Ast 448 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 ∇µT = 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] = − ∇2 + 3K H + ∇ ∇ 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 ∇ 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 ∇2Q(0) = −k2Q(0) S , 2 (±1) 2 (±1) ∇ Qi = −k Qi V , 2 (±2) 2 (±2) ∇ Qij = −k Qij T , • Vector and tensor modes satisfy divergence-free and transverse-traceless conditions
i (±1) ∇ Qi = 0 i (±2) ∇ 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 ∇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 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 = √ (eˆ1 ± ieˆ2)iexp(ik · x) i 2 r3 Q(±2) = − (eˆ ± ieˆ ) (eˆ ± ieˆ ) exp(ik · x) ij 8 1 2 i 1 2 j
where eˆ3 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
√ (∓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 ∇µG = 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˙ − nˆ · ∇A q a 2 T ij L i Source Terms (m) • Temperature source terms Sl (rows ±|m|; flat assumption
(0) ˙ (0) (0) ˙ (0) (0) 2 ˙ (0) τ˙Θ0 − HL τv˙ b + B τP˙ − 3 HT √ (±1) (±1) 0τv ˙ + B˙ (±1) τP˙ (±1) − 3 H˙ b 3 T (±2) ˙ (±2) 0 0τP ˙ − HT
where τ˙ ≡ neσT a terms are Thomson scattering sources with 1 √ P (m) ≡ (Θ(m) − 6E(m)) 10 2 2 • Polarization source terms are generated through Thomson scattering from temperature quadrupoles
dσ 3 T = |Eˆ 0 · Eˆ|2σ , dΩ 8π T Polarized Source Term
. • Heuristically, incoming e'
photon electric field accelerates e- Thomson electron in the same direction e' Scattering Θ and radiates out a photon whose polarization is given by projection e e of this direction in transverse plane • Consider scattering by 90 degrees: photons coming in from the left/right supply one polarization state, in/out of page the other • A quadrupole temperature anisotropy in left/right vs top/bottom leads to net linear polarization • Polarization source term √ (m) (m) (m) E` = −τ˙ 6P δ`,2 B` = 0 Quadrupole Source Term
. k ê2
ê1 v Gravity Waves (Tensors) Compression (Scalars) Vorticity m=0 m=1 (Vectors) m=2 v
• Each type leads to quadrupoles with different azimuthal symmetry, polarization aligned with cold lobe • For the vector and tensor cases, the breaking of azimuthal symmetry leads to B-mode polarization Gravitational Wave Observability • A gravitational wave makes a quadrupolar (transverse-traceless) distortion to metric • Just like the scale factor or spatial curvature, a temporal variation in its amplitude leaves a residual temperature variation in CMB photons – here anisotropic • Before recombination, anisotropic variation is eliminated by scattering • Gravitational wave temperature effect drops sharply at the horizon scale at recombination - distorts the spectrum ˙ • Source to polarization goes as τ/˙ HT and peaks at the horizon not damping scale • B modes since symmetry of plane wave broken by the transverse nature of gravity wave polarization Truncated Hierarchy • CMBFast introduced the hybrid truncated hierarchy, integral solution technique • Formal integral solution contains sources that are not external to system but defined through the Boltzmann hierarchy itself • Solution: recall the fluid approximation where interactions suppress all but the ` = 0 (density) and ` = 1 (velocity) terms • CMBFast extends this idea by solving a truncated hierarchy of equations, e.g. out to ` = 25 with non-reflecting boundary conditions
• For completeness, we explicitly derive the scattering source term via polarized radiative transfer in the last part of the notes Polarized Radiative Transfer
• Define a specific intensity “vector”: Iν = (Θk, Θ⊥, U, V ) where
Θ = Θk + Θ⊥, Q = Θk − Θ⊥
dI ν =τ ˙(S − I ) dη ν ν
. • Thomson collision e'
based on differential cross section e- Thomson e' Scattering dσ 3 Θ T = |Eˆ 0 · Eˆ|2σ , dΩ 8π T e e Polarized Radiative Transfer • Eˆ 0 and Eˆ denote the incoming and outgoing directions of the electric field or polarization vector.
