Properties of Dark Matter Haloes and their Correlations: the Lesson from
  Principal Component Analysis
  We study the correlations between the structural parameters of dark matter
haloes using Principal Component Analysis (PCA). We consider a set of eight
parameters, six of which are commonly used to characterize dark matter halo
properties: mass, concentration, spin, shape, overdensity, and the angle
($\Phi_L$) between the major axis and the angular momentum vector. Two
additional parameters ($\x_{off}$ and $\rho_{rms}$) are used to describe the
degree of `relaxedness' of the halo. We find that we can account for much of
the variance of these properties with halo mass and concentration, on the one
hand, and halo relaxedness on the other. Nonetheless, three principle
components are usually required to account for most of the variance. We argue
that halo mass is not as dominant as expected, which is a challenge for halo
occupation models and semi-analytic models that assume that mass determines
other halo (and galaxy) properties. In addition, we find that the angle
$\Phi_L$ is not significantly correlated with other halo parameters, which may
present a difficulty for models in which galaxy disks are oriented in haloes in
a particular way. Finally, at fixed mass, we find that a halo's environment
(quantified by the large-scale overdensity) is relatively unimportant.
