Lagrangian perturbations and the matter bispectrum I: fourth-order model for non-linear clustering

Abstract

We investigate the Lagrangian perturbation theory of a homogeneous and\nisotropic universe in the non-relativistic limit, and derive the solutions up\nto the fourth order. These solutions are needed for example for the\nnext-to-leading order correction of the (resummed) Lagrangian matter\nbispectrum, which we study in an accompanying paper. We focus on flat\ncosmologies with a vanishing cosmological constant, and provide an in-depth\ndescription of two complementary approaches used in the current literature.\nBoth approaches are solved with two different sets of initial conditions---both\nappropriate for modelling the large-scale structure. Afterwards we consider\nonly the fastest growing mode solution, which is not affected by either of\nthese choices of initial conditions. Under the reasonable approximation that\nthe linear density contrast is evaluated at the initial Lagrangian position of\nthe fluid particle, we obtain the nth-order displacement field in the so-called\ninitial position limit: the nth order displacement field consists of 3(n-1)\nintegrals over n linear density contrasts, and obeys self-similarity. Then, we\nfind exact relations between the series in Lagrangian and Eulerian perturbation\ntheory, leading to identical predictions for the density contrast and the\npeculiar-velocity divergence up to the fourth order.\n