# The general solution at large scale for second order perturbations in a scalar field dominated universe

## TL;DR

This paper derives the general solution for second order perturbations in a flat universe dominated by a scalar field, providing explicit formulas and a new conserved quantity for super-horizon scales.

## Contribution

It presents the first explicit general solution for second order scalar perturbations in a scalar field universe and introduces a new conserved quantity at second order.

## Key findings

- Explicit general solution for second order perturbations
- New conserved quantity for super-horizon perturbations
- Applicable to arbitrary scalar field potentials

## Abstract

In this paper we consider second order perturbations of a flat Friedmann-Lema\^{i}tre universe whose stress-energy content is a single minimally coupled scalar field with an arbitrary potential. We derive the general solution of the perturbed Einstein equations in explicit form for this class of models when the perturbations are in the super-horizon regime. As a by-product we obtain a new conserved quantity for long wavelength perturbations of a single scalar field at second order.