Lecture 4: Cosmological perturbations

Houjun Mo

February 17, 2004

So far we have discussed how a given perturbation evolves with time. Inorder to describe the cosmic structures, we need to know the properties of thecosmological perturbations.

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Linear power spectra

Since we can write δ(k, t) ∝ T(k, t)δ(k, ti), the power spectrum can be written as

P(k, t)≡ 〈|δ(k, t)|2〉 ∝ T2(k, t)P(k, ti) ,

Thus if the initial power spectrum P(k, ti) and the transfer function are known, wecan obtain the power spectrum at any subsequent time.

The power spectrum is the most important property of the cosmic density field.If the cosmic density field is Gaussian, the power spectrum completely specifiesthe statistical properties of the cosmic density field.

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The initial power spectrum

The initial power spectrum is assumed to have the form:

P(k, ti) = Aikn (n = 1),

which is consistent with the predition of inftation theory and CMB observations.

Since inftation theory is not complete, the amplitude Ai has to be determinedthrough observations. Note that to determine Ai, we only need to specify P(k, ti)for any given k. There are therefore many ways to normalize the power spectrum.

e.g. (1) by CMB observations (e.g. COBE); (2) by cluster abundance

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For P(k) with a given shape, the amplitude is fixed if we know the value of P(k)at any k, or the value of any statistic that depends only on P(k).

Normalization is usually represented by the value of σ8 ≡ σ(8h−1Mpc), where

σ2(R) =1

2π2

∫k3P(k)

9[sin(kR)−kRcos(kR)]2

(kR)6

dkk

,

where P(k) is the linear power spectrum at the present time t0.

Reason: the variance of counts-in-cells for normal galaxies is about unity inspheres of radii R= 8h−1Mpc.

If we write δgal = bδ with b = constant,then σ(8h−1Mpc) = σgal(8h−1Mpc)/b≈ 1/b, where b is a bias parameter.

δ(x;R) =∫

δ(y)W(x−y;R)d3y, δk(R) = δkW(k;R)

σ2(R) = (1/V)∫

δ2(x,R)d3x =∫

d3kδ2kW

2(k;R)

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The values of σ8 in different cosmogonies

Model Ωm,0 Ων,0 ΩΛ,0 h σ8(COBE) σ8(cluster)SCDM 1 0 0 0.5 1.3 0.6HDM 1 1 0 0.5 1.3 –MDM 1 0.3 0 0.5 0.5 0.5

OCDM 0.3 0 0 0.7 0.5 0.9ΛCDM 0.3 0 0.7 0.7 1.0 1.1

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The power spectra for variouscosmogonic models. The initialpower spectrum is assumed to bescale invariant (i.e. n = 1), and thespectra are normalized to reproducethe COBE observations of CMBanisotropies.

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Cosmic density field

For a given cosmology, the density field at a cosmic time t, is described by

δ(x, t) or δk(t).

How to specify a linear density field? to specify δ(x) for all x or to specify δk forall k? NO!

• We consider the cosmic density field to be the realization of a random process,which is descibed by a probability distribution function:

Px(δ1,δ2, · · ·,δN) dδ1dδ2 · · · dδN, (N → ∞)

Thus, we emphsize the properties of Px, rather than the exact form of δ(x).

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• The form of Px(δ1,δ2, · · ·,δN): is determined if we know all of its moments:⟨δ`1

1 δ`22 · · ·δ`N

N

⟩≡

∫δ`1

1 δ`22 · · ·δ`N

N Px(δ1,δ2, · · ·,δN)dδ1dδ2 · · · dδN,

where (`1, `2, · · ·, `N) = 0,1,2, · · ·.

In real space:

〈δ(x)〉= 0, ξ(x) = 〈δiδ j〉 , where x≡ |xi −x j| .

In Fourier space:

〈δk〉= 0, P(k)≡Vu⟨|δk|2

⟩≡Vu 〈δkδ−k〉=

∫ξ(x)exp(ik ·x)d3x ,

In general, it is quite difficult to describe a random field.

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Gaussian Random Fields

• In real space:

P (δ1,δ2, · · ·,δn) =exp(−Q )

[(2π)ndet(M )]1/2; Q ≡ 1

2∑i, j

δi

(M −1

)i j

δ j,

where Mi j ≡ 〈δiδ j〉. For a homogeneous and isotropic field, all the multivariatedistribution functions are invariant under spatial translation and rotation, andso are completely determined by the two-point correlation function ξ(x)!

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• In Fourier space:δk = Ak + iBk = |δk|exp(iϕk).

Since δ(x) is real, we have Ak = A−k, Bk =−B−k, and so we need only Fouriermodes with k in the upper half space to specify δ(x). It is then easy to provethat, for k in the upper half space,

〈AkAk′〉= 〈BkBk′〉=12V−1

u P(k)δ(D)kk ′ ; 〈AkBk′〉= 0,

Thus As a result, the multivariate distribution functions of Ak and Bk arefactorized according to k, each factor being a Gaussian:

P (αk)dαk =1

[πV−1u P(k)]1/2

exp

[− α2

k

V−1u P(k)

]dαk,

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In terms of |δk| and ϕk, the distribution function for each mode,P (Ak)P (Bk)dAk dBk, can be written as

P (|δk|,ϕk)d|δk|dϕk = exp

[− |δk|2

2V−1u P(k)

]|δk|d|δk|V−1

u P(k)dϕk

2π.

