Cosmological Perturbations from Inflation

Cosmological Perturbations from Inflation Misao Sasaki Kyoto University

1. Introduction 2 Horizon problem ds 2 = dt 2 + a 2 (t)d x 2 + Einstein eqs. ä a = 4πG (ρ + 3p) ρ + 3p > 0 decelerated expansion 3 If a t n, then n(n 1) < 0 0 < n < 1 ds 2 = a 2 (η) ( dη 2 + d x 2), dη = dt a. (η: conformal time maintains causality) F H A I A J dη = ±dx : light ray = I J I? = J J A H E C I K H B =? A η = dt a 0 for t 0

Solution to the horizon problem F H A I A J 3 Existence of a stage a t n the early universe ρ + 3p < 0 n > 1 in = I J I? = J J A H E C I K H B =? A t 0 dt a = dη =!! Entropy problem (= flatness problem) Entropy within the curvature radius: N γ conserved ( ) 3 ( ) a 3 ( ) 3 T0 N γ = n γ 1 Ω 0 3/2 T0 > 10 87 K H 0 H 0 T 0 10 4 ev Where does this big number come from? Huge entropy production in the early universe H 0 10 33 ev

2. Single-field slow-roll inflation 4 Universe dominated by a scalar field: ρ = 1 φ 2 + V (φ) 2 p = 1 φ 2 V (φ) 2 ρ + 3p = 2( φ 2 V (φ)) if φ 2 < V (φ) = ä a = 4πG (ρ + 3p) > 0 3 accelerated expansion Chaotic inflation (or Creation of Universe from nothing) (Linde, Vilenkin, Hartle-Hawking, ) ρ initial m 4 pl ( 10 19 GeV ) 4 quantum gravitational if V (φ) m 2 pl, then φ m pl

Equations of motion: 5 φ + 3H φ friction +V (φ) = 0 (H m pl initially in chaotic inflation) φ V (slow roll (1)) 3H Ḣ = 4πG(ρ + p) = 4πG φ 2 H 2 = 8πG ( ) 1 φ 2 + V (φ) 3 2 φ 3H φ 1 H 2 8πG V (φ) (potential dominated (2)) 3 The slow-roll condition (1) is satisfied, provided that V 9H V 2 24πGV 1, Ḣ 3H V 2 2 48πGV 1 2 Slow-roll inflation assumes that the above two are fulfilled. (Note that these are not necessary but sufficient conditions.) Ḣ H 2 3 φ 2 2V (φ) 1 There are models that violate either or both of the above two conditions. (Need special care in the calculation of perturbations)

6 e-folding number of inflation a e N N(φ) = tf t H dt = φf φ H φ dφ 8πG φ φ f ( V φ V dφ = 2πG(φ2 φ 2 f ) 2π m pl ) 2 slow roll V = 1 2 m2 φ 2 For V (φ) (10 15 GeV) 4, N(φ) 60 solves horizon & flatness problems N(φ) 60 at φ 3m pl for V = 1 2 m2 φ 2 Slow roll ends at φ 0.2m pl Reheating (entropy generation)

3. Generation of cosmological perturbations Action: S = d 4 x g ( 1 16πG R 1 ) 2 gµν φ,µ φ,ν V (φ) Cosmological perturbations are generated from quantum (vacuum) fluctuations of the inflaton φ and the metric g µν.. 7 Scalar-type (density) perturbations g µν and φ: ds 2 = a 2 [ (1 + 2A)dη 2 2 i B dηdx j + φ(t, x i ) = φ(t) + χ(t, x i ) ((1 + 2R)δ ij + 2 i j H T )dx i dx j ], A : Lapse function ( time coordinate) perturbation B : Shift vector ( space coordinate) perturbation Scalar perturbation has 2 degrees of coordinate gauge freedom. ( R : Spatial curvature (potential) perturbation δ (3) R = 4 ) (3) a 2 R H T : Shear of the metric ( traceless part of the extrinsic curvature) No dynamical degree of freedom in the metric itself.

8 Action expanded to 2nd order (in the Hamiltonian form) cf. Garriga, Montes, MS & Tanaka (1998) S 2 = ( ) dη d 3 x P a Q a H s A C A B C B a H s = 1 2a 2P 2 χ 4πGφ P R χ +, = d/dη, C A = φ P χ + (Hamiltonian constraint), C B = P HT (Momentum constraint), Q a = {R, H T, χ}, P a = {Momentum conjugate to Q a }. Gauge transformation [ξ µ = (T, i L) ] is generated by C A and C B : { } δ g Q = Q, (T C A + LC B ) d 3 x ( Q = {Q a, P a } ) P.B.

