 sȧ 2 = 4π 3 Gr2 sρ(t) 2 (t)+u. (ȧ ) 2 = 8πG 2U ρ(t)+ 3 rs 22 (t). Since ρ(t) 1/ 3 (t), we see tht if U is negtive the r.h.s. of this eqution will eventully hit zero, fter which the expnsion reverses. Although this derivtion describes n isolted sphere, Newton s iron-sphere theorem tells us tht it should lso describe ny sphericl volume of homogeneous nd isotropic universe, since the grvittionl effects of sphericlly symmetric externl mtter distribution cncel. A generliztion of the iron-sphere theorem turns out to hold in GR s well. 1
The Friedmnn Eqution in GR A proper derivtion of the Friedmnn eqution begins by inserting the Friedmnn-Robertson- Wlker metric into the Einstein Field Eqution. Since GR yields the Newtonin limit, we should expect the smll scle behvior to resemble tht of our Newtonin derivtion bove, nd it does, with two importnt chnges. First, the mss density ρ(t) is replced by the totl energy density ǫ(t)/c 2, which includes rest mss energy nd other forms of energy (e.g., energy of photons, or therml energy of toms). [In most texts, this totl energy density is just written s ρ(t) nd understood to include ll contributions, not just rest mss. I will try to stick with Ryden s more pedgogicl nottion here.] Second, the potentil energy term is intimtely tied to the curvture of spce. The GR form of the Friedmnn Eqution is (ȧ ) 2 = 8πG 3 ǫ(t) c 2 kc2 1 R0 2 2 (t), where R 0 is the present vlue of the curvture rdius nd k = +1, 0, or -1 is the curvture index in the FRW metric. While the precise form of the lst term is not obvious without GR derivtion, it mkes resonble sense tht positive spce curvture is ssocited with stronger grvity nd thus with negtive binding energy. If k 0, nd the energy density is positive, then the r.h.s. is lwys positive, nd n expnding universe continues to expnd forever. If mtter is the dominnt form of energy, then dilution implies ǫ(t) 1/ 3 (t). If k = +1, then the r.h.s. must eventully rech zero, fter which the expnsion will reverse. Thus, positive spce curvture corresponds to bound universe. However, form of energy for which ǫ(t) flls more slowly thn 1/ 2 (t), such s cosmologicl constnt, cn chnge this utomtic correspondence. The Criticl Density nd the Density Prmeter, Ω Substituting H(t) = ȧ/ llows us to write the Friedmnn eqution in terms of the Hubble prmeter, H 2 (t) = 8πG 3 ǫ(t) c kc2 1 2 R0 2 2 (t). From this eqution, we cn see tht spce is flt (k = 0) if the men density of the universe equls the criticl density ρ c (t) = ǫ c(t) c 2 = 3H2 (t) 8πG. The two cosmologicl equtions most worth memorizing re H = ȧ/ nd this definition of the criticl density. Together they re the Friedmnn eqution for flt universe. 2
The present dy vlue of the criticl density is ρ c,0 = ǫ c,0 c 2 = 3H2 0 8πG ( ) 2 = 9.2 10 30 gcm 3 H 0 70km s 1 Mpc 1 ( ) 2 = 1.4 10 11 M Mpc 3 H 0 70km s 1 Mpc 1. Cosmologists frequently describe the energy density of the universe in terms of the density prmeter Ω ǫ ǫ c = ǫ c 2 8πG 3H 2, the rtio of the totl energy density ǫ to the criticl energy density. Substituting this definition into the Friedmnn eqution yields H 2 = ΩH 2 kc2 R 2 0 2 (t) kc 2 = 1 Ω(t) = H 2 (t) 2 (t)r0 2. If Ω = 1, then it equls one t ll times, since the r.h.s. of this eqution lwys vnishes. In other cses, the vlue of Ω chnges with time, but if Ω > 1 it is lwys > 1, nd if Ω < 1 it is lwys < 1, becuse the r.h.s. cnnot chnge sign. At the present dy, we cn solve this eqution to get R 0 = c H 0 1 Ω 0 1/2. If Ω 0 is very close to one, then the curvture rdius is lrge compred to the Hubble rdius c/h 0, nd curvture effects on this scle re smll. Evolution of Energy Density: The Fluid Eqution The Friedmnn eqution determines (t) if we know H 0 nd the energy density ǫ(t) s function of time. (By compring ǫ 0 to the criticl density, we cn determine whether k = +1, 0, or 1, nd we cn use our lst eqution to determine R 0.) If the only energy contribution is from non-reltivistic mtter, then ǫ(t) = ǫ 0 (/ 0 ) 3, since expnsion of the universe simply dilutes the density of prticles. For the more generl cse, let s turn to the first lw of thermodynmics, de = P dv +dq, the chnge of internl energy of volume of fluid is the sum of P dv work nd dded het. The expnsion of homogeneous universe is dibtic, s there is no plce for het to come from, nd no friction to convert energy of bulk motion into rndom motions of prticles. 3
