Excursion set halo mass function and bias in a stochastic barrier model of ellipsoidal collapse

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We use the excursion set formalism to compute the properties of the halo mass distribution for a stochastic barrier model which encapsulates the main features of the ellipsoidal collapse of dark matter halos. Non-Markovian corrections due to the sharp filtering of the linear density field in real space are computed with the path-integral technique introduced by Maggiore and Riotto [M. Maggiore and A. Riotto, Astrophys. J. 711, 907 (2010).]. Here, we provide a detailed derivation of the results presented in [P. S. Corasaniti and I. Achitouv, Phys. Rev. Lett. 106, 241302 (2011).] and extend the mass function analysis to higher redshift. We also derive an analytical expression for the linear halo bias. We find a remarkable agreement with $N$-body simulation data from Tinker et al. [J. L. Tinker et al., Astrophys. J. 688, 709 (2008).] with differences $\ensuremath{\lesssim}5%$ over the range of mass probed by simulations. The excursion set solution from Monte Carlo generated random walks shows the same level of agreement, thus confirming the validity of the path-integral approach for the barrier model considered here. Similarly, the analysis of the linear halo bias shows deviations no greater than 20%. Overall these results indicate that the excursion set formalism in combination with a realistic modeling of the conditions of halo collapse can provide an accurate description of the halo mass distribution.