Abstract:
Super-homogeneity is a property that is supposed to be satisfied by matter fluctuations in all standard theoretical models of structure formation, such as LCDM and its variants. This is a global condition on the correlation properties of the matter density field, which can be understood as a consistency constraint in the framework of FRW cosmology, and it corresponds to a very fine tuned balance between negative and positive correlations of density fluctuations and to the fastest possible decay of the normalized mass variance on large scales. By considering several galaxy samples, we discuss that these are characterized by the presence of large amplitude fluctuations with spatial extension limited only the size of the current samples. There is therefore a tension between the standard prediction of super-homogeneity and the detection of large scale inhomogeneities in the matter distribution at scales of the order of 100 Mpc/h. We discuss the theoretical implications of these results with respect to models of structure formation and to future galaxy and CMBR data, emphasizing the central role of the super-homogeneity property in the current description of fluctuations in FRW models.

Abstract:
A distribution of points that satisfies the property of local isotropy is not necessarily homogeneous: homogeneity is implied by the condition of local isotropy together with the assumption of analyticity or regularity. Here we show that the evidence of dipole saturation in galaxies (and clusters) catalogues, together with a monotone growth of the monopole, is an evidence of isotropy but not of homogeneity. This is fully compatible with a fractal structure which has the property of local isotropy, but it is non-analytic and non-homogeneous.

Abstract:
The fact that galaxy distribution exhibits fractal properties is well established since twenty years. Nowadays, the controversy concerns the range of the fractal regime, the value of the fractal dimension and the eventual presence of a cross-over to homogeneity. Fractal properties maybe studied with methods which do not assume homogeneity a priori as the standard statistical methods do. We show that complementary to the adoption of new methods of analysis there are important theoretical implications for the usual scenario of galaxy formation. For example, we focus on the concept of bias and we show that it needs a basic revision even if future redshift surveys will be able to identify an eventual tendency to homogenization.

Abstract:
The collapse of an isolated, uniform and spherical cloud of self-gravitating particles represents a paradigmatic example of a relaxation process leading to the formation of a quasi-stationary state in virial equilibrium. We consider several N-body simulations of such a system, with the initial velocity dispersion as a free parameter. We show that there is a clear difference between structures formed when the initial virial ratio is b_0 =2K_0/W_0 < b_0^c ~ -1/2 and b_0> b_0^c. These two sets of initial conditions give rise respectively to a mild and violent relaxation occurring in about the same time scale: however in the latter case the system contracts by a large factor, while in the former it approximately maintains its original size. Correspondingly the resulting quasi equilibrium state is characterized by a density profile decaying at large enough distances as ~1/r^4 or with a sharp cut-off. The case b_0b_0^c is the ejection of particles and energy, which is not captured by such a theoretical approach: for this case we introduce a simple physical model to explain the formation of the power-law density profile. This model shows that the behavior n(r) ~1/r^4 is the typical density profile that is obtained when the initial conditions are cold enough that mass and energy ejection occurs. In addition, we clarify the origin of the critical value of the initial virial ratio b_0^c.

Abstract:
There is a general agreement that galaxy structures exhibit fractal properties, at least up to some small scale. However the presence of an eventual crossover towards homogenization, as well as the exact value of the fractal dimension, are still a matter of debate. I summarize the main points of the this discussion, considering also some galaxy surveys which have recently appeared. Further I discuss the implications for the standard picture of the observed fractal behaviour in galaxy distribution. In particular I consider the co-existence of fractal structures and the linear Hubble-law within the same scales. This fact represents a challenge for the standard cosmology where the linear Hubble law is a strict consequence of homogeneity of the expanding universe. Finally I consider the comparison of CDM-like models with the data noting that the simulations are not able to reproduce the observed properties of galaxy correlations.

Abstract:
We study the statistical properties of the gravitational field generated by galaxy distribution observed bythe Sloan Digital Sky Survey (DR7). We characterize the probability density function of gravitational force fluctuations and relate its limiting behaviors to the correlation properties of the underlying density field. In addition, we study whether the PDF converges to an asymptotic shape within sample volumes. We consider several volume-limited samples of the Sloan Digital Sky Survey and we compute the gravitational force probability density function (PDF). The gravitational force is computed in spheres of varying radius as is its PDF. We find that (i) the PDF of the force displays features that can be understood in terms of galaxy two-point correlations and (ii) density fluctuations on the largest scales probed, i.e. r~100 Mpc/h, still contribute significantly to the amplitude of the gravitational force. Our main conclusion is that fluctuations in the gravitational force field generated by galaxy structures are also relevant on scales ~ 100 Mpc/h. By assuming that the gravitational fluctuations in the galaxy distribution reflect those in the whole matter distribution, and that peculiar velocities and accelerations are simply correlated, we may conclude that large-scale fluctuations in the galaxy density field may be the source of the large-scale flows recently observed.

