Cosmological perturbation theory at three-loop order

Transcription

1 Cosmological perturbation theory at three-loop order Grenoble, based on and with Diego Blas, Thomas Konstandin

2 Outline Large-scale structure Non-linear corrections Three loop result Padé resummation

3 Large-scale structure SDSS-III (sdss3.org) BOSS (red), SDSS (white/yellow)

4 Large-scale structure Power spectrum of density contrast δ(x, z) = ρ(x, z)/ ρ(z) 1 δ(k, z)δ(k, z) = δ (3) (k + k )P(k, z) SDSS 2003 (Tegmark et al) BOSS DES ( ); LSST, Euclid ( 2020): (sub-)percent at BAO scales

5 Large-scale structure cross-check prediction from CMB for ΛCDM improve constraints for extended/non-standard models extract information on late-time effects (expansion history, neutrino mass) Audren, Lesgourgues, Bird et. al Planck H(z)/[D A (z)b(z)] 2 P g (k ref, µ, z) M ν = 0.21 ev M ν = 0 shot noise z=0.5, µ= k ref (h/mpc) Neff Planck+WP+highL Planck+WP+highL+BAO Σmν [ev] Euclid prospect ρ ν { a 4 T >m ν m ν a 3 T <m ν M ν = m ν { 0.056eV (nh) 0.095eV (ih) Σm ν [ev] Basse, Bjaelde, Hamann, Hannestad, Wong ml N eff

6 Large-scale structure Euclid forecast vs theoretical errors Audren, Lesgourgues, Bird et. al relative error z=0.5, µ=0 observational theoretical error error [M ν = 0.05 ev] / [massless] k ref (h/mpc) { 25meV fiducial (2%th. err. at k = 0.4h/Mpc, z = 0.5) σ(m ν) 14meV th. err. / = 10, k max = 0.6h/Mpc theoretical uncertainties from bias, non-linear evolution,...

7 Large-scale structure Desirable to develop a (fast but reliable) method for theoretical prediction of the power spectrum for a given set of parameters Understand onset of non-linearities Weakly non-linear regime perturbation theory M [M ] Cosmic Cluster Galactic CDM Unknown small scale behavior 2 (k) non-linear (simulation) linear (analytic) Baryon Acoustic Oscillations k [Mpc 1 ] ADM WDM(8keV) 2 (k, z) = 4πk 3 P(k, z) Kuhlen, Vogelsberger, Angulo

8 Idea Initial conditions at z0 103 set by linear dynamics of the photon-baryon-dm fluid as imprinted in the CMB (for a given model) e.g. CAMB, CLASS Plin Horizon Run 2 P(k,z=0) [(h/mpc)-3] k [h/mpc] 1 Goal: compute P(k, z 0) from P(k, z0 ) beyond linear approximation on scales khorizon k. knl (governed by Newtonian dynamics)

9 Basic formalism for large scale structure Particle picture (comoving coord., conf. time dτ = dt/a) x i (τ), p i (τ) = amu i (τ), dp i /dτ = am Φ(x i )

10 Basic formalism for large scale structure Particle picture (comoving coord., conf. time dτ = dt/a) x i (τ), p i (τ) = amu i (τ), dp i /dτ = am Φ(x i ) Number of particles dn = f (τ, x, p)d 3 xd 3 p f (τ, x, p) = f (τ + dτ, x + p dτ, p am Φdτ) am

11 Basic formalism for large scale structure Particle picture (comoving coord., conf. time dτ = dt/a) x i (τ), p i (τ) = amu i (τ), dp i /dτ = am Φ(x i ) Number of particles dn = f (τ, x, p)d 3 xd 3 p f (τ, x, p) = f (τ + dτ, x + p dτ, p am Φdτ) am Vlasov eq 0 = ( τ + p am x am Φ ) f (τ, x, p) p (Vlasov)

