Note on gauge invariance of second order cosmological perturbations * *Z.C. and Q.-H.Z. are supported by the National Natural Science Foundation of China (12075249, 11675182, 11690022), and S.W. is supported by the grants from the Institute of High Energy Physics (Y954040101, H95120P0U7), the Key Research Program of the Chinese Academy of Sciences (XDPB15), and the science research grants from the

Abstract

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.