The excursion set theory based on spherical or ellipsoidal gravitationalcollapse provides an elegant analytic framework for calculating the massfunction and the large-scale bias of dark matter haloes. This theory assumesthat the perturbed density field evolves stochastically with the smoothingscale and exhibits Markovian random walks in the presence of a density barrier.Here we derive an analytic expression for the halo bias in a new theoreticalmodel that incorporates non-Markovian extension of the excursion set theorywith a stochastic barrier. This model allows us to handle non-Markovian randomwalks and to calculate perturbativly these corrections to the standardMarkovian predictions for the halo mass function and halo bias. Our modelcontains only two parameters: kappa, which parameterizes the degree ofnon-Markovianity and whose exact value depends on the shape of the filterfunction used to smooth the density field, and a, which parameterizes thedegree of stochasticity of the barrier. Appropriate choices of kappa and a inour new model can lead to a closer match to both the halo mass function andhalo bias in the latest N-body simulations than the standard excursion settheory.
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