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. 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 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, valid for a generic filter function and arbitrary non-Gaussian theories. We perform explicitly the computation of the halo mass function with a tophat filter in coordinate space, 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 deve loped in this paper to compute from first principles the effect of non-Gaussianities on the halo mass function.