1 heory of Cosmological Perturbations Part I gauge-invariant formalism Misao Sasaki PDF files available at http://www2.yukawa.kyoto-u.ac.jp/~misao/pdf/cp/
1. Formulation Background spacetime Friedmann-Lematre-Robertson-Walker metric K = ±1, 0): ds 2 = dt 2 + a 2 t)dσ 2 K ; dσ 2 K = γ ij dx i dx j = dχ 2 + sinh2 K χ) K dω 2 2) 2 Conformal time coordinate: dη = dt a ds 2 = a 2 η) ) dη 2 + dσk 2 Energy momentum tensor in the perfect fluid form: µν = ρu µ u ν + pg µν + u µ u ν ) ; u ν dx ν = dt = adη. Friedmann equation: G 0 0 = R 0 0 1 2 R = 8πG 0 0) ȧ ) 2 + K a a = 8πG ρ, ρ + 3ȧρ + p) = 0 2 3 a ) a 2 + K = 8πG a 3 ρ a2, ρ + 3 a ρ + p) = 0 a ȧ = da ) dt a = da ). dη
3 Hereafter, we use the symbols: H = ȧ a, H = a a Spatial harmonics scalar harmonics + k 2 ) Y k = 0 γ ij D i D j ). k 2 K) for K 0 k 2 : continuous) k 2 = nn + 2)K for K > 0 n = 0, 1, 2, ) Examples in the case of K = 0: Y k = e ik x ; k 2 = k 2, or Y k = j l kχ)y lm Ω) ; k = k, l, m).
4 vector harmonics index k omitted) scalar type: Y i = k 1 D i Y, vector transverse) type: D i Y 1)i = 0, ) + k 2 Y 1)i = 0 ; + k 2 2K) ) Y i = 0. k 2 = nn + 2) 1)K for K > 0 n = 1, 2, ), k 2 2K) for K 0. tensor harmonics scalar traceless) type: Y ij = k 2 D i D j + 1 ) 3 γ ij Y, + k 2 6K ) Y ij = 0. vector traceless) type with index σ, k): σ = 1, 2): Y 1) ij = 1 ) D i Y 1) j + D j Y 1) i, + k 2 4K ) Y 1) ij = 0. 2k tensor transverse-traceless) type with index σ, k): σ = 1, 2): D j Y 2) ij = Y 2)i i = 0, ) + k 2 Y 2) ij = 0 ; k 2 = nn + 2) 2)K for K > 0 n = 2, 3, ), k 2 3K) for K 0.
Perturbation variables scalar type perturbations metric: g 00 = a 2 1 + 2Aη)Y ), g 0j = a 2 BY j = a 2B ) k D jy, 5 matter: g ij = a 2 [1 + 2H L Y )γ ij + 2H Y ij ] [ ] = a 2 1 1 + 2RY )γ ij + 2H k 2D id j Y ; R H L + 1 3 H. µ ν = ρũ µ ũ ν + τ µ ν ; ρ = ρ1 + δ Y ), ṽ i ũi u = v Y i = v ) 0 k Di Y u 0 = a) 1 ũ0 = u 0 1 AY ) from ũ µ ũ µ = 1 ) τ i j = p [ ] δj1 i + π L Y ) + π Y i j
geometrical meaning 3 + 1)-decomposition: 6 ds 2 = a 2 η)dŝ 2 ; dŝ 2 = Ñ 2 dη 2 + γ ij dx i + Ñ i dη)dx j + Ñ j dη). ~i N dη xi=const. ~ Ndη η + dη =const. η =const. extrinsic curvature: ñ µ dx µ = Ñdη hypersurface normal : Ñ = 1 + AY lapse function, Ñ i = BY i shift vector. ˆK ij = H kb)y ij = kσ g Y ij σ g 1 k H B shear of η =const. hypersurface.
expansion : 7 θ 1 a 4 µ a 3 ñ µ ) = 3 a a 1 + K gy ) = 3H 2 a 1 + K gy ) K g A + H R 1 k ) 3 σ g 3-curvature: In particular, δ s ˆRi jmn = D m δ s Γ i jn D n δ s Γ i jm ; δ s Γ i jm = 1 2 γil D m δγ lj + D j δγ lm D l δγ jm ), s ˆR = 6K + 4k 2 3K) δγ ij = 2H L γ ij + 2H Y ij. H L + 1 ) 3 H Y = 6K + 4k 2 3K)RY. R intrinsic curvature perturbation potential).