12 Basic formalism for large scale structure Particle picture (comoving coord., conf. time dτ = dt/a) x i (τ), p i (τ) = amu i (τ), dp i /dτ = am Φ(x i ) Number of particles dn = f (τ, x, p)d 3 xd 3 p f (τ, x, p) = f (τ + dτ, x + p dτ, p am Φdτ) am Vlasov eq 0 = ( τ + p am x am Φ ) f (τ, x, p) p (Vlasov) Poisson eq 2 Φ(x, τ) = 3 2 ΩmH2 δ(x, τ) (Poisson)

13 Basic formalism for large scale structure Particle picture (comoving coord., conf. time dτ = dt/a) x i (τ), p i (τ) = amu i (τ), dp i /dτ = am Φ(x i ) Number of particles dn = f (τ, x, p)d 3 xd 3 p f (τ, x, p) = f (τ + dτ, x + p dτ, p am Φdτ) am Vlasov eq 0 = ( τ + p am x am Φ ) f (τ, x, p) p (Vlasov) Poisson eq 2 Φ(x, τ) = 3 2 ΩmH2 δ(x, τ) (Poisson) Moments m d 3 p f = ρ = ρ(1 + δ), m d 3 p p am f = ρu

14 Basic formalism for large scale structure Moments of Vlasov eq. δ(x, τ) τ + {(1 + δ(x, τ)u(x, τ)} = 0 (continuity) u(x, τ) τ stress tensor σ ij = m d 3 p p i p j a 2 m 2 f ρu i u j vorticity w = u, divergence θ = u + Hu + u u = Φ 1 ρ j(σ ij ρ) (Euler)

15 Basic formalism for large scale structure Moments of Vlasov eq. δ(x, τ) τ + {(1 + δ(x, τ)u(x, τ)} = 0 (continuity) u(x, τ) τ stress tensor σ ij = m d 3 p p i p j a 2 m 2 f ρu i u j vorticity w = u, divergence θ = u + Hu + u u = Φ 1 ρ j(σ ij ρ) (Euler) solution of linearized eqs for σ ij = 0 (pressureless perfect fluid PPF) ( δlin (x, τ) ) θ lin (x, τ)/h ( ) δ0(x) = [D +(τ)a + D (τ)b] θ 0(x)/H 0 w lin (x, τ) = w 0(x) a 0/a(τ) with k-indep. growing/decaying mode (D + a, D a 3/2 in EdS)

16 Basic formalism for large scale structure Moments of Vlasov eq. δ(x, τ) τ + {(1 + δ(x, τ)u(x, τ)} = 0 (continuity) u(x, τ) τ stress tensor σ ij = m d 3 p p i p j a 2 m 2 f ρu i u j vorticity w = u, divergence θ = u + Hu + u u = Φ 1 ρ j(σ ij ρ) (Euler) solution of linearized eqs for σ ij = 0 (pressureless perfect fluid PPF) ( δlin (x, τ) ) θ lin (x, τ)/h ( ) δ0(x) = [D +(τ)a + D (τ)b] θ 0(x)/H 0 w lin (x, τ) = w 0(x) a 0/a(τ) with k-indep. growing/decaying mode (D + a, D a 3/2 in EdS) (D + + HD + = 3 2 ΩmH2 D +) P lin (k, τ) = D +(τ) 2 P(k, τ 0) + O(D )

17 Validity of the linear and PPF approximations Linear vs N-body PPF vs non-ideal fluid 1.4 z PkPnowigglek k hmpc black=linear, red=horizon Run 2 corr. to P δδ (P θθ ) from non-zero σ ij N-body data from Kim et al Pueblas, Scoccimarro

18 Non-linear corrections Continuity, Euler, Poisson for PPF ηψ a(k, η) + Ω ab Ψ b = δ (3) (k k 1 k 2)γ abc (k 1, k 2)Ψ b (k 1, η)ψ c(k 2, η) k 1,k 2 ( ) where η = ln a, Ψ = δ, γ θ/h 121 = k k ( ) 1, γ k = k2 k 1 k 2, k 2 1 k2 Ω = 3/2 1/2 2

