Evolution of cosmological perturbations

Houjun Mo

January 27, 2004

Although the universe is homogeneous and isotripic on very large scales, itcontains 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 throughgravitational instability.

The initial perturbations are believed to be generated by quantum fluctuationsduring the inflationary phase.

.

• What is gravitational instability? According to Jeans, a self-gravitatinghomogeneous gas is unstable on large scales. This instability was originallyconsidered for a steady space, but we will see that it also exist in an expandingspace.

• 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).

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Newtonian Theory of Small Perturbations1. Ideal Fluid

Consider a small fixed volume at x where the density and velocity of the fluidare ρ and u. The time evolution is given by the equation of continuity (whichdescribes mass conservation), Euler’s equation (the equation of motion) andPoisson’s equation (describing the gravitational field):

∂ρ∂t

+∇r · (ρu) = 0 (continuity) ,

∂u∂t

+(u ·∇r)u =−∇rPρ

−∇rΦ (Euler) ,

∇2r Φ = 4πGρ (Poisson),

Φ(r): gravitational potential; r : proper coordinates; ∂/∂t for fixed r .

To complete the description, these equations must be supplemented by theequation of state to specify the pressure P(ρ).

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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 →1a

∇x;∂∂t→ ∂

∂t− a

ax ·∇x.

The proper velocity, u = r , at a point x is

u = a(t)x+v , v≡ ax ,

where v is the peculiar velocity.

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3. Equations for perturbation quantities

Expressing ρ asρ(x, t) = ρ(t) [1+δ(x, t)] ,

we can write∂δ∂t

+1a

∇ · [(1+δ)v] = 0 ,

∂v∂t

+aa

v+1a(v ·∇)v =−∇φ

a− ∇P

aρ(1+δ),

∇2φ = 4πGρa2δ ,

where φ≡Φ+aax2/2, ∇≡ ∇x and ∂/∂t is for fixed x.

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 perturbationquantities δ, v.

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The effect of a smooth background

If there is a smooth background of zero-mass particles (photons or neutrinos) orof vacuum energy (the cosmological constant), both the continuity equation andEuler equation retain their forms, but Poisson equation is now

∇2r Φ = 4πG(ρ+ρr +ρv) .

ρr and ρv: the equivalent mass densities of the background. In these cases,δ (defined against the mean density of the non-relativistic fluid, rather than thetotal mean density), v and φ still obey the early equations. The effect of ρr andρv is to change the general expansion, i.e. the form of a(t).

In order to remain uniform, the background should not be perturbed significantlyby the pertubations in the non-relativistic component. True for a background ofvacuum energy, but only approximately true for relativistic background on scales ct, where relativistic particles cannot cluster.

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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 extraequation 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

TdSdt

=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 theevolution is adiabatic, then dS/dt = 0.

/.

For an ideal nonrelativistic monatomic gas, the combined first and second law ofthermodynamics applied to a unit mass is

T dS= d(

32

Pρ

)+Pd

(1ρ

).

Using P = (ρ/µ)kT (µ: mean molecular weight) gives

P ∝ ρ5/3exp

(23

µkS

).

Thus∇Pρ

=1ρ

[(∂P∂ρ

)S

∇ρ+(

∂P∂S

)ρ

∇S

]= c2

s∇δ+23

T∇S.

where c2s = (∂P/∂ρ)S.

/.

The Euler equation can then be written

∂v∂t

+aa

v+1a(v ·∇)v =−∇φ

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 specificentropy.

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 notchange with time.

Thus, for adiabatic evolution, ∇S in the above equation can be replaced by itsinitial value. Note that initially isentropic perturbations remain isentropic duringadiabatic evolution.

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Small perturbations

In special cases where both δ and v are small so that the nonlinear terms in theperturbation equations can be neglected:

∂δ∂t

+1a

∇ ·v = 0 ,∂v∂t

+aa

v =−∇φa− c2

s

a∇δ− 2T

3a∇S.

