Resummation methods in cosmological perturbation theory: next-to-leading computations Stefano Anselmi email: stefano.anselmi@pd.infn.it Padova University Paris, September 20th, 2011
Contents Motivations: beyond linear theory Dark Matter hydrodynamics The non-linear propagator Leading and next-to-leading resummations Numerical results Future prospects and conclusions 2
We need to go beyond perturbation theory The crucial role played by the gravitational instability makes the dark matter density field non-linear on the relevant range of scales. The study of the LSS requires to go beyond the linear theory. N-body simulations work also in this mildly non linear regime but they need very large volumes and high resolutions, with the consequence that due to time limitations, only few cosmologies have been investigated so far. Perturbation theory improves on the linear one for z > 2, but fails at smaller redshifts Resummation methods needed. A new promising semi-analytical approach: Resummed Perturbation Theory The general idea is describing how, in some noticeable cases, classes of diagrams can be resummed to all orders in perturbation theory. 3
Dark matter hydrodynamics DM is described by a fluid-like equation where the only driving force is gravity. The fluid is pressurless (no interacting particle) and the velocities are not relativistic. Neglecting the stress tensor (single stream approximation) one gets the Euler- Poisson system. δ τ + [(1 + δ)v p ] = 0, v p τ + H v p + (v )v p = φ, 2 φ = 3 2 Ω mh 2 density contrast: δ = δρ/ ρ matter density: Ω m where conformal time: dτ = dt/a(τ) peculiar velocity: v p conformal expansion rate: H = dlog a/dτ peculiar potential: φ In this approximation neglecting deviations from a coherent fluid flow (does not work in the strong gravitational regime). 4
Equations of motion in nice Compact Form Assuming EdS cosmology, the hydrodynamical equations for density and velocity perturbations can be written in a compact form in Fourier space ( δab η + Ω ab ) ϕb (k,η) = e η γ abc (k, p, q)ϕ b (p,η)ϕ c (q,η) Crocce and Scoccimarro, 2006 (repeated indices/momenta are summed/integrated over) where ( ϕ1 (k,η) ϕ 2 (k,η) velocity divergence ) e η ( δ(k, η) θ(k,η)/h ) scale factor, η = log a (, Ω = a in 1 1 3/2 3/2 ) The mode-mode coupling is given by the vertex function. The only non zero components are γ 121 (k, p, q) = γ 112 (k, q, p) = 1 2 δ D(k + p + q)α(p,q), γ 222 (k, p, q) = δ D (k + p + q)β(p,q) (p+q) 2 p q 2p 2 q 2 (p+q) p p 2 Without changing the equations structure there is a simple way to extend this approach to ΛCDM cosmology. 5
Linear approximation and linear propagator Crocce and Scoccimarro, 2006 The linear approximation consists in neglecting mode-mode coupling in fluid equations: α=β=0 This approximation holds at early times and/or on large scale. In order to solve the linear hydrodynamical equations (compact form) we have ( δab ηa + Ω ab ) gbc (η a, ) = δ ac δ D (η a ) so that ϕ 0 a (k,η a) = g ab (η a, )ϕ 0 b (k,) is the solution of the linear equation The linear retarder propagator step function ( 3 2 ) g ab (η a, ) = [ B + A e 5/2(η a ) ] ab θ(η a ) B = 1 5 A = 1 5 ( 3 2 2 2 3 3 ) 6
QFT-like formulation and Feynman dagrams The previous results allow us to give a QFT-like formulation of the problem Matarrese and Pietroni, 2008 In this way we take into account both the dynamics of the system and the statistical properties (here we will suppose Guassian initial conditions [encoded in the initial power spectrum]). Following the standard (Quantum Field Theory) procedure one can read out the Feynman Rules three fundamental building blocks Now we can recast the cosmological perturbation series in a more appropriate way. Basically it works by means of a new fundamental object: the non-linear propagator 7
