Evolution of cosmological perturbations

Transcription

1 Evolution of cosmological perturbations Houjun Mo January 27, 2004 Although the universe is homogeneous and isotripic on very large scales, it contains structures, such as galaxies, clusters of galaxies, superclusters. The standard cosmology does not explain the origin of the structure formation It provides the condition of the growth of density perturbations through gravitational instability. The initial perturbations are believed to be generated by quantum fluctuations during the inflationary phase.

2 What is gravitational instability? According to Jeans, a self-gravitating homogeneous gas is unstable on large scales. This instability was originally considered for a steady space, but we will see that it also exist in an expanding space. The two aspects of the problem: (1) The generation and properties of the initial perturbations (2) The growth of perturbations with time We will first consider (2) and then (1).

3 Newtonian Theory of Small Perturbations 1. Ideal Fluid Consider a small fixed volume at x where the density and velocity of the fluid are ρ and u. The time evolution is given by the equation of continuity (which describes mass conservation), Euler s equation (the equation of motion) and Poisson s equation (describing the gravitational field): ρ t + r (ρu) = 0 (continuity), u t + (u r)u = rp ρ rφ (Euler), 2 rφ = 4πGρ (Poisson), Φ(r): gravitational potential; r: proper coordinates; / t for fixed r. To complete the description, these equations must be supplemented by the equation of state to specify the pressure P(ρ).

4 2. Fluid equations in an expanding background In this case, it is useful to use the comoving coordinates x defined as r = a(t)x. Note that r 1 a x; t t ȧ a x x. The proper velocity, u = ṙ, at a point x is where v is the peculiar velocity. u = ȧ(t)x + v, v aẋ,

5 3. Equations for perturbation quantities Expressing ρ as we can write ρ(x,t) = ρ(t)[1 + δ(x,t)], δ t + 1 [(1 + δ)v] = 0, a v t + ȧ a v + 1 (v )v = φ a a 2 φ = 4πGρa 2 δ, where φ Φ + aäx 2 /2, x and / t is for fixed x. P aρ(1 + δ), For a given cosmology [which specifies a(t)], and a given equation of state, the above set of equations can in principle be solved to give the perturbation quantities δ, v.

6 The effect of a smooth background If there is a smooth background of zero-mass particles (photons or neutrinos) or of vacuum energy (the cosmological constant), both the continuity equation and Euler equation retain their forms, but Poisson equation is now 2 rφ = 4πG(ρ + ρ r + ρ v ). ρ r and ρ v : the equivalent mass densities of the background. In these cases, δ (defined against the mean density of the non-relativistic fluid, rather than the total mean density), v and φ still obey the early equations. The effect of ρ r and ρ v is to change the general expansion, i.e. the form of a(t). In order to remain uniform, the background should not be perturbed significantly by the pertubations in the non-relativistic component. True for a background of vacuum energy, but only approximately true for relativistic background on scales ct, where relativistic particles cannot cluster.

7 The equation of state In general, the equation of state is P = P(ρ,S), where S is the specific entropy. Since a new quantity (S) is introduced, an extra equation is needed to complete the set of equations. By definition, ds = dq/t (dq: the amount of heat added to a fluid element of unit mass), so T ds dt = H C ρ, where H and C are the heating and cooling rates per unit volume, respectively, which are given processes such as radiative emission and absorption. If the evolution is adiabatic, then ds/dt = 0.

8 For an ideal nonrelativistic monatomic gas, the combined first and second law of thermodynamics applied to a unit mass is T ds = d ( 3 2 ) P + Pd ρ ( ) 1 ρ Using P = (ρ/µ)kt (µ: mean molecular weight) gives Thus P ρ = 1 ρ where c 2 s = ( P/ ρ) S. [ ( P ) ρ P ρ 5/3 exp S ρ + ( ) 2 µ 3 k S ( ) ] P S S ρ.. = c 2 s δ T S.

