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.

Abstract. The Local Group environment at 1-10 Mpc expands linearly and smoothly, as if ruled by uniform matter, while observations show on the same scales the very lumpy local galaxy universe. This enigma in cosmology has also been demonstrated by high-resolution N-body CDM simulations. We suggest that the homogeneous dark energy component, revealed by SNIa observations, may resolve the problem of the local cold Hubble flow within the highly non-uniform environment. Linear density perturbations on a homogeneous background with the equation of state pQ = wρQc 2 are decaying for w < −1/3. Exact non-linear Einstein's equations for a spherically symmetric matter concentration, show that there is a zero-mass surface where the positive mass of the local cloud is compensated by the negative dark energy mass, and beyond this surface dark energy dominates dynamically. In such regions the velocity dispersion is adiabatically cooling, and this may explain why the Hubble law starts on the outskirts of the Local Group, with the same H0 as globally and with a remarkably small velocity dispersion.