Linear Perturbation Theory

Lecture 2: Linear Perturbation Theory

Structure Formation and the Dark Sector

Wayne Hu Trieste, June 2002 Outline • Covariant Perturbation Theory • Scalar, Vector, Tensor Decomposition • Linearized Einstein-Conservation Equations • Dark (Multi) Components • Gauge • Applications:

Bardeen Curvature Baryonic wiggles Scalar Fields Parameterizing dark components Transfer function Massive neutrinos Sachs-Wolfe Effect Dark energy COBE normalization Covariant Perturbation Theory • Covariant = takes same form in all coordinate systems • Invariant = takes the same value in all coordinate systems • Fundamental equations: Einstein equations, covariant conservation of stress-energy tensor:

Gµν = 8πGTµν µν ∇µT = 0 • Preserve general covariance by keeping all degrees of freedom: 10 for each symmetric 4×4 tensor

1 2 3 4 5 6 7 8 9 10 Metric Tensor • Expand the metric tensor around the general FRW metric

2 2 g00 = −a , gij = a γij .

where the “0” component is conformal time η = dt/a and γij is a 2 spatial metric of constant curvature K = H0 (Ωtot − 1). • Add in a general perturbation (Bardeen 1980)

g00 = −a−2(1 − 2A) , g0i = −a−2Bi , ij −2 ij ij ij g = a (γ − 2HLγ − 2HT ) .

i • (1) A ≡ a scalar potential; (3) B a vector shift, (1) HL a ij perturbation to the spatial curvature; (6) HT a trace-free distortion to spatial metric = (10) Matter Tensor • Likewise expand the matter stress energy tensor around a homogeneous density ρ and pressure p:

0 T 0 = −ρ − δρ , 0 T i = (ρ + p)(vi − Bi) , i i T0 = −(ρ + p)v , i i i T j = (p + δp)δ j + pΠ j ,

• (1) δρ a density perturbation; (3) vi a vector velocity, (1) δp a

pressureperturbation;(5)Πij an anisotropic stress perturbation • So far this is fully general and applies to any type of matter or coordinate choice including non-linearities in the matter, e.g. cosmological defects. Counting DOF’s

20 Variables (10 metric; 10 matter) −10 Einstein equations −4 Conservation equations +4 Bianchi identities −4 Gauge (coordinate choice 1 time, 3 space) —— 6 Degrees of freedom • Without loss of generality these can be taken to be the 6 components of the matter stress tensor • For the background, specify p(a) or equivalently w(a) ≡ p(a)/ρ(a) the equation of state parameter. Scalar, Vector, Tensor • In linear perturbation theory, perturbations may be separated by their transformation properties under rotation and translation. • The eigenfunctions of the Laplacian operator form a complete set

∇2Q(0) = −k2Q(0) S , 2 (±1) 2 (±1) ∇ Qi = −k Qi V , 2 (±2) 2 (±2) ∇ Qij = −k Qij T , and functions built out of covariant derivatives and the metric

(0) −1 (0) Qi = −k ∇iQ , 1 Q(0) = (k−2∇ ∇ − γ )Q(0) , ij i j 3 ij 1 Q(±1) = − [∇ Q(±1) + ∇ Q(±1)] , ij 2k i j j i Spatially Flat Case • For a spatially ﬂat background metric, harmonics are related to plane waves:

Q(0) = exp(ik · x)

(±1) −i Q = √ (eˆ1 ± ieˆ2)iexp(ik · x) i 2 s 3 Q(±2) = − (eˆ ± ieˆ ) (eˆ ± ieˆ ) exp(ik · x) ij 8 1 2 i 1 2 j

where eˆ3 k k. • For vectors, the harmonic points in a direction orthogonal to k suitable for the vortical component of a vector • For tensors, the harmonic is transverse and traceless as appropriate for the decompositon of gravitational waves Perturbation k-Modes • For the kth eigenmode, the scalar components become

