Non-Gaussian Correlations Outside the Horizon II: The General Case

The results of a recent paper [0808.2909] are generalized. A more detailed proof is presented that under essentially all conditions, the non-linear classical equations governing matter and gravitation in cosmology have ``adiabatic'' solutions in which, far outside the horizon, in a suitable gauge, the reduced spatial metric $g_{ij}({\bf x},t)/a^2(t)$ becomes a time-independent function ${\cal G}_{ij}({\bf x})$, and all perturbations to the other metric components and to all matter variables vanish. The corrections are of order $a^{-2}$, and their ${\bf x}$-dependence is now explicitly given in terms of ${\cal G}_{ij}({\bf x})$ and its derivatives. The previous results for the time-dependence of the corrections to $g_{ij}({\bf x},t)/a^2(t)$ in the case of multi-scalar field theories are now shown to apply for any theory whose anisotropic inertia vanishes to order $a^{-2}$. Further, it is shown that the adiabatic solutions are attractive as $a$ becomes large for the case of single field inflation and now also for thermal equilibrium with no non-zero conserved quantities, and the $O(a^{-2})$ corrections to the other dynamical variables are explicitly calculated in both cases.