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 $P\left(k\right)\mathrm{}],$ 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 $\xi \mathrm{}\left(r\right)\mathrm{}\sim -r^{-4},$ and a sub-Poissonian normalized variance in spheres ${\sigma }^2\mathrm{}\left(R\right)\mathrm{}\sim R^{-4}\ln R.\mathrm{}$ We note in particular that this latter behavior is at the limit of the most rapid decay $\left(\sim R^{-4}\right)$ 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 $P\left(0\right)=0,$ 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.