Gauge transformation properties 8 x x µ x µ = x µ + ξ { µ η = η + Y, x i = x i + LY i. his induces a gauge transformation, x ξ x x=x+ ξ x ḡ µν = g µν ξ µ;ν + ξ ν;µ ) [ ] in general, Q{A} = Q {A} L ξ Q {A}. {A} spacetime indices)
metric: ) H = a a 9 Ā = A H lapse ) B = B + L + k H L = H L k σ g = σ g k 3 L H R = R H H = H + kl σ g = 1 ) k H B shear R = H L + 1 ) 3 H curvature matter: δ = δ + 31 + w)h v = v + L w p ) ρ v B) = v B) k ) π L = π L + 3c 2 1 + w w c w H 2w p ρ π = π ~ Ndη ~i N dη xi=const. v i dη η + dη =const. u µ η =const. v i ~ + N i )d η =v-b )Y idη Blue quantities depend only on the choice of time-slicing.
gauge-invariant variables 4 metric variables 2 gauge variables = 2 degrees of freedom) 10 Ψ A 1 ) σ k g + H σ g, σ g = 1 ) k H B Φ R H k σ g. R = H L + 1 ) 3 H s δ + 3 H k 1 + w)σ g V v 1 k H = v B) σ g shear of 4-velocity ũ µ Γ π L c2 w w δ pγ = δp c2 wδρ ) entropy perturbation Π π anisotropic stress perturbation. Ψ = A, Φ = R, s = δ, V = v B on σ g = 0 hypersurface. Shear-free slicing or Newton slicing
Other popular choices: 11 R c R H k v B) = Φ H k V, σ g) c = V, A c A 1 k [v B) + Hv B)] = Ψ 1 k [V + H V ] δ + 3 H k 1 + w)v B) = s + 3 H k 1 + w)v, Velocity-orthogonal or comoving slicing v B = 0) ζ R + 1 31 + w) δ = R c + 1 31 + w) Curvature perturbation on uniform density slices δ = 0) f δ + 31 + w)r = + 31 + w)r c = 31 + w)ζ Density perturbation on flat slices R = 0)
vector type perturbations 12 Ñ 1) j = B 1) Y 1) j, γ 1) ij = γ ij + 2H 1) Y 1) ij. gauge transformation: ṽ 1)j = aũ [ 1)j = v 1) Y 1)j ], τ 1)i j = p δj i + π 1) Y 1) ij. x j = x j + L 1) Y 1)j gauge-invariant variables: { B1) = B 1) + L 1), H 1) = H 1) + kl 1). v 1) = v 1) + L 1), π 1) = π 1). 2 metric variables 1 gauge variable = 1 degree of freedom) σ 1) g = 1 k H1) B 1), V 1) = v 1) B 1), Π 1) = π1). tensor type perturbations [ ] γ 2) ij = γ ij + 2H 2) Y 2) ij. τ 2)i j = p δj i + π 2) Y 2) ij Both H 2) and π 2) are gauge-invariant by themselves..
2. Einstein equations in terms of gauge-invariant variables Scalar type perturbations Einstein equations: δg µ ν = 8πGδ µ ν : δg 0 0 = 2 a 2 [ 3H 2 A + H kσ g 3H R k 2 3K)R ] Y δg 0 j = 2 a [kh A 2 kr + Kσ g ] Y j δg i j = 2 [ ) 2H + H 2 k2 A + H A + k ) σ a 2 3 3 g + 2H σ g R 2HR 1 ] 3 k2 3K)R δjy i 13 + 1 a 2 [ k 2 A + kσ g + 2Hσ g ) k 2 R ] Y i j δ 0 0 = ρ δ Y, δ 0 j = ρ + p)v B)Y j, δ i j = pγ + c 2 wρ δ)δ i jy + pπ Y i j. N.B. pγ + c 2 wρ δ = δp) he above equations depend only on the choice of time slicing. ie, they are spatially gauge-invariant.
