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

We present a physical model of the Mass Accretion Histories (MAH) of haloes in concordance with the {\it observed} cosmic star formation rate density (CSFRD). We model the MAHs of dark matter haloes using a Gamma ($\Gamma$) functional form: $M_h(T) = \frac{M_0}{f_{0}} \, \times \frac{\gamma(\alpha_h, ~\beta_h \times (T-Th))}{\Gamma(\alpha_h)}$, where $M_0$ is the halo mass at present time, $T$ is time, $\alpha_h$ and $\beta_h$ are parameters we explore, $f_{0}$ is the percentage of the mass of the halo at z = 0 with respect to the final mass of the halo achieved at $T = \infty$. We use the MAHs of haloes obtained from cosmological simulations and analytical models to constrain our model. $f_{0}$ 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 $<f_{0}>$ of haloes in the Universe is $ > 0.95$ pointing to the direction that the cosmic MAH/CSFRD is saturated at our era. The average $<\beta_{h}>$ parameter (the depletion rate of the available dark matter for halo growth) is related to the dynamical timescales of haloes. The $\alpha$ parameter is a power-law index of $M_{0}$ and represents the early growth a halo experiences before the expansion of the Universe starts to slow it down. Finally, $T_{h}$ (the time that marks the co-evolution/growth of galaxies and haloes after the Big Bang) is found to be 150-300 million years.