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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.
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