• Thomson scattering by 90 deg: Θ⊥ → Θ⊥ but Θk does not scatter • More generally if β is the scattering angle cos2 β 0 0 0 3 Z 0 1 0 0 0 0 Sν = dΩ Iν 8π 0 0 cos β 0 0 0 0 cos β
• But to calculate Stokes parameters in a fixed coordinate system must rotate into the scattering basis, scatter and rotate back out to the fixed coordinate system Thomson Collision Term • The U → U 0 transfer follows by writing down the polarization vectors in the 45◦ rotated basis
ˆ 1 ˆ ˆ ˆ 1 ˆ ˆ E1 = √ (Ek + E⊥), E2 = √ (Ek − E⊥) 2 2 • Define the temperature in this basis
ˆ ˆ 2 0 ˆ ˆ 2 0 Θ1 ∝ |E1 · E1| Θ1 + |E1 · E2| Θ2 1 1 ∝ (cos β + 1)2Θ0 + (cos β − 1)2Θ0 4 1 4 2 ˆ ˆ 2 0 ˆ ˆ 2 0 Θ2 ∝ |E2 · E2| Θ2 + |E2 · E1| Θ1 1 1 ∝ (cos β + 1)2Θ0 + (cos β − 1)2Θ0 4 2 4 1 0 0 or Θ1 − Θ2 ∝ cos β(Θ1 − Θ2) Scattering Matrix • Transfer matrix of Stokes state T ≡ (Θ, Q + iU, Q − iU)
T ∝ S(β)T0
2 1 2 1 2 cos β + 1 − 2 sin β − 2 sin β 3 S(β) = − 1 sin2 β 1 (cos β + 1)2 1 (cos β − 1)2 4 2 2 2 1 2 1 2 1 2 − 2 sin β 2 (cos β − 1) 2 (cos β + 1) normalization factor of 3 is set by photon conservation in scattering Scattering Matrix • Transform to a fixed basis, by a rotation of the incoming and outgoing states T = R(ψ)T where 1 0 0 R(ψ) = 0 e−2iψ 0 0 0 e2iψ
giving the scattering matrix
R(−γ)S(β)R(α) = (1) √ 0 0 q 3 −2 q 3 2 Y2 (β, α) + 2 5Y0 (β, α) − 2 Y2 (β, α) − 2 Y2 (β, α) r √ 1 4π 0 2iγ −2 2iγ 2 2iγ − 6 2Y2 (β, α)e 3 2Y2 (β, α)e 3 2Y2 (β, α)e 2 5 √ 0 −2iγ −2 −2iγ 2 −2iγ − 6 −2Y2 (β, α)e 3 −2Y2 (β, α)e 3 −2Y2 (β, α)e (2) Addition Theorem for Spin Harmonics • Spin harmonics are related to rotation matrices as r 2` + 1 Y m(θ, φ) = D` (φ, θ, 0) s ` 4π −ms Note: for explicit evaluation sign convention differs from usual (e.g. Jackson) by (−1)m • Multiplication of rotations
X ` ` ` Dmm00 (α2, β2, γ2)Dm00m(α1, β1, γ1) = Dmm0 (α, β, γ) m00 • Implies r X 2` + 1 Y m∗(θ0, φ0) Y m(θ, φ) = (−1)s1−s2 Y −s1 (β, α)eis2γ s1 ` s2 ` 4π s2 ` m Sky Basis • Scattering into the state (rest frame)
Z dnˆ0 C [T] =τ ˙ R(−γ)S(β)R(α)T(nˆ0) , in 4π 2 Z dnˆ0 1 Z X =τ ˙ (Θ0, 0, 0) + τ˙ dnˆ0 P(m)(nˆ, nˆ0)T(nˆ0) . 4π 10 m=−2 where the quadrupole coupling term is P(m)(nˆ, nˆ0) =