(0) (0) A(x) = A(k) Q , HL(x) = HL(k) Q , δρ(x) = δρ(k) Q(0) , δp(x) = δp(k) Q(0) , the vectors components become

1 1 X (m) (m) X (m) (m) Bi(x) = B (k) Qi , vi(x) = v (k) Qi , m=−1 m=−1 and the tensors components

2 X (m) (m) HT ij(x) = HT (k) Qij , m=−2 2 X (m) (m) Πij(x) = Π (k) Qij , m=−2 Homogeneous Einstein Equations • Einstein (Friedmann) equations:

1 da!2 8πG = ρ a dt 3 1 d2a 4πG = − (ρ + 3p) a dt2 3 so that w ≡ p/ρ < −1/3 for acceleration µ • Conservation equation ∇ Tµν=0implies

ρ˙ a˙ = −3(1 + w) ρ a Homogeneous Einstein Equations • Counting exercise:

20 Variables (10 metric; 10 matter) −17HomogeneityandIsotropy −2 Einstein equations −1 Conservation equations +1 Bianchi identities —— 1 Degree of freedom

• without loss of generality choose ratio of homogeneous & isotropic component of the stress tensor to the density w(a) = p(a)/ρ(a). Covariant Scalar Equations • Einstein equations (suppressing 0) superscripts (Hu & Eisenstein 1999):

1 a˙ 1 (k2 − 3K)[H + H + (kB − H˙ )] L 3 T a k2 T a˙ = 4πGa2 δρ + 3 (ρ + p)(v − B)/k , Poisson Equation a 1 d a˙ k2(A + H + H ) + + 2 (kB − H˙ ) L 3 T dη a T = 8πGa2pΠ , a˙ 1 K A − H˙ − H˙ − (kB − H˙ ) a L 3 T k2 T = 4πGa2(ρ + p)(v − B)/k , " # a¨ a˙ 2 a˙ d k2 d a˙ 1 2 − 2 + − A − + (H˙ + kB) a a a dη 3 dη a L 3 1 = 4πGa2(δp + δρ) . 3 Covariant Scalar Equations • Conservation equations: continuity and Navier Stokes

d a˙ a˙ + 3 δρ + 3 δp = −(ρ + p)(kv + 3H˙ ) , dη a a L d a˙ (v − B) 2 K + 4 (ρ + p) = δp − (1 − 3 )pΠ + (ρ + p)A , dη a k 3 k2 µν • Equations are not independent since ∇µG = 0 via the Bianchi identities. • Related to the ability to choose a coordinate system or “gauge” to represent the perturbations. Covariant Scalar Equations • DOF counting exercise

8 Variables (4 metric; 4 matter) −4 Einstein equations −2 Conservation equations +2 Bianchi identities −2 Gauge (coordinate choice 1 time, 1 space) —— 2 Degrees of freedom

• without loss of generality choose scalar components of the stress tensor δp, Π . Covariant Vector Equations • Einstein equations

2 (±1) ˙ (±1) (1 − 2K/k )(kB − HT ) = 16πGa2(ρ + p)(v(±1) − B(±1))/k , d a˙ + 2 (kB(±1) − H˙ (±1)) dη a T = −8πGa2pΠ(±1) .

• Conservation Equations

d a˙ + 4 [(ρ + p)(v(±1) − B(±1))/k] dη a 1 = − (1 − 2K/k2)pΠ(±1) , 2 • Gravity provides no source to vorticity → decay Covariant Vector Equations • DOF counting exercise

8 Variables (4 metric; 4 matter) −4 Einstein equations −2 Conservation equations +2 Bianchi identities −2 Gauge (coordinate choice 1 time, 1 space) —— 2 Degrees of freedom