9 The Euler equation can then be written v t + ȧ a v + 1 (v )v = φ a a c2 s a δ (1 + δ) 2T 3a S (1 + δ). The last term in the above equation is due to the spatial fluctuation in the specific entropy. If the perturbation is isentropic, i.e. there is no fluctuation in the specific entropy, this term is zero. The growth of structure is said to be adiabatic, if the specific entropy does not change with time. Thus, for adiabatic evolution, S in the above equation can be replaced by its initial value. Note that initially isentropic perturbations remain isentropic during adiabatic evolution.

10 Small perturbations In special cases where both δ and v are small so that the nonlinear terms in the perturbation equations can be neglected: δ t + 1 a v = 0, v t + ȧ a v = φ a c2 s 2T δ a 3a S. T the background temperature, c s : the sound speed evaluated using the background quantities. For polytropic processes, P ρ γ and c 2 s = γp/ρ.

11 Operating on both sides of the linear Euler equation gives v a 1. The curl of v dies off with the expansion and can be neglected at late times! Differentiating the equation of δ once with respect to t and using the Euler and Poisson equations: 2 δ t 2 + 2ȧ a δ t = 4πGρδ + c2 s δ + 2 a T S. a 2 2

12 Equations in Fourier space In Fourier space: d 2 δ k dt + 2ȧ [ dδ k = 4πGρ k2 c 2 ] s δ 2 a dt a 2 k 2 T S 3a 2k2 k, where δ k (t) and S k (t) are related to δ(x,t) and S(x,t) by f (x,t) = f k (t)exp(ik x); f k (t) = 1 f (x,t)exp( ik x)d 3 x, k V u V u : volume of a large box on which perturbations are (assumed) periodical. Being curl-free, v can be written as the gradient of a potential: v = Ψ and so v k = ikψ k. The Fourier transformation of the continuity equation then gives δ t + 1 a v = 0, v k = iak k δ 2 k.

13 Isentropic Perturbations and the Jeans Length For isentropic perturbations (S k = 0), d 2 δ k dt + 2ȧ 2 a dδ k dt = [ 4πGρ k2 c 2 ] s δ a 2 k The right-hand side define a characteristic proper length (Jeans length),λ J, λ J 2πa k J = c s π Gρ. For perturbations with k k J (i.e. λ J λ J ), the pressure term can be neglected and the matter behaves like a pressureless fluid. For k k J, the gravity term can be neglected, and the equation of δ k is that for a damped oscillator. Thus, only perturbations with k k J can grow with time.

14 After recombination when matter decoupled with photons, the sound spead is c s = (5kT /3m p ) 1/2, and the Jeans lengthand the associated mass are [ ] z λ J 0.01(Ω m,0 h 2 ) 1/2 kpc, 1000 M J = 4π 3 ρ m (z)λ3 J (Ω m,0 h 2 ) 1/2 M. This mass is about that of a globular cluster.

15 Before recombination, electrons and photons are tightly coupled via Thomson scattering, and matter and photons act like a single fluid with ρ = nm p c 2 + a γ T 4, P = (1/3)a γ T 4 : [ ] c s = c 1/2 3ρ m (z) 3 4 ρ γ (z) + 1. At the time when matter and photons have equal energy density, 4000(Ω m,0 h 2 ), the Jeans mass is M J (Ω m,0 h 2 ) 2 M. Thus, before recombination, all adiabatic perturbations with scales smaller than supercluster scales cannot grow. When baryons decoupled from radiation, its Jeans mass decreases by about 10 orders of magnitude to globular cluster scales. z

16 Specific Solutions Pressureless Fluid For isentropic perturbations in a pressureless fluid (or when k k J ): d 2 δ k dt 2 + 2ȧ a dδ k dt = 4πGρ m δ k, If δ 1 (t) and δ 2 (t) are two solutions then δ 2 δ1 δ 1 δ2 a 2. This is true even if the pressure term is included. Thus, if one solution is known, the other one can be obtained by solving this first-order differential equation.

