Note on gauge invariance of second order cosmological perturbations

TL;DR

This paper investigates the gauge invariance of second order cosmological perturbations, revealing infinite families of gauge invariant variables and providing explicit equations of motion in various gauge-invariant forms.

Contribution

It introduces a framework for understanding gauge invariance at second order and derives explicit relations among different gauge invariant variables.

Findings

Infinite families of gauge invariant variables identified at both first and second orders.

Conversion formulas among different gauge invariant families are finite and basis-based.

Explicit equations of motion for second order perturbations in Newtonian, synchronous, and hybrid gauges provided.

Abstract

We study the gauge invariant cosmological perturbations up to second order. We show that there are infinite families of gauge invariant variables at both of the first and second orders. The conversion formulae among different families are shown to be described by a finite number of bases that are gauge invariant. For the second order cosmological perturbations induced by the first order scalar perturbations, we explicitly represent the equations of motion of them in terms of the gauge invariant Newtonian, synchronous and hybrid variables, respectively.