1 Chapter 4. COSMOLOGICAL PERTURBATION THEORY 4.1. NEWTONIAN PERTURBATION THEORY Newtonian gravity is an adequate description on small scales (< H ...

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∇r P − ∇r Φ ρ

∇r2 Φ = 4πGρ

(C) Continuity equation [conservation of mass] (E) Euler equation

[conservation of momentum]

“F=ma”

(P ) Poisson equation

We wish to consider small fluctuations around a homogeneous background: ρ(t, r) = ρ¯(t) + δρ(t, r) , . . . and “linearise” the fluid equations.

1) Static space without gravity ¯ = 0. The background solution is ρ¯ = const., P¯ = const. and u The linearised evolution equations are ∂t δρ = −∇r ·(¯ ρu) ρ¯ ∂t u = −∇r δP

(C) (E)

Combining ∂t (C) and ∇r ·(E), we get ∂t2 δρ − ∇r2 δP = 0 .

speed of sound

↓ For adiabatic fluctuations (see later), we have δP = c2s δρ and hence ∂t2 − c2s ∇r2 δρ = 0

wave equation

⇒ Solution: δρ = Aei(ωt−k·r) , where ω = cs k, with k ≡ |k|. • “fluctuations oscillate”

2) Static space with gravity For Φ 6= 0, we get ∂t2 − c2s ∇r2 δρ = 4πG¯ ρ δρ ↑ from ρ¯ ∇r2 δΦ ⇒ Solution: δρ = Aei(ωt−k·r) , where ω 2 = c2s k 2 − 4πG¯ ρ ≡ c2s k 2 − kJ2 ↑ pressure

↑

↑

gravity

Jeans’ scale: kJ ≡

• On small scales, k > kJ , “fluctuations oscillate” • On large scales, k < kJ , “fluctuations grow exponentially”

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p

4πG¯ ρ/cs

3) Expanding space In an expanding space, we have r = a(t)x and u ≡ r˙ = Hr + v Rather than labelling events by t and r, it is convenient to use t and x: • For spatial gradients, this means using ∇r = a−1 ∇x

(?)

• For time derivatives, this means using ∂ ∂ ∂x ∂ ∂(a−1 (t)r) = + · ∇x = + · ∇x ∂t r ∂t x ∂t r ∂t x ∂t x ∂ − Hx · ∇x (??) = ∂t x Notation alert: we will drop the subscripts x from now on!

With this in mind, we look at the fluid equations in an expanding universe: • Continuity equation: Substituting (?) and (??) into (C), we get 1 ∂ − Hx · ∇ ρ¯(1 + δ) + ∇ · ρ¯(1 + δ)(Hax + v) = 0 , ∂t a where δ ≡ δρ/¯ ρ is the density contrast. – At zeroth-order, we find ∂ ρ¯ + 3H ρ¯ = 0 , ∂t where we used ∇x · x = 3. As expected, we have ρ¯ ∝ a−3 . – At first-order, we get ∂ ρ¯ ∂δ ρ¯ + 3H ρ¯ δ + ρ¯ + ∇·v =0 . ∂t ∂t a | {z } =0 3

Hence, we find 1 δ˙ + ∇ · v = 0 a

where δ˙ ≡

(C)

∂δ ∂t

. x

• Euler equation: Similar manipulations of the Euler equation lead to v˙ + Hv = −

1 1 ∇δP − ∇δΦ a¯ ρ a

(E)

For δP = δΦ = 0, this implies v ∝ a−1 (cf. Ch. 1). • Poisson equation: It takes no work to show that ∇2 δΦ = 4πGa2 ρ¯ δ

(P )

Combining ∂t (C) with ∇·(E) and (P ), we get c2s 2 ¨ ˙ ρδ . δ + 2H δ − 2 ∇ δ = 4πG¯ a ↑ Hubble friction

The Jeans’ scale kJ (t) ≡

p 4πG¯ ρ(t)/cs (t) is time-dependent.

• k > kJ : “fluctuations oscillate with decreasing amplitude” • k < kJ : “fluctuations grow as a power law” Ex: Show that long-wavelength fluctuations in a matter-dominated universe, 4πG¯ ρ = 23 H 2 Ωm , have the following power-law solutions δ = a , a−3/2 .

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4.2. RELATIVISTIC PERTURBATION THEORY GR is required on large scales and for relativistic fluids. • Perturb metric:

gµν = g¯µν + δgµν

• Perturb matter:

ρ = ρ¯ + δρ , . . .