19 Non-linear corrections Continuity, Euler, Poisson for PPF ηψ a(k, η) + Ω ab Ψ b = δ (3) (k k 1 k 2)γ abc (k 1, k 2)Ψ b (k 1, η)ψ c(k 2, η) k 1,k 2 ( ) where η = ln a, Ψ = δ, γ θ/h 121 = k k ( ) 1, γ k = k2 k 1 k 2, k 2 1 k2 Ω = 3/2 1/2 2 Linear propagator g(η, η 0) = Θ(η η 0) exp( η Ω) = Θ(η η ( 0) e η η 0 }{{} A + e 3/2(η η0) ) }{{} B η 0 D + D

20 Non-linear corrections Continuity, Euler, Poisson for PPF ηψ a(k, η) + Ω ab Ψ b = δ (3) (k k 1 k 2)γ abc (k 1, k 2)Ψ b (k 1, η)ψ c(k 2, η) k 1,k 2 ( ) where η = ln a, Ψ = δ, γ θ/h 121 = k k ( ) 1, γ k = k2 k 1 k 2, k 2 1 k2 Ω = 3/2 1/2 2 Linear propagator g(η, η 0) = Θ(η η 0) exp( Integral equation Ψ(k, η) = g(η, η 0)Ψ(k, η 0)+ η Ω) = Θ(η η ( 0) e η η 0 }{{} A + e 3/2(η η0) ) }{{} B η 0 D + D dη g(η, η ) δ (3) (k k 1 k 2)γΨΨ η η 0 k 1,k 2 η

21 Non-linear corrections Continuity, Euler, Poisson for PPF ηψ a(k, η) + Ω ab Ψ b = δ (3) (k k 1 k 2)γ abc (k 1, k 2)Ψ b (k 1, η)ψ c(k 2, η) k 1,k 2 ( ) where η = ln a, Ψ = δ, γ θ/h 121 = k k ( ) 1, γ k = k2 k 1 k 2, k 2 1 k2 Ω = 3/2 1/2 2 Linear propagator g(η, η 0) = Θ(η η 0) exp( Integral equation Ψ(k, η) = g(η, η 0)Ψ(k, η 0)+ η Ω) = Θ(η η ( 0) e η η 0 }{{} A + e 3/2(η η0) ) }{{} B η 0 D + D dη g(η, η ) δ (3) (k k 1 k 2)γΨΨ η η 0 k 1,k 2 η iterative solution in powers of initial condition Ψ(k, η 0) Ψ (1) (k, η) = g(η, η 0)Ψ(k, η 0)

22 Non-linear corrections Continuity, Euler, Poisson for PPF ηψ a(k, η) + Ω ab Ψ b = δ (3) (k k 1 k 2)γ abc (k 1, k 2)Ψ b (k 1, η)ψ c(k 2, η) k 1,k 2 ( ) where η = ln a, Ψ = δ, γ θ/h 121 = k k ( ) 1, γ k = k2 k 1 k 2, k 2 1 k2 Ω = 3/2 1/2 2 Linear propagator g(η, η 0) = Θ(η η 0) exp( Integral equation Ψ(k, η) = g(η, η 0)Ψ(k, η 0)+ η Ω) = Θ(η η ( 0) e η η 0 }{{} A + e 3/2(η η0) ) }{{} B η 0 D + D dη g(η, η ) δ (3) (k k 1 k 2)γΨΨ η η 0 k 1,k 2 η iterative solution in powers of initial condition Ψ(k, η 0) Ψ (1) (k, η) = g(η, η 0)Ψ(k, η 0) η Ψ (2) (k, η) = Ψ (1) + dη g(η, η ) δ (3) (k k 1 k 2)γΨ (1) Ψ (1) η η 0 k 1,k 2...

