A numerical study on the dimension of an extremely inhomogeneous matter distribution

Abstract

We have developed an algorithm that numericaly computes the dimension of an extremely inhomogeneous matter distribution, given by a discrete hierarchical metric. With our results it is possible to analise how the dimension of the matter density tends to d = 3 , as we consider larger samples.Most authors believe that the large scale distribution of matter should be a constant, as predicted by Einstein's cosmological principle, but there is also a belief that such an idea should be further checked against observation [1]. Some authors argued quite convincingly that the large scale distribution of matter should be fractal, quite the oposite to that expected by the cosmological principle. In a previous work [10], we have presented a study on a toy model given by the hierarchical metricwhere the metric is defined on all integers, depending on their decomposition in terms of powers of 2 aswhere k, , m and n are integers. We obtained the following expression for the matter density,by means of the Einstein equations. Computing the Christoffel symbols and subsequently the curvature tensor for this metric requires some care, since we are not dealing with derivatives of functions, but differences of functions defined on a discrete space.We speculated whether such a matter distribution could be described by a fractal. As it turned out, from our preliminary analisis (see [10]), the dimension of the matter density tends slowly to d = 3.Considering the relationwhere K is some constant, and the constant d is the fractal dimension of the matter density [11], it is easy to see that it impliesfor large r. We have numerically computed N(r) and the plot ln N(r) × ln r, in figure (1), showing that equation (5), although valid at very large r), is a good approximation for the behavior of the data.