• without loss of generality choose vector components of the stress tensor Π(±1). Covariant Tensor Equation • Einstein equation

d2 a˙ d + 2 + (k2 + 2K) H(±2) = 8πGa2pΠ(±2) . dη2 a dη T • DOF counting exercise

4 Variables (2 metric; 2 matter) −2 Einstein equations −0 Conservation equations +0 Bianchi identities −0 Gauge (coordinate choice 1 time, 1 space) —— 2 Degrees of freedom • wlog choose tensor components of the stress tensor Π(±2). Arbitrary Dark Components • Total stress energy tensor can be broken up into individual pieces • Dark components interact only through gravity and so satisfy separate conservation equations • Einstein equation source remains the sum of components. • To specify an arbitrary dark component, give the behavior of the stress tensor: 6 components: δp, Π(i), where i = −2, ..., 2. • Many types of dark components (dark matter, scalar fields, massive neutrinos,..) have simple forms for their stress tensor in terms of the energy density, i.e. described by equations of state. • An equation of state for the background w = p/ρ is not sufficient to determine the behavior of the perturbations. Gauge • Metric and matter fluctuations take on different values in different coordinate system • No such thing as a “gauge invariant” density perturbation! • General coordinate transformation:

η˜ = η + T x˜i = xi + Li

free to choose (T,Li) to simplify equations or physics. Decompose these into scalar and vector harmonics.

• Gµν and Tµν transform as tensors, so components in different frames can be related Gauge Transformation • Scalar Metric: a˙ A˜ = A − T˙ − T , a B˜ = B + L˙ + kT , k a˙ H˜ = H − L − T , L L 3 a

H˜T = HT + kL , • Scalar Matter (Jth component):

δρ˜J = δρJ − ρ˙J T ,

δp˜J = δpJ − p˙J T ,

v˜J = vJ + L˙ , • Vector:

˜(±1) (±1) ˙ (±1) ˜ (±1) (±1) (±1) (±1) (±1) ˙ (±1) B = B + L , HT = HT + kL , v˜J = vJ + L , Gauge Dependence of Density Background evolution of the density induces a density fluctuation • from a shift in the time coordinate Common Scalar Gauge Choices • A coordinate system is fully speciﬁed if there is an explicit prescription for (T,Li) or for scalars (T,L) • Newtonian:

B˜ = H˜T = 0 Ψ ≡ A˜ (Newtonian potential)

Φ ≡ H˜L (Newtonian curvature)

L = −HT /k 2 T = −B/k + H˙ T /k

Good: intuitive Newtonian like gravity; matter and metric algebraically related; commonly chosen for analytic CMB and lensing work Bad: numerically unstable Example: Newtonian Reduction

• In the general equations, set B = HT = 0:

a˙ (k2 − 3K)Φ = 4πGa2 δρ + 3 (ρ + p)v/k a k2(Ψ+Φ)=8πGa2pΠ

so Ψ = −Φ if anisotropic stress Π = 0 and

d a˙ a˙ + 3 δρ + 3 δp = −(ρ + p)(kv + 3Φ˙ ) , dη a a d a˙ 2 K + 4 (ρ + p)v = kδp − (1 − 3 )p kΠ + (ρ + p) kΨ , dη a 3 k2 • Competition between stress (pressure and viscosity) and potential gradients Relativistic Term in Continuity • Continuity equation contains relativistic term from changes in the spatial curvature – perturbation to the scale factor • For w=0 (matter), simply density dilution; for w=1/3 (radiation) density dilution plus (cosmological) redshift

• a.k.a. ISW effect – photon redshift from change in grav. potential Common Scalar Gauge Choices • Comoving:

˜ 0 B =v ˜ (Ti = 0)

HT = 0 ξ = A˜

ζ = H˜L (Bardeen curvature) T = (v − B)/k

L = −HT /k

Good: ζ is conserved if stress ﬂuctuations negligible, e.g. above the horizon if |K| H2

a˙ δp 2 3K p ζ˙ + Kv/k = − + 1 − Π → 0 a ρ + p 3 k2 ρ + p Bad: explicitly relativistic choice Common Scalar Gauge Choices • Synchronous:

