The accretion history of dark matter haloes - I. The physical origin of the universal function

Understanding the universal accretion history of dark matter haloes is the first step towards determining the origin of their structure. We use the extended Press-Schechter formalism to derive the halo mass accretion history from the growth rate of initial density perturbations. We show that the halo mass history is well described by an exponential function of redshift in the high-redshift regime. However, in the low-redshift regime the mass history follows a power law because the growth of density perturbations is halted in the dark energy dominated era due to the accelerated expansion of the Universe. We provide an analytic model that follows the expression M(z) = M₀(1 + z)^{af (M0)}e^{-f (M0)z}, where M0 = M(z = 0), a depends on cosmology and f(M0) depends only on the linear matter power spectrum. The analytic model does not rely on calibration against numerical simulations and is suitable for any cosmology. We compare our model with the latest empirical models for the mass accretion history in the literature and find very good agreement. We provide numerical routines for the model online (available at https://bitbucket.org/astroduff/commah).