The Halo Mass Function from the Excursion Set Method. I. First Principle Derivation for the Non-markovian Case of Gaussian Fluctuations and Generic Filter

A classic method to compute the mass function of dark matter halos is the excursion set method, where the density perturbation evolves 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. To date, however, analytical results were only obtained if the density perturbation is smoothed with a sharp filter in momentum space: the dynamics is then markovian, and the probability satisfies the Fokker-Planck equation, with an “absorbing barrier” boundary condition. For different filters (such as the tophat filter in real space necessary to compare with N-body simulations), or when non-Gaussianity is present, the dynamics becomes non-markovian, the probability does not satisfy a local diffusion equation, and even the notion of absorbing barrier may be ill-defined. We develop an approach from first principles for computing analytically the halo mass function, formulating the problem in terms of a path integral with boundaries. Our method can be applied when the density is smoothed with a generic filter function, and to arbitrary non-Gaussian theories. When we restrict to a sharp filter in k-space and to gaussian fluctuations, we recover the usual results of the excursion set method. As soon as we consider a different filter function, already for gaussian fluctuations a complicated structure emerges. Beside “markovian” terms, we now have “memory” terms that reflect the non-markovianity of the dynamics. We develop a formalism for evaluating perturbatively the non-markovian corrections, and we perform explicitly the computation of the halo mass function with a tophat filter in coordinate space, to first order in the non-markovian corrections, finding full agreement with existing Monte Carlo simulations. These results put excursion set theory on firmer mathematical foundation and confirm that excursion set theory does not reproduce well the results of N-body simulations when combined with the spherical collapse model with fixed collapse barrier. In paper II of this series we show that this discrepancy disappears when one properly takes into account the fact that the collapse barrier is itself of stochastic nature, and in paper III we use the formalism developed in this paper to compute from first principles the effect of non-Gaussianities on the halo mass function. Subject headings: cosmology:theory — dark matter:halos — large scale structure of the universe