T the background temperature, cs: the sound speed evaluated using thebackground quantities. For polytropic processes,

P ∝ ργ and c2s = γP/ρ .

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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 andPoisson equations:

∂2δ∂t2

+2aa

∂δ∂t

= 4πGρδ+c2s

a2∇2δ+

23

Ta2

∇2S.

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Equations in Fourier space

In Fourier space:

d2δk

dt2+

2aa

dδk

dt=

[4πGρ− k2c2

s

a2

]δk−

23

Ta2

k2Sk ,

where δk(t) and Sk(t) are related to δ(x, t) and S(x, t) by

f (x, t) = ∑k

fk(t)exp(ik ·x) ; fk(t) =1Vu

∫f (x, t)exp(−ik ·x)d3x ,

Vu: 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 sovk = ikΨk. The Fourier transformation of the continuity equation then gives

∂δ∂t

+1a

∇ ·v = 0 , → vk =iakk2

δk .

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Isentropic Perturbations and the Jeans Length

For isentropic perturbations (Sk = 0),

d2δk

dt2+

2aa

dδk

dt=

[4πGρ− k2c2

s

a2

]δk

The right-hand side define a characteristic proper length (Jeans length),λJ,

λJ ≡2πakJ

= cs

√π

Gρ.

For perturbations with k kJ (i.e. λJ λJ), the pressure term can be neglectedand the matter behaves like a pressureless fluid. For k kJ, the gravity term canbe neglected, and the equation of δk is that for a damped oscillator. Thus, onlyperturbations with k kJ can grow with time.

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After recombination when matter decoupled with photons, the sound spead iscs = (5kT/3mp)1/2, and the Jeans lengthand the associated mass are

λJ ≈ 0.01(Ωm,0h2)−1/2

[1+z1000

]−1

kpc ,

MJ =4π3

ρm(z)λ3J ≈ 1.3×106(Ωm,0h

2)−1/2M.

This mass is about that of a globular cluster.

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Before recombination, electrons and photons are tightly coupled via Thomsonscattering, and matter and photons act like a single fluid with ρ = nmpc2 +aγT4, P = (1/3)aγT4:

cs =c√3

[34

ρm(z)ργ(z)

+1

]−1/2

.

At the time when matter and photons have equal energy density, z ≈4000(Ωm,0h2), the Jeans mass is

MJ ≈ 9×1016(Ωm,0h2)−2M.

Thus, before recombination, all adiabatic perturbations with scales smaller thansupercluster scales cannot grow. When baryons decoupled from radiation, itsJeans mass decreases by about 10 orders of magnitude to globular clusterscales.

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Specific Solutions

Pressureless Fluid

For isentropic perturbations in a pressureless fluid (or when k kJ):

d2δk

dt2+

2aa

dδk

dt= 4πGρmδk ,

If δ1(t) and δ2(t) are two solutions then

δ2δ1−δ1δ2 ∝ 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.

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Recall that the Hubble’s ‘constant’, H(t)≡ a/a, obeys

dHdt

+H2(t) =−4πG3

(ρm +ρv)

Since ρm ∝ a−3 and ρv = constant, differentiating the above equation once withrespect to t gives

d2Hdt2

+2aa

H(t) = 4πGρmH .

Thus, both δ(t) and H(t) obey the same equation. Since H(t) decreases witht, δ− ∝ H(t) gives the decaying mode of δ(t). The growing mode can then beobtained

δ+ ∝ H(t)∫ t

0

dt ′

a2(t ′)H2(t ′)∝ H(t)

∫ ∞

z

(1+z′)E3(z′)

dz′ .

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• For an Einstein-de Sitter universe,

δ+ ∝ t2/3; δ− ∝ H(t) ∝ t−1,

• For Ωm,0 < 1 and ΩΛ,0 = 0,

δ+ ∝ 1+3x

+3(1+x)1/2

x3/2ln

[(1+x)1/2−x1/2

],

where x≡ (Ω−1m,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.

