Causality constraints on fluctuations in cosmology: A study with exactly solvable one-dimensional models

A well-known argument in cosmology gives that the power spectrum (or structure function) P(k) of mass density fluctuations produced from a uniform initial state by physics which is causal (i.e. moves matter and momentum only up to a finite scale) has the behaviour P(k) proportional to k(4) at small k. Noting the assumption of analyticity at k = 0 of P(k) in the standard derivation of this result, we introduce a class of solvable one-dimensional models which allows us to study the relation between the behaviour of P(k) at small k and the properties of the probability distribution f(1) for the spatial extent I of mass and momentum-conserving fluctuations. We find that the k(4) behaviour is obtained in the case that the first six moments of f(l) are finite. Interestingly, the condition that the fluctuations be localised-taken to correspond to the convergence of the first two moments of f(l)-imposes only the weaker constraint P(k) proportional to k(n) with n anywhere in the range 0 < n less than or equal to 4. We interpret this result to suggest that the causality bound will be loosened in this way if quantum fluctuations are permitted.