Cosmology 018 - Tutorial 8 Jue 5, 018 I this ote I have largely followed Ladau & Lifshitz Fluid Mechaics (1987, Weiberg s Graviatios ad Cosmology (197 ad otes by Yacie Ali-Haïmoud. We have mostly covered the homogeeous uiverse so far. Clearly the uiverse is very much ihomogeeous - there are stars, galaxies, clusters of galaxies, people to talk about all that, et cetera. We ll be iterested i the course i relatively simple aspects of the ihomogeeous uiverse - roughly how galaxies ad clusters were formed ad the aisotropies of the CMB. For a simplified picture oe ca start from a cosmic soup which icludes photos, electros ad baryos. Recall, whe we are iterested i the epochs of matter-radiatio equality, recombiatio ad photo decouplig (durig the last oe the state of perturbatios is imprited o the CMB. These correspod to temperatures of 0.5 ev, i.e. baryos ad electros are o-relativistic. 1 No-relativistic fluid dyamics Our ed game goal here is to uderstad some aspects of imperfect relativistic fluid dyamics ad i the process gai some ituitio o fluid dyamics. We will start with o-relativistic equatios for a ideal fluid i a o-expadig uiverse, ρ t + (ρv = 0, v t + (v v = p ρ Φ. (1 These are the Euler equatios which were derived earlier i the course 1. Expadig, ρ = ρ(1 + δ, with δ 1, v δ, Φ = Φ 0 + φ with φ Φ 0, ad the equatio p = c s ρ, oe fids δ + v = 0, v + c s δ = φ. ( Expadig i k modes, essetially replacig ik, the first equatio becomes δ + ik v = 0 ad the secod equatio combied with that ad the Poisso equatio, Φ = 4πGρ, becomes δ + c s(k k Jδ = 0, (3 where k J 4πG ρ/c s H/c s. This is the familiar result derived i the lecture. We fid oscillatory perturbatios behaviour for k > k J, smaller scales, meaig pressure wis. For k < k J, larger scales, the behaviour is expoetial - perturbatios grow via a gravitatioal istability. Clearly oe recovers soud waves i the limit k k J. 1 By writig dow the eergy-mometum tesor of a ideal fluid i a boosted form ad takig the o-relativistic limit, or simply doig the exercise i Galilea relativity. Alteratively, oe ca cosider a ifiitesimal fluid elemet ad write dow the mass coservatio equatio ad the mometum equatio ( Newto s d law. 1
1.1 Dampig by diffusio Ideed, i some scearios the ideal fluid assumptio breaks dow. As discussed i the lectures, sice particles have some radom velocity, they have some mea-free path (mfp, depedig o the rate of iteractios. The comovig mea-free path is give by λ mfp = 1 a e σ T. (4 O scales comparable or smaller to that, the fluid is viscous. Let us quatify the correctio to the Euler equatio usig a crude estimate, goig from the microscopic physics to the macroscopic. Cosider a fluid parcel whose velocity chages due to iteractios i equilibrium dv dt ecσ T (v v mfp. (5 The pre-factor comes as dv/dt v/τ v/(l mfp /c cv/(1/σ T e = e cσ T v Expadig v(x v v(x + x i x + 1 i x v i x j x i x, (6 j therefore, assumig the field is uiform locally such that the liear term (the drag is egligible, where α = O(1. Reisertig to Eq. (5, v(x mfp = v(x + αλ mfp v, (7 dv dt ecσ T λ mfp v cλ mfp v. (8 This should remid you of the heat equatio, u/ t = D u. It is clear the that photos work to smooth out perturbatios. Thus, photos behave as a viscous fluid o small eough scales, (Let us drop the fudge factor α or absorb it ito λ mfp v = φ p ρ + cλ mfp v, (9 which traslates to (agai usig the cotiuity equatio ad the speed of soud relatio δ + cλ mfp k δ + c 3 k δ 0, (10 where we eglected the gravitatioal potetial (which holds assumig there s separatio of scales large eough to allow lookig at λ 1 mfp k k J. With the WKB approximatio, [ η ] δ exp i ωdη, ω ω. (11 Recall, λ mfp depeds o time, or i this case - the coformal time. This leads to ω = c k 3 + cλ mfpk iω, (1