Energy-momentum conservation: 14 δ 0 ν ;ν ) = 0 : δ + 3H [ c 2 w w)δ + wγ ] + k1 + w)v B) + 1 + w)3r kσ g ) = 0. δ j ν ;ν ) = 0 : v B) + H1 3c 2 w)v B) = ka + k c2 wδ + wγ 1 + w 2k 3 hese are also spatially gauge-invariant. w 1 + w 1 3Kk ) π 2. o obtain gauge-invariant equations, it is simplest to set a gauge time-slicing) condition that fixes the time slices completely. gauge-invariance complete gauge-fixing Gauge-invariant quantities are NO necessarily directly related to observables Advantage of gauge-invariant formalism is because one does not have to choose a particular gauge while dealing only with physical degrees of freedom, not because the values of the gauge-invariant variables are physical by themselves.
15 Example: choose σ g = 0 Newton slicing), so that A = Ψ, R = Φ, δ = s, v B = V. δg 0 0 = 8πGδ 0 0 : δg 0 j = 8πGδ 0 j : 1 a 2 [ 3HHΨ Φ ) k 2 3K)Φ ] = 4πGρ s, 1) k a 2 [HΨ Φ ] = 4πGρ + p)v, 2) δg i j = 8πGδ i j) traceless : k2 a 2 [Ψ + Φ] = 8πpΠ. 3) δg i j = 8πGδ i j) trace : ) 1 [2H + H 2 k2 Ψ + H Ψ a 2 3 Φ 2HΦ 1 ] 3 k2 3K)Φ = 4πGp [ Γ + c 2 wρ s ]. 4) Combining Eqs. 1) and 2) [Hamiltonian and momentum constraints] gives k 2 3K Φ = 4πGρ [ a 2 s + 3 Hk ] 1 + w)v = 4πGρ. 5) Eqs. 3) and 5) algebraically determine Φ and Ψ in terms of and Π. Combining Eqs. 1), 3) and 4), one can derive 2nd order differential equation for Φ. Alternatively, one may appeal to the energy-momentum conservation laws. contracted Bianchi identities)
16 ν δ j ;ν ) = 0 on shear-free Newton) slices : V + H1 3c 2 w)v = kψ + k c2 w s + wγ 1 + w s = 3 Hk ) 1 + w)v 2k 3 w 1 + w 1 3Kk 2 ) π V + H V = kψ + k c2 w + wγ 1 + w 2k w 1 3Kk ) π 3 1 + w 2 6) Ψ = Φ 8πGρ ) a2 w 4πGρ a2 Π k 2, Φ = k 2 3K δ 0 ν ;ν ) = 0 on comoving slices : + 3H [ c 2 w w) + wγ ] + 1 + w) 3R c + kv ) = 0 ; R c = Φ H ) k V k a [HΨ 2 Φ ] = 4πGρ + p)v, V =, 3wH = 1 3Kk ) [1 + w)kv + 2H wπ 2 ] 7) Combining Eqs. 6) and 7) gives a 2nd order differential equation for.
17 + 1 + 3c 2 w 2w) ) H + [ c 2 sk 2 3K) 4πGρa 2 1 + 3w)1 w) + 6w c 2 w) ) + 34w 3c 2 w)k ] = S c [ ] S c = k 2 δp)c c 2 3K) sδρ) c 2 1 3Kk ) H wπ ρ 2 +2 [3w 2 2w + 3c 2w)H 2 + 4w c 2w)K + c2 wk 2 ] 1 3Kk ) 3 2 Master equation for equation for sound waves) Π In the Newtonian, non-relativistic limit w 1, c 2 s 1), + H + c 2 sk 2 4πGρ a 2) S c, + 2H c 2 + s k 2 ) 4πGρ S c a 2 a 2 dispersion relation: ωeff 2 = c2 sk 2 4πGρ Jeans instability: for c 2 w p /ρ = p/ ρ c 2 s) ) k k gravitationally unstable for a < 4πGρ a c 2 s a 2 J or λ > λ J πc 2 s Gρ
Vector type perturbations 18 δg 0 j = 8πGδ 0 j : δg i j = 8πGδ i j : σ g 1) + 2Hσ g 1) k2 2K σ 1) 2a 2 g = 8πG1 + w)ρ V 1). = 8πρa 2w k Π1). Either combining these two or directly from the momentum conservation, For Π 1) δ i µ ;µ ) = 0 : = 0 adiabatic), σ 1) g 1 a 2, V 1) Vorticity Ω: V 1) + 1 3c 2 w)h V 1) = k2 2K 2k 1 ρ1 + w)a 4 1 a 1 3w 1 ρ w/1+w) a his is just the vorticity conservation law. w 1 + w Π1). ) for w = const. Ω µν P α µ u [α;β] P β ν ; P µν = g µν + u µ u ν. Ω ij = a k V 1) Z ij Z ij = 1 ) k D [jy 1) i], Ω ij Ω ij = k2 a 2 V 1) 2 Z ij Z ij ρ w/1+w) S Ω =const. S: area a 2 ).
