The accretion history of dark matter haloes - II. The connections with the mass power spectrum and the density profile

We explore the relation between the structure and mass accretion histories of dark matter haloes using a suite of cosmological simulations. We confirm that the formation time, defined as the time when the virial mass of the main progenitor equals the mass enclosed within the scale radius, correlates strongly with concentration. We provide a semi-analytic model for halo mass history that combines analytic relations with fits to simulations. This model has the functional form, M(z) = M-0(1 + z)(alpha)e(beta z), where the parameters alpha and beta are directly correlated with concentration. We then combine this model for the halo mass history with the analytic relations between alpha, beta and the linear power spectrum derived by Correa et al. to establish the physical link between halo concentration and the initial density perturbation field. Finally, we provide fitting formulae for the halo mass history as well as numerical routines. We derive the accretion rate as a function of halo mass, and demonstrate how the halo mass history depends on cosmology and the adopted definition of halo mass.