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(c), where xi(r(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(c) characterizes the behavior of the mass variance, which decays for r > r(c) as r(-4), the fastest decay of any mass distribution. The length-scale r(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.