• Linearise evolution equations. 4.2.1. PERTURBED METRIC The most general perturbations around FRW are h i 2 2 2 i i j ds = a (τ ) (1 + 2A)dτ − 2Bi dx dτ − (δij + hij )dx dx , where A(τ, x), etc. Convention: Latin indices are raised and lowered with δij : e.g. hi i = δ ij hij . Write

ˆi Bi ≡ ∂i B + B scalar

ˆi = 0 vector : ∂ i B

hij ≡ 2Cδij + 2∂hi ∂ji E + 2∂(i Eˆj) + 2Eˆij scalar

scalar

vector ∂ i Eˆi = 0

tensor ∂ i Eˆij = Eˆ i i = 0

where ∂hi ∂ji E ≡ ∂i ∂j − 13 δij ∇2 E ∂(i Eˆj) ≡ 21 (∂i Eˆj + ∂j Eˆi ) 10 = 4 scalar modes : A, B, C, E ˆi , Eˆi + 4 vector modes : B + 2 tensor modes : Eˆij Theorem: At first order, scalars, vectors, and tensors don’t mix! ⇒ We can treat them separately.

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The Gauge Problem Perturbations depend on the choice of coordinates (“gauge choice”). Consider an unperturbed FRW universe, ds2 = a2 (τ ) dτ 2 − δij dxi dxj , ρ = ρ¯(t) , · · · . • A change of spatial coordinates, xi 7→ x˜i ≡ xi + ξ i (τ, x), implies dxi = d˜ xi − ∂τ ξ i dτ − ∂k ξ i dxk , which results in fictitious metric perturbations i j ds2 = a2 (τ ) dτ 2 − 2ξi0 d˜ x d˜ x + O(ξ 2 ) xi dτ − δij + 2∂(i ξj) d˜ ↑ ↑ “fake perturbations” (gauge modes)

• Similarly, a change of time slicing, τ 7→ τ + ξ 0 (τ, x), induces fictitious density perturbations ρ(τ ) 7→ ρ(τ + ξ 0 ) = ρ¯(τ ) + ρ¯ 0 ξ 0 . We need a way to identify “true perturbations”. Gauge Transformations Consider ˜ µ ≡ X µ + ξ µ (τ, x) , X µ 7→ X

where

ξ0 ≡ T ,

ξ i ≡ ∂ iL .

To see how the metric transforms, note that ˜ X ˜ α dX ˜β . ds2 = gµν (X)dX µ dX ν = g˜αβ (X)d ˜ α = (∂ X ˜ α /∂X µ )dX µ (and similarly for dX ˜ β ), we find Writing dX ˜β ˜ α ∂X ∂X ˜ . g˜αβ (X) gµν (X) = ∂X µ ∂X ν

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As an example, let us take µ = ν = 0: 2 ˜ α ∂X ˜β ∂X to first order ˜ −−−−−−−−→ ∂ τ˜ g˜00 (X) ˜ . g˜αβ (X) g00 (X) = ∂τ ∂τ ∂τ We can write this as 2 a2 (τ ) 1 + 2A = 1 + T 0 a2 (τ + T ) 1 + 2A˜ 2 = 1 + 2T 0 + · · · a(τ ) + a0 T + · · · 1 + 2A˜ = a2 (τ ) 1 + 2HT + 2T 0 + 2A˜ + · · · , where H ≡ a0 /a. This implies that A 7→ A˜ = A − T 0 − HT .

Ex: Show that ˜ = B + T − L0 B 7→ B 1 C 7→ C˜ = C − HT − ∇2 L , 3 E 7→ E˜ = E − L .

Gauge-Invariant Variables Ex: Show that Ψ ≡ A + H(B − E 0 ) + (B − E 0 )0 , 1 Φ ≡ −C − H(B − E 0 ) + ∇2 E , 3 are gauge-invariant. The Bardeen potentials Ψ and Φ represent “true perturbations” that cannot be removed by a coordinate transformation.

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Gauge Fixing An alternative solution of the gauge problem is to • fix the gauge, • follow all perturbations (metric and matter), • compute observables. Two popular gauges are: • Newtonian gauge:

B = E ≡ 0.

• Spatially flat gauge:

C = E ≡ 0.

