Excursion Set Theory for Correlated Random Walks

Abstract

We present a new method to compute the first crossing distribution in excursion set theory for the case of correlated random walks. We use a combination of the path integral formalism of Maggiore & Riotto, and the integral equation solution of Zhang & Hui, and Benson et al. to find a numerically robust and convenient algorithm to derive the first crossing distribution in terms of a perturbative expansion around the limit of an uncorrelated random walk. We apply this methodology to the specific case of a Gaussian random density field filtered with a Gaussian smoothing function. By comparing our solutions to results from Monte Carlo calculations of the first crossing distribution we demonstrate that our method accurate for power spectra $P(k)\propto k^n$ for $n=1$, becoming less accurate for smaller values of $n$. It is therefore complementary to the method of Musso & Sheth, which will therefore be more useful for standard $\Lambda$CDM power spectra. Our approach is quite general, and can be adapted to other smoothing functions, and also to non-Gaussian density fields.

15 pages, 7 figures, Accepted for publication in MNRAS. Some corrections following comments from referee

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• Cosmology with Galaxy Clusters: Systematic Effects in the Halo Mass Function
• Statistics of Dark Matter Halos in the Excursion Set Peak Framework
• Accurate halo mass functions from the simplest excursion set theory
• Approximate methods for the generation of dark matter halo catalogs in the age of precision cosmology
• Excursion Sets with a "Perfect" Collapse Model