On the non-linear scale of cosmological perturbation theory

On the non-linear scale of cosmologicalperturbation theoryMathias Garny (DESY Hamburg / CERN)Cosmo Cambridge, 02.09.13based on 1304.1546 and 1309.xxxxwith Diego Blas, Thomas KonstandinMathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

OutlineStandard Perturbation TheoryThree loop resultPadé resummationPadé improved PTMathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Standard Perturbation TheoryPoisson/Euler/Continuity eq. for density contrast δ = ρ/¯ρ − 1 and pec.velocity (neglect vorticity/viscosity → talks by Mercolli, Zaldarriaga)Expand solution in δ 0(k) = δ(k, z 0 ∼ 10 3 ), for EdS / growing modeδ(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)in=1= D +(z)δ 0(k) + . . .( )e.g. F2 s (q 1, q 2) = 5 + 1 q 1·q 2 q17 2 q 1 q 2 q 2+ q 2q 1+ 2 (q 1·q 2 ) 27 q1 2q2 2Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Standard Perturbation TheoryPoisson/Euler/Continuity eq. for density contrast δ = ρ/¯ρ − 1 and pec.velocity (neglect vorticity/viscosity → talks by Mercolli, Zaldarriaga)Expand solution in δ 0(k) = δ(k, z 0 ∼ 10 3 ), for EdS / growing modeδ(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)in=1= D +(z)δ 0(k) + . . .( )e.g. F2 s (q 1, q 2) = 5 + 1 q 1·q 2 q17 2 q 1 q 2 q 2+ q 2q 1+ 2 (q 1·q 2 ) 27 q1 2q2 2Power spectrum 〈δ(k, z)δ(k ′ , z)〉 = δ (3) (k + k ′ )P(k, z) is obtained usingWick theorem in terms of P 0(q) = P(q, z 0) for Gaussian ICP(k, z) =P 1−loopP lin{ }} { { }} {D +(z) 2 P 0(k) + D +(z) 4 (2P 13 + P 22)+ D +(z) 6 (2P 15 + 2P 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|)Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Standard Perturbation Theory100Plin1-loop2-loopP(k,z=0) [(h/Mpc) -3 ]1010.10.010.001 0.01 0.1 1 10k [h/Mpc]SPT breaks down at small scales / late times [P L−loop ∝ D 2L+2+ ∼ ( 11+z) 2L+2]initial spectrum from CAMB (ΛCDM - WMAP5)Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Standard Perturbation TheoryMany approaches have been developed to improve behaviour at large k(closure, LPT, RPT, eikonal, ...) see e.g. Carlson, White, Padmanabhan 0905.0479Enhanced contributions (∝ k 2n , n ≤ L) related to interaction with softmodes cancel out completely in equal-time correlators (like the PS), asexpected from Galilean invariancesee e.g. Bertschinger, Jain 95; Frieman, Scoccimarro 95; Anselmi, Pietroni 12; Pietroni, Peloso 13;Sugiyama, Futamase 13; Blas, MG, Konstandin 13; Carrasco, Foreman, Green, Senatore 13; . . .Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Standard Perturbation TheoryMany approaches have been developed to improve behaviour at large k(closure, LPT, RPT, eikonal, ...) see e.g. Carlson, White, Padmanabhan 0905.0479Enhanced contributions (∝ k 2n , n ≤ L) related to interaction with softmodes cancel out completely in equal-time correlators (like the PS), asexpected from Galilean invariancesee e.g. Bertschinger, Jain 95; Frieman, Scoccimarro 95; Anselmi, Pietroni 12; Pietroni, Peloso 13;Sugiyama, Futamase 13; Blas, MG, Konstandin 13; Carrasco, Foreman, Green, Senatore 13; . . .