N 9 Reduction to unconstrained variables a lá Faddeev-Jackiw (1988) 1. Solve C A = φ P χ + = 0 with respect to P χ and insert it into S 2. Also, insert C B = P HT = 0 into S 2. 2. The resulting S 2 is a functional of {P R, R, χ}: S 2 = S 2 [P R, R, χ] 3. Since S 2 is gauge-invariant, S2 must be written solely in terms of gaugeinvariant variables. Indeed, we find S 2 = S 2 [P c, R c ] ; P c P R + a2 (3) 4πGφ χ, R c R H φ χ This is in fact the same as choosing χ = 0 gauge (called comoving slicing). i.e., R c is the curvature perturbation on the comoving hypersurface. D? A A H C O B M E A 4

10 Useful geometrical gauge-invariant variables: H a a = ah δρ ζ := R + curvature perturbation on uniform density slices 3(ρ + P ) (ρ = const.) R c := R + H(v + B) curvature perturbation on comoving slices (hypersurface normal n µ = u µ ) ζ = R c + O(k 2 /H 2 a 2 ) Φ := R + H(B H T ) Ψ := A + 1 a [a(b H T )] curvature perturbation on Newtonian slices (hypersurface shear = 0) Newton potential in Newtonian slices ζ expressed in terms of Φ and Ψ: ζ = Φ 3H Φ 3H 2 Ψ a 2 2 Φ 3Ḣ ζ R c Φ 3H Φ 3H 2 Ψ 3Ḣ on superhorizon scales

S 2 in the Lagrangean form: S 2 = dη d 3 x a2 φ 2 2H 2 ( R 2 c ( R c ) 2) ; H a a = a H Equation of motion (for Fourier modes: (3) k 2 ) R c + 2z z R c + k2 R c = 0 ; z aφ H = a φ H ( a for slow-roll inflation). 11 For k < H ( k/a < H), R c { z 1 decaying mode 0 growing mode Growing mode of R c stays constant on super-horizon scales. This holds for adiabatic perturbations in general cosmological models. (i.e., the existence of a constant mode) But this does not mean that R c is constant on super-horizon scales.

Inflaton perturbation on flat slicing 12 Alternatively, in terms of χ on R = 0 hypersurface (flat slicing), S 2 = S 2 [χ F ] = χ F χ φ H R = φ H R c dη d 3 x a2 2 ( ) χ 2 F ( χ F ) 2 a 2 m 2 eff χ2 F ; m 2 eff = { a 2 (φ /H) } a 4 (φ /H) = 2 φ V + 16πG d dt χ F minimally coupled almost massless scalar in de Sitter space ( 2 φ V H2, 16πG(V /H) 6Ḣ H 2 for slow-roll inflation.) N.B. the sufficient conditions for slow roll were 2 φ V 3H2 and Ḣ 3H 2. de Sitter approximation for the background: H = const., a(η) = 1 Hη ( < η < 0) This is a good approximation for k > H (sub-horizon scale) modes. ( V H )

Canonical quantization π(η, x) = δs 2 [χ F ] δχ F (η, x), [χ F (η, x), π(η, x )] = iδ( x x ) d 3 k ( ) ˆχ F = â (2π) 3/2 k χ k (η) e i k x + h.c. ; [â k, â ] = δ( k k k ) ( ) χ k + 2Hχ k + k 2 + m 2 eff a2 χ k = 0 ; χ k χ k χ k χ k = i ( k 2 ) χ k + 3H χ k + a + 2 m2 eff χ k = 0 ; χ k χ k χ k χ k = i a 3 (in terms of the cosmic proper time t) de Sitter approximation: slow roll m 2 eff H2 massless χ k H (i kη) e iη (2k) 3/2 k/h k/h 0 1 2ka e ikη H 2k 3 e iα k a 2 13 δφ 2 k on flat slice = χ 2 F k 4πk3 (2π) 3 χ k 2 ( ) H 2 for k H 2π

de Sitter approximation breaks down at k H. i.e., the time-variation of χ k on super-horizon scales cannot be neglected. However, the corresponding k-mode of R c becomes constant on superhorizon scales. R c,k (η) R c,k (η k ) = H φ χ k(η k ) H2 (t k ) 2k 3 φ(tk ) e iα k. 14 log L N ( t k ) L = 1 H t k t f log a t = t k η = η k k = H(η k ) horizon crossing time