m∗ 0 m q 3 m∗ 0 m q 3 m∗ 0 m Y2 (nˆ ) Y2 (nˆ) − 2 2Y2 (nˆ ) Y2 (nˆ) − 2 −2Y2 (nˆ ) Y2 (nˆ) √ m∗ 0 m m∗ 0 m m∗ 0 m − 6Y2 (nˆ ) 2Y2 (nˆ) 3 2Y2 (nˆ ) 2Y2 (nˆ) 3 −2Y2 (nˆ ) 2Y2 (nˆ) , √ m∗ 0 m m∗ 0 m m∗ 0 m − 6Y2 (nˆ ) −2Y2 (nˆ) 3 2Y2 (nˆ ) −2Y2 (nˆ) 3 −2Y2 (nˆ ) −2Y2 (nˆ) expression uses angle addition relation above. We call this term
CQ. Scattering Matrix • Full scattering matrix involves difference of scattering into and out of state
C[T] = Cin[T] − Cout[T] • In the electron rest frame Z dnˆ0 C[T] =τ ˙ (Θ0, 0, 0) − τ˙T + C [T] 4π Q which describes isotropization in the rest frame. All moments have −τ e suppression except for isotropic temperature Θ0. Transformation into the background frame simply induces a dipole term Z dnˆ0 C[T] =τ ˙ nˆ · v + Θ0, 0, 0 − τ˙T + C [T] b 4π Q 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. Separate Universes • Geometry of the gauge or time slicing and spatial threading • For perturbations larger than the horizon, a local observer should just see a different (separate) FRW universe • Scalar equations should be equivalent to an appropriately remapped Friedmann equation • Unit normal vector N µ to constant time hypersurfaces µ µ Nµdx = N0dη, N Nµ = −1, to linear order in metric
N0 = −a(1 + AQ),Ni = 0 N 0 = a−1(1 − AQ),N i = −BQi • Expansion of spatial volume per proper time is given by 4-divergence
k 3 ∇ N µ ≡ θ = 3H(1 − AQ) + BQ + H˙ Q µ a a L Shear and Acceleration
• Other pieces of ∇νNµ give the vorticity, shear and acceleration
1 ∇ N ≡ ω + σ + θP − a N ν µ µν µν 3 µν µ ν with
Pµν = gµν + NµNν α β ωµν = Pµ Pν (∇βNα − ∇αNβ) 1 1 σ = P αP β(∇ N + ∇ N ) − θP µν 2 µ ν β α α β 3 µν α aµ = (∇αNµ)N
ν projection PµνN = 0, trace free antisymmetric vorticity, symmetric shear and acceleration Shear and Acceleration
• Vorticity ωµν = 0, σ00 = σ0i = 0 = a0 • Remaining perturbed quantities are the spatial shear and acceleration
σij = a(H˙ T − kB)Qij
ai = −kAQi
• A convenient choice of coordinates might be shear free ˙ HT − kB = 0 • A alone is related to the perturbed acceleration Separate Universes • So the e-foldings of the expansion are given by dτ = (1 + AQ)adη
Z 1 N = dτ θ 3 Z a˙ 1 = dη + H˙ Q + kBQ a L 3
Thus if kB can be ignored as k → 0 then HL plays the role of a local change in the scale factor, more generally B plays the role of Eulerian → Lagrangian coordinates.