(There is cvet to this sttement: when prticles nnihilte, such s electrons nd positrons, this dds het nd mkes the expnsion temporrily non-dibtic. This mtters t some specific epochs in the very erly universe.) Therefore, de +P dv = 0 = Ė +P V = 0. For sphere of comoving rdius r s, V = 4π 3 r3 s 3 (t), V = 4π ( ) 3 r3 32ȧ s = V ( ) 3ȧ, nd E = Vǫ. Therefore Ė = V ǫ+ Vǫ = V ( ) ǫ+3ȧ ǫ. Together with Ė +P V = 0, we get V ( ) ǫ+3ȧ ǫ+3ȧ P = 0 nd thus ǫ+3ȧ (ǫ+p) = 0. This fluid eqution describes the evolution of energy density in n expnding universe. To solve this eqution, we need n dditionl eqution of stte relting P nd ǫ. Suppose we write this in the form P = wǫ. In principle, w could chnge with time, but we will ssume tht ny time derivtives of w re negligible compred to time derivtives of ǫ. This is resonble if the eqution of stte is determined by microphysics tht is not directly tied to the expnsion of the universe. The fluid eqution then implies ǫ ǫ = 3(1+w)ȧ, with solution For non-reltivistic mtter, ( ) 3(1+w) ǫ =. ǫ 0 0 w = P/ǫ mv2 th mc 2 v2 th c 2 1, where v th is the therml velocity of prticles. To ner-perfect pproximtion, w = 0, implying ǫ 3, in line with our simple dilution rgument. For rdition (i.e., photons), w = 1/3, implying ǫ 4. 4
This behvior lso follows from simple rgument: the number density of photons flls s n 3, nd the energy per photon flls s hν 1 becuse of cosmologicl redshift. The fluid eqution will led us to some less obvious conclusions when we consider drk energy. The Accelertion Eqution If we multiply our stndrd version of the Friedmnn eqution by 2, we get Tke the time derivtive divide by 2ȧ nd substitute from the fluid eqution ȧ 2 = 8πG 3c 2 ǫ2 kc2. R 2 0 2ȧä = 8πG 3c 2 ( ǫ 2 +2ǫȧ ), ä = 4πG ( 3c 2 ǫ ȧ ) +2ǫ, ǫ ȧ = 3(ǫ+P) to get ä = 4πG 3c 2 (ǫ+3p). We see tht if ǫ nd P re positive, the expnsion of the universe decelertes. Higher P produces stronger decelertion for given ǫ, e.g., rdition-dominted universe decelertes fster thn mtter-dominted universe. Theppernce of ǫ/c 2 +3P/c 2 is specific exmple of more generl phenomenonin GR: pressure ppers in the stress-energy tensor, nd it therefore hs grvittionl effect. (It must, becuse the division between energy density nd pressure depends on the stte of motion of the observer.) With the stress-energy tensor of n idel fluid, the Newtonin limit of GR yields Poisson eqution 2 Φ = 4πG(ρ+3P/c 2 ). The Cosmologicl Constnt, Vcuum Energy, nd Cosmic Accelertion A negtive energy density would be pretty bizrre. Negtive pressure sounds bizrre too, but it s not quite s crzy. For fluid to hve negtive pressure mens tht it hs tension it tkes work to expnd the fluid, insted of tking work to compress it. Suppose the universe is pervded by form of energy tht is constnt density, in spce nd time. Such n energy could conceivbly rise s consequence of the quntum vcuum, which is not relly empty but filled with virtul prticles. Expnding volume increses its energy, so tht energy must hve come from doing P dv work on the volume, which implies tht the pressure must be negtive (the volume resists stretching. ) In terms of our fluid eqution, we see tht ǫ = const. implies w = 1, nd thus P = ǫ. 5
From the ccelertion eqution, we see tht ny fluid with P < ǫ/3 cuses ccelertion of the universe, insted of decelertion. Such fluid hs, in GR, repulsive grvity. This discussion mkes cosmologicl ccelertion sound lmost resonble. The problem is tht no one knows how to clculte the quntum vcuum energy from first principles. In the bsence of complete theory of quntum grvity, the most resonble guess is tht ǫ vc E p /l 3 p, where l p = ( hg/c 3 ) 1/2 = 1.6 10 35 m is the Plnck length nd E p = ( hc 5 /G) 1/2 = 2.0 10 9 J is the Plnck energy. This estimte exceeds the criticl density by 120 orders of mgnitude! Since the only nturl number close to 10 120 is zero, the generl expecttion t lest until the mid 1990s ws tht correct clcultion of quntum vcuum energy would produce cncelltions tht mke the nswer exctly zero. It s possible tht vcuum energy relly does hve the vlue required to produce the observed cosmic ccelertion. It s lso possible tht the fundmentl vcuum energy is zero, nd tht ccelertion of the Universe is cused by some other negtive pressure fluid, or by brekdown of GR on cosmologicl scles. Einstein introduced (in 1917) the cosmologicl constnt Λ with different conception, s modifiction of the curvture term of the Field Eqution rther thn n dditionl contribution to the stress-energy tensor. However, the grvittionl effects of Einstein s cosmologicl constnt re identicl to those of form of energy with P = ǫ, which remins constnt in spce nd time s the universe expnds. Einstein needed the repulsive grvity of the Λ-term to llow sttic universe, countercting the grvittionl ttrction of mtter. He bndoned the ide when the universe ws found to be expnding. But tody we need cosmologicl constnt, or something like it, to explin how the universe cn ccelerte. 6