A Two-Parameter Matching Scheme for Massive Galaxies and Dark Matter Haloes

Halo Abundance Matching has been used to construct a one-parameter mapping between galaxies and dark matter haloes by assuming that halo mass and galaxy luminosity (or stellar mass) are monotonically related. While this approach has been reasonably successful, it is known that galaxies must be described by at least two parameters, as can be seen from the two-parameter Fundamental Plane on which massive early-type galaxies lie. In this paper, we derive a connection between initial dark matter density perturbations in the early universe and present-day virialized dark matter haloes by assuming simple spherical collapse combined with conservation of mass and energy. We find that $z = 0$ halo concentration, or alternatively the inner slope of the halo density profile $\alpha$, is monotonically and positively correlated with the collapse redshift of the halo. This is qualitatively similar to the findings of some previous works based on numerical simulations, with which we compare our results. We then describe how the halo mass and concentration (or inner slope $\alpha$) can be used as two halo parameters in combination with two parameters of early-type galaxies to create an improved abundance matching scheme.