17 Recall that the Hubble s constant, H(t) ȧ/a, obeys dh dt + H2 (t) = 4πG 3 (ρ m + ρ v ) Since ρ m a 3 and ρ v = constant, differentiating the above equation once with respect to t gives d 2 H dt + 2ȧ 2 a H(t) = 4πGρ m H. Thus, both δ(t) and H(t) obey the same equation. Since H(t) decreases with t, δ H(t) gives the decaying mode of δ(t). The growing mode can then be obtained t δ + H(t) 0 dt a 2 (t )H 2 (t ) H(t) z (1 + z ) E 3 (z ) dz.

18 For an Einstein-de Sitter universe, δ + t 2/3 ; δ H(t) t 1, For Ω m,0 < 1 and Ω Λ,0 = 0, δ x + 3(1 + x)1/2 x 3/2 ] ln [(1 + x) 1/2 x 1/2, where x (Ω 1 m,0 1)/(1 + z). Note that δ + (1 + z) 1 as x 0 and δ + 1 as x. In general, the growing mode can be obtained from numerically.

19 An approximation (Carroll et al 1992): δ + D(z) g(z) (1 + z), g(z) 5 { } 1 2 Ω m(z) Ω 4/7 m (z) Ω Λ (z) + [1 + Ω m (z)/2][1 + Ω Λ (z)/70] Inserting the growing mode of δ into the expression for v k : v k = ik k 2Haδ k f (Ω m ), f (Ω m ) dlnd(z) dln(1 + z) Ω0.6 m to good approximation.

20 Perturbations in Two Nonrelativistic Components Consider isentropc perturbations in two nonrelativistic components, one is pressureless (e.g. cold dark matter) and the other is with pressure (e.g. baryons). If ρ B ρ DM, then δ B obeys δ B + 2ȧ a δ B + k2 c 2 sa a 3 δ B = 4πG ρ 0 a3 0 a 3 δ DM, If c 2 sa = constant (polytropic fluid with P ρ 4/3 ) and a(t) t 2/3 (i.e. for EdS), a special solution is δ B (k,t) = δ DM(k,t) 1 + k 2 /kj 2, with k J = ( ) 1/2 3 Ha. 2 c s The perturbations in baryon with scale smaller than the Jeans length are suppressed with respect to that in the cold dark matter, because of the pressure in baryons.

21 Acoustic waves For k k J, the density perturbations of baryons behave as acoustic waves (driven by pressure). Consider the equations for isentropic perturbations in a single fluid. Suppose that the time scale we are interested in is much shorter than the Hubble time so that the expansion of the universe can be neglected. In this case: δ k = k 2 φ k k 2 c 2 sδ k, where a prime denotes / τ (τ = t/a). This is the equation of motion for a forced oscillator.

22 If φ k = constant over the time of interest, then δ k (τ) = [ δ k (0) + φ ] k cos(kc c 2 s τ) + 1 δ s kc k(0)sin(kc s τ) φ k. s c 2 s δ k oscillates around a zero-point φ k /c 2 s, with a frequency ω = kc s, and with amplitude and phase set by the initial conditions δ k (0) and δ k (0). The corresponding velocity perturbations: (from the continuity equation): v k (t) = ik k 2δ k = ic [ sk δ k (0) + φ ] k sin(kc k c 2 s τ) + ik s k k(0)cos(kc 2δ s τ). There is a difference of π/2 in phase between the v and δ.

23 Acoustic waves in pre-recombination era In the pre-recombination era. photons and baryons are tightly coupled and can be considered as a single fluid with a sound speed: c s = c 3(1 +R ), where R 3ρ B 4ρ γ. The acoustic waves here are driven by the photon pressure, and for a given mode, the oscillation frequency, amplitude and zero-point all depend on the ratio R. The acoustic waves in the photon/baryon fluid at the epoch of decoupling can give rise to oscillations in the CMB power spectrum. The amplitudes and separations between peaks (or valleys) of such oscillations can therefore be used to constrain the baryonic density in the universe.

24 Collisional Damping Although photons and baryons are tightly coupled to each other by Compton scattering before recombination, the coupling is imperfect, because the photon mean-free path, λ = (σ T n e ) 1, is nonzero. Because of the imperfect coupling, photons can diffuse from high density to low density regions, thereby damping the perturbations in the photon distribution. Since the acoustic waves in the pre-recombination era are driven by photon pressure, the photon diffusion also leads to damping of the acoustic oscillation in the photon/baryon fluid. Such damping is called Silk damping.

