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.