A˜ = B˜ = 0 1 η ≡ −H˜ − H˜ L L 3 T

hT = H˜T or h = 6HL Z −1 −1 T = a dηaA + c1a Z L = − dη(B + kT ) + c2

Good: stable, the choice of numerical codes

Bad: residual gauge freedom in constants c1, c2 must be speciﬁed as an initial condition, intrinsically relativistic. Common Scalar Gauge Choices • Spatially Unperturbed:

H˜L = H˜T = 0

L = −HT /k A˜ , B˜ = metric perturbations a˙ −1 1 T = H + H a L 3 T Good: eliminates spatial metric in evolution equations; useful in inflationary calculations (Mukhanov et al) Bad: intrinsically relativistic. • Caution: perturbation evolution is governed by the behavior of stress fluctuations and an isotropic stress fluctuation δp is gauge dependent. Hybrid “Gauge Invariant” Approach • With the gauge transformation relations, express variables of one gauge in terms of those in another – allows a mixture in the equations of motion • Example: Newtonian curvature above the horizon. Conservation of the Bardeen-curvature ζ=const. implies:

3 + 3w Φ = ζ 5 + 3w e.g. calculate ζ from inflation determines Φ for any choice of matter content or causal evolution. • Example: Scalar field (“quintessence” dark energy) equations in comoving gauge imply a sound speed δp/δρ = 1 independent of potential V (φ). Solve in synchronous gauge (Hu 1998). Transfer Function Example • Example: Transfer function transfers the initial Newtonian curvature to its value today (linear response theory) Φ(k, a = 1) T (k) = Φ(k, ainit) • Conservation of Bardeen curvature: Newtonian curvature is a constant when stress perturbations are negligible: above the horizon during radiation and dark energy domination, on all scales during matter domination • When stress fluctuations dominate, perturbations are stabilized by the Jeans mechanism • Hybrid Poisson equation: Newtonian curvature, comoving density

perturbation ∆ ≡ (δρ/ρ)com implies Φ decays

(k2 − 3K)Φ = 4πGρ∆ ∼ η−2∆ Transfer Function Example

• Freezing of ∆ stops at ηeq

−2 −2 Φ ∼ (kηeq) ∆H ∼ (kηeq) Φinit

−2 −1 • Transfer function has a k fall-off beyond keq ∼ ηeq

k–2

T ( k ) 0.1

wiggles

0.01 0.0001 0.001 0.01 0.1 1 k (h–1 Mpc) Gauge and the Sachs-Wolfe Effect • Going from comoving gauge, where the CMB temperature perturbation is initially negligible by the Poisson equation, to the Newtonian gauge involves a temporal shift δt = Ψ t • Temporal shift implies a shift in the scale factor during matter domination δa 2 δt = a 3 t • CMB temperature is cooling as T ∝ a • Induced temperature ﬂuctuation δT δa 2 = − = − Ψ T a 3 Gauge and the Sachs-Wolfe Effect • Add the gravitational redshifta photon suffers climbing out of the gravitational potential

δT ! δT 1 = Ψ + = Ψ T obs T 3 COBE Normalization

•Sachs-Wolfe Effect relates the COBE detection to the gravitational potential on the last scattering surface

Last Scattering Surface

1 [Θ + Ψ](nˆ, x)= Ψ(x + Dnˆ,η ) D 3 ∗

D=η0η x, η − 0 0 jl(kD)Yl

η∗ •Decompose the angular and spatial information into normal modes: spherical harmonics for angular, plane waves for spatial

m ` 4π m ikx G (nˆ, x, k)=(i) Y (nˆ)e · . ` − s2`+1 ` COBE Normalization cont.

•Multipole moment decomposition for each k d3k Θ(n, x)= Θ(m)(k)Gm(x, k, n) (2π)3 ` ` Z X`m •Power spectrum is the integral over k modes (m) (m) 3 dk Θ` ∗Θ` C =4π ` 3 D 2 E (2π) m (2` +1) Z X •Fourier transform Sachs-Wolfe source 3 1 d k ik (Dnˆ+x) [Θ+Ψ](nˆ,x)= Ψ(k, η )e · 3 ∗ 3Z (2π) •Decompose plane wave

exp(ikD nˆ)= ( i)` 4π(2` +1)j(kD)Y 0(n), · − ` ` X` q COBE Normalization cont.