25 The scale on which the Silk damping is effective is the typical distance a photon can diffuse in a Hubble time. The mean number of steps a photon takes over a time t is N = ct/λ, where λ is the mean step length of a random walk. Thus λ d = (N/3) 1/2 λ = (ct/3σ T n e ) 1/2. Applied to the pre-recombination epoch (z 1000), this defines a mass scale M d 4πλ 3 d/ (Ω B h 2 ) 5/4 M. Perturbations with masses below M d in the baryonic component are expected to be damped exponentially in the pre-recombination era.

26 A problem then arises: how can galaxies (which have masses much smaller that M ) form if perturbations on galactic scales are damped out? Two possibilities: If one assumes the universe to be dominated by baryonic matter, the formation of galaxies (and smaller structures) must be through the fragmentations of structures of masses larger than M (a process which is not well understood). Alternatively, if the universe is dominated by dark matter (which is not subject to Silk damping), baryons can catch up with the perturbations in the dark matter component after they have decoupled from the photons. Baryon-dominated models have many difficulties in matching with observations, and so the second option is the more attractive one.

27 Perturbations on a Relativistic Background In the presence of a uniform relativistic background, the scale factor a obeys (ȧ a ) 2 = 8πG 3 (ρ m + ρ r ), where ρ m a 3, ρ r a 4. Defining a new time variable, ζ ρ m /ρ r a, then d 2 δ k (2 + 3ζ) dδ k + dζ2 2ζ(1 + ζ) dζ = 3 2 δ k ζ(1 + ζ). The two solution of this equation are δ ζ; δ ( ) [ (1 + ζ) 1/2 ] ζ + 1 ln 3(1 + ζ) 1/2. (1 + ζ) 1/2 1 Mészáros (1974) effect: perturbations in nonrelativistic component cannot grow if the relativistic component dominates. (i.e. ζ 1). Can grow only at z < z eq.

28 Collisionless gas: free streaming damping Must based on the distribution function: dn = f (x,p,t)d 3 xd 3 p, (1) where p i = L/ ẋ i is the canonical momentum conjugate to the comoving coordinates x i. To obtain p, we use the Lagrangian of a particle with mass m in an expanding universe: L = 1 2 m(aẋ + ȧx)2 mφ(x,t), (2) which can be transformed into L = 1 2 ma2 ẋ 2 mφ (3)

29 by a canonical transformation L L dx/dt with X = maȧx 2 /2. It then follows that the canonical momentum and the equation of motion are p = ma 2 ẋ = mav and dp dt = m φ. (4) According to Liouville s theorem, the phase-space density f is a constant along a particle trajectory for a collisionless gas and so obeys the Vlasov equation: f t + 1 f ma2p f m φ p = 0. (5) This equation is just a result of conservation of particle number: the rate of change in particle number in a unit phase-space volume is equal to the net flux of particles across its surface. A common practice in solving the Vlasov equation is to consider the p moments (or the velocity moments) of f. Generally, if Q is a quantity which depends only

30 on p, the average value of Q in the neighborhood of x is Q 1 n Q f d 3 p, (6) where n f d 3 p = ρa 3 (1 + δ)/m (7) is the comoving number density of particles at x, and for simplicity, we assume the density of the universe to be dominated by the collisionless particles in consideration. Multiplying (5) by Q and integrating over p we get t [n Q ] + 1 ma2 [n Qp ] + mn φ Q p = 0, (8) where we have assumed f = 0 for p and so the surface terms have been

31 neglected. Seting Q = m in (8) and using (7) we obtain δ t + 1 [(1 + δ) v ] = 0, (9) a which is just a result of mass conservation. obtained by setting Q = v i : The equation of motion can be t [(1 + δ) v i ] + ȧ a (1 + δ) v i = 1 + δ a φ 1 [(1 + δ) v i v j ]. (10) x i a x j Notice that v is the average velocity of particles in the neighborhood of x and can be much smaller than the typical velocity of individual particles. In principle, one can set Q = v i v j and obtain the dynamical equation for v i v j which, in turn, will depends on the third velocity moment. As a result, the complete dynamics is given by infinite number of equations of velocity moments. In practice, we can truncate the hierarchy by making some assumptions. If the velocity stress v i v j