Abstract:
In order to investigate whether galaxy structures are compatible with the predictions of the standard LCDM cosmology, we focus here on the analysis of several simple and basic statistical properties of the galaxy density field. Namely, we test whether, on large enough scales (i.e., r>10 Mpc/h), this is self-averaging, uniform and characterized by a Gaussian probability density function of fluctuations. These are three different and clear predictions of the LCDM cosmology which are fulfilled in mock galaxy catalogs generated from cosmological N-body simulations representing this model. We consider some simple statistical measurements able to tests these properties in a finite sample. We discuss that the analysis of several samples of the Two Degree Field Galaxy Redshift Survey and of the Sloan Digital Sky Survey show that galaxy structures are non self-averaging and inhomogeneous on scales of ~100 Mpc/h, and are thus intrinsically different from LCDM model predictions. Correspondingly the probability density function of fluctuations shows a "fat tail" and it is thus different from the Gaussian prediction. Finally we discuss other recent observations which are odds with LCDM predictions and which are, at least theoretically, compatible with the highly inhomogeneous nature of galaxy distribution. We point out that inhomogeneous structures can be fully compatible with statistical isotropy and homogeneity, and thus with a relaxed version of the Cosmological Principle.

Abstract:
Standard models of galaxy formation predict that matter distribution is statistically homogeneous and isotropic and characterized by (i) spatial homogeneity for r<10 Mpc/h, (ii) small-amplitude structures of relatively limited size (i.e., r<100) Mpc/h and (iii) anti-correlations for r > r_c ~ 150 Mpc/h (i.e., no structures of size larger than r_c). Whether or not the observed galaxy distribution is interpreted to be compatible with these predictions depend on the a-priori assumptions encoded in the statistical methods employed to characterize the data and on the a-posteriori hypotheses made to interpret the results. We present strategies to test the most common assumptions and we find evidences that, in the available samples, galaxy distribution is spatially inhomogeneous for r<100 Mpc/h but statistically homogeneous and isotropic. We conclude that the observed inhomogeneities pose a fundamental challenge to the standard picture of cosmology but they also represent an important opportunity which may open new directions for many cosmological puzzles.

Abstract:
The evolution of an isolated over-density represents a useful toy model to test the accuracy of a cosmological N-body code in the non linear regime as it is approximately equivalent to that of a truly isolated cloud of particles, with same density profile and velocity distribution, in a non expanding background. This is the case as long as the system size is smaller than the simulation box side, so that its interaction with the infinite copies can be neglected. In such a situation, the over-density rapidly undergoes to a global collapse forming a quasi stationary state in virial equilibrium. However, by evolving the system with a cosmological code (GADGET) for a sufficiently long time, a clear deviation from such quasi-equilibrium configuration is observed. This occurs in a time t_{LI} that depends on the values of the simulation numerical parameters such as the softening length and thetime-stepping accuracy, i.e. it is a numerical artifact related to the limited spatial and temporal resolutions. The analysis of the Layzer-Irvine cosmic energy equation, confirms that this deviation corresponds to an unphysical dynamical regime. By varying the numerical parameters of the simulation and the physical parameters of the system we show that the unphysical behaviour originates from badly integrated close scatterings of high velocity particles. We find that, while the structure may remain virialized in the unphysical regime, its density and velocity profiles are modified with respect to the quasi-equilibrium configurations, converging however to well defined shapes, the former characterised by a Navarro Frenk White-type behaviour.

Abstract:
We study the statistical properties of the Luminous Red Galaxies sample from the Sloan Digital Sky Survey. In particular we test, by determining the probability density function (PDF) of galaxy (conditional) counts in spheres, whether statistical properties are self-averaging within the sample. We find that there are systematic differences in the shape of the PDF and in the location of its peak, signaling that there are major systematic effects in the data which make the estimation of volume average quantities unreliable within this sample. We discuss that these systematic effects are related to the fluctuating behavior of the redshift counts which can be originated by intrinsic fluctuations in the galaxy density field or by observational selection effects. The latter possibility implies that more than 20 % of the galaxies have not been observed and that such a selection should not be a smooth function of redshift.