The full propagator: exact time evolution equation The full propagator gives us the cross-correlation between the perturbation at a given time and length scale and the initial condition at the same scale. P ab (k;η a,0) = G ac (k;η a,0)p 0 cb (k) full propagator S.A., Matarrese and Pietroni, 2011 - Arxiv: 1011.4477 initial PS It is not an observable quantity but it is a fundamental ingredient for the computation of the equal time non-linear power spectrum P ab (k;η,η) The evolution equations in times (or scale) are useful tools to study this quantity. The full propagator obeys the exact evolution equation: ηa G ab (k,η a, ) = Ω ac G cb (k,η a, ) + dsσ ac (k,η a, s)g cb (k, s, ) is the full two-point 1PI function: 8
Leading resummation for the propagator We solve, in an approximate way, the propagator evolution equation. PROCEDURE Compute the propagator evolution equation in the large momentum limit Compute the evolution equation in the small momentum limit These results automatically match and allow us to extend the evolution equation for any momentum value 9
Large momentum limit - Factorization Property S.A., Matarrese and Pietroni, 2011 - Arxiv: 1011.4477 In the large-k limit we resum all the infinite series of the dominant diagrams Each of the 2n vertices contributes a factor: u c γ acb (k, q,q k) large k 1 k q 2 q 2 δ ab Each loop integral decouples and the diagrams at n-loop order k 2n we get the remarkable FACTORIZATION PROPERTY n 1 j=0 (n j) dsσ ac (k; η a, s)g ( j) cb (k; s,) large k G (n 1) (k; η ab a, ) dsσ (1) ac (k; η a, s) u c 4 ( 1 1 ) 10
Large momentum result Summing over all the loop orders, in the large-k limit we get ηa G ab (k; η a, ) = Ω ac G cb (k; η a, ) +G ab (k; η a, ) dsσ (1) ac (k; η a, s) u c where dsσ (1) ac (k; η large k a, s) u c k 2 σ 2 e η a (e η a e ), (for a = 1,2) velocity dispersion: σ 2 1 3 d 3 q P0 (q) q 2 This result can be integrated to get G ab (k; η a, ) = g ab (η a, ) e ( k 2 σ 2 (e η a e ) 2 ) This matches with the diagrammatic results found by (Crocce and Scoccimarro, 2006), obtained resumming the dominant diagrams in the large-k limit. 2 By means of the evol. equation we can go beyond this result. 11
Small momentum limit - THE SAME EVOLUTION EQ. In the small-k limit (large scale) higher order contributions to the propagator are suppressed, and linear perturbation theory is recovered. We can compute the evolution equation at one-loop level Modulo terms at least of two-loop order we get ηa G ab (k; η a, ) = Ω ac G cb (k; η a, ) +G ab (k; η a, ) dsσ (1) ac (k; η a, s) u c In the large and small momentum limit we achieve the SAME EVOLUTION EQUATION 12
Next-to-leading resummation - large momentum limit S.A., Matarrese and Pietroni, 2011 - Arxiv: 1011.4477 In the leading resummation we have partially renormalized the vertex and the propagator but we did not renomalize the PS. Now we take into account also the PS renormalization (in this way we avoid the double counting problem). Large momentum behavior We consider the dominant diagrams replacing the linear PS with the non-linear one.... + + +... = non linear PS Since we have: A c γ acb (k, q,q k) large k 1 k q A 2 2 q 2 δ ab where A c = ( A1 A 2 ) Each loop integral decouples; Diagrams at n-loop order k 2n ; Only 22 is selected P nl 13
Extended factorization -> NEW Evolution Equation FACTORIZATION PROPERTY STILL HOLDS (Now the upper indices count the number of non-linear PS in a given contribution to and G) n 1 j=0 (n j) dsσ ac (k; η a, s)g ( j) cb (k; s,) large k G (n 1) (k; η ab a, ) dsσ PSnl ac (k; η a, s)u c 4 Summing over all the loop orders we get this NEW EVOLUTION EQUATION ηa G ab (k; η a, ) = Ω ac G cb (k; η a, ) +G ab (k; η a, ) dsσ PSnl ac (k; η a, s) u c Integrating the diff. eq. we get G ab (k; η a, ) = g ab (η a, ) e ( k 2 σ 2 nl (η a, ) (eη a e ) 2 ) 2 velocity-velocity component of the PS 1 3 ds 1 s1 ds 2 e s 1+s 2 d 3 q Pnl 22 (q; s 1, s 2 ) q 2 14