32 is small so that the right-hand side of (10) is dominated by the gravitational term, then to first order in δ (note that v δ in the linear regime), (9) and (10) can be combined to give 2 δ t + 2ȧ δ = 4πGρδ. (11) 2 a t This equation is the same as fluid case with c s = 0. Thus, on scales where the velocity stress is negligible, collisionless gas can be treated as ideal fluid with zero pressure. In general, however, the fluid treatment is not valid for collisionless gas because of particle free-streaming. Another way to solve the Vlasov equation is to consider the evolution of the distribution function f itself. In general, we can write f = f 0 + f 1, (12) where f 0 is the unperturbed distribution function and f 1 is the perturbation. Notice that f 0 is independent of x in a homogeneous and isotropic background.

33 The comoving number density of particles at x is n f d 3 p, and so the mass density at x is ρ(x,t) = m a 3 [ ] f (x,p,t)d 3 mn0 p = ρ(t) ρa + δ(x,t), (13) 3 where n 0 f 0 d 3 p is the mean number density of particles (in comoving units), and δ(x,t) = m f ρa 3 1 d 3 p (14) is the density contrast to the background. The gravitational potential φ due to the density perturbation is given by the Poisson equation. For a homogeneous and isotropic background, f 0 depends only on the magnitude of p, and since φ is a first-order perturbation, the unperturbed distribution function f 0 obeys ( ) f0 t q = 0, (15)

34 where q = ( i p 2 i ) 1/2 is the magnitude of p. [Notice that p is reserved to denote ( g i j p i p j ) 1/2, which is equal to q/a in a flat universe.] As we have seen in 2.?, the unperturbed particle distribution function has the form f 0 = [ e E/kT (a) ± 1 ] 1, where E = p 2 /2m a 2 T (a) for nonrelativistic particles, and E = p a 1 T (a) for relativistic particles. We see that f 0 is independent of a for fixed q (or fixed p), instead of for fixed p. To the first order in perturbation quantities, the equation for f 1 is f 1 t + 1 ma 2p f 1 m φ f 0 p = 0 (16) or, in terms of Fourier transforms, is f k ξ + ik p ( m f k(p,ξ) = ma 2 ik f ) 0 φ k (t), (17) p

35 where dξ = dt/a 2. This equation can be written in the form [ ( )] ( ik p f k exp ξ m ξ = ma 2 ik f ) ( ) 0 ik p φ k exp p m ξ Integrating both sides from some initial time ξ i to ξ, we get [ f k (p,ξ) = f k (p,ξ i )exp ik p +mik ] m (ξ ξ i) ( ) f0 ξ dξ a 2 (ξ )φ k (ξ )exp p ξ i [ ik p ] m (ξ ξ ). (18). (19) Since the gravitational potential φ depends on f 1 through Poisson s equation, equation (19) is an integral equation for f k and can be solved iteratively. The first term on the right-hand side of (19) is a kinematic term due to the propagation of the initial condition, which can be neglected if the initial condition is set at

36 an early time when the perturbation amplitudes are much smaller than the ones we are concerned at an later time. The second term describes the dynamical evolution of the perturbation due to gravitational interaction. Inserting (19) into (14), it is straightforward to show that the dynamical part of δ k is given by δ k (ξ) = mk2 ρa 3 ξ ξ i dξ (ξ ξ )a 2 (ξ )φ k (ξ )G [k(ξ ξ )/m], (20) where G is the Fourier transform of f 0 : G(s) = d 3 p f 0 (p)e ip s. (21) Since f 0 depends only on q (the amplitude of p), the angular part of the

37 integration over p can be carried out, giving δ k (ξ) = 4πkm2 ρa 3 ξ ξ i dξ a 2 (ξ )φ k (ξ )I k (ξ ξ ), (22) where I k (ξ ξ ) 0 [ kq(ξ ξ ] ) dqq f 0 (q)sin m. (23)

