Recursive Solutions of Lagrangian Perturbation Theory

TL;DR

This paper derives general recurrence relations for Lagrangian perturbation theory in cosmology, enabling more flexible and accurate solutions for the evolution of self-gravitating fluids in an expanding universe.

Contribution

It generalizes previous limited recurrence relations in LPT, allowing inclusion of any initial conditions and providing simplified, accurate time-dependent relations.

Findings

Derived general recurrence relations for LPT

Identified fastest-growing modes in the relations

Provided simplified approximate relations for time dependence

Abstract

In the standard perturbation theory (SPT) of self-gravitating Newtonian fluid in an expanding universe, recurrence relations for higher-order solutions are well known and play an important role both in practical applications and in theoretical investigations. The recurrence relations in Lagrangian perturbation theory (LPT), however, had not been known for a long time. Recently, two different kinds of recurrence relations in LPT have been proposed in limited cases. In this paper, we generalize those methods, and most generally derive the recurrence relations, which are capable to include any initial condition in general models of cosmology. The fastest-growing modes in the general relations are identified, and simplified recurrence relations with accurate approximation for the time-dependence are obtained.