Perturbation theory of LSS in the $Λ$CDM Universe: exact time evolution and the two-loop power spectrum

Abstract

We derive exact analytic solutions for density and velocity fields to all orders in Eulerian perturbation theory for ΛCDM cosmology. In particular, we show that density and velocity field kernels can be written in a separable form in time and momenta at each perturbative order. The kernel solutions are built from an analytic basis of momentum operators and their time-dependent coefficients, which solve a set of recursive differential equations. We also provide an exact closed perturbative solution for such coefficients, expanding around the (quasi-)EdS approximation. We find that the perturbative solution rapidly converges towards the numerically obtained solutions and its leading order result suffices for any practical requirements. To illustrate our findings, we compute the exact two-loop dark matter density and velocity power spectra in ΛCDM cosmology. We show that the difference between the exact ΛCDM and the (quasi-)EdS approximated result can reach the level of several percent. This deviation can be partially mitigated by exploiting the degeneracy with the EFT counterterms. As an additional benefit of our algorithm for the solutions of time-dependent coefficients, the computational complexity of power spectra loops in ΛCDM is reduced to the same level of the EdS case. In performing the two-loop computation, we devise an explicit method to implement the so-called IR cancellations, as well as the cancellations arising as a consequence of mass and momentum conservation.

Minimal basis for exact time dependent kernels in cosmological perturbation theory and application to ΛCDM and w 0 wa CDM

“…The only non-vanishing elements of γ a bc are γ 1 21 = α(k 1 , k 2 ) and γ 2 22 = β(k 1 , k 2 ) and the matrix is defined as…”

“…1. We first introduce a non-minimal basis, following the lines of [11,21], which is convenient to obtain in an algorithmic way. We refer to this as "naive" basis, see eq.…”

“…While a workaround has been suggested in ref. [21], it is clearly preferred to eliminate the redundancy of the set of basis functions. This reduces the number of required terms and makes symmetry constraints manifest for arbitrary time-dependent cosmologies.…”

“…(1.1) also for higher n, following for example the algorithm from [11], see e.g. [21]. Alternatively, one may also resort to computing the entire kernel numerically at higher order, as done for example in [5,6,22] for the one-loop bispectrum (requiring n = 4) and two-loop power spectrum (requiring n = 5) within ΛCDM.…”

“…That is, the "naive" basis functions H X n,i generically approach a non-vanishing value for k 2 /q 2 ≪ 1. The correct scaling is only recovered when summing over all i [21]. This feature occurs for kernels with n ≥ 4.…”