•Extract multipole moment, assume a constant potential (0) Θ` 1 = Ψ(k, η )j`(kD) 2` +1 3 ∗ 1 = Ψ(k, η )j (kD) 3 0 ` •Construct angular power spectrum

dk 2 1 2 C`=4π j` (kD) ∆Ψ Z k 9 •For scale invariant potential (n=1), integral reduces to

∞ dx 2 1 j` (x)= Z0 x 2`(`+1) •Log power spectrum = Log potential spectrum / 9 `(` +1) 1 C = ∆2 (n=1) 2π ` 9 Ψ COBE Normalization cont.

•Relate to density fluctuations: Poisson equation and Friedmann eqn. k2Ψ= 4πGa2δρ −3 = H2Ω2 δ −2 0 m •Power spectra relation 4 2 9 H0 2 2 ∆Ψ = Ωm∆δ 4 k •In terms of density fluctuation at horizon and transfer function n+3 2 2 k 2 ∆δ δH T (k) ≡ H0 •For scale invariant potential

`(` +1) 1 C = Ω2 δ2 (n=1) 2π ` 4 m H

COBE Normalization cont.

•Some numbers `(` +1) 1 C = Ω2 δ2 (n=1) 2π ` 4 m H

2 28µK 10 = 10− ! 2.726 106µK ! ≈ × 5 1 δ (2 10− )Ω− H ≈ × m •Detailed calculation from Bunn & White (1997) including decay of potential in low density universe and tilt

2 5 0.785 0.05ln Ωm 0.95(n 1) 0.169(n 1) δ =1.94 10− Ω− − e− − − − H × m Matter Power Spectrum Combine the transfer function with the COBE normalization •

101

] 1 2 no on-linear scale -1

P ( k ) / 2 π 10 3

( k ) [ = 10-2 2 ∆

10-3

0.01 0.1 1 k (h Mpc-1) Matter Power Spectrum Usually plotted as the power spectrum, not log-power spectrum • 105 3 104 M p c ) - 1

P ( k ) h 103

0.01 0.1 1 k (h Mpc-1) Galaxy Power Spectrum Data Galaxy clustering tracks the dark matter – but as a biased tracer • 105 2dF PSCz <1994 3 104 Mpc) -1

P ( k ) h 103

0.01 0.1 1 k (h Mpc-1) Galaxy Power Spectrum Data Each galaxy population has different linear bias in linear regime • 105 2dF PSCz <1994 3 104 norm Mpc)

-1 adjusted

P ( k ) h 103

0.01 0.1 1 k (h Mpc-1) Acoustic Oscillations • Example: Stabilization accompanied by acoustic oscillations • Photon-baryon system - under rapid scattering

d a˙ a˙ + 3 δρ + 3 δp = −(ρ + p )(kv + 3Φ˙ ) , dη a γb a γb γb γb γb d a˙ + 4 [(ρ + p )v /k] = δp + (ρ + p)Ψ , dη a γb γb γb γb

or with Θ = δργ/ργandR=3ρb/ργ

k Θ˙ = − v − Φ˙ 3 γb R˙ 1 v˙ = − v + kΘ + kΨ γb 1 + R γb 1 + R • Forced oscillator equation – see Zaldarriaga’s talks • Anisotropic stresses and entropy generation through non-adiabatic stress Silk damps ﬂuctuations Silk 1968 Acoustic Peaks in the Matter ¥ Baryon density & velocity oscillates with CMB

¥ Baryons decouple at τ / R ~ 1, the end of Compton drag epoch Decoupling: δ (drag) ∼ V (drag), but not frozen ¥ . b b δ ¥ Continuity: b =ÐkVb Velocity Overshoot Dominates: δ ∼ V (drag) kη >> δ (drag) ¥ b b b ¥ Oscillations π/2 out of phase with CMB ¥ Infall into potential wells (DC component)