4.2.2. PERTURBED MATTER In a homogeneous universe, T¯µ ν = (¯ ρ + P¯ )U¯ µ U¯ν − P¯ δνµ , where U¯ µ = a−1 δ0µ for a comoving observer. In a perturbed universe, T µ ν = T¯µ ν + δT µ ν = T¯µ ν + (δρ + δP )U¯ µ U¯ν + (¯ ρ + P¯ )(δU µ U¯ν + U¯ µ δUν ) − δP δνµ − Πµ ν ↑ anisotropic stress (mostly negligible) µ

µ

ν

To derive δU , we consider gµν U U = 1: • At first order, this implies δgµν U¯ µ U¯ ν + 2U¯µ δU µ = 0. • Using U¯ µ = a−1 δ µ and δg00 = 2a2 A, we find δU 0 = −Aa−1 . 0

• Writing δU i = v i /a, where v i ≡ dxi /dτ , we get U µ = a−1 [1 − A, v i ] . Ex: Show that

Uµ = a[1 + A, −(vi + Bi )] .

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Hence, we find T 0 0 = ρ¯ + δρ T i 0 = (¯ ρ + P¯ )v i ≡ q i

←

3-momentum density

T 0 j = −(¯ ρ + P¯ )(vj + Bj ) T i j = −(P¯ + δP )δji In a multi-component universe, we have Tµν = δρ =

X

δP =

X

qi =

X

P

I

I Tµν and hence

δρI

I

δPI

I

qIi

(note: velocities don’t add!)

I

Gauge Transformations ˜ µ = X µ + (T, ∂ i L) implies X → 7 X µ

˜β ∂X µ ∂ X ˜ . T˜α β (X) T ν (X) = ν α ˜ ∂X ∂X µ

δρ 7→ δρ − T ρ¯ 0 ,

and

δP 7→ δP − T P¯ 0 , v 7→ v + L0 ,

where vi ≡ ∂i v .

A gauge-invariant density fluctuation is ρ¯∆ ≡ δρ + ρ¯ 0 (v + B) . comoving-gauge density contrast

Popular “matter gauges” are: • Uniform density gauge: • Comoving gauge:

δρ ≡ 0

+

q≡0

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B ≡ 0.

Adiabatic Fluctuations are predicted by inflation. What are they? Consider a mixture of fluids, with ρI , PI , etc. Adiabatic perturbations are perturbations induced by a common, local shift in time of all background quantities: e.g. δρI (τ, x) = ρ¯I (τ + δτ (x)) − ρ¯I (τ ) = ρ¯I0 δτ (x) ↑ same for all I. Hence, δτ =

δρI δρJ = 0 , for all I and J. 0 ρ¯I ρ¯J

Using ρ¯I0 = −3H(1 + wI )¯ ρI and δI ≡ δρI /¯ ρI , we find δI δJ = . 1 + wI 1 + wJ 4 ⇒ Adiabatic perturbations satisfy δr = δm . 3 ⇒ Whatever dominates the background also dominates fluctuations: X δρtot = ρ¯tot δtot = ρ¯I δI I ↑ similar for all I.

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4.2.3. LINEARISED EVOLUTION EQUATIONS See: www.damtp.cam.ac.uk/user/db275/Cosmology/CPT.nb (thanks to Yi Wang)

We will work in Newtonian gauge: 1 + 2Ψ 0 2 gµν = a , 0 −(1 − 2Φ)δij and ignore anisotropic stress: Πij = 0. • Inverse metric g µν

1 = 2 a

1 − 2Ψ 0 0 −(1 + 2Φ)δ ij

+ O(Ψ2 , Φ2 ) .

• Christoffel symbols At first order, we find 1 1 Γ000 = g 00 ∂0 g00 = 2 (1 − 2Ψ)∂0 [a2 (1 + 2Ψ)] 2 2a = H + Ψ0 . Ex: Show that

Γ00i = ∂i Ψ , Γi00 = δ ij ∂j Ψ , Γ0ij = Hδij − [Φ0 + 2H(Φ + Ψ)] δij , Γij0 = Hδji − Φ0 δji , i Γijk = −2δ(j ∂k) Φ + δjk δ il ∂l Φ .

