Second order cosmological perturbations: new conserved quantities and the general solution at super-horizon scale

The study of long wavelength scalar perturbations, in particular the existence of conserved quantities when the perturbations are adiabatic, plays an important role in e.g. inflationary cosmology. In this paper we present some new conserved quantities at second order and relate them to the curvature perturbation in the uniform density gauge, $\zeta$, and the comoving curvature perturbation, ${\cal R}$. We also, for the first time, derive the general solution of the perturbed Einstein equations at second order, which thereby contains both growing and decaying modes, for adiabatic long wavelength perturbations for a stress-energy tensor with zero anisotropic stresses and zero heat flux. The derivation uses the total matter gauge, but results are subsequently translated to the uniform curvature and Poisson (longitudinal, zero shear) gauges.