⇒ RPT ‘as good as’ SPT for large-k for equal-time correlators (but veryuseful for unequal-time corr. as e.g. propagator) e.g. Sugiyama, Spergel 13Cancellations (leading + sub-leading terms) can be made manifest forloop integrand⇒ crucial for Monte-Carlo integrationBlas, MG, Konstandin 13; Carrasco, Foreman, Green, Senatore 13Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Three loop100Plin1-loop2-loop3-loop k ref =k3-loop log measureP(k,z=0) [(h/Mpc) -3 ]1010.10.010.001 0.01 0.1 1 10k [h/Mpc]Diego Blas, MG, Thomas Konstandin 1309.xxxx (power spectrum); see also Bernardeau, Taruya, Nishimichi,1211.1571 (propagator)Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Loop expansion of PS in the limit of small kFor small k Fry AJ421 (1994) 21P 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) 6P L−loop (k, z) ∝ k 2 P lin (k, z)for small kMathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

One, two and three loop normalized to k 2 P lin (k, z)10001-loop2-loop3-loop log measureP(k,z=0)/P lin /k 2 [(h/Mpc) -2 ]1001010.001 0.01 0.1 1 10k [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 %-levelMathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Loop expansion of PS in the limit of small kUsing property F s 2L+1 ∝ k 2 /q 2 for k ≪ q i and q =max(q i )P L−loop → P small−kL−loop= − 244π325 k2 P lin (k, z) × C L ×∫ ∞0dqP lin (q, z) σ 2L−2l (q, z)with coeff. C L (C 1 = 1, C 2 ≃ 0.71, C 3 ≃ 1.05) and scale-dep. varianceσ 2 l (q, z) ≡ 4π∫ q0dp p 2 P lin (p, z)Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Loop expansion of PS in the limit of small kUsing property F s 2L+1 ∝ k 2 /q 2 for k ≪ q i and q =max(q i )P L−loop → P small−kL−loop= − 244π325 k2 P lin (k, z) × C L ×∫ ∞0dqP lin (q, z) σ 2L−2l (q, z)with coeff. C L (C 1 = 1, C 2 ≃ 0.71, C 3 ≃ 1.05) and scale-dep. varianceσ 2 l (q, z) ≡ 4π∫ q0dp p 2 P lin (p, z)Estimate for Eisenstein-Hu spectrum with n s ≃ 1P small−kL−loop ∝ k 2 (3L − 1)!P lin (k, z) × C L × D +(z) 2L2 3L⇒ Loop expansion is divergent series even at small k and for any zMathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Loop expansion of PS in the limit of small kUsing property F s 2L+1 ∝ k 2 /q 2 for k ≪ q i and q =max(q i )P L−loop → P small−kL−loop= − 244π325 k2 P lin (k, z) × C L ×∫ ∞0dqP lin (q, z) σ 2L−2l (q, z)with coeff. C L (C 1 = 1, C 2 ≃ 0.71, C 3 ≃ 1.05) and scale-dep. varianceσ 2 l (q, z) ≡ 4π∫ q0dp p 2 P lin (p, z)Estimate for Eisenstein-Hu spectrum with n s ≃ 1P small−kL−loop ∝ k 2 (3L − 1)!P lin (k, z) × C L × D +(z) 2L2 3L⇒ Loop expansion is divergent series even at small k and for any zTerms decrease up to a certain order L max(z), then increaseTypical behaviour of an asymptotic series (e.g. loop exp. in QED)Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Loop expansion of PS in the limit of small kUsing property F s 2L+1 ∝ k 2 /q 2 for k ≪ q i and q =max(q i )P L−loop → P small−kL−loop= − 244π325 k2 