23 Loop expansion of the power spectrum Classical-statistical average over IC s yields perturbative expansion of Ψ(k, η)ψ( k, η) in terms of n-point fctns Ψ(k 1, η 0) Ψ(k n, η 0)

24 Loop expansion of the power spectrum Classical-statistical average over IC s yields perturbative expansion of Ψ(k, η)ψ( k, η) in terms of n-point fctns Ψ(k 1, η 0) Ψ(k n, η 0) Can be expressed in terms of initial power spectrum (n = 2) for Gaussian IC s (Wick theorem) Feynman rules g ab (η, η) (η, a) ( η, b) (k 1, b) Pab 0 γ abc (k 1, k 2 ) (k) (k 1 + k 2, a) (k, a) ( k, b) (k 2, c)

25 Loop expansion of the power spectrum Classical-statistical average over IC s yields perturbative expansion of Ψ(k, η)ψ( k, η) in terms of n-point fctns Ψ(k 1, η 0) Ψ(k n, η 0) Can be expressed in terms of initial power spectrum (n = 2) for Gaussian IC s (Wick theorem) Feynman rules g ab (η, η) (η, a) ( η, b) (k 1, b) Pab 0 γ abc (k 1, k 2 ) (k) (k 1 + k 2, a) (k, a) ( k, b) (k 2, c) Loop expansion (only retarded prop, arrows=time-flow)

26 Loop expansion of the power spectrum Classical-statistical average over IC s yields perturbative expansion of Ψ(k, η)ψ( k, η) in terms of n-point fctns Ψ(k 1, η 0) Ψ(k n, η 0) Can be expressed in terms of initial power spectrum (n = 2) for Gaussian IC s (Wick theorem) Feynman rules g ab (η, η) (η, a) ( η, b) (k 1, b) Pab 0 γ abc (k 1, k 2 ) (k) (k 1 + k 2, a) (k, a) ( k, b) (k 2, c) Loop expansion (only retarded prop, arrows=time-flow) Fastest growing contributions P L loop [D +(z) 2 P 0(q i )] L+1 P(k, z) = P 1 loop P lin {}}{{}}{ D +(z) 2 P 0(k) + D +(z) 4 (2P 13 + P 22) + D +(z) 6 (2P P 24 + P 33) +... e.g. P 22(k) = 2 d 3 q F s 2 (q, k q) 2 P 0(q)P 0( k q ), F 2 = 5 7 γ γ222

27 Two loop 100 Plin 1-loop 2-loop P(k,z=0) [(h/mpc) -3 ] k [h/mpc] SPT breaks down at small scales / late times [P L loop D 2L+2 + ( 1 1+z ) 2L+2 ] initial spectrum from CAMB (ΛCDM - WMAP5)

28 Two loop z PkPnowigglek k hmpc black=linear, 1L(dashed), 2L(dotdashed), red=horizon Run 2

29 Two loop z PkPnowigglek k hmpc black=linear, 1L(dashed), 2L(dotdashed), red=horizon Run 2

30 Expansion parameter? Enhancement from soft loops, vertex γ k q i /q 2 i k 2 σ 2 d(z) for soft q i k ( ) 2 ( ) 2 ( ) 2 d 3 k q 1 k q P lin(q, z) O(10) q z k NL At L-loop (k 2 σ 2 d) l with l L (resummation RPT Crocce, Scoccimarro 05) k k

31 Expansion parameter? Enhancement from soft loops, vertex γ k q i /q 2 i k 2 σ 2 d(z) for soft q i k ( ) 2 ( ) 2 ( ) 2 d 3 k q 1 k q P lin(q, z) O(10) q z k NL At L-loop (k 2 σ 2 d) l with l L (resummation RPT Crocce, Scoccimarro 05) k k Power spectrum at 1-loop, for large k k 2 σ 2 d(z)p lin (k, z), 1 2 k 2 σ 2 d(z)p lin (k, z) Leading terms (l = L) cancel Bertschinger, Jain 95; Bernardeau, de Rijt, Vernizzi 11 Expected from Galilean invariance Frieman,Scoccimarro 95; Pietroni,Peloso 13