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An approximation (Carroll et al 1992):

δ+ ∝ D(z) ∝g(z)

(1+z), g(z)≈ 5

2Ωm(z)

Ω4/7

m (z)−ΩΛ(z)+ [1+Ωm(z)/2][1+ΩΛ(z)/70]−1

Inserting the growing mode of δinto the expression for vk:

vk =ikk2

Haδk f (Ωm) ,

f (Ωm)≡− d lnD(z)d ln(1+z)

≈Ω0.6m

to good approximation.

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Perturbations in Two Nonrelativistic Components

Consider isentropc perturbations in two nonrelativistic components, one ispressureless (e.g. cold dark matter) and the other is with pressure (e.g.baryons). If ρB ρDM, then δB obeys

δB +2aa

δB +k2c2

saa3

δB = 4πGρ0a

30

a3δDM ,

If c2sa = constant (polytropic fluid with P ∝ ρ4/3) and a(t) ∝ t2/3 (i.e. for EdS), a

special solution is

δB(k, t) =δDM(k, t)1+k2/k2

J, with kJ =

(32

)1/2 Hacs

.

The perturbations in baryon with scale smaller than the Jeans length aresuppressed with respect to that in the cold dark matter, because of the pressurein baryons.

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Acoustic waves

For k kJ, the density perturbations of baryons behave as acoustic waves(driven by pressure).

Consider the equations for isentropic perturbations in a single fluid. Supposethat the time scale we are interested in is much shorter than the Hubble time sothat the expansion of the universe can be neglected. In this case:

δ′′k =−k2φk−k2c2sδk ,

where a prime denotes ∂/∂τ (τ = t/a). This is the equation of motion for a forcedoscillator.

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If φk = constant over the time of interest, then

δk(τ) =[

δk(0)+φk

c2s

]cos(kcsτ)+

1kcs

δ′k(0)sin(kcsτ)−φk

c2s.

δk oscillates around a zero-point −φk/c2s, with a frequency ω = kcs, and with

amplitude and phase set by the initial conditions δk(0) and δ′k(0).

The corresponding velocity perturbations: (from the continuity equation):

vk(t) =ikk2

δ′k =−icskk

[δk(0)+

φk

c2s

]sin(kcsτ)+

ikk2

δ′k(0)cos(kcsτ) .

There is a difference of π/2 in phase between the v and δ.

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Acoustic waves in pre-recombination era

In the pre-recombination era. photons and baryons are tightly coupled and canbe considered as a single fluid with a sound speed:

cs =c√

3(1+R ), where R ≡ 3ρB

4ργ.

The acoustic waves here are driven by the photon pressure, and for a givenmode, the oscillation frequency, amplitude and zero-point all depend on the ratioR .

The acoustic waves in the photon/baryon fluid at the epoch of decoupling cangive rise to oscillations in the CMB power spectrum. The amplitudes andseparations between peaks (or valleys) of such oscillations can therefore beused to constrain the baryonic density in the universe.

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Collisional Damping

Although photons and baryons are tightly coupled to each other by Comptonscattering before recombination, the coupling is imperfect, because the photonmean-free path, λ = (σTne)−1, is nonzero.

Because of the imperfect coupling, photons can diffuse from high density to lowdensity regions, thereby damping the perturbations in the photon distribution.Since the acoustic waves in the pre-recombination era are driven by photonpressure, the photon diffusion also leads to damping of the acoustic oscillationin the photon/baryon fluid. Such damping is called Silk damping.

/.

The scale on which the Silk damping is effective is the typical distance a photoncan diffuse in a Hubble time. The mean number of ‘steps’ a photon takes over atime t is N = ct/λ, where λ is the mean ‘step length’ of a random walk. Thus

λd = (N/3)1/2λ = (ct/3σTne)1/2 .

