The Halo Mass Function from Excursion Set Theory. I. Gaussian
  fluctuations with non-markovian dependence on the smoothing scale
  A classic method for computing the mass function of dark matter halos is
provided by excursion set theory, where density perturbations evolve
stochastically with the smoothing scale, and the problem of computing the
probability of halo formation is mapped into the so-called first-passage time
problem in the presence of a barrier. While the full dynamical complexity of
halo formation can only be revealed through N-body simulations, excursion set
theory provides a simple analytic framework for understanding various aspects
of this complex process. In this series of paper we propose improvements of
both technical and conceptual aspects of excursion set theory, and we explore
up to which point the method can reproduce quantitatively the data from N-body
simulations. In paper I of the series we show how to derive excursion set
theory from a path integral formulation. This allows us both to derive
rigorously the absorbing barrier boundary condition, that in the usual
formulation is just postulated, and to deal analytically with the non-markovian
nature of the random walk. Such a non-markovian dynamics inevitably enters when
either the density is smoothed with filters such as the top-hat filter in
coordinate space (which is the only filter associated to a well defined halo
mass) or when one considers non-Gaussian fluctuations. In these cases, beside
``markovian'' terms, we find ``memory'' terms that reflect the non-markovianity
of the evolution with the smoothing scale. We develop a general formalism for
evaluating perturbatively these non-markovian corrections, and in this paper we
perform explicitly the computation of the halo mass function for gaussian
fluctuations, to first order in the non-markovian corrections due to the use of
a tophat filter in coordinate space.
