THE HALO MASS FUNCTION FROM EXCURSION SET THEORY. I. GAUSSIAN FLUCTUATIONS WITH NON-MARKOVIAN DEPENDENCE ON THE SMOOTHING SCALE

Abstract

A classic method for computing the mass function of dark matter halos is\nprovided by excursion set theory, where density perturbations evolve\nstochastically with the smoothing scale, and the problem of computing the\nprobability of halo formation is mapped into the so-called first-passage time\nproblem in the presence of a barrier. While the full dynamical complexity of\nhalo formation can only be revealed through N-body simulations, excursion set\ntheory provides a simple analytic framework for understanding various aspects\nof this complex process. In this series of paper we propose improvements of\nboth technical and conceptual aspects of excursion set theory, and we explore\nup to which point the method can reproduce quantitatively the data from N-body\nsimulations. In paper I of the series we show how to derive excursion set\ntheory from a path integral formulation. This allows us both to derive\nrigorously the absorbing barrier boundary condition, that in the usual\nformulation is just postulated, and to deal analytically with the non-markovian\nnature of the random walk. Such a non-markovian dynamics inevitably enters when\neither the density is smoothed with filters such as the top-hat filter in\ncoordinate space (which is the only filter associated to a well defined halo\nmass) or when one considers non-Gaussian fluctuations. In these cases, beside\n``markovian'' terms, we find ``memory'' terms that reflect the non-markovianity\nof the evolution with the smoothing scale. We develop a general formalism for\nevaluating perturbatively these non-markovian corrections, and in this paper we\nperform explicitly the computation of the halo mass function for gaussian\nfluctuations, to first order in the non-markovian corrections due to the use of\na tophat filter in coordinate space.\n