Statistical analyses of finite sample distributions usually assume that fluctuations are self-averaging, i.e. statistically similar in different regions of the given sample volume. By using the scale-length method, we test whether this assumption is satisfied in several samples of the Sloan Digital Sky Survey Data Release Six. We find that the probability density function (PDF) of conditional fluctuations, if filtered on large enough spatial scales (i.e., r > 30 Mpc/h), shows relevant systematic variations in different subvolumes of the survey. Instead for scales of r < 30 Mpc/h, the PDF is statistically stable, and its first moment presents scaling behavior with a negative exponent around one. Thus while up to 30 Mpc/h galaxy structures have well-defined power-law correlations, on larger scales it is not possible to consider whole sample average quantities as meaningful and useful statistical descriptors. This situation stems from galaxy structures corresponding to density fluctuations that are too large in amplitude and too extended in space to be self-averaging on such large scales inside the sample volumes: galaxy distribution is inhomogeneous up to the largest scales, i.e. r ≈ 100 Mpc/h probed by the SDSS samples. We show that cosmological corrections, such as K-corrections and standard evolutionary corrections, do not qualitatively change the relevant behaviors. We consider in detail the relation between several statistical measurements generally used to quantify galaxy fluctuations and the scale-length analysis by discussing how the breaking of self-averaging properties makes it impossible to have a reliable estimation of average fluctuations amplitude, variance, and correlations for r > 30 Mpc/h. Finally we show that the large-amplitude galaxy fluctuations observed in the SDSS samples are at odds with the predictions of the standard ΛCDM model of structure formation.

We consider the conditional galaxy density around each galaxy, and study its fluctuations in the newest samples of the Sloan Digital Sky Survey Data Release 7. Over a large range of scales, both the average conditional density and its variance show a nontrivial scaling behavior, which resembles to criticality. The density depends, for 10 ≤ r ≤ 80 Mpc/h, only weakly (logarithmically) on the system size. Correspondingly, we find that the density fluctuations follow the Gumbel distribution of extreme value statistics. This distribution is clearly distinguishable from a Gaussian distribution, which would arise for a homogeneous spatial galaxy configuration. We also point out similarities between the galaxy distribution and critical systems of statistical physics.1 We use H 0 = 100h km/sec/Mpc, with 0.4 ≤ h ≤ 0.7, for the Hubble's constant.

We discuss the large-scale properties of standard cold dark-matter cosmological models characterizing the main features of the power spectrum, of the two-point correlation function, and of the mass variance. Both the real-space statistics show a very well-defined behavior on large enough scales, for their amplitudes to become smaller than unity. The correlation function, in the range 0 < ξ(r) < 1, is characterized by a typical length scale r c , where ξ(r c ) = 0, which is fixed by the physics of the early universe. Beyond this scale it becomes negative, going to zero with a tail proportional to −(r −4 ). These anti-correlations thus represent an important observational challenge for verifying models in real space. The same length scale r c characterizes the behavior of the mass variance, which decays for r > r c as r −4 , the fastest decay of any mass distribution. The length-scale r c defines the maximum extension of (positively correlated) structures in these models. These are the features expected for the dark-matter field: however galaxies, which represent a biased field, may differ in their behaviors, which we analyze. We then discuss the detectability of these real-space features by considering several estimators of the two-point correlation function. By making tests on numerical simulations, we emphasize the important role of finite size effects, which should always be controlled for careful measurements.

The properties of the galaxy distribution at large scales are usually studied using statistics which are assumed to be self-averaging inside a given sample. We present a new analysis able to quantitatively map galaxy large-scale structures while testing for the stability of average statistical quantities in different sample regions. We find that the newest samples of the Sloan Digital Sky Survey provide unambiguous evidence that galaxy structures correspond to large-amplitude density fluctuations at all scales limited only by sample sizes. The two-point correlations properties are self-averaging up to approximately 30 Mpc/h and are characterized by a fractal dimension D=2.1±0.1. Then at all larger scales probed density fluctuations are too large in amplitude and too extended in space to be self-averaging inside the considered volumes. These inhomogeneities are compatible with a continuation of fractal correlations but incompatible with: i) a homogeneity scale smaller than 100 Mpc/h, ii) predictions of standard theoretical models, iii) mock galaxy catalogs generated from cosmological N-body simulations.