End of Drag Epoch Velocity Overshoot + Infall

Hu & Sugiyama (1996) Features in the Power Spectrum ¥ Features in the linear power spectrum ¥ Break at sound horizon ¥ Oscillations at small scales; washed out by nonlinearities

nonlinear scale (arbitrary norm.) ) k (

P 0.1 Eisenstein & Hu (1998) numerical

0.01 0.1 k (h Mpc-1) Combining Features in LSS + CMB ¥ Consistency check on thermal history and photonÐbaryon ratio → Ð1 ¥ Infer physical scale lpeak(CMB) kpeak(LSS) in Mpc

ΛCDM 8

(arbitrary norm.) 4 k3Pγ(k) Power 2

0.05 0.1 Ð1 Eisenstein, Hu & Tegmark (1998) k (Mpc ) Hu, Eisenstein, Tegmark & White (1998) Combining Features in LSS + CMB ¥ Consistency check on thermal history and photonÐbaryon ratio → Ð1 ¥ Infer physical scale lpeak(CMB) kpeak(LSS) in Mpc Ð1 → ¥ Measure in redshift survey kpeak(LSS) in h Mpc h

ΛCDM 8

(arbitrary norm.) 4 k3Pγ(k) h Power 2

Pm(k) 0.05 0.1 Ð1 Eisenstein, Hu & Tegmark (1998) k (Mpc ) Hu, Eisenstein, Tegmark & White (1998) Parameterizing Dark Components equation of state sound speed viscosity 2 2 wg ceff cvis Prototypes: • Cold dark matter 0 0 0 (WIMPs) • Hot dark matter 1/3→0 (light neutrinos) • Cosmological constant –1 arbitrary arbitrary (vacuum energy) Exotica: • Quintessence variable 1 0 (slowly-rolling scalar field) • Decaying dark matter 1/3→0→1/3 (massive neutrinos) • Radiation backgrounds 1/3 1/3 0→1/3 (rapidly-rolling scalar field, NBR) scale • Ultra-light fuzzy dark mat. 0 dependent 0

Hu (1998) Massive Neutrinos • Relativistic stresses of a light neutrino slow the growth of structure • Neutrino species with cosmological abundance contribute to matter 2 as Ωνh = mν/94eV, suppressing power as ∆P/P ≈ −8Ων/Ωm Massive Neutrinos • Current data from 2dF galaxy survey indicates mν<1.8eV assuming a ΛCDM model with parameters constrained by the CMB.

105 2dF 3 104 M p c ) - 1

P ( k ) h 103

0.01 0.1 1 k (h Mpc-1) Dark Energy Stress & Smoothness ¥ Raising equation of state increases redshift of dark energy domination and raises large scale anisotropies ¥ Lowering the sound speed increases clustering and reduces ISW effect at large angles

10Ð10 Total ISW Effect

c eff =1

Ð11 c Power 10 eff =1/3

w=Ð2/3

w=Ð1

10Ð12 10 100 Coble et al. (1997) Hu (1998); Hu (2001) l Caldwell et al. (1998) LensingÐCMB Temperature Correlation ¥ Any correlation is a direct detection of a smooth energy density component through the ISW effect ¥ Show dark energy smooth >5-6 Gpc scale, test quintesence

"Perfect" Planck 10Ð9

10Ð10 cross power

10Ð11 10 100 1000 Hu (2001); Hu & Okamoto (2001) L Summary • In linear theory, evolution of fluctuations is completely defined once the stresses in the matter fields are specified. • Stresses and their effects take on simple forms in particular coordinate or gauge choices, e.g. the comoving gauge. • Gauge covariant equations can be used to take advantage of these simplifications in an arbitrary frame. • Curvature (potential) fluctuations remain constant in the absence of stresses. • Evolution can be used to test the nature of the dark components, e.g. massive neutrinos and the dark energy by measuring the matter power spectrum. • Problem: luminous tracers of the matter clustering are biased – next lecture.