38 Free Streaming Damping If (k p/m) ξ, i.e. (a/k) vt, the phases in the dynamical part of (19) changes rapidly with ξ and so the integration over ξ makes little contribution even if the perturbation φ k was big in the past. For the same reason, the contribution of the kinematic part to the density perturbation is also small at later time, because it is an integration over p. In this case, particles originally in the crests can move to the troughs and vice versa within the time available, and density perturbations are damped out with time. This is free-streaming damping; it results from the streaming motion of collisionless particles. The proper length scale below which the free-streaming damping becomes important is of the order vt, where t is the age of the universe, and v is the typical particle velocity at t. More precisely, the proper distance streamed by a particle before time t can be written as t λ FS = a(t) 0 v(t ) a(t ) dt. (24)

39 The particle peculiar velocity scales with a as v c at t < t nr and as v a 1 at t > t nr (see 2.?), where t nr is the time when the particle becomes non-relativistic. We will assume that the universe is radiation dominated before t nr, i.e. t nr < t eq, as is almost always true in real applications. Assuming also a(t) t 1/2 at t < t eq and a(t) t 2/3 at t > t eq, it is straightforward to show that λ FS a(t) (2ct nr /a nr )[a/a nr ] (t < t nr ) (2ct nr /a nr )[1 + ln(a/a nr )] (t nr < t < t eq ) (2ct nr /a nr )[5/2 + ln(a eq /a nr )] (t > t eq ). (25) Thus, for a species which has become non-relativistic, the maximum freestreaming length at present time is λ FS (t 0 ) ( a0 a nr ) ( 5 (2ct nr ) 2 + ln a ) eq a nr. (26)

40 For light neutrinos with T ν /T = (4/11) 1/3, and assuming kt ν (t nr ) m ν c 2 /3, we get ( mν ) 1 ( λ FS (t 0 ) 20Mpc, MFS mν ) 2 M. (27) 30MeV 30MeV Thus, if the universe is dominated by light neutrinos, all perturbations with masses smaller than that of a typical supercluster are damped out in the linear regime, and the first objects to form are superclusters.

41 The Zel dovich Approximation Given that all fluctuations were small at early times (e.g. z 1000), so only the growing mode is present with significant amplitude at recent epochs. In this case, the linear evolution of density perturbations reduces to δ(x,a) = D(a)δ i (x), where δ i (x) is the perturbation at some initial time t i. Thus the density field grows self-similarly with time. This is true also for the gravitational acceleration and the peculiar velocity. Substituting the above equation into the Poisson equation gives φ(x,a) = D(a) a ψ(x) where 2 ψ = 4πGρ m a 3 δ i (x). In an EdS universe, D a, and φ is independent of a.

42 Integrating the linearized Euler s equation, v + (ȧ/a)v = φ/a, for fixed x: v = ψ a D a dt. By definition, D(a) satisfies δ+(2ȧ/a) δ = 4πGρ m δ, so that (D/a)dt = Ḋ/4πGρ m a. We have Ḋ v = 4πGρ m a 2 ψ(x) = 1 Ḋ 4πGρ m ad φ. Thus v is proportional to the current gravitational acceleration. Since v = aẋ, integrating the above equation once and to the first order of perturbation so that ψ(x) can be replaced by ψ(x i ) (x i is the initial position of the mass element): x = x i D(a) 4πGρ m a 3 ψ(x i).

43 This formulation of linear perturbation for pressureless fluid is due to Zel dovich (1970). It is a Lagrangian description because it specifies the growth of structure using the properties of the density field at the initial position x i. Zel dovich suggested that this formulation could be used to extrapolate the evolution of structure into the regime when the displacements are not small. This procedure is known as the Zel dovich approximation. This approximation is kinematic: particles move in straight lines, with the distance travelled proportional to D. The density field is given by mass conservation, 1 + δ = x x i 1 = 1 (1 λ 1 D)(1 λ 2 D)(1 λ 3 D). λ 1 λ 2 λ 3 : eigenvalues of ψ/4πgρ m a 3. In linear case, λ 1 D 1, δ(x) = D(a)(λ 1 +λ 2 +λ 3 ) = D(a)δ i (x). Zel dovich proposed that this solution applies even for λ 1 D(a) 1. In this case, the density will become infinite at a time when λ 1 D(a) = 1. The first nonlinear structure to form will then be pancakes.

