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 $Λ$CDM power spectra. Our approach is quite general, and can be adapted to other smoothing functions, and also to non-Gaussian density fields.