Causality constraints on fluctuations in cosmology: A study with exactly solvable one-dimensional modelsA. Gabrielli1, M. Joyce2, B. Marcos3 and P. Viot4
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 in final form 2 February 2004)
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 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 k4 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 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.
02.50.Ey - Stochastic processes.
05.70.-a - Thermodynamics.
98.80.-k - Cosmology.
© EDP Sciences 2004