Scaling and fluctuations in galaxy distribution : two tests to probe large scale structure

We present a brief introduction to the statistical properties of systems with large fluctuations. We point out that for such systems the relevant statistical quantities are scaling exponents and the nature of fluctuations is completely different from the one belonging to system with small fluctuations. We then propose two test to be performed on galaxy counts data as a function of apparent magnitude, which may clarify in a direct way the nature of the large scale galaxy clustering In Fig.1 it is shown an homogenous distribution with poissonian fluctuations in the two-dimensional Euclidean space. In such a distribution the average density is a well defined property if it measured on scales larger than the mean particle separation Λ. Even if there are some small amplitude fluctuations up to a certain scale, or up to the sample’s size, the average density is still a well defined property. For example in Fig.2 it is shown a distribution where the fluctuation structures are extended over the whole sample. In such a case the average density is still a well defined property, and it can be measured at large enough scales (r > λ0), when the amplitude of the fluctuations become smaller than the average density itself, i.e. when δN(r)/〈N(r)〉 ≤ 1. These two systems belong to the class of distributions with small amplitude fluctuations. The fact that the average density is well defined implies that any one point property is well defined. For example in a ball of radius r > λ0 > Λ centered in a randomly chosen point, in both the distributions, the number of points is constant, a part a small fluctuations whci vanishes when r is farther increased. In these case when the average density is well-defined, it is possible to study the statistical properties of the fluctuations around the it. For example, given an occupied point, the number of points at distance r is given by (Peebles, 1980) Np(< r) = 〈n〉V (r) + 〈n〉 ∫