32 Expansion parameter? Enhancement from soft loops, vertex γ k q i /q 2 i k 2 σ 2 d(z) for soft q i k ( ) 2 ( ) 2 ( ) 2 d 3 k q 1 k q P lin(q, z) O(10) q z k NL At L-loop (k 2 σ 2 d) l with l L (resummation RPT Crocce, Scoccimarro 05) k k Power spectrum at 1-loop, for large k k 2 σ 2 d(z)p lin (k, z), 1 2 k 2 σ 2 d(z)p lin (k, z) Leading terms (l = L) cancel Bertschinger, Jain 95; Bernardeau, de Rijt, Vernizzi 11 Expected from Galilean invariance Frieman,Scoccimarro 95; Pietroni,Peloso 13 Subleading terms cancel as well (1 l L) Blas, MG, Konstandin 13 k 2 σd 2 spurious scale for equal-time correlators Sugiyama, Spergel 13

33 Expansion parameter k ( ) 2 σl 2 (k, z) 4π dq q 2 1 P lin (q, z) O(1) ln 3 (e + k/k NL) z Large k P 1 loop (k) ( σl 2 (k, z) k k + 0.1[k k ] 2) P lin (k, z) ( P 2 loop (k) σl 4 (k, z) k k [k k ] [k k ] [k k ] 4 )P lin (k, z) Small k P 1 loop (k) k2 P lin (k, z) 4π 3 P 2 loop (k) k2 P lin (k, z) 4π 3 0 dqp lin (q, z)σ 0 l (q, z) 0 dqp lin (q, z) J(q) }{{} σ 2 l (q,z)

34 Two loop 100 linear one-loop two-loop 10 P / Mpc k / Mpc -1

35 Cancellation of IR-enhanced terms Cancellation can be made manifest for loop integrand Blas, MG, Konstandin 13; Carrasco, Foreman, Green, Senatore 13 (k, k1,..., kn) B n l=0 (k, k1,..., kn) B n l= sum 1-loop 2 P13 P22 k 2 k sum 2 P17 2 P26 2 P35 P44 k 6 3-loop k [h/mpc] sum 2 P15 2 P24 P33 k 4 k sum 2 P19 2 P28 2 P37 2 P46 P55 k 8 2-loop 4-loop (non-av) k [h/mpc] very helpful for Monte-Carlo integration (2L: 5d, 3L: 8d)

36 Three loop 100 Plin 1-loop 2-loop 3-loop k ref =k 3-loop log measure P(k,z=0) [(h/mpc) -3 ] k [h/mpc] Diego Blas, MG, Thomas Konstandin (power spectrum); see also Bernardeau, Taruya, Nishimichi, (propagator)

37 Three loop z PkPnowigglek khmpc black=linear, 1L(dashed), 2L(dotdashed), 3L(diamonds), red=horizon Run 2

38 Loop expansion of PS in the limit of small k For small k Fry AJ421 (1994) 21 P L loop (k, z) (2L + 1)! 2 L 1 L! P lin (k, z) F2L+1(k, s q 1, q 1,..., q L, q L ) q 1 q L P lin (q 1, z) P lin (q L, z) Using property F s 2L+1 k 2 for k q i Goroff, Grinstein, Rey, Wise AJ311 (1986) 6 P L loop (k, z) k 2 P lin (k, z) for small k

39 One, two and three loop normalized to k 2 P lin (k, z) loop 2-loop 3-loop log measure P(k,z=0)/P lin /k 2 [(h/mpc) -2 ] k [h/mpc] P L loop (k, z) k 2 P lin (k, z) up to k = 0.003, 0.06, 0.08 h/mpc for 1, 2, 3-loop at %-level