Applied to the pre-recombination epoch (z∼ 1000), this defines a mass scale

Md ≡ 4πλ3d/3∼ 1012(ΩBh2)−5/4M .

Perturbations with masses below Md in the baryonic component are expected tobe damped exponentially in the pre-recombination era.

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A problem then arises: how can galaxies (which have masses much smaller that∼ 1011M) 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 thefragmentations of structures of masses larger than 1014M (a process whichis not well understood).

• Alternatively, if the universe is dominated by dark matter (which is not subjectto Silk damping), baryons can catch up with the perturbations in the darkmatter 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.

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Perturbations on a Relativistic Background

In the presence of a uniform relativistic background, the scale factor a obeys(aa

)2

=8πG

3(ρm +ρr) ,

where ρm ∝ a−3, ρr ∝ a−4. Defining a new time variable, ζ≡ ρm/ρr ∝ a, then

d2δk

dζ2+

(2+3ζ)2ζ(1+ζ)

dδk

dζ=

32

δk

ζ(1+ζ).

The two solution of this equation are

δ+ ∝ 1+32

ζ ; δ− ∝(

1+32

ζ)

ln

[(1+ζ)1/2+1(1+ζ)1/2−1

]−3(1+ζ)1/2 .

Meszaros (1974) effect: perturbations in nonrelativistic component cannot growif the relativistic component dominates. (i.e. ζ 1). Can grow only at z< zeq.

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Collisionless gas: free streaming damping

Must based on the distribution function:

dN = f (x,p, t)d3xd3p, (1)

where pi = ∂L/∂xi is the canonical momentum conjugate to the comovingcoordinates xi. To obtain p, we use the Lagrangian of a particle with mass min an expanding universe:

L =12

m(ax+ ax)2−mΦ(x, t), (2)

which can be transformed into

L =12

ma2x2−mφ (3)

/.

by a canonical transformation L → L −dX/dt with X = maax2/2. It then followsthat the canonical momentum and the equation of motion are

p = ma2x = mav anddpdt

=−m∇φ . (4)

According to Liouville’s theorem, the phase-space density f is a constant alonga particle trajectory for a collisionless gas and so obeys the Vlasov equation:

∂ f∂t

+1

ma2p ·∇ f −m∇φ · ∂ f

∂p= 0 . (5)

This equation is just a result of conservation of particle number: the rate ofchange in particle number in a unit phase-space volume is equal to the net fluxof 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

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on p, the average value of Q in the neighborhood of x is

〈Q〉 ≡ 1n

∫Q f d3p, (6)

where

n≡∫

f d3p = ρa3(1+δ)/m (7)

is the comoving number density of particles at x, and for simplicity, we assumethe density of the universe to be dominated by the collisionless particles inconsideration. Multiplying (5) by Q and integrating over p we get

∂∂t

[n〈Q〉]+ 1ma2

∇ · [n〈Qp〉]+mn∇φ ·⟨

∂Q∂p

⟩= 0, (8)

where we have assumed f = 0 for p → ∞ and so the surface terms have been

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neglected. Seting Q = m in (8) and using (7) we obtain

∂δ∂t

+1a

∇ · [(1+δ)〈v〉] = 0 , (9)

which is just a result of mass conservation. The equation of motion can beobtained by setting Q = vi:

∂∂t

[(1+δ)〈vi〉]+aa(1+δ)〈vi〉=−1+δ

a∂φ∂xi

− 1a

∂∂x j

[(1+δ)〈viv j〉] . (10)

Notice that 〈v〉 is the average velocity of particles in the neighborhood of x andcan be much smaller than the typical velocity of individual particles. In principle,one can set Q = viv j and obtain the dynamical equation for 〈viv j〉 which, in turn,will depends on the third velocity moment. As a result, the complete dynamicsis given by infinite number of equations of velocity moments. In practice, we cantruncate the hierarchy by making some assumptions. If the velocity stress 〈viv j〉

/.