Small momentum limit with a generic non linear PS In the opposite small-k limit the linear perturbation theory works. To recover this limit we need that the approximation considered for the one-particle irreducible function is such that Σ ab (k; η a, s) k 0 0 On the other hand, naively substituting the leading contribution (to the 1PI) in the evolution equation (the higher contributions are subdominant) Σ ab Σ PSnl ab 4 = dsσ PSnl k 0 ac (k; η a, s) u c the factorization property still holds but one gets 1 3 δ a1 ds e η a+s [ ] d 3 q g 2d (η a s)p nl 1d (q;η a, s) g 1d (η a s)p nl 2d (q;η a, s) that is non zero in general - therefore in the small k limit we must address this problem 15
Considering the one-loop power spectrum If we consider P nl (k;η a, s) P 1-loop (k;η a, s) we get these diagrams Σ PS1l ab (k;η a, s) : (I) (II) (III) (IV) s' s'' 4 16 16 8 s! a Taking the small-k limit one finds lim k 0 ΣPS1l ab (k;η a, s) 0 we do not retrieve the linear theory Indeed the correct limit is recovered when we also take into account the remaining 2-loop diagrams for Σ 2loop ac 16
Adding contributions to one particle irreducible Matching with the high k limit remaining 2-loop diagrams for Σ 2loop ac (V) 16 s s' (VI) s s' s'' 16 s''! a! a Σ 2l rest ac (k; η a, s) : (VII) 16 (VIII) 16 (IX) 16 Diagrams (V) and (VI) are already taken into account in the large-k limit In the large-k limit diagrams (VII) and (VIII) and (IX) are not involved in the next to leading resummation we are interested in. Therefore this approximation gives us both the correct momentum limits so the evolution eq. Σ ac (k; η a, s) Σ PS1l ac (k; η a, s) + lim Σ 2l rest ac (k; η a, s) k 0 = Σ PS1l ac (k; η a, s) lim Σ PS1l ac (k; η a, s) k 0 ηa G ab (k; η a, ) = Ω ac G cb (k; η a, ) +G ab (k; η a, ) ds Σ ac (k; η a, s) u c 17
Considering a resummed power spectrum We consider the Time Renormalization Group (TRG) power spectrum. With such PS we achieve these kind of contributions (M. Pietroni - 2008)... + +... In order to get the different-time power spectrum. We take advantage from this approximation contribution given by initial non-gaussianities P TRG ab (q;η a, s) G leading-order ac (q;η a, s)p TRG cb (q; s, s) + P NG ab (k;η, s) We have again a non-vanishing small-k limit for the 1PI. We use the previous prescription to incorporate the relevant corrections. Σ ac (k; η a, s) Σ PTRG ac (k; η a, s) lim Σ PTRG ac (k; η a, s) k 0 encoded in the Bispectrum at time s (given by non-linear behavior of gravity) Again we get that the usual Evolution eq. holds in both the limits considered 18
Numerical implementation - Density propagator We consider a ΛCDM cosmology close to the best fit model Initial conditions: z = 100 (well inside the linear regime) Density propagator: G1 = G11 + G12 density propagator plot 1.0 Relative difference respect to the leading resummed propagator 0.8 Black lines: leading resummed propagator (linear PS) Red lines: propagator with 1-loop PS 0.5 Blu lines: propagator with TRG-PS Propagators 0.5 0.4 Prop. with 1 loop PS 0.4 0.6 Prop. with TRG PS G 1 0.3 0.3 0.4 z 1 G 1 G 1 0.2 z 0 G 2 G 2 0.2 0.2 z 0 0.1 0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 k h Mpc 0.0 0.0 0.1 0.2 0.3 0.4 BAO scales k h Mpc Weaker damping for the propagator computed with the next-to-leading resummations at intermediate and large momentum values z 1 0.0 19
Velocity propagator Velocity propagator: G2 = G21 + G22 velocity propagator plot 1.0 0.8 Black lines: leading resummed propagator (linear PS) Red lines: propagator with 1-loop PS 0.5 Blu lines: propagator with TRG-PS Propagators Relative difference respect to the leading resummed propagator Propagators Prop. with 1 loop PS Prop. with TRG PS 0.4 Prop. with 1 loop PS Prop. with TRG PS 0.6 0.3 G 2 0.4 z 1 z 0 G 2 G 2 0.2 z 0 0.2 z 0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 k h Mpc z 1 0.1 0.0 0.0 0.1 0.2 0.3 0.4 BAO scales k h Mpc z 1 The new corrections show again a weaker damped propagator For the velocity propagator the effect is stronger than for the density one 20
