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Large deviation principle at play in large scale structure cosmology

Sorbonne Universités, UPMC Université Paris 6 et CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis boulevard Arago, 75014 Paris, France and CEA—CNRS, UMR 3681, Institut de Physique Théorique, F-91191 Gif-sur-Yvette, France

Phys. Rev. D 94, 063520 – Published 15 September, 2016

DOI: https://doi.org/10.1103/PhysRevD.94.063520

Abstract

We present an application of large deviation theory to the problem of structure growth on large scale structure cosmology. Starting from Gaussian distributed overdensities on concentric spherical shells, we show that a large deviation principle holds for the densities on the corresponding shells after gravitational evolution if no shell crossing happens. As consequences of the large deviation principle we obtain the cumulant-generating function for the nonlinear densities, and present formulas to compute the cumulant-generating function for general window functions.

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To be precise, we should require two different limits to hold, one for open sets and one for closed. The definition given, however, is consistent if we consider sets $A$ for which $inf_{α \in A^{o}} I (α) = inf_{α \in A} I (α) = inf_{α \in \bar{A}} I (α)$ , where $A^{o}$ is the interior of $A$ , and $\bar{A}$ its closure. This condition will hold for most of the “naturally conceivable” sets $A$ , but would fail, for example, for the ternary Cantor set.
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Large deviation principle at play in large scale structure cosmology

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