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 ) ∝ 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 ( l ) for the spatial extent l 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 ) ∝ k n with n anywhere in the range 0 < n ⩽ 4. We interpret this result to suggest that the causality bound will be loosened in this way if quantum fluctuations are permitted.