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