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KIPMU Set 2: Perturbation Theory KIPMU Set 2: Perturbation Theory Wayne Hu 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 rµT = 0 • Preserve general covariance by keeping all free variables: 10 for each symmetric 4×4 tensor 1 2 3 4 5 6 7 8 9 10 Metric Tensor • Useful to think in a 3 + 1 language since there are preferred spatial surfaces where the stress tensor is nearly homogeneous • In general this is an Arnowitt-Deser-Misner (ADM) split • Specialize to the case of a nearly 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). 6K (3)R = a2 Metric Tensor • First define the slicing (lapse function A, shift function Bi) g00 = −a−2(1 − 2A) ; g0i = −a−2Bi ; A defines the lapse of proper time between 3-surfaces whereas Bi defines the threading or relationship between the 3-coordinates of the surfaces • This absorbs 1+3=4 free variables in the metric, remaining 6 is in the spatial surfaces which we parameterize as ij −2 ij ij ij g = a (γ − 2HLγ − 2HT ) : ij here (1) HL a perturbation to the scale factor; (5) HT a trace-free distortion to spatial metric Curvature Perturbation • Curvature perturbation on the 3D slice 4 2 δ[(3)R] = − r2 + 3K H + r r Hij a2 L a2 i j T • When HT = 0 we will often loosely refer to HL as the “curvature perturbation” • It is easier to work with a dimensionless quantity • First example of a 3-scalar - SVT decomposition Matter Tensor • Likewise expand the matter stress energy tensor around a homogeneous density ρ and pressure p: 0 T 0 = −ρ − δρ; i i T0 = −(ρ + p)v ; 0 T i = (ρ + p)(vi − Bi) ; i i i T j = (p + δp)δ j + pΠ j ; • (1) δρ a density perturbation; (3) vi a vector velocity, (1) δp a pressure perturbation; (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. scalar fields, cosmological defects, exotic dark energy. Counting Variables 20 Variables (10 metric; 10 matter) −10 Einstein equations −4 Conservation equations +4 Bianchi identities −4 Gauge (coordinate choice 1 time, 3 space) —— 6 Free Variables • Without loss of generality these can be taken to be the6 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. Homogeneous Einstein Equations • Einstein (Friedmann) equations: 1 da2 K 8πG 1 a_ 2 = − + ρ [= ] a dt a2 3 a a 1 d2a 4πG 1 d a_ = − (ρ + 3p) [= ] a dt2 3 a2 dη a so that w ≡ p/ρ < −1=3 for acceleration µ • Conservation equation r Tµν = 0 implies ρ_ a_ = −3(1 + w) ρ a overdots are conformal time but equally true with coordinate time Homogeneous Einstein Equations • Counting exercise: 20 Variables (10 metric; 10 matter) −17 Homogeneity and Isotropy −2 Einstein equations −1 Conservation equations +1 Bianchi identities —— 1 Free Variables without loss of generality choose ratio of homogeneous & isotropic component of the stress tensor to the density w(a) = p(a)/ρ(a). Scalar, Vector, Tensor • In linear perturbation theory, perturbations may be separated by their transformation properties under 3D rotation and translation. • The eigenfunctions of the Laplacian operator form a complete set r2Q(0) = −k2Q(0) S ; 2 (±1) 2 (±1) r Qi = −k Qi V ; 2 (±2) 2 (±2) r Qij = −k Qij T ; • Vector and tensor modes satisfy divergence-free and transverse-traceless conditions i (±1) r Qi = 0 i (±2) r Qij = 0 ij (±2) γ Qij = 0 Vector and Tensor Quantities • A scalar mode carries with it associated vector (curl-free) and tensor (longitudinal) quantities • A vector mode carries and associated tensor (trace and divergence free) quantities • A tensor mode has