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 becombined to give

∂2δ∂t2

+2aa

∂δ∂t

= 4πGρδ . (11)

This equation is the same as fluid case with cs = 0. Thus, on scales wherethe velocity stress is negligible, collisionless gas can be treated as ideal fluidwith zero pressure. In general, however, the fluid treatment is not valid forcollisionless gas because of particle free-streaming.

Another way to solve the Vlasov equation is to consider the evolution of thedistribution function f itself. In general, we can write

f = f0+ f1 , (12)

where f0 is the unperturbed distribution function and f1 is the perturbation.Notice that f0 is independent of x in a homogeneous and isotropic background.

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The comoving number density of particles at x is n≡∫

f d3p, and so the massdensity at x is

ρ(x, t) =ma3

∫f (x,p, t)d3p = ρ(t)

[mn0

ρa3+δ(x, t)

], (13)

where n0 ≡∫

f0d3p is the mean number density of particles (in comoving units),and

δ(x, t) =m

ρa3

∫f1d3p (14)

is the density contrast to the background. The gravitational potential φ due to thedensity perturbation is given by the Poisson equation. For a homogeneous andisotropic background, f0 depends only on the magnitude of p, and since φ is afirst-order perturbation, the unperturbed distribution function f0 obeys(

∂ f0∂t

)q

= 0 , (15)

/.

where q = (∑i p2i )

1/2 is the magnitude of p. [Notice that p is reserved to denote(−gi j pi 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 f0 =

[eE/kT(a)±1

]−1,

where E = p2/2m∝ a−2 ∝ T(a) for nonrelativistic particles, and E = p ∝ a−1 ∝ T(a)for relativistic particles. We see that f0 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 f1 is

∂ f1∂t

+1

ma2p ·∇ f1−m∇φ · ∂ f0

∂p= 0 (16)

or, in terms of Fourier transforms, is

∂ fk∂ξ

+ik ·p

mfk(p,ξ) = ma2

(ik · ∂ f0

∂p

)φk(t) , (17)

/.

where dξ = dt/a2. This equation can be written in the form

∂∂ξ

[fk exp

(ik ·p

mξ)]

= ma2

(ik · ∂ f0

∂p

)φk exp

(ik ·p

mξ)

. (18)

Integrating both sides from some initial time ξi to ξ, we get

fk(p,ξ) = fk(p,ξi)exp

[−ik ·p

m(ξ−ξi)

]+mik ·

(∂ f0∂p

)∫ ξ

ξi

dξ′a2(ξ′)φk(ξ′)exp

[−ik ·p

m(ξ−ξ′)

]. (19)

Since the gravitational potential φ depends on f1 through Poisson’s equation,equation (19) is an integral equation for fk and can be solved iteratively. The firstterm on the right-hand side of (19) is a kinematic term due to the propagationof the initial condition, which can be neglected if the initial condition is set at

/.

an early time when the perturbation amplitudes are much smaller than the oneswe are concerned at an later time. The second term describes the dynamicalevolution of the perturbation due to gravitational interaction.

Inserting (19) into (14), it is straightforward to show that the dynamical part of δk

is given by

δk(ξ) =−mk2

ρa3

∫ ξ

ξi

dξ′(ξ−ξ′)a2(ξ′)φk(ξ′)G [k(ξ−ξ′)/m] , (20)

where G is the Fourier transform of f0:

G(s) =∫

d3p f0(p)e−ip·s. (21)

Since f0 depends only on q (the amplitude of p), the angular part of the

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integration over p can be carried out, giving

δk(ξ) =−4πkm2

ρa3

∫ ξ

ξi

dξ′a2(ξ′)φk(ξ′)Ik(ξ−ξ′) , (22)

where

Ik(ξ−ξ′)≡∫ ∞

0dqq f0(q)sin

[kq(ξ−ξ′)

m

]. (23)