44 Relativistic Perturbation Theory A relativistic approach is required in the early universe when the horizon size is small, the scales of many perturbations are larger than the horizon size. In this case, Newtonian theory is no longer valid. Basic principle: T µ ν ;ν x ν [ gtµ ν ] 1 2 R µν = 8πG c 4 g αβ g x T αβ = 0 µ ( T µν 1 ) 2 g α µνt α Write quantities as the sums of the background and perturbations, e.g. g µν = g µν + δg µν δg µν 1 and solve for perturbation quantities, δ, v, and δg µν for a given background.

45 Complications arising from gauge freedom: In GR one is allowed to choose different coordinate systems. The question is how to distinguish physical perturbations in the metric from the change in metric due to coordinate transformations. For example, dl 2 = dx dx 2 2, and dl 2 = dx x 2 1dx 2 2 can both be used to represent the metric of a 2-dimensional flat surface. In cosmology, if we choose a time coordinate which is not the cosmic time but changes from place to place, the densities in different places will be different at a given coordinate time even the universe is uniform. If the scale of the perturbation in consideration is much smaller than the horizon, the clocks at different places can be synchronized, so that there is no difficulty in distinguishing true perturbations from coordinate wrinkles. But for perturbations with scales much larger than the horizon size, problems may arise.

46 Possible solutions Choose a coordinate system which satisfies certain gauge conditions (so that there is no gauge freedom). For example, synchronous gauge: ds 2 = c 2 dt 2 a 2 (t)(δ i j + h i j )dx i dx j Gauge-invariant formalism: use combinations of quantities which are invariant under coordinate transformations to describe the evolution of perturbations, and make interpretations of perturbation quantities in a coordinate system corresponding to the measurement setup.

47 The evolution of adiabatic perturbations in a CDM universe with Ω m,0 = 1, Ω B,0 = 0.05, h = 0.5. The scale factor is normalized at the present time. Decoupling: a 10 3 Matter/radiation equality: a 10 4

48 Linear Transfer Functions In the linear regime, equations describing the evolutions of perturbations are all linear in the pertubation quantities, and so each Fourier component evolves independently, and so we can write δ(k,t) = δ k (t) e iϕ k, and the phase ϕ k is independent of time. Thus, the evolution of a perturbation can be described by a linear transfer function: T (k,t) = K A δ k (t) A(k), where A(k) is the initial amplitude and K A is a normalization to make T (k) = 1 for k 0. T (k,t) depends both on cosmology and the matter content of the universe, and can be calculated using the perturbation theory described before.

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50 Some analytical fitting formulae: Adiabatic HDM model (neutrinos): T (k) = exp( 3.9q 2.1q 2 ), q k/k ν, where 2π/k ν = 41(m ν /30eV) 1 Mpc: the mean streaming length of neutrinos. Adiabatic Cold Dark Matter Models: T (k) = ln( q) 2.34q [ q + (16.1q) 2 + (5.46q) 3 + (6.71q) 4] 1/4, where q 1 ( ) k Γ hmpc 1 and Γ = Ω 0 h Γ : the shape parameter characterizing the horizon scale at t eq

51 The shape of the transfer function For CDM model: Superhorizon perturbations remain constant: δ CDM A(k); Subhorizon perturbations grow logarithmically with time in radiationdominated era; Subhorizon perturbations grow with time as δ CDM k 2 aa(k), in matterdominated era. The characteristic length scale is the horizon-size at radiation/matter equality: k e = 2π(ct e ) 1 Ω m,0 h 2. Thus: T (k) { 1 for k/ke 1 (k/k e ) 2 ln(k/k e ) for k/k 2 1

52 Including baryons: [ Γ = Ω 0 hexp Ω B,0 (1 + ] 2h/Ω m,0 )