15 Curvature perturbation spectrum (say, at η = η f ) R 2 c k 4πk3 (2π) 3P R c (k; η) = 4πk3 (2π) R c,k(η) 2 = 3 Since dn = Hdt, N φ = Ḣ φ ( N R 2 c k = φ ) H 2 2π t=t k = ( H 2 2π φ ) 2 ( ) N 2 φ δφ t=t k t=t k on flat slice That is, for single-field slow-roll inflation, R c = δn t=tk = N φ δφ (δφ = H ) on flat slice t=tk 2π Only the knowledge of the homogeneous background is sufficient to predict the perturbation spectrum. δn-formula If R 2 c k k n 1 n = 1 : scale-invariant (Harrison-Zeldovich) spectrum n = 1 ɛ (ɛ 1) for chaotic inflation (V (φ) φ p ).

Large angle CMB anisotropy ( ) δt η0 ( γ, η 0 ) = (ζ r + Θ) (η dec, x(η dec )) + dη η Θ(η, x(η)) T η dec (Sachs-Wolfe) (Integrated Sachs-Wolfe) 16 ζ r curvature perturbation on ρ photon = const. surfaces Θ Ψ Φ x ( η dec ) γ o η0 η dec dη is along the photon path. Last Scattering Surface

For a dust-dominated universe at decoupling, 17 SW: ζ r + Θ 1 5 ζ 2 5 S dr ζ (initial) adiabatic curvature perturbation S dr = δρ d 3 δρ r entropy perturbation ρ d 4 ρ r For standard adiabatic perturbations, 5ρ + 3p R c (= ζ ) 3(ρ + p) Ψ 5 3 Ψ, ζ r + Θ 1 3 Ψ

CMB Observation vs Inflation Model COBE-DMR: ApJ Lett. 464 (1996); WMAP: astro-ph/0306132 18 (δt ) 2 T 10 10 at θ 10 Ψ 2 k 10 10 at k 0 a 0 = H 0 1 3000 Mpc 1 10 28 cm For V = 1 2 m2 φ 2, Ψ 2 k 0 ( 3 5 ) 2 R 2 c = k 0 ( ) 3 2 ( H 2 ) 2 5 m 10 13 GeV V (10 16 GeV) 4 2π φ k0a =H m2 N 2 (φ) m 2 pl k0a =H power-law index: n WMAP = 0.93 ± 0.03 (for scalar perturbations) Slight deviation from scale invariant spectrum (n = 1)

Tensor type perturbation 19 ds 2 = dt 2 + a 2 (t) (δ ij + h ij ) dx i dx j h ij Transverse-Traceless δ 2 S G = 1 ( d 4 x a 3 ḣ ij 1 ) 64πG a 2( h ij) 2 = 1 ( d 4 x a 3 ϕ 2 ij 2 1 ) a 2( ϕ ij) 2 ; ϕ ij := ϕ ij massless scalar (2 degrees of freedom) ( ) H 2 ϕ 2 ij k = 2 2π T S = tensor scalar h2 ij 2 φ = 24 V T ( ) H 2 = 8 h 2 ij k = 2 32πG 2π contribute to CMB anisotropy R 2 c k0 =ah 1 32πG h ij π H 2 m 2 pl slow roll T S 1. S 0.13 for V = 1 2 m2 φ 2 (small but non-negligible)

Model dependence 20 power-law inflation V (φ) exp[λφ/m pl ] dilaton in string theories? a t α (α = 16π λ ) 2 T n < 1, 0.1 S hybrid inflation supergravity-motivated? e.g., V (φ, ψ) = 1 ( M 2 λψ 2) 2 1 + 4λ 2 m2 φ 2 + 1 2 g2 φ 2 ψ 2 a e Ht, H 2 8πG 3 V 0 when ψ = 0, φ > M/g. n > 1, T S can be large or small.

4. Perturbation spectrum in non-slow-roll inflation Curvature perturbation on comoving slice: ( H 2 ) R c final R c (t k ) 2π φ k=ah Leach, MS, Wands & Liddle (2001) for slow-roll inflation What if slow-roll condition is violated? 21 Reconsideration of EOM for R c : R c + 2z z R c + k2 R c = 0 ; = d dη, Two independent solutions for k 2 0: z = a φ H = aφ H, H = a a. u(η) const. growing mode η dη v(η) decaying mode (η z 2 (η end of inflation) ) η v 0 as η η by definition, but u is arbitrary. In slow-roll inflation, v(η) v(η k ) for η > η k ( kη k/h 1) v may not decay right after horizon crossing in general.

