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.