Note on gauge invariance of second order cosmological perturbations *

We study the gauge invariant cosmological perturbations up to the second order. We demonstrate that there are infinite families of gauge invariant variables at both the first and second orders. The conversion formulae among different families are verified 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 their equations of motion in terms of the gauge invariant Newtonian, synchronous and hybrid variables, respectively.

Automated summary

This study demonstrates the existence of infinite families of gauge invariant variables in cosmological perturbations and verifies conversion formulae among them. The equations of motion for cosmological perturbations induced by perturbations of different orders are explicitly represented.