Long-wavelength approximation u(η) = u n (η) k 2n, u n+1 + 2z z u n+1 = u n, To O(k 2 ), n=0 u 0 = const. 22 u u 0 + [C 1 + C 2 D 0 (η) + F (η)] u 0 ; η D 0 (η) = 3H(η k ) dη z2 (η k ) z 2 (η ) η F (η) = k 2 dη η η η z 2 (η )dη. z 2 (η ) η k lowest order decaying mode F 0 as η η (F behaves also like decaying mode) Convenient choice for the growing mode : C 1 = 0, u(η) C 2 = F k D k ; F k = F (η k ), D k = D 0 (η k ) [ 1 F k D 0 (η) D k ] + F (η) u 0 ; u(η k ) = u(η ), u (η k ) = 3H k F k D k u(η k ) ( ) H k = H(η k )

23 Decaying mode accurate to O(k 2 ): v(η) = u(η) D(η) D(η k ) ; D(η) 3H k η η dη z2 (η k )u 2 (η k ) z 2 (η )u 2 (η ). General solution for R c : R c (η) = αu(η) + βv(η) ; α + β = 1, R c (η k ) = u(η k ). R c (η ) = α u(η ) = α u(η k ) = α R c (η k ) R c (η k) = u (η k ) 3(1 α)h ku(η k ) D(η k ) α 1 + D [ k R ] c u 3H k R c u η=η k. R c (η ) (1 F k ) R c (η k ) + D k 3H k R c (η k) R c (η ) R c (η k ) if F k 1 and/or D k 1

24 Leach-Liddle model Leach & Liddle (2000) Quartic potential with vacuum energy: V (φ) = M [ ] 4 1 + B 64π2 φ 4 4 m 4 pl ; B = 0.55 Inflation suspended at φ m pl

To summarize, Spectral formula for non-slow-roll inflation: k R c (η ) = (1 F k ) R c (η k ) + D k 3H k η D k = 3H k dη z2 (η k ) η k z 2 (η ), F k = k 2 η η k dη z 2 (η ) η ( ) d kdη R c(η k ) η k z 2 (η )dη, z 2 = ( a φ H ) 2 25 R c (η ) in linear combination of R c (η k) and R c (η k ) with coefficients expressed in terms of background quantities. For slow-roll case, we have D k 1, F k 0 for k/h k 0.1 R c (η ) R c (η k ). For non-slow-roll case, an enhancement of R c can occur: A sharp dip appears in the spectrum at F k 1. A kink appears at D k 1.

26 5. Extension to multi-field inflation MS & Tanaka (1998) n-component scalar field: S φ = d 4 x g 2 ( ) g µν µ φ ν φ + V ( φ), φ φ = δ pq φ p φ q (p, q = 1, 2,, n). Homogeneous background solutions: (N as a time coordinate; a = e +N ) G 0 0 = T 0 0 : (units : 8πG = 1) ( H 2 1 1 ) φ 2 N = 1 6 3 V ; φ N d φ dn Scalar [ field equation : H d dn H d dn + d ] 3H2 φ p + V p = 0. dn

General solution is parametrized by 2n parameters: { φ = φ(λ α ), π = π(λ α )} ( π = He 3N φn ) 27 λ α = {N, λ a } (α = 1 2n, a = 2 2n) F? I J λ α can be regarded as a new set of phase space coordinates for the homogeneous solutions. solution is labeled by λ a (a = 2, 3,, 2n) B

Metric perturbation: (expanded in spherical harmonics) [ ds 2 = a 2 (η) (1 + 2A Y )dη 2 2B Y j dηdx j ] + ((1 + 2H L Y )δ ij + 2H T Y ij )dx i dx j, 28 Scalar field perturbation: φ φ(η) + δ φ Y ( (3) +k 2 )Y = 0, Y j = k 1 j Y, Y ij = k 2 i j Y + 1 3 δ ijy. Perturbed e-folding number Ñ: η ( Ñ = N + R 1 ) 3 kσ g Y dη, η 0 ( = d ) dη R = H L + H T 3 kσ g = H T kb : curvature perturbation on Σ(η) : shear of Σ(η) δn = 0 for R 1 3 kσ g = 0 (H L = B = 0) Constant e-fold gauge

29 Super-horizon scale perturbation on R 1 3 kσ g = 0 slices (constant e-fold (δn = 0) gauge at k 2 /a 2 H 2 ) A & δ φ δh H = A = H2 φ N δ φ N + V p δφ p A is subject to δ φ. [ 2V H d ( H d ) + 3H 2 d ] δφ p + V p qδφ q + 2V p A H 2 φ p N dn dn dn ( φ N δ φ N ) = 0. This turns out to be the same as the equation for φ/ λ α. Hence, δ φ = c α φ λ α, λ α = {N, λ a } ; φ λ φ 1 N : time-translation mode δ φ in this gauge is completely decsribed by the knowledge of (a congruence of) background solutions.