ad i the limit k λ 1 mfp, ω ± ck 3 + i cλ mfpk. (13 We obtai a geeral expressio for the perturbatio evolutio, ( [ kηc δ(k, η = δ 0 (k cos exp 1 η ] ( ] kηc 3 k c λ mfp(η dη = δ 0 (k cos exp [ k 3 kd, (14 where η η k D c λ mfp (η dη = λ mfp(η dη, (15 η scat where η scat λ mfp /c is the mea time betwee scatterig evets of a photo, which ca therefore be expressed as 1/ η scat = dn scat /dη, from which we obtai the origial estimate from the lecture, k D λ D = N scat λ mfp. (16 We fid what was expected all alog - perturbatios are damped o scales smaller tha the characteristic radom walk scale of photos. Relativistic fluid mechaics Let us begi by recosiderig a relativistic ideal fluid. tesor It is described by the eergy-mometum T µν = (ρ + pu µ u ν + pη µν wu µ u ν + pη µν. (17 I the rest frame u 0 = 1, u i = 0. A geeralizatio of the cotiuity equatio ca be writte as µ (u µ = 0, (18 where is the proper umber desity of particles. The equatios of motio are derived from the coservatio equatio which reads Multiplyig by u ν, recallig u ν u ν = 1, we fid µ T µν = 0, (19 µ T µν = u ν µ (wu µ + wu µ µ u ν + ν p = 0. (0 0 = µ (wu µ u ν ν p = µ uµ u ν ν p = u µ µ Ivokig the thermodyamics relatio d [ u µ µ p = u µ µ 1 ] µp. (1 ( σ = T d + 1 dp, ( 3
which leads to the expressio of a adiabatic flow ( σ u µ µ = 0. (3 Alteratively, usig Eq. (18, this ca be writte as a cotiuity of etropy µ (σu µ = 0. Let us ow take the coservatio equatio (Eq. (0 ad rewrite it as T µν x ν leadig to the relativistic geeralizatio of Euler s equatios, u µ u α T αν x ν = 0, (4 wu ν u µ x ν = p x µ u µu ν p x ν. (5 For isetropic flow, d(σ/ = 0, we have d(w/ = (1/dp, therefore which gives, together with Eq. (5, u α = p x µ x. (6 µ x α u µ = x µ. (7 For a steady flow, assumig time idepedece of everythig, we fid ( γwv γ(v + = 0. (8 Scalar multiplicatio by v leads to the coclusio that alog streamlies γw = cost, (9 which is the relativistic geeralizatio of Beroulli s equatio. I the o relativistic case γ = 1 1 v 1 + v, (30 w = ρ + p m + p, (31 or simply γw m + mv + p cost, (3 v + p ρ = cost. (33 This ca be geeralized with potetial eergy term gz for example i a uiform gravitatio field. 4
.1 Relativistic imperfect fluid Fially, we would like to briefly discuss how to treat relativistic imperfect fluids. We ca icorporate the imperfectess by addig terms to the eergy-mometum tesor ad the particle curret vector, T µν = pη µν + (p + ρu µ u ν + T µν, N µ = u µ + N µ. (34 First, we pick the approach that the particle flux N i vaishes i a comovig frame. Therefore, i a comovig frame T 00 ρ, N 0, N i 0, u 0 1 ad u i 0. This leads to the costraits T 00 = N 0 = N i = 0. Therefore i a geeral Loretz frame u µ u ν T µν = 0, N µ = 0. (35 The rest of the challege is to figure out which forms of T µν satisfy these coditios ad the secod law of thermodyamics, writte as ( ( σ 1 ( ρ T d = pd + d. (36 Developig these argumets more, askig what etropy is produced by fluid motios, oe fids (See Weiberg G&C (197.11 the form where T µν = ηh µγ H νδ W γδ χ (H µγ u ν + H νγ u µ µν uγ Q γ ζh x, (37 γ H µν η µν + u µ u ν, (38 Q µ T x + T u µ µ x β uβ, (39 W µν u µ x + u ν ν x µ 3 η u γ µν x. γ (40 For cosmology, whe a fluid has a very short mea free time τ χ = 4 3 at 3 τ, η = 4 15 at 4 τ, ζ = 4aT 4 τ [ 1 3 ( ] p, (41 ρ where a is the Stefa-Boltzma costat. Clearly all the parameters which parameterize viscosity ad heat coductio are proportioal to the mea free path. The secod beig the the eergy trasport T i0 vaishes i a comovig frame. 5