*Europhys. Lett.*,

**66**(1) , pp. 1-7 (2004)

DOI: 10.1209/epl/i2003-10150-y

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

**A. Gabrielli**

^{1}, M. Joyce^{2}, B. Marcos^{3}and P. Viot^{4}^{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)

** Abstract **

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
*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
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.

**PACS**

02.50.Ey - Stochastic processes.

05.70.-a - Thermodynamics.

98.80.-k - Cosmology.

**©**

*EDP Sciences 2004*