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

¹ “E. Fermi” Center - Via Panisperna 89 A, Compendio del Viminale 00184 Roma, Italy
² Laboratoire de Physique Nucléaire et de Hautes Energies, Université de Paris VI 4 Place Jussieu, Tour 33 -RdC, 75252 Paris Cedex 05, France
³ Laboratoire de Physique Théorique, Université de Paris XI, Bâtiment 211 91405 Orsay, France
⁴ Laboratoire de Physique Théorique des Liquides 4 Place Jussieu, 75252 Paris Cedex 05, France

Received: 27 October 2003
Accepted: 2 February 2004

Abstract

A well-known argument in cosmology gives that the power spectrum (or structure function) 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 at small k. Noting the assumption of analyticity at of 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 at small k and the properties of the probability distribution for the spatial extent l of mass and momentum-conserving fluctuations. We find that the k⁴ behaviour is obtained in the case that the first six moments of are finite. Interestingly, the condition that the fluctuations be localised —taken to correspond to the convergence of the first two moments of — imposes only the weaker constraint with n anywhere in the range . We interpret this result to suggest that the causality bound will be loosened in this way if quantum fluctuations are permitted.