Causality and the superhorizon power spectrum

A well known argument in cosmology gives that the power spectrum of mass fluctuations produced from a uniform initial state by physics which is causally connected up to a finite scale has the behaviour $P(k) \propto k^4$ at small $k$. We point out the implicit, and {\it a priori} unjustified, assumptions made in the standard demonstration of this result. Introducing a class of one dimensional models describing the generation of fluctuations by a mass and momentum conserving process, we show how the analyticity properties of $P(k)$ at small $k$ are determined by the convergence properties of the probability distribution $f(l)$ for the spatial extent $l$ of the fluctuations. Taking $f(l)$ to have finite mean and variance, which we argue to be an appropriate definition of causality in the context of the early universe, we find that the small $k$ behaviour $P(k) \propto k^n$ with $n$ anywhere in the range $0< n \leq 4$ can be obtained.