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PERTURBED CONSERVATION EQUATIONS ∇µ T µ ν = 0

Consider

= ∂µ T µ ν + Γµµα T α ν − Γαµν T µ α . • The ν = 0 component leads to the continuity equation: – At zeroth-order, we find ρ¯ 0 = −3H(¯ ρ + P¯ ) . – At first-order, we get δρ 0 = − 3H(δρ + δP )

3Φ0 (¯ ρ + P¯ )

+

dilution by expansion

−

perturbed expansion:

∇·q fluid flow

aeff ≡ a(1 − Φ) For adiabatic perturbations, we can write this as δ 0 + (1 + w) (∇·v − 3Φ 0 ) + 3H(c2s − w)δ = 0

(C)

For non-relativistic matter (w = c2s 1) on sub-Hubble scales (Φ0 ∇·v ← Ch. 5), this reproduces the Newtonian continuity equation δ 0 + ∇·v ≈ 0 . • The ν = i component leads to the Euler equation: P¯ 0 ∇δP + 3H 0 v − ρ¯ ρ¯ + P¯

0

v = − Hv redshifting

relativistic correction

− ∇Ψ

pressure

gravity

For adiabatic perturbations, we get 0

v + H(1 −

3c2s )v

c2s = − ∇δ − ∇Ψ 1+w

(E)

For non-relativistic matter (w = c2s 1), this reproduces the Newtonian Euler equation.

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PERTURBED EINSTEIN EQUATIONS Γλµν (Φ, Ψ) −→ Rµν

1 −→ Gµν ≡ Rµν − gµν R 2

Ex: Show that (see lecture notes) G00 = 3H2 + 2∇2 Φ − 6HΦ0 G0i = 2∂i (Φ0 + HΨ) Gij = −(2H0 + H2 )δij + ∂i ∂j (Φ − Ψ) + ∇2 (Ψ − Φ) + 2Φ00 + 2(2H0 + H2 )(Φ + Ψ) + 2HΨ0 + 4HΦ0 δij

• Consider the trace-free part of the ij-Einstein equation: ∂hi ∂ji (Φ − Ψ) = 0

⇒

Φ=Ψ .

• Next, look at the 00-equation: G00 = 8πGT00 3H2 + 2∇2 Φ − 6HΦ0 = 8πGg0µ T µ 0 = 8πG g00 T 0 0 + g0i T i 0

= 8πGa2 (1 + 2Φ)(¯ ρ + δρ) = 8πGa2 ρ¯(1 + 2Φ + δ) . – At zeroth-order, we recover the Friedmann equation: 3H2 = 8πGa2 ρ¯. – At first-order, we get ∇2 Φ = 4πGa2 ρ¯δ + 3H(Φ0 + HΦ)

(?)

• Moving on to the 0i-equation: G0i = 8πGT0i

⇒

Φ0 + HΦ = −4πGa2 q

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(??)

• Substituting (??) into (?), we find ∇2 Φ = 4πGa2 ρ¯ ∆

Poisson equation

where ∆ is the comoving-gauge density contrast. • Finally, we look at the trace of the ij-equation: Gi i = 8πGT i i . – At zeroth-order, we recover the second Friedmann equation: 2H0 + H2 = −8πGa2 P¯ . – At first-order, we get an evolution equation for Φ: Φ00 + 3HΦ0 + (2H0 + H2 )Φ = 4πGa2 δP .

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4.3. CONSERVED CURVATURE PERTURBATION A very important quantity is the comoving curvature perturbation R. It is “conserved” on superhorizon scales (k H = aH) and therefore provides the link between fluctuations created by inflation (Ch. 6) and those observed in the late universe (Ch. 5). Consider the perturbed 3-metric γij = a2 [(1 + 2C)δij + 2Eij ], and the corresponding 3-curvature 1 a2 R(3) = −4∇2 C − ∇2 E . 3 • The comoving curvature perturbation is 1 2 R≡ C− ∇ E 3 qi = Bi = 0 or, in gauge-invariant form 1 R = C − ∇2 E + H(v + B) . 3 • Evaluated in Newtonian gauge, this is H(Φ0 + HΦ) R = −Φ + Hv = −Φ − . 4πGa2 (¯ ρ + P¯ ) • Using the Einstein equations, we can show that (see lecture notes) ¯0 P¯ 0 2 P 2 0 2 ¯ −4πGa (¯ ρ + P )R = 4πGa H δP − 0 δρ + H 0 ∇ Φ ρ¯ ρ¯ The first term on the r.h.s. vanishes for adiabatic perturbations and we get 2 d ln R k kH ∼ −−−−−→ 0 d ln a H i.e. R is “conserved on superhorizon scales”!

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