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Free–Streaming Damping

If (k ·p/m) ξ, i.e. (a/k) vt, the phases in the dynamical part of (19) changesrapidly with ξ′ and so the integration over ξ′ makes little contribution even if theperturbation φk was big in the past. For the same reason, the contribution of thekinematic part to the density perturbation is also small at later time, because itis an integration over p. In this case, particles originally in the crests can moveto the troughs and vice versa within the time available, and density perturbationsare damped out with time. This is free-streaming damping; it results from thestreaming motion of collisionless particles. The proper length scale below whichthe free-streaming damping becomes important is of the order vt, where t is theage of the universe, and v is the typical particle velocity at t. More precisely, theproper distance streamed by a particle before time t can be written as

λFS = a(t)∫ t

0

v(t ′)a(t ′)

dt ′ . (24)

/.

The particle peculiar velocity scales with a as v∼ c at t < tnr and as v ∝ a−1 att > tnr (see §2.?), where tnr is the time when the particle becomes non-relativistic.We will assume that the universe is radiation dominated before tnr, i.e. tnr < teq,as is almost always true in real applications. Assuming also a(t) ∝ t1/2 at t < teqand a(t) ∝ t2/3 at t > teq, it is straightforward to show that

λFS

a(t)≈

(2ctnr/anr) [a/anr] (t < tnr)(2ctnr/anr) [1+ ln(a/anr)] (tnr < t < teq)(2ctnr/anr) [5/2+ ln(aeq/anr)] (t > teq)

. (25)

Thus, for a species which has become non-relativistic, the maximum free-streaming length at present time is

λFS(t0)∼(

a0

anr

)(2ctnr)

(52

+ lnaeq

anr

). (26)

/.

For light neutrinos with Tν/T = (4/11)1/3, and assuming kTν(tnr)∼mνc2/3, we get

λFS(t0)≈ 20Mpc( mν

30MeV

)−1, MFS ≈ 4×1015

( mν

30MeV

)−2M . (27)

Thus, if the universe is dominated by light neutrinos, all perturbations withmasses smaller than that of a typical supercluster are damped out in the linearregime, and the first objects to form are superclusters.

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The Zel’dovich Approximation

Given that all fluctuations were small at early times (e.g. z∼ 1000), so only thegrowing mode is present with significant amplitude at recent epochs. In thiscase, the linear evolution of density perturbations reduces to

δ(x,a) = D(a)δi(x) ,

where δi(x) is the perturbation at some initial time ti.

Thus the density field grows self-similarly with time. This is true also forthe gravitational acceleration and the peculiar velocity. Substituting the aboveequation into the Poisson equation gives

φ(x,a) =D(a)

aψ(x) where ∇2ψ = 4πGρma3δi(x) .

In an EdS universe, D ∝ a, and φ is independent of a.

/.

Integrating the linearized Euler’s equation, v+(a/a)v =−∇φ/a, for fixed x:

v =−∇ψa

∫Da

dt .

By definition, D(a) satisfies δ+(2a/a)δ = 4πGρmδ, so that∫(D/a)dt = D/4πGρma.

We have

v =− D4πGρma2

∇ψ(x) =− 14πGρma

DD

∇φ .

Thus v is proportional to the current gravitational acceleration. Since v = ax,integrating the above equation once and to the first order of perturbation so that∇ψ(x) can be replaced by ∇ψ(xi) (xi is the initial position of the mass element):

x = xi−D(a)

4πGρma3∇ψ(xi) .

/.

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 structureusing the properties of the density field at the initial position xi. Zel’dovichsuggested that this formulation could be used to extrapolate the evolution ofstructure into the regime when the displacements are not small. This procedureis 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∂xi

∣∣∣∣−1

=1

(1−λ1D)(1−λ2D)(1−λ3D).