P lin (k, z) × C L ×∫ ∞0dqP lin (q, z) σ 2L−2l (q, z)with coeff. C L (C 1 = 1, C 2 ≃ 0.71, C 3 ≃ 1.05) and scale-dep. varianceσ 2 l (q, z) ≡ 4π∫ q0dp p 2 P lin (p, z)Estimate for Eisenstein-Hu spectrum with n s ≃ 1P small−kL−loop ∝ k 2 (3L − 1)!P lin (k, z) × C L × D +(z) 2L2 3L⇒ Loop expansion is divergent series even at small k and for any zTerms decrease up to a certain order L max(z), then increaseTypical behaviour of an asymptotic series (e.g. loop exp. in QED)Partial sum up to smallest term yields best result, with error of order thesmallest term (e.g. P 2−loop /P lin ≃ 6% at z = 0, k = 0.1h/Mpc)Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Padé ansatzGoal: improve convergence to go to %-accuracyIdea: resummation in small-k limitP small−k (k, z) = − 244π315 k 2 P lin (k, z) ×∫ ∞0dqP 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−1L=1Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Padé ansatzGoal: improve convergence to go to %-accuracyIdea: resummation in small-k limitP small−k (k, z) = − 244π315 k 2 P lin (k, z) ×∫ ∞0dqP 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 ansatzK padenmK(x) =∞∑C L x L−1L=1(x) ≡ 1 + ∑ ni=1 a ix i1 + ∑ mj=1 b jx jMatch for small x, using coeff. up to three loop C 1 = 1, C 2 ≃ 0.71, C 3 ≃ 1.05Two-loop matching (using C 1, C 2): n, m = 0, 1Three-loop matching: either n, m = 0, 2 or n, m = 1, 1Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Result for Padé resummed small-k limit1.15Correction to Pk,z rel. to oneloopPk,zP 1loop k,z1.101.051 loop2 loop3 loopK 01padeK 02padeK 11pade1.000 2 4 6 8 10redshift zblack=SPT, solid=Padé resummed resultMathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Padé improved PTwhereP(k, z) = P lin (k, z) + P padesmall−k(k, z)+ P sub1−loop(k, z) + P sub2−loop(k, z) + P sub3−loop(k, z) + . . . ,P subL−loop(k, z) ≡ P L−loop (k, z) − P small−kL−loop (k, z)Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Padé improved PT vs N-body1.4z⩵0.375PkPnowigglek1.3 1.2 1.1 1.0 0.9 0.00 0.05 0.10 0.15 0.20 0.25 0.30khMpcblack=SPT, blue=Padé improved PT, red=N-body Horizon Run 2Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Padé improved PT vs N-body1.4z⩵0PkPnowigglek1.31.21.11.0 0.90.00 0.05 0.10 0.15 0.20 0.25 0.30khMpcblack=SPT, blue=Padé improved PT, red=N-body Horizon Run 2Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

ConclusionThree loop larger than one loop at z = 0 at all scalesExpected for Eisenstein-Hu-like spectrumLoop expansion exhibits behaviour of asymptotic seriesPadé resummation in small k limitImproved perturbative expansion with better convergence properties atBAO scales, good agreement with N-body dataMathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Basic formalism for large scale structureDensity contrast ρ(x, τ) = ¯ρ(τ)(1 + δ(x, τ)), pec. velocity u(x, τ)1st and 2nd moment of Vlasov eq. for f (x, p, τ), neglect multi-streaming(pressure, stress/viscosity → talks by Mercolli, Zaldarriaga)∂δ(x, τ)∂τ+ ∇ · {(1 + δ(x, τ)u(x, τ)} = 