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

TL;DR

This paper develops a fourth-order Lagrangian perturbation theory for non-linear matter clustering in flat cosmologies, providing detailed solutions and relations to Eulerian theory, crucial for accurate bispectrum modeling.

Contribution

It derives the fourth-order solutions in Lagrangian perturbation theory and establishes exact relations with Eulerian predictions, advancing non-linear structure formation models.

Findings

Derived fourth-order displacement fields in Lagrangian perturbation theory.

Established exact relations between Lagrangian and Eulerian perturbation series.

Provided insights into the initial position limit approximation for large-scale structure modeling.

Abstract

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