λ1 ≥ λ2 ≥ λ3: eigenvalues of ∇∇ψ/4πGρma3. In linear case, λ1D 1, δ(x) =D(a)(λ1+λ2+λ3) = D(a)δi(x). Zel’dovich proposed that this solution applies evenfor λ1D(a) ∼ 1. In this case, the density will become infinite at a time whenλ1D(a) = 1. The first nonlinear structure to form will then be pancakes.

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Relativistic Perturbation Theory

• A relativistic approach is required in the early universe when the horizon sizeis small, the scales of many perturbations are larger than the horizon size. Inthis case, Newtonian theory is no longer valid.

• Basic principle:

Tµν

;ν ≡∂

∂xν

[√−gTµ

ν]− 12

√−g

∂gαβ

∂xµTαβ = 0

Rµν =8πGc4

(Tµν−

12

gµνTαα)

Write quantities as the sums of the background and perturbations, e.g. gµν =gµν +δgµν |δgµν| 1and solve for perturbation quantities, δ, v, and δgµν for a given background.

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Complications arising from gauge freedom:

In GR one is allowed to choose different coordinate systems. The question ishow to distinguish physical perturbations in the metric from the change in metricdue to coordinate transformations. For example,

dl2 = dx21+dx2

2 , and dl2 = dx21+x2

1dx22

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 butchanges from place to place, the densities in different places will be different ata 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 indistinguishing true perturbations from coordinate ‘wrinkles’. But for perturbationswith scales much larger than the horizon size, problems may arise.

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Possible solutions

• Choose a coordinate system which satisfies certain gauge conditions (so thatthere is no gauge freedom). For example, synchronous gauge:

ds2 = c2dt2−a2(t)(δi j +hi j )dxidxj

• Gauge-invariant formalism: use combinations of quantities which are invariantunder coordinate transformations to describe the evolution of perturbations,and make interpretations of perturbation quantities in a coordinate systemcorresponding to the measurement setup.

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The evolution of adiabaticperturbations in a CDM universewith Ωm,0 = 1, ΩB,0 = 0.05, h = 0.5.The scale factor is normalized at thepresent time.Decoupling: a∼ 10−3

Matter/radiation equality: a∼ 104

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Linear Transfer Functions

In the linear regime, equations describing the evolutions of perturbations areall linear in the pertubation quantities, and so each Fourier component evolvesindependently, and so we can write δ(k, t) = |δk(t)|eiϕk , and the phase ϕk isindependent of time. Thus, the evolution of a perturbation can be describedby a linear transfer function:

T(k, t) =KA|δk(t)||A(k)|

,

where A(k) is the initial amplitude and KA is a normalization to make T(k) = 1 fork→ 0.

T(k, t) depends both on cosmology and the matter content of the universe, andcan be calculated using the perturbation theory described before.

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Some analytical fitting formulae:

• Adiabatic HDM model (neutrinos):

T(k) = exp(−3.9q−2.1q2) , q≡ k/kν ,

where 2π/kν = 41(mν/30eV)−1Mpc: the mean streaming length of neutrinos.

• Adiabatic Cold Dark Matter Models:

T(k) =ln(1+2.34q)

2.34q

[1+3.89q+(16.1q)2+(5.46q)3+(6.71q)4

]−1/4,

where

q≡ 1Γ∗

(k

hMpc−1

)and Γ∗ = Ω0h

Γ∗: the shape parameter characterizing the horizon scale at teq

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The shape of the transfer function

For CDM model:

• Superhorizon perturbations remain constant: δCDM ∝ A(k);

• Subhorizon perturbations grow logarithmically with time in radiation-dominated era;

• Subhorizon perturbations grow with time as δCDM ∝ k2aA(k), in matter-dominated era.

The characteristic length scale is the horizon-size at radiation/matter equality:ke = 2π(cte)−1 ∝ Ωm,0h2.

Thus:

T(k) ∝

1 for k/ke 1(k/ke)−2 ln(k/ke) for k/k2 1

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Including baryons:

Γ∗ = Ω0hexp[−ΩB,0(1+

√2h/Ωm,0)

]

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