0 (continuity)∂u(x, τ)∂τ+ Hu + u · ∇u = −∇Φ (Euler)∇ 2 Φ(x, τ) = 3 2 ΩmH2 δ(x, τ) (Poisson)neglect vorticity ∇ × u ≈ 0 ⇒ sufficient to use θ = ∇ · usolution of linearized eqs (D + ∼ a, D − ∼ a −3/2 in EdS)Power spectrum in Fourier spaceδ lin (x, τ) = D +(τ)δ 0(x) + O(D −)〈δ(k, τ)δ(k ′ , τ)〉 = δ (3) (k + k ′ )P(k, τ)assume Gaussian IC described by P 0(k) = P(k, τ 0)Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Expansion parameter∫ kσl 2 (k, z) ≡ 4π dq q 2 P lin (q, z)0Large kP 1−loop (k)P 2−loop (k)∼∼()1.14P lin (k, z) − 0.55k∂ k P lin (k, z) + 0.1[k∂ k ] 2 P lin (k, z) σl 2 (k, z)(2.14P lin (k, z) − 1.62k∂ k P lin (k, z) + 0.55[k∂ k ] 2 P lin (k, z))− 0.082[k∂ k ] 3 P lin (k, z) + 0.005[k∂ k ] 4 P lin (k, z) σl 4 (k, z)Small kP 1−loop (k) → − 61105 k 2 P lin (k, z) 4π 3∫ ∞P 2−loop (k) → − 44764143325 k 2 P lin (k, z) 4π 30dqP lin (q, z)∫ ∞where J(q) = 4π ∫ q0 dp p2 g(p/q)P lin (p, z) ∼ σ 2 l (q, z)Mathias Garny (DESY Hamburg / CERN)0dqP lin (q, z)J(q)On the non-linear scale of cosmological perturbation theory

Result for Padé resummed small-k limitz⩵0z⩵310 3 khMpc10 3 khMpck Plink KΣ l2 k,z10 210 110 010 110 210 310 410 510 610 71 loopK 11padeK 01pade2 loop3 looppadeK 02k Plink KΣ l2 k,z10 210 110 010 110 210 310 410 510 610 2 10 1 10 0 10 1 10 2 10 7padepadeK 11 K 011 loop2 looppadeK 023 loop10 2 10 1 10 0 10 1 10 2Integrand kernel k P lin (k)K L (σ 2 l (k, z)) for the power spectrum as obtained inSPT at one-loop (black dashed), two loops (black dot-dashed), three loops(black dotted). The solid lines are the integrand kernels obtained after Padéresummation, K pade01 (green), K pade02 (blue) and K pade11 (magenta).Mathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Padé improved PT vs N-body1.4z⩵01.4z⩵0.375PkPnowigglek1.3 1.2 1.1 1.0 0.90.00 0.05 0.10 0.15 0.20 0.25 0.301.4khMpcz⩵0.833PkPnowigglek1.3 1.2 1.1 1.0 0.9 0.00 0.05 0.10 0.15 0.20 0.25 0.301.4khMpcz⩵1.75PkPnowigglek1.31.2 1.1 1.0 PkPnowigglek1.31.2 1.1 1.0 0.90.00 0.05 0.10 0.15 0.20 0.25 0.30khMpc0.90.00 0.05 0.10 0.15 0.20 0.25 0.30khMpcMathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory

Enhancement from soft loops q i ≪ kScale σ d related to F n ∝ k · q i /q 2 ik 2 σ 2 d(z)≡Power spectrum at 1-loop, for large k∫for soft q i ≪ kd 3 (k · q)2q Pq 4 lin (q, z) = 4π ∫3 k2dq P lin (q, z)P 22 → k 2 σ 2 d(z)P lin (k, z), 2P 13 → −k 2 σ 2 d(z)P lin (k, z)At L-loop ∼ (k 2 σ 2 d) l with l ≤ L (resummation → RPT Crocce, Scoccimarro 05)Leading terms (l = L) cancel when summing over all contributionsBertschinger, Jain 95Subleading terms cancel as well (1 ≤ l ≤ L) Blas, MG, Konstandin 13Expected from Galilean invariance Frieman,Scoccimarro 95; Pietroni,Peloso 13⇒ SPT as good as RPT for equal-time correlators Sugiyama, Spergel 13Cancellation can be made manifest for loop integrand Blas, MG, Konstandin 13;Carrasco, Foreman, Green, Senatore 13⇒ very helpful for Monte-Carlo integrationMathias Garny (DESY Hamburg / CERN)On the non-linear scale of cosmological perturbation theory