40 Loop expansion of PS in the limit of small k Using property F s 2L+1 k 2 /q 2 for k q i and q =max(q i ) P L loop P small k L loop = 244π 325 k2 P lin (k, z) C L 0 dq P lin (q, z) }{{} q 3 with coeff. C L (C 1 = 1, C , C ) and scale-dep. variance σ 2 l (q, z) 4π q 0 dp p 2 P lin (p, z) D +(z) 2 ln 3 (q/q 0) 2L 2 σl (q, z)

41 Loop expansion of PS in the limit of small k Using property F s 2L+1 k 2 /q 2 for k q i and q =max(q i ) P L loop P small k L loop = 244π 325 k2 P lin (k, z) C L 0 dq P lin (q, z) }{{} q 3 with coeff. C L (C 1 = 1, C , C ) and scale-dep. variance σ 2 l (q, z) 4π q 0 dp p 2 P lin (p, z) D +(z) 2 ln 3 (q/q 0) Estimate for Eisenstein-Hu spectrum with n s 1 P small k L loop k 2 (3L 1)! P lin (k, z) C L D +(z) 2L 2 3L Loop expansion is divergent series even at small k and for any z 2L 2 σl (q, z)

42 Loop expansion of PS in the limit of small k Using property F s 2L+1 k 2 /q 2 for k q i and q =max(q i ) P L loop P small k L loop = 244π 325 k2 P lin (k, z) C L 0 dq P lin (q, z) }{{} q 3 with coeff. C L (C 1 = 1, C , C ) and scale-dep. variance σ 2 l (q, z) 4π q 0 dp p 2 P lin (p, z) D +(z) 2 ln 3 (q/q 0) Estimate for Eisenstein-Hu spectrum with n s 1 P small k L loop k 2 (3L 1)! P lin (k, z) C L D +(z) 2L 2 3L Loop expansion is divergent series even at small k and for any z Terms decrease up to a certain order L max(z), then increase Typical behaviour of an asymptotic series (e.g. loop exp. in QED) 2L 2 σl (q, z)

43 Loop expansion of PS in the limit of small k Using property F s 2L+1 k 2 /q 2 for k q i and q =max(q i ) P L loop P small k L loop = 244π 325 k2 P lin (k, z) C L 0 dq P lin (q, z) }{{} q 3 with coeff. C L (C 1 = 1, C , C ) and scale-dep. variance σ 2 l (q, z) 4π q 0 dp p 2 P lin (p, z) D +(z) 2 ln 3 (q/q 0) Estimate for Eisenstein-Hu spectrum with n s 1 P small k L loop k 2 (3L 1)! P lin (k, z) C L D +(z) 2L 2 3L Loop expansion is divergent series even at small k and for any z Terms decrease up to a certain order L max(z), then increase Typical behaviour of an asymptotic series (e.g. loop exp. in QED) 2L 2 σl (q, z) Partial sum up to smallest term yields best result, with error of order the smallest term (e.g. P 2 loop /P lin 6% at z = 0, k = 0.1h/Mpc)

44 Padé ansatz Goal: improve convergence to go to %-accuracy Idea: resummation in small-k limit P small k (k, z) = 244π 315 k 2 P lin (k, z) 0 dqp lin (q, z) K(σ 2 l (q, z)) where the integrand kernel K is given by a series in x σ 2 l (q, z), K(x) = C L x L 1 L=1

45 Padé ansatz Goal: improve convergence to go to %-accuracy Idea: resummation in small-k limit P small k (k, z) = 244π 315 k 2 P lin (k, z) 0 dqp lin (q, z) K(σ 2 l (q, z)) where the integrand kernel K is given by a series in x σ 2 l (q, z), Padé ansatz K pade nm K(x) = C L x L 1 L=1 (x) 1 + n i=1 a ix i 1 + m j=1 b jx j Match for small x, using coeff. up to three loop C 1 = 1, C , C Two-loop matching (using C 1, C 2): n, m = 0, 1 Three-loop matching: either n, m = 0, 2 or n, m = 1, 1