Thus, for a Gaussian field, different Fourier modes are mutually independent,so are the real and imaginary parts of individual modes. This, in turn, impliesthat the phases ϕk of different modes are mutually independent and haverandom distribution over the interval between 0 and 2π.

P(k) is the only function we need!

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Is the cosmic density field a Gaussain field?

Gaussian field is the first choice, becaue

• It is the simplest

• It is quite general because of central-limit theorem

• There is no observational contradiction

• It is predicted by inflation

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Realizations of Cosmological Density Field

Particularly simple for a Gaussian field.

• Start with an array of Fourier modes (each is characterized by k)

• Assign each mode a random amplitude |δk| and a random phase ϕk accordingthe distribution function.

• Obtain the density perturbation field, δ(x), via the Fast Fourier Transform(FFT).

Clearly, the resulting perturbation fields differ in different realizations!

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Gravitational Clustering

A cosmological density field is generally unstable. Because of gravitationalinteraction, regions with initial density above the mean grow more overdense inthe passage of time, while underdense regions tend to grow more underdense.This kind of gravitational instability in the linear regime has been described indetail.

In the linear regime, the perturbations in different Fourer mode evolveindependently, and so gravitational instability enhance the contrast of thefluctuations in a cosmological density field but does not change the correlationsbetween different modes.

In the nonlinear regime, the evolution of the density field is generallycomplicated, because the equations of motion are nonlinear, and it is in generalvery difficult to come up with accurate analytical description. Because of this,the exact evolution of the density field is usually solved by means of N-bodysimulations.

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Hierarchical Clustering

In the linear regime, the density perturbation δ(x, t) grows with time as δ(x, t) ∝D(t), and so the power variance. σ2(R; t) ∝ D2(t). For P(k) ∝ kn, σ2(R) ∝ R−(n+3),and so

σ2(R; t) =[

RR?(t)

]−(n+3)

=[

MM?(t)

]−(n+3)/3

,

whereM?(t) ∝ [D(t)]6/(n+3) , R?(t) ∝ [D(t)]2/(n+3) ,

are the fiducial mass and length scales on which σ = 1 at time t.

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Nonlinear structures with mass ∼ M begin to form in abundance when σ(M; t)∼1. The t-dependence of M?(t) can be used to understand how structures developwith time.

Since D(t) increases with t, the mass scale for nonlinearity, M?(t), increases witht for n > −3. In this case, structure formation proceeds in a “bottom-up” fashion.On the other hand, for models where the effective spectral index is smaller than−3, the first structures to collapse are large pancakes, and smaller structureswill then have to form from the fragmentation of larger structures. giving rise toa “top-down” scenario of structure formation.

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N-Body Simulations

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The Origin of Cosmological Perturbations

The answer to this question is still open.

Broadly speaking, two classes of models have been proposed. One is basedon inflation models and the other based on topological defects produced duringphase transitions in the early universe.

Roughly speaking, inflation models prodict isentropic, Gaussian pertubationswith spectra close to the Harison-Zel’dovich spectrum, while defect modelsgenerally predict non-Gaussian perturbations.

The inflationary scenario seems to be quite successful

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Perturbations from Inflation, heuristic arguments

The arguments consist of the following four important points.

1. Since the universe is assumed to have gone through a phase of very fastexpansion (inflation), the structures today all have sizes smaller than thehorizon size during inflation, and so inflation provides the conditions forgenerating the initial perturbations in a causal way.

2. If the scalar field driving inflation has negligible self-coupling (as is assumedin most models), different modes in the quantum fluctuations of the field areindependent of each other, the density perturbations generated are expectedto be Gaussian.

3. The perturbations produced during the inflation phase are those in the energydensity of the scalar field. When the inflation is over and as the energy

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density in the scalar field is converted into photons and other particles, wedo not expect any segregation between photons and other particles. Theperturbations are therefore expected to be isentropic.

4. Since the space during inflation (which is a de Sitter space) is invariant undertime translation, the perturbations generated by inflation are expected to bescale-invariant.

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Scale-free spectrum

Consider two perturbations: generation times, t1 and t2 Since the same atgeneration, we have at some late time t,

λ1

λ2=

exp(Ht)/exp(Ht1)exp(Ht)/exp(Ht2)

= eH(t2−t1)

Time at horizon crossing:

λ1a[tH(λ1)]a(t)

= ctH(λ1)∼c

H[tH(λ1)]∼ c

H[tH(λ2)]∼ λ2a[tH(λ2)]

a(t)

Thus, λ1/λ2 = eH[tH(λ2)−tH(λ1)] Hence tH(λ2)− t2 ∼ tH(λ1)− t1. Amplitudes must bethe same at horizon crossing, scale free!

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Zeldovich spectrum

P(k) ∝ kn (n = 1)

∆2 ∝ k3P(k) ∝ k3+n

Since φk ∝ δk/k2,∆2

φ ∝ ∆2/k4 ∝ kn−1

which is independent of k if n = 1.

For superhorizon perturbation, δ ∝ 1/(ρa2) ∝ t2/3

At horizon crossing ra(tH) = ctH, and so tH ∝ r3.

Thus,

∆(tH) ∝ r−(n+3)/2t2/3H ∝ r−(n+3)/2r2 = r(1−n)/2

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