Extension and estimation of correlations in Cold Dark Matter models

TL;DR

This paper analyzes the large-scale properties of cold dark matter models, focusing on the correlation function and mass variance, and discusses the observational challenges in verifying these features in real data.

Contribution

It characterizes the real-space behavior of cold dark matter models, especially the correlation length and anti-correlations, and examines how these features can be detected considering finite size effects.

Findings

Correlation function has a characteristic scale r_c where ; beyond this, it becomes negative.

Mass variance decays as r^{-4} beyond the scale r_c, indicating maximum structure size.

Finite size effects significantly impact the measurement of correlation functions in simulations.

Abstract

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 have a very well defined behavior on large enough scales, where their amplitudes become smaller than unity. The correlation function, in the range 0<\xi(r)<1, is characterized by a typical length-scale r_c, at which \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 represent thus an important observational challenge to verify 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 for 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: galaxies, which represent a biased field, however may have differences with respect to these 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.