Causal perturbation theory in general FRW cosmologies II: superhorizon behaviour and initial conditions

We describe how causality and energy-momentum conservation constrain the superhorizon behaviour of perturbation variables in a general FRW spacetime. The effect of intrinsic curvature upon the horizon scale is discussed and `white noise' results for the power spectra of generic `causally-seeded' superhorizon perturbations are obtained. We present a detailed derivation of the superhorizon energy density power spectrum for curved universes, and provide elegant mathematical arguments for a curved universe analogue of the Bessel function addition theorem. This yields physical insight into both the familiar $k^4$ result for flat space and the $K \not= 0$ case: curvature effects are seen to be evident from as early as half the curvature scale. We consider the implications of these results for the physically and numerically significant issue of setting consistent initial conditions.