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 normalized variance in spheres σ2M(R)∼R−4lnR. In particular this latter behavior 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.