Modelling the mass accretion histories of dark matter haloes using a gamma formalism

Abstract

ABSTRACT We present a physical model of the mass accretion histories (MAH) of haloes in concordance with the observed cosmic star formation rate density (CSFRD). We model the MAHs of dark matter haloes using a Gamma (Γ) functional form: $M_h(T) = \frac{M_0}{f_{0}} \, \times \frac{\gamma (\alpha _h, ~\beta _h \times (T-Th))}{\Gamma (\alpha _h)}$, where M0 is the halo mass at present time, T is time, αh and βh are parameters we explore, and f0 is the percentage of the mass of the halo at z = 0 with respect to the final mass of the halo achieved at T = ∞. We use the MAHs of haloes obtained from cosmological simulations and analytical models to constrain our model. f0 can be described by a power-law ($f_{0} = 1- c \times M_{0}^{d}$). Haloes with small masses have already on average attained most of their final masses. The average < f0 > of haloes in the Universe is >0.95 pointing to the direction that the cosmic MAH/CSFRD is saturated at our era. The average < βh > parameter (the depletion rate of the available dark matter for halo growth) is related to the dynamical time-scales of haloes. The α parameter is a power-law index of M0 and represents the early growth a halo experiences before the expansion of the Universe starts to slow it down. Finally, Th (the time that marks the co-evolution/growth of galaxies and haloes after the big bang) is found to be 150–300 million years.