Extension and estimation of correlations in cold dark matter models

We discuss the large-scale properties of standard cold dark-matter cosmological models characterizing the main features of the power spectrum, of the two-point correlation function, and of the mass variance. Both the real-space statistics show a very well-defined behavior on large enough scales, for their amplitudes to become smaller than unity. The correlation function, in the range $0<\xi(r)<1$, is characterized by a typical length scale $r_{\rm c}$, where $\xi(r_{\rm c})=0$, which is fixed by the physics of the early universe. Beyond this scale it becomes negative, going to zero with a tail proportional to -(r-4). These anti-correlations thus represent an important observational challenge for verifying models in real space. The same length scale $r_{\rm c}$ characterizes the behavior of the mass variance, which decays for $r>r_{\rm c}$ as r-4, the fastest decay of any mass distribution. The length-scale $r_{\rm c}$ defines the maximum extension of (positively correlated) structures in these models. These are the features expected for the dark-matter field: however galaxies, which represent a biased field, may differ in their behaviors, which we analyze. We then discuss the detectability of these real-space features by considering several estimators of the two-point correlation function. By making tests on numerical simulations, we emphasize the important role of finite size effects, which should always be controlled for careful measurements.