only a tensor (trace and divergence free) • These are built from the mode basis out of covariant derivatives and the metric (0) −1 (0) Qi = −k riQ ; 1 Q(0) = (k−2r r + γ )Q(0) ; ij i j 3 ij 1 Q(±1) = − [r Q(±1) + r Q(±1)] ; ij 2k i j j i Perturbation k-Modes • For the kth eigenmode, the scalar components become (0) (0) A(x)= A(k) Q ;H L(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 ;v i(x)= v (k) Qi ; m=−1 m=−1 and the tensors components 2 2 X (m) (m) X (m) (m) HT ij(x)= HT (k) Qij ; Πij(x)= Π (k) Qij ; m=−2 m=−2 • Note that the curvature perturbation only involves scalars 4 1 δ[(3)R] = (k2 − 3K)(H(0) + H(0))Q(0) a2 L 3 T Spatially Flat Case • For a spatially flat background metric, harmonics are related to plane waves: Q(0) = exp(ik · x) (±1) −i Q = p (e^1 ± ie^2)iexp(ik · x) i 2 r3 Q(±2) = − (e^ ± ie^ ) (e^ ± ie^ ) exp(ik · x) ij 8 1 2 i 1 2 j where e^3 k k. Chosen as spin states, cf. polarization. • For vectors, the harmonic points in a direction orthogonal to k suitable for the vortical component of a vector Spatially Flat Case • Tensor harmonics are the transverse traceless gauge representation • Tensor amplitude related to the more traditional h+[(e1)i(e1)j − (e2)i(e2)j] ; h×[(e1)i(e2)j + (e2)i(e1)j] as p (∓2) h+ ± ih× = − 6HT (±2) • HT proportional to the right and left circularly polarized amplitudes of gravitational waves with a normalization that is convenient to match the scalar and vector definitions 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 Free Variables without loss of generality choose scalar components of the stress tensor δp, Π . Covariant Scalar Equations • Einstein equations (suppressing 0) superscripts 1 a_ a_ a_ (k2 − 3K)[H + H ] − 3( )2A + 3 H_ + kB = L 3 T a a L a = 4πGa2δρ; 00 Poisson Equation 1 d a_ k2(A + H + H ) + + 2 (kB − H_ ) L 3 T dη a T = 8πGa2pΠ ; ij Anisotropy Equation a_ 1 K A − H_ − H_ − (kB − H_ ) a L 3 T k2 T = 4πGa2(ρ + p)(v − B)=k ; 0i Momentum Equation " # 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) ; ii Acceleration Equation 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 rµG = 0 via the Bianchi identities. • Related to the ability to choose a coordinate system or “gauge” to represent the perturbations. 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 Free Variables 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 Free Variables wlog choose tensor components of the stress tensor Π(±2). Photon Moments • The photon stress-energy tensor is given by moments of distribution function Z d3q qµq T µ = g ν f ν (2π)3 E(q) (0) • ` = 0 Boltzmann moment is continuity equation: Θ0 = δργ=4ργ (m) (m) • ` = 1 moment is Navier-Stokes equation with Θ1 = vγ and 5 Θ(0) = (1 − 3K=k2)1=2Π(0) 2 12 γ and similarly up to normalization for vector and tensor cases (m) • Either extract the source S` from these associations or by noting that the geodesic equation gives the redshifting term q_ a_ 1 = − − ninjH_ − H_ + niB_ − n^ · rA q a 2 T ij L i Source Terms (m) • Temperature source terms Sl (rows ±|mj; flat assumption 0 1 (0) _ (0) (0) _ (0) (0) 2 _ (0) τ_Θ0 − HL τv_ b + B τP_ − 3 HT p B (±1) (±1) C B 0τv _ + B_ (±1) τP_ (±1) − 3 H_ C @ b 3 T A (±2) _ (±2) 0 0τP _ − HT where τ_ ≡ neσT a terms are Thomson scattering sources with 1 p P (m) ≡ (Θ(m) − 6E(m)) 10 2 2 • Polarization source terms are generated through Thomson scattering from temperature quadrupoles dσ 3 T = jE^ 0 · E^j2σ ; dΩ 8π T Polarized Source Term .
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