Cut-off dependence with the one-loop PS Propagator computed with the one-loop PS results cut-off dependent Limit of one-loop approximation: PS takes unphysical negative values at large k s (especially when the time difference is very different) Relative difference respect to the propagator computed with linear PS 0.5 density propagator 0.5 velocity propagator cut-off: qmax = 2h/Mpc 0.4 0.4 0.3 z 0 0.3 z 0 G 1 G 1 G 2 G 2 0.2 0.2 0.1 z 1 0.1 z 1 0.0 0.0 0.1 0.2 0.3 0.4 k h Mpc 0.0 0.0 0.1 0.2 0.3 0.4 k h Mpc cut-off: qmax = 1h/Mpc The cut-off dependence is quite strong for the velocity-propagator. Propagator computation with one-loop PS is unreliable at low redshift. The TRG-PS (always positive) does not give rise to a cut-off dependence 21
Effective velocity dispersion To deeper understand our results we define an effective velocity dispersion σ 2 e f f,a (k;η,0) = In the high-k limit it reduces to Black lines: leading resummed propagator (linear PS) Red lines: propagator with 1-loop PS Blu lines: propagator with TRG-PS 2 k 2 (e η 1) 2 ln(g a(k;η,0)), (a = 1,2) σ 2 nl (η,0) = 2 η 0 ds 1 s1 0 ds 2 e s 1+s 2 d 3 q Pnl 22 (q;s 1,s 2 ) q 2 3(e η 1) 2 0.007 density propagator Propagators Prop. CS approximation 0.012 velocity propagator Propagators Prop. CS approximation 0.006 Prop. with 1 loop PS Prop. with TRG PS Prop. with 1 loop PS Prop. with TRG PS 0.010 2 Σ eff,1 0.005 2 Σ eff,2 0.008 0.004 z 0 0.006 z 0 0.003 0.005 0.01 0.05 0.1 0.5 1. k h Mpc 0.005 0.01 0.05 0.1 0.5 1. k h Mpc BAO scales BAO scales 22
Considerations For G1 the subleading affects considered here are larger than 1% for k 0.10h/Mpc at z = 0 For G2 they are larger than 1% for k 0.07h/Mpc at z = 0 Well inside the BAO range, we achieve k = 0.2h/Mpc at z = 0 4.0 % for G1 7.8 % for G2 These effects should be taken into account to achieve a percent computation of PS The next-to-leading propagator results enhanced w.r.t. the one computed with the leading contributions. (velocity PS is smaller than the linear one) At large-k σ 2 depends only on P22 - this trend persists at smaller k 23
Short term goal: the power spectrum The full power spectrum is governed by this equation P ab = P I ab + P II ab where P I exhibits a strong propagator dependence P I ab (k;η,η) = G ac(k;η,0)g bd (k;η,0)p 0 cd (k) η η P II ab (k;η,η) = ds 1 ds 2 G ac (k;η, s 1 )G bd (k;η, s 2 )Φ cd (k; s 1, s 2 ) 0 0 is the full two-point 1PI function: In the same spirit of the G computation we plan to consider the following approximation for 2 non-linear PS We have to implement our extended propagator into the PS computation and compare the results with N-body simulations. 24
Conclusions Resummation methods useful tools to study the (mild) non-linear regime. In this context the non-linear propagator gains a fundamental role. We improved the prop. computations considering next-to-leading resummations. We resummed the corrections given by a generic non-linear PS. We analyzed two different approximated non-linear power spectra: the one-loop PS and the TRG-PS (resummed). The latter does not exhibit a cut-off dependence, the former does (unphysical behavior). The effect of these new corrections is to enhance the prop. w.r.t. the one computed with only the leading resummation. New corrections are significant in the BAO scale: in principle they should be taken into account in the PS computation. Reconsidering the comparison of non-linear propagator computation with N-body simulations is needed to test our predictions. 25