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

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Abstract
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 {\it negative} power law tail \xi(r) \sim - r^{-4} and a {\it sub-poissonian} normalised variance in spheres \sigma^2(R) \sim R^{-4} \ln R. We note in particular that this latter behaviour is at the limit of the most rapid decay (\sim R^{-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 ``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 glass-like. They do not have either ``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.

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