• Change in HL between separate universes related to change in number of e-folds: so called δN approach, simplifying equations by using N as time variable to absorb local scale factor effects ˙ • We shall see that for adiabatic perturbations p(ρ) that HL = 0 outside horizon for an appropriate choice of slicing – plays an important role in simplifying calculations Separate Universes ˙ • Choosing coordinates where HL + kB/3 = 0 (defines the slicing), the e-folding remains unperturbed, we get that the 00 Einstein equations at k → 0 are
a˙ 2 1 k2 − 3K 4πG − A + (H + H /3) = a2δρ a 3 a2 L T 3 which is to be compared to the Friedmann equation K 8πG H2 + = ρ a2 3 Noting that H = H¯ (1 − AQ) and using the perturbation to (3)R
δK 8πG 2δHH¯ + = δρQ a2 3 2 k2 − 3K 8πG −2AQH¯ 2 + (H + H /3)Q = δρQ 3 a2 L T 3 a˙ 2 1 k2 − 3K 4πG − A + (H + H /3) = δρ a 3 a2 L T 3 Separate Universes • And the space-space piece
" # a¨ a˙ 2 a˙ d 4πG 2 − 2 + A = a2(δp + 3δρ) a a a dη 3
which is to be compared with the acceleration equation
d 4πG (aH) = − a2(p + 3ρ) dη 3 again expanding H = H¯ (1 − AQ) and also dη = (1 + AQ)dη¯
d d (aH) = (1 − AQ) (aH¯ )[1 − AQ] dη dη¯ d d a˙ a˙ d ≈ (aH¯ ) − 2AQ + AQ dη¯ dη¯ a a dη¯ Separate Universes ˙ • Finally the continuity equation (using slicing with HL = −kB/3) a˙ δρ˙ + 3 (δρ + δp) = −(ρ + p)k(v − B) a is to be compared to
dρ/dη = −3(aH)(ρ + p) which again with the substitutions becomes
d (1 − AQ) (¯ρ + δρQ) = −3(aH)(1 − AQ)[¯ρ +p ¯] − 3(aH)[δρ + δp]Q dη¯ d a˙ δρ = −3 (δρ + δp) dη¯ a ˙ • δρ/ρ constant in HL + kB/3 = 0 slicing outside horizon where peculiar velocity cannot move matter (cf. Newtonian gauge below). • Note also that v − B has a special interpretation as well: setting i i v = B gives a comoving slicing since N ∝ v , Ni ∝ vi − Bi = 0 Separate Universes • There are other possible choices what to hold fixed on constant time slices besides N = ln a. While separate universe statements still hold a must be perturbed and the simplest gauge to see these identifications with the Friedmann equations changes. • More generally the analysis of the normal to constant time surfaces has identified geometric quantities associated with the metric perturbations ˙ • Uniform efolding: HL + kB/3 = 0 ˙ • Shear free: HT − kB = 0 • Zero acceleration, coordinate and proper time coincide: A = 0 ˙ • Uniform expansion: −3HA + (3HL + kB) = 0 • Comoving: v = B 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 - corresponds to a choice of slicing and threading in ADM. • Decompose these into scalar T , L(0) and vector harmonics L(±1). Gauge
• gµν and Tµν transform as tensors, so components in different frames can be related
∂xα ∂xβ g˜ (˜η, x˜i) = g (η, xi) µν ∂x˜µ ∂x˜ν αβ ∂xα ∂xβ = g (˜η − T Q, x˜i − LQi) ∂x˜µ ∂x˜ν αβ • Fluctuations are compared at the same coordinate positions (not same space time positions) between the two gauges • For example with a TQ perturbation, an event labeled with η˜ =const. and x˜ =const. represents a different time with respect to the underlying homogeneous and isotropic background Gauge Transformation • Scalar Metric: a˙ A˜ = A − T˙ − T, a B˜ = B + L˙ + kT, k a˙ H˜ = H − L − T, L L 3 a 1 1 a˙ H˜ = H + kL, H˜ + H˜ = H + H − T T T L 3 T L 3 T a curvature perturbation depends on slicing not threading • Scalar Matter (Jth component):
δρ˜J = δρJ − ρ˙J T,
δp˜J = δpJ − p˙J T,
v˜J = vJ + L˙ , density and pressure likewise depend on slicing only Gauge Transformation • Vector:
B˜(±1) = B(±1) + L˙ (±1), ˜ (±1) (±1) (±1) HT = HT + kL , (±1) (±1) ˙ (±1) v˜J = vJ + L , • Spatial vector has no background component hence no dependence on slicing at first order Tensor: no dependence on slicing or threading at first order • Gauge transformations and covariant representation can be extended to higher orders • A coordinate system is fully specified if there is an explicit prescription for (T,Li) or for scalars (T,L) Slicing Common choices for slicing T : set something geometric to zero • Proper time slicing A = 0: proper time between slices corresponds to coordinate time – T allows c/a freedom • Comoving (velocity orthogonal) slicing: v − B = 0, matter 4 velocity is related to N ν and orthogonal to slicing - T fixed ˙ • Newtonian (shear free) slicing: HT − kB = 0, expansion rate is isotropic, shear free, T fixed ˙ • Uniform expansion slicing: −(˙a/a)A + HL + kB/3 = 0, perturbation to the volume expansion rate θ vanishes, T fixed (3) • Flat (constant curvature) slicing, δ R = 0,(HL + HT /3 = 0), T fixed
• Constant density slicing, δρI = 0, T fixed Threading • Threading specifies the relationship between constant spatial coordinates between slices and is determined by L
Typically involves a condition on v, B, HT • Orthogonal threading B = 0, constant spatial coordinates orthogonal to slicing (zero shift), allows δL = c translational freedom • Comoving threading v = 0, allows δL = c translational freedom.
