Marcello Musso, Ravi K. Sheth
Insight into a number of interesting questions in cosmology can be obtained by studying the first crossing distributions of physically motivated barriers by random walks with correlated steps: higher mass objects are associated with walks that take fewer steps before crossing the barrier. We show how to write the first crossing distribution as a formal series, ordered by the minimum number of times a walk upcrosses the barrier. Since walks with many upcrossings are negligible if the walk has not taken too many steps, the leading order term in this series is the most relevant for understanding the massive objects of most interest in cosmology. For walks associated with Gaussian random fields, this first term only requires knowledge of the bivariate distribution of the walk height and slope, and provides an excellent approximation to the first crossing distribution for all barriers and smoothing filters of current interest. We show that this simplicity survives when extending the approach to the case of non-Gaussian random fields. Although this second part of our analysis is motivated by the possibility that the primordial fluctuation field is non-Gaussian, our results are general. In particular, they do not assume the non-Gaussianity is small, so they may be viewed as the solution to an excursion set analysis of the late-time, nonlinear fluctuation field rather than the initial one.
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http://arxiv.org/abs/1305.0724
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