Glass-like universe: Real-space correlation properties of standard cosmological models

After reviewing the basic relevant properties of stationary stochastic processes (SSP), defining basic terms and quantities, we discuss the properties of the so-called Harrison-Zeldovich like spectra. These correlations, usually characterized exclusively in k space [i.e., in terms of power spectra P(k)], are a fundamental feature of all current standard cosmological models. Examining them in real space we note their characteristics to be a negative power law tail xi(r)similar to-r(-4), and a sub-Poissonian normalized variance in spheres sigma(2)(R)similar toR(-4)ln R. We note in particular that this latter behavior is at the limit of the most rapid decay (similar toR(-4)) of this quantity possible for any stochastic distribution (continuous or discrete). This very particular characteristic is usually obscured in cosmology by the use of Gaussian spheres. In a simple classification of all SSP into three categories, we highlight with the name superhomogeneous the properties of the class to which models such as this, with P(0)=0, belong. In statistical physics language they are well described as glass-like. They have neither scale-invariant features, in the sense of critical phenomena, nor fractal properties. We illustrate their properties with some simple examples, in particular that of a shuffled lattice.