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

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 $P(k) \propto k^4$ 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.
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http://hal.in2p3.fr/in2p3-00150305
Contributor : Dominique Girod <>
Submitted on : Wednesday, May 30, 2007 - 9:19:31 AM
Last modification on : Wednesday, September 16, 2020 - 4:04:48 PM

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A. Gabrielli, M. Joyce, B. Marcos, P. Viot. Causality constraints on fluctuations in cosmology: a study with exactly solvable one dimensional models. EPL - Europhysics Letters, European Physical Society/EDP Sciences/Società Italiana di Fisica/IOP Publishing, 2004, 66, pp.1-7. ⟨10.1209/epl/i2003-10150-y⟩. ⟨in2p3-00150305⟩

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