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

### Abstract

7 pages, 2 figures; revised version with considerable changes to presentation (including modified title), results and conclusions unchanged; version to appear in Europhys. LettA 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 $P(k) \propto k^4$ 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 {\it 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 $P(k) \propto k^n$ with $n$ anywhere in the range $0< n \leq 4$. We interpret this result to suggest that the causality bound will be loosened in this way if quantum fluctuations are permitted

Topics: 02.50.Ey, 05.70.-a, 9.8.80.-k, [PHYS.ASTR.CO] Physics [physics]/Astrophysics [astro-ph]/Cosmology and Extra-Galactic Astrophysics [astro-ph.CO], [PHYS.COND.CM-SM] Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], [PHYS.HPHE] Physics [physics]/High Energy Physics - Phenomenology [hep-ph], [PHYS.HTHE] Physics [physics]/High Energy Physics - Theory [hep-th], [SDU.ASTR] Sciences of the Universe [physics]/Astrophysics [astro-ph]
Publisher: European Physical Society/EDP Sciences/Società Italiana di Fisica/IOP Publishing
Year: 2004
DOI identifier: 10.1209/epl
OAI identifier: oai:HAL:in2p3-00150305v1
Provided by: Hal-Diderot
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