• Isotropic threading HT = 0, fully fixes L Total Matter Gauge • Total matter: (comoving slicing, isotropic threading)
˜ 0 B =v ˜ (Ti = 0)
H˜T = 0 ξ = A˜
R = H˜L (comoving curvature) ∆= δ˜ (total density pert) T = (v − B)/k
L = −HT /k
Good: Algebraic relations between matter and metric; comoving curvature perturbation obeys conservation law Bad: Non-intuitive threading involving v Total Matter Gauge • Euler equation becomes an algebraic relation between stress and potential
2 3K (ρ + p)ξ = −δp + 1 − pΠ 3 k2 • Einstein equation lacks momentum density source
a˙ K ξ − R˙ − kv = 0 a k2 Combine: R is conserved if stress fluctuations negligible, e.g. above the horizon if |K| H2 a˙ δp 2 3K p R˙ + Kv/k = − + 1 − Π → 0 a ρ + p 3 k2 ρ + p “Gauge Invariant” Approach • Gauge transformation rules allow variables which take on a geometric meaning in one choice of slicing and threading to be accessed from variables on another choice • Functional form of the relationship between the variables is gauge invariant (not the variable values themselves! – i.e. equation is covariant) • E.g. comoving curvature and density perturbations
1 a˙ R = H + H − (v − B)/k L 3 T a a˙ ∆ρ = δρ + 3(ρ + p) (v − B)/k a Newtonian (Longitudinal) Gauge • Newtonian (shear free slicing, isotropic threading):
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 Newtonian (Longitudinal) Gauge
• Newtonian (shear free) slicing, isotropic threading B = HT = 0 :
a˙ (k2 − 3K)Φ=4 πGa2 δρ + 3 (ρ + p)v/k Poisson + Momentum a k2(Ψ+Φ) = 8 πGa2pΠ Anisotropy
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 • Newtonian competition between stress (pressure and viscosity) and potential gradients • Note: Poisson source is the density perturbation on comoving slicing Newtonian-Total Matter Hybrid • With the gauge in(or co)variant approach, express variables of one gauge in terms of those in another – allows a mixture in the equations of motion • Example: Newtonian curvature and comoving density
(k2 − 3K)Φ = 4πGa2ρ∆ ordinary Poisson equation then implies Φ approximately constant if stresses negligible. • Example: Exact Newtonian curvature above the horizon derived through comoving curvature conservation Gauge transformation a˙ v Φ = R + a k Hybrid “Gauge Invariant” Approach Einstein equation to eliminate velocity
a˙ Ψ − Φ˙ = 4πGa2(ρ + p)v/k a Friedmann equation with no spatial curvature
a˙ 2 8πG = a2ρ a 3
With Φ˙ = 0 and Ψ ≈ −Φ a˙ v 2 = − Φ a k 3(1 + w) Newtonian-Total Matter Hybrid Combining gauge transformation with velocity relation
3 + 3w Φ= R 5 + 3w Usage: calculate R from inflation determines Φ for any choice of matter content or causal evolution. • Example: Scalar field (“quintessence” dark energy) equations in total matter gauge imply a sound speed δp/δρ = 1 independent of potential V (φ). Solve in synchronous gauge. Synchronous Gauge • Synchronous: (proper time slicing, orthogonal threading )
A˜ = B˜ = 0 1 η ≡ −H˜ − H˜ T L 3 T hL ≡ 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,
threading conditions breaks down beyond linear regime if c1 is fixed to CDM comoving. Synchronous Gauge • The Einstein equations give
K v η˙ − (h˙ + 6η ˙ ) = 4πGa2(ρ + p) , T 2k2 L T k a˙ h¨ + h˙ = −8πGa2(δρ + 3δp) , L a L 1 a˙ −(k2 − 3K)η + h˙ = 4πGa2δρ T 2 a L [choose (1 & 2) or (1 & 3)] while the conservation equations give