46 Result for Padé resummed small-k limit 1.15 Correction to Pk,z rel. to oneloop Pk,zP 1loop k,z loop 2 loop 3 loop K 01 pade K 02 pade K 11 pade redshift z black=spt, solid=padé resummed result

47 Padé improved PT where P(k, z) = P lin (k, z) + P pade small k (k, z) + P sub 1 loop(k, z) + P sub 2 loop(k, z) + P sub 3 loop(k, z) +..., P sub L loop(k, z) P L loop (k, z) P small k L loop (k, z)

48 Padé improved PT vs N-body PkPnowigglek 1.4 z khmpc black=spt, blue=padé improved PT, red=n-body Horizon Run 2 (solid=linear, dashed=1l, dotdashed=2l, diamonds=3l)

49 Padé improved PT vs N-body 1.4 z 0 PkPnowigglek khmpc black=spt, blue=padé improved PT, red=n-body Horizon Run 2 (solid=linear, dashed=1l, dotdashed=2l, diamonds=3l)

50 What is going on? k Plink KΣ l 2 k,z z pade pade K 11 K 01 1 loop 2 loop 3 loop pade K khmpc k Plink KΣ l 2 k,z z 3 pade K 11 pade K 01 1 loop 2 loop pade K 02 3 loop khmpc Padé damps UV-sensitivity

51 Conclusion Three loop larger than one loop at z = 0 at all scales Expected for realistic ( ΛCDM) spectrum Loop expansion exhibits behaviour of asymptotic series Padé resummation in small k limit Improved perturbative expansion with better convergence properties at BAO scales, good agreement with N-body data

52 Conclusion Three loop larger than one loop at z = 0 at all scales Expected for realistic ( ΛCDM) spectrum Loop expansion exhibits behaviour of asymptotic series Padé resummation in small k limit Improved perturbative expansion with better convergence properties at BAO scales, good agreement with N-body data Many open questions Justification of Padé ansatz Origin of asymptotic behaviour Corrections from non-ideal fluid...

53 Padé improved PT vs N-body PkPnowigglek PkPnowigglek z khmpc z PkPnowigglek PkPnowigglek 1.4 z khmpc z khmpc khmpc

54 Standard Perturbation Theory Expand solution in δ 0(k) = δ(k, z ), for EdS / growing mode δ(k, z) = (D +(z)) q n δ (3) (k q i )F n(q 1,..., q n)δ 0(q 1) δ 0(q n) i n=1 = D +(z)δ 0(k) +... ( ) e.g. F2 s (q 1, q 2) = q 1 q 2 q1 7 2 q 1 q 2 q 2 + q 2 q (q 1 q 2 ) 2 7 q1 2q2 2

55 Standard Perturbation Theory Expand solution in δ 0(k) = δ(k, z ), for EdS / growing mode δ(k, z) = (D +(z)) q n δ (3) (k q i )F n(q 1,..., q n)δ 0(q 1) δ 0(q n) i n=1 = D +(z)δ 0(k) +... ( ) e.g. F2 s (q 1, q 2) = q 1 q 2 q1 7 2 q 1 q 2 q 2 + q 2 q (q 1 q 2 ) 2 7 q1 2q2 2 Power spectrum δ(k, z)δ(k, z) = δ (3) (k + k )P(k, z) is obtained using Wick theorem in terms of P 0(q) = P(q, z 0) for Gaussian IC P(k, z) = P 1 loop P lin {}}{{}}{ D +(z) 2 P 0(k) + D +(z) 4 (2P 13 + P 22) + D +(z) 6 (2P P 24 + P 33) +... e.g. P 22(k) = 2 d 3 q F s 2 (q, k q) 2 P 0(q)P 0( k q )