Perturbation theory of large scale structure in the ΛcDM Universe: Exact time evolution and the two-loop power spectrum

We derive exact analytic solutions for density and velocity fields to all orders in Eulerian standard 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-)Einstein-de Sitter (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 (at redshift zero, for wave numbers k<1 h/Mpc). This deviation can be partially mitigated by exploiting the degeneracy with the effective field theory 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 comparable to 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.