30 R & kσ g From δg 0 j = δt 0 j and traceless part of δg i j = δt i j, N R = c α dn W (α) N b a 3 H, kσ g = 3cα W (α) ; a ( 2 W (α) = a3 H 3 d φ N 2V dn χ (α) φ N d χ ) (α) = const., dn χ (α) φ λ α where W (1) = 0 and N b is an arbitrary constant. (One can always choose λ a such that W (2) 0 and W (a) = 0 for a = 3 2n.) Perturbation in the constant e-fold gauge is parametrized by 2n + 1 parameters {N b, c α }: δ φ = c α φ N λ, A = A(δ φ), R = c α dn W α (α) a 3 H, N b one gauge degree of freedom. kσ g = 3cα W (α) a 2.

Gauge mode An infinitesimal change of time slicing: N N δ g N, R R + δ g N, kσ g kσ g + O(k 2 ), δ φ δ φ + φ N δ g N. The condition R 1 3 kσ g = 0 is maintained for δ g N = const. in the limit k 2 /a 2 H 2. (δφ, R, kσ g ) = (c φ N, c, 0) is pure gauge. Time-Translation Mode Necessary to construct gauge-invariant quantities such as R c. Construction of R c Comoving slice condition φ N (δφ) c = 0 defines a surface in the phase space as ( ) ( ) φ N (N, 0) φ(n + N, λ a ) φ(n, 0) = φ N (N, 0) δ φ + φ N N = 0. 31 N(λ α ) = φ N δ φ φ 2 N N depends on x µ only through its dependence on λ α.

Gauge transformation to the comoving slice: 32 R c (λ α ) = R(λ α ) + N(λ α ) ; λ α = λ α (x µ ) φ = R N δ φ φ 2 N χ F := δ φ φ N R = φ N χ F φ 2 N = c 1 φ N + c a ( φ λ a W (a) φ N ; N N b dn a 3 H χ F δ φ on flat slice (contains all the information) convenient quantity for evaluating quantum fluctuations Change of N b is absorbed by the redefinition of c 1. ). c 1 adiabatic growing mode amplitude c a W (a) (= c 2 W (2) with W (a) = 0 for a 3) adiabatic decaying mode amplitude The rest (c a ; a 3) are entropy perturbations Generally R c varies in time even on super-horizon scales

Slow roll limit: φ 2 N 1, D φ N dn φ N φ p N = V p 3H = (ln V 2 ) p, H 2 = 1 3 V. 2n-d phase space n-d configuration space Slow roll kills all the decaying modes: δ φ = χ F, R = kσ g = 0 on δn = 0 slice δn = 0 slice becomes equivalent to flat (R = 0) slice in slow-roll limit. R c = N = φ N χ F φ 2 N where χ F = C α φ λ α (α = 1 n) R c expressed in terms of ( χ F ) N=Nk (N k : horizon crossing time) [ N R c (N) = φ p χp F ]N=Nk Ca φ N φ [ λ a ] λ ; a Ca = φ p χp F (a = 2 n) N=N k φ 2 N χ F N=Nk : to be evaluated by quantization R c = N at the end of inflation (H 2 = 1 3 V =const. at N = N f) φ N [ ] [ ] φ ln V N N = = 0 R λa λ a c (N e ) = φ p χp F = N k φ p δφp N k 33

34 5. Summary Super-horizon scale perturbations are described solely by (a congruence of) homogeneous background solutions. Correlations among various quantities can be easily calculated. (e.g., correlation between entropy and adiabatic perturbations) Curvature perturbation R c may vary in time on super-horizon scales either for multi-field or non-slow-roll inflation. R c N in the slow-roll case. Large enhancement on super-horizon scales can occur even for singlefield inflation. Simple (slow-roll) models predict scale-invariant spectrum, but other, more complicated spectral shapes are possible. Tensor perturbations may be non-negligible.

35 On-going and future observations SDSS 10 6 galaxies MAP, PLANK CMB anisotropy map with resolution of θ 10 Inflaton potential may be determined Understanding of physics of the early universe ( extreme high energy physics)