Real space statistical properties of standard cosmological models

After reviewing some basic relevant properties of stationary stochastic processes (SSP), we discuss the properties of the so-called Harrison-Zeldovich like spectra of mass density perturbations. These correlations are a fundamental feature of all current standard cosmological models. Examining them in real space we note they imply a sub-poissonian normalised variance in spheres σ2 M(R) ∼ R−4 lnR. In particular this latter behaviour is at the limit of the most rapid decay (∼ R−4) of this quantity possible for any stochastic distribution (continuous or discrete). In a simple classification of all SSP into three categories, we highlight with the name “super-homogeneous” the properties of the class to which models like this, with P(0) = 0, belong. In statistical physics language they are well described as lattice or glass-like. We illustrate their properties through two simple examples: (i) the “shuffled” lattice and the One Component Plasma at thermal equilibrium.