d a˙ a˙ 1 + 3 δρ + 3 δp = −(ρ + p )(kv + h˙ ) , dη a J a J J J J 2 L d a˙ v 2 K + 4 (ρ + p ) J = δp − (1 − 3 )p Π . dη a J J k J 3 k2 J J Synchronous Gauge • Lack of a lapse A implies no gravitational forces in Navier-Stokes equation. Hence for stress free matter like cold dark matter, zero velocity initially implies zero velocity always. • Choosing the momentum and acceleration Einstein equations is ˙ good since for CDM domination, curvature ηT is conserved and hL is simple to solve for. • Choosing the momentum and Poisson equations is good when the equation of state of the matter is complicated since δp is not involved. This is the choice of CAMB.
Caution: since the curvature ηT appears and it has zero CDM source, subtle effects like dark energy perturbations are important everywhere Spatially Flat Gauge • Spatially Flat (flat slicing, isotropic threading):
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: non-intuitive slicing (no curvature!) and threading • Caution: perturbation evolution is governed by the behavior of stress fluctuations and an isotropic stress fluctuation δp is gauge dependent. Uniform Density Gauge • Uniform density: (constant density slicing, isotropic threading)
HT = 0 ,
ζI ≡ HL
BI ≡ B
AI ≡ A δρ T = I ρ˙I
L = −HT /k
Good: Curvature conserved involves only stress energy conservation; simplifies isocurvature treatment Bad: non intuitive slicing (no density pert! problems beyond linear regime) and threading Uniform Density Gauge • Einstein equations with I as the total or dominant species
a˙ 2 a˙ a˙ (k2 − 3K)ζ − 3 A + 3 ζ˙ + kB = 0 , I a I a I a I a˙ K v − B A − ζ˙ − B = 4πGa2(ρ + p) I , a I I k I k
• The conservation equations (if J = I then δρJ = 0)
d a˙ a˙ + 3 δρ + 3 δp = −(ρ + p )(kv + 3ζ˙ ) , dη a J a J J J J I d a˙ v − B 2 K + 4 (ρ + p ) J I = δp − (1 − 3 )p Π + (ρ + p )A . dη a J J k J 3 k2 J J J J I Uniform Density Gauge • Conservation of curvature - single component I
˙ a˙ δpI 1 ζI = − − kvI . a ρI + pI 3
• Since δρI = 0, δpI is the non-adiabatic stress and curvature is
constant as k → 0 for adiabatic fluctuations pI (ρI ). • Note that this conservation law does not involve the Einstein equations at all: just local energy momentum conservation so it is valid for alternate theories of gravity
• Curvature on comoving slices R and ζI related by
1 ρ∆I ζI = R + . 3 (ρI + pI ) comoving and coincide above the horizon for adiabatic fluctuations Uniform Density Gauge • Simple relationship to density fluctuations in the spatially flat gauge
1 δρ˜I ζI = . 3 (ρI + pI ) flat • For each particle species δρ/(ρ + p) = δn/n, the number density fluctuation
• Multiple ζJ carry information about number density fluctuations between species
• ζJ constant component by component outside horizon if each
component is adiabatic pJ (ρJ ). Vector Gauges • Vector gauge depends only on threading L • Poisson gauge: orthogonal threading B(±1) = 0, leaves constant L translational freedom (±1) • Isotropic gauge: isotropic threading HT = 0, fixes L • To first order scalar and vector gauge conditions can be chosen separately • More care required for second and higher order where scalars and vectors mix