19 ensor type gravitational wave) perturbations For K = 0, Π 2) δg i j = 8πGδ i j : H 2) Ḧ 2) + 2H H 2) + k 2 + 2K)H 2) = 8πG p a 2 Π 2) + 3H Ḣ2) + k2 + 2K H 2) a 2 = 8πG p Π 2) = 0, this is the same as the field equation for a massless minimal scalar: ϕ + 3H ϕ + k2 a 2 ϕ = 0. On superhorizon scales k 2 H 2 a 2 = H 2 ), const. H 2) η dη a 2 growing mode decaying mode tensor perturbation a homogeneous, anisotropic universe On subhorizon scales k 2 H 2 a 2 = H 2 ), H 2) 1 a eikη ρ GW Ḣ2) 2 1 a 4
3. Adiabatic perturbations on superhorizon scales Spatially flat universe K = 0) is a good approximation in the early universe. hen, ρ a 3 ) + 1 + 3c 2 w)hρ a 3 ) + [c 2sk 2 32 ] 1 + w)h2 ρ a 3 ) = ρ a 3 S c. 20 Here, c 2 w p /ρ, c 2 s p/ ρ) comoving δp) c = c 2 sδρ) c + entropy perturbation Also, all cosmologically relevant scales exceed Hubble horizon. ln L phys λ = 2πa k a, H 1 ρ { 1/2 a 3/2 for dust w = 0) a 2 for radiation w = 1/3) λ for a 0 H 1 k = const. L ~ a) L = 1/H ~ a2 in rad-dom universe) ln a
21 For S c = 0 adiabatic perturbation), one general solution in the limit c 2 sk 2 0 is ρ a 3 H 1 a H a = 1 decaying mode 2 H a 3 Using the Wronskian for 2 independent solutions, the other solution is found as ρ a 3 H a η 0 1 H a 2 1 + w)a 2 dη η 0 t 1 + w)a 2 dη = 1 1 + w)adt growing mode H a 3 0 η 2 for w = const. ) Conservation of growing mode amplitude Let us set = C 1 k 2 H a 2 η 0 1 + w)a 2 dη his gives, from the Hamiltonian and momentum constraints, Φ = 3 2 k = 3 2 2 C H 1 a 2 H 2 η R c = Φ H k V = C 1. 0 1 + w)a 2 dη, H k V = 2 1 3 1 + w)h Φ + H Φ)
22 he growing mode amplitude of R c stays constant on superhorizon scales. he condition for the linear perturbation theory to be valid is R c = C 1 1. linear theory is applicable up to t = 0 singularity if only the growing mode is present. For the decaying mode, where = C 2 k 2 H a 2, R c = C 2 ηf η η f = R c = H c2 s + wγ c 1 + w c 2 sk 2 1 + w)a 2dη Γ c δp) c c 2 sδρ) c ρ { for w 1/3 0 finite) for w < 1/3 ) = 0 Decaying mode amplitude for R c is not constant. he standard lore that R c = const. on superhorizon scales is not strictly correct. It is correct only if the decaying mode can be ignored.
23 Growing mode solution for several other variables assuming w =const.) a η 2/3w+1) 2 ), H = 3w + 1)η ζ = R c + 31 + w) C 1 ζ = R on uniform density slice) is also constant. his is true not only for GR but also for any metric theory. Wands et al. 2000) 3w + 1)2 1 + w) 23w + 5) σ g ) c = V 3w + 1 C 1 kη) 2, A c = R c = c2 s 1 + w 3w + 1)2 w 23w + 5) C 1 kη) 2, 3w + 5 C 1 kη, 31 + w) Ψ = Φ 3w + 5 C 61 + w) 1, s 2Φ 3w + 5 C 1 Φ curvature perturbation on Newton slices) or Ψ Newton potential) stays constant for w =const., but the amplitude changes when w changes. In particular, during inflation when w 1, the amplitude of Φ stays very small, but it grows significantly at the end of inflation. During inflation, quasi-nonlinear perturbations C 1 1) can be studied in Newton gauge.