The accretion history of dark matter haloes – III. A physical model for the concentration–mass relation

We present a semi-analytic, physically motivated model for dark matter halo concentration as a function of halo mass and redshift. The semi-analytic model combines an analytic model for the halo mass accretion history (MAH), based on extended Press–Schechter (EPS) theory, with an empirical relation between concentration and formation time obtained through fits to the results of numerical simulations. Because the semi-analytic model is based on EPS theory, it can be applied to wide ranges in mass, redshift and cosmology. The resulting concentration–mass (c–M) relations are found to agree with the simulations, and because the model applies only to relaxed haloes, they do not exhibit the upturn at high masses or high redshifts found by some recent works. We predict a change of slope in the z ∼ 0 c–M relation at a mass-scale of 1011 M⊙. We find that this is due to the change in the functional form of the halo MAH, which goes from being dominated by an exponential (for high-mass haloes) to a power law (for low-mass haloes). During the latter phase, the core radius remains approximately constant, and the concentration grows due to the drop of the background density. We also analyse how the c–M relation predicted by this work affects the power produced by dark matter annihilation, finding that at z = 0 the power is two orders of magnitude lower than that obtained from extrapolating best-fitting c–M relations. We provide fits to the c–M relations as well as numerical routines to compute concentrations and MAHs.