*Fluctuation field of the cosmic microwave background radiation measured by the WMAP satellite. Different colors refer to different temperature fluctuations with respect to the mean temperature of T~2.73 Kelvin and have an amplitude of hundreds of micro Kelvin.*

In the theoretical models now used by most cosmologists to describe the universe (the so-called “standard cold dark matter (CDM)” and its variants) fluctuations are described as small fluctuations about a well defined mean density. The initial conditions for such models (i.e. very early in the history of the universe) is specified by the so-called Harrison-Zeldovich (HZ) power spectrum. It is here that the concept of “super-homogeneity” is relevant, as these models describe fluctuations which are in fact of this type: Standard type models are thus characterized by surface quadratic fluctuations (of the mass in spheres) and, for the particular form of primordial cosmological spectra, by a negative power-law in the reduced correlation function at large separations. Super-homogeneous distributions are particular systems which in statistical physics are in general described by a dynamics which at thermal equilibrium give rise to such configurations. We may identify three different but related problems associated to such distributions:

### Discretization:

Generation of particle distributions with the very specific properties we discussed above. This is a problem which is of both heuristic and practical interest, because the numerical simulation of formation of structures by gravity (cosmological N-body simulations) require the generation of distributions of particles which represent the continuous density field at the initial time.

### Sampling and bias:

Because of the highly irregular nature of structure at small scales, standard models with super-homogeneousfeatures cannot be used evenat zeroth order to describe these observed structures. This does not mean, however, that these models cannot describe successfully galaxy structures: but to establish whether they can, it must first be shown from observations that there is a clear crossover toward homogeneity i.e. a scale beyond which the average density becomes a well-defined (i.e. sample-independent) positive quantity. These models then predict that, on much larger scales (beyond the turn-over scale in the power spectrum (PS)), galaxy structures should present the super-homogeneous character of the HZ type PS. Indeed this should in principle be a critical test of the paradigm linking the measurements of CMBR on large scales to the distribution of matter. Observationally a crucial question is the feasibility of measuring the transition between these regimes directly in galaxy distributions. With large forthcoming galaxy surveys (e.g. SDSS) it may be possible to do so, but this is a question which must address exactly the statistics of these surveys and the exact nature of the signal in any given model. This is an issue we will address in a specific way by considering the possible estimators of correlations (or power spectrum) which can optimize the signal. In particular we will look at one of the important elements in it: the galaxy distribution is a discrete set of objects whose properties are related in a non-trivial way to the ones of the underlying continuous field. To understand the relation between the two, one has to consider the additional effects related to sampling the continuous field. This is intimately related to the problem of “biasing” between the distribution of visible and dark matter. Sampling a super-homogeneous fluctuation field may change the nature of correlations. The reason can be found in the property of super-homogeneity of such distribution: the sampling, as for instance in the so-called “bias model” (selection of highest peaks of the fluctuations field) necessarily destroys the surface nature of the fluctuations, as it introduces a volume (Poisson-like) term in the variance. The “primordial” form of the power spectrum is thus not apparent in that which one would expect to measure from objects selected in this way. This conclusion should hold for any generic model of bias. If a linear amplification is obtained in some regime of scales (as it can be in certain phenomenological models of bias) it is necessarily a result of a fine-tuning of the model parameters. The study of different samplings and of the correlation features invariant under sampling represents an important issue in relation to the comparison of observations of galaxy structures, or distributions given by N-body simulations with primordial fluctuations (CMBR anisotropies) and theoretical models.

### Statistical properties of CMB fluctuations

The CMB fluctuation field represent an interesting playground for studies of correlation properties. Standard analyses aim to measure the power spectrum of density fluctuation. A study of real space correlation properties may be more efficinet to make a local analysis of correlation of fluctuation. This may help to characterize the real space features of the “acoustic peaks” in stardard models and make tests to look for possible angular anisotropies

## Some References

A. Gabrielli, F. Sylos Labini , R. Durrer “Biasing in Gaussian random fields and galaxy correlations” Astrophys. J. Lett. 531 , L1-L4 (2000)

A. Gabrielli, M. Joyce, and F. Sylos Labini “The Glass-like Universe: Real-space correlation properties of standard cosmological models” , Physical Review D65 , 083523 – 083523-18, (2002)

A. Gabrielli, B. Jancovici, M. Joyce, J. Lebowitz, L. Pietronero and F. Sylos Labini “Generation of primordial fluctuations from Statistical Mechanical models” Phys.Rev. D67 , 043406-043513 (2003)

R. Durrer, A. Gabrielli, M. Joyce and F. Sylos Labini , “Bias and the power spectrum beyond the turn-over” , Astrophys.J.Lett, 585 , L1-L4 (2003)

M. Joyce, P. W. Anderson, M. Montuori, L. Pietronero and F. Sylos Labini “Fractal Cosmology in an Open Universe” Europhys.Letters 50 , 416-422 (2000)

L. Pietronero, A. Gabrielli and F. Sylos Labini “Statistical Physics for Cosmic Structures” Physica A, 306 , 395-401 (2002).

F. Sylos Labini & A. Gabrielli “Complexity in cosmic structures” , Physica A, 338 , Issues 1-2, 44-49, (2004).

A. Gabrielli, F. Sylos Labini, M. Joyce and L. Pietronero “Statistical physics for cosmic structures” Springer-Verlag (2004).