Many surveys have been carried out to chart the positions of galaxies in large regions of the universe around us, and many more surveys which go deeper into the universe are currently underway or are planned for the future. These surveys give us detailed information about the distribution of matter in the universe, and identifying the salient features that characterize this distribution has been a very important problem in cosmology. The statistical properties, the geometry and the topology are some of the features that have been used to characterize the distribution of galaxies, and a large variety of tools have been developed and used for this purpose.
The correlation functions which characterize the statistical properties of the distributions have been widely applied to quantify galaxy clustering. Of the various correlation functions (2-point, 3- point, etc...) the galaxy-galaxy two point correlation function is very well determined on small scales (Peebles 1993 and references therein) and it has been found to have the form
This power-law form of the two point correlation function suggests that the universe exhibits a scale invariant behaviour on small scales . The two point correlation function becomes steeper at larger scales . It is, however, not very well determined on very large scales where the observations are consistent with the correlation function being equal to zero. The standard cosmological model and the correlation function analysis are both based on the underlying assumption that the universe is homogeneous on very large scales and the indication that the correlation function vanishes at very large scales is consistent with this.
Fractal characterization is another way of quantifying the gross features of the galaxy distribution. Fractals have been invoked to describe many physical phenomena which exhibit a scale invariant behaviour and it is very natural to use fractals to describe the clustering of galaxies on small scales where the correlation function analysis clearly demonstrates a scale invariant behaviour.
Coleman & Pietronero (Coleman & Pietronero, 1992) applied the fractal analysis to galaxy distributions and concluded that it exhibits a self-similar behaviour up to arbitrarily large scales. Their claim that the fractal behaviour extends out to arbitrarily large scales implies that the universe is not homogeneous on any scale and hence it is meaningless to talk about the mean density of the universe. These conclusions are in contradiction with the Cosmological Principle and the entire framework of cosmology, as we understand today, will have to be revised if these conclusions are true.
On the other hand, several others (Martinez & Jones, 1990 , Borgani, 1995) have applied the fractal analysis to arrive at conclusions that are more in keeping with the standard cosmological model. They conclude that while the distribution of galaxies does exhibit self similarity and scaling behaviour, the scaling behaviour is valid only over a range of length scales and the galaxy distribution is homogeneous on very large scales. Various other observations including the angular distribution of radio sources and the X-ray background testify to the universe being homogeneous on large scales (Wu et al. 1998; Peebles 1998).
Recent analysis of the ESO slice project (Guzzo 1998) also indicates that the universe is homogeneous over large scales. The fractal analysis of volume limited subsamples of the SSRS2 (Cappi et al. 1998) studies the spatial behaviour of the conditional density at scales up to . Their analysis is unable to conclusively determine whether the distribution of galaxies is fractal or homogeneous and it is consistent with both the scenarios. A similar analysis carried out for the APM-Stromlo survey (Labini & Montuori, 1997) seems to indicate that the distribution of galaxies exhibits a fractal behaviour with a dimension of on scales up to . In a more recent paper (Amendola & Palladino, 1999) the fractal analysis has been applied to volume limited subsamples of the Las Campanas Redshift Survey. This uses the conditional density to probe scales up to . They find evidence for a fractal behaviour with dimension on scales up to . They also conclude that there is a tendency to homogenization on larger scales () where the fractal dimension has a value , but the scatter in the results is too large to conclusively establish homogeneity and rule out a fractal universe on large scales.
In this paper we study the scaling properties of the galaxy distribution in the Las Campanas Redshift Survey (LCRS) (Shectman et al. 1996). This is the deepest redshift survey available at present. Here we apply the multi-fractal analysis (Martinez & Jones, 1990 , Borgani, 1995) which is based on a generalization of the concept of a mono-fractal. In a mono-fractal the scaling behaviour of the point distribution is the same around each point and the whole distribution is characterized by a single scaling index which corresponds to the fractal dimension. A multi-fractal allows for a sequence of scaling indices known as the multi-fractal spectrum of generalized dimensions. This allows for the possibility that the scaling behaviour is not the same around each point. The spectrum of generalized dimensions tells how the scaling properties of the galaxy distribution changes from the very dense regions (clusters) to the sparsely populated regions (voids) in the survey.
In this paper we compute the spectrum of generalized dimensions ( vs q) by calculating the Minkowski-Bouligand dimension (Borgani, 1995) for both volume limited and magnitude limited subsamples of the LCRS. We also investigate how the spectrum of generalized dimensions depends on the length scales over which it is measured and whether the distribution of galaxies in the LCRS exhibits homogeneity on very large scales or if the fractal nature extends to arbitrarily large scales..
We next present a brief outline of the organization of the paper. Sect. 2 describes the method we adopt to compute the spectrum of generalized dimensions. In Sect. 3 we describe the basic features of the LCRS and discuss the issues related to the processing of the data so as to bring it into a form usable for our purpose. Sect. 4 gives the details of the method of analysis specifically in the context of LCRS. The discussion of the results are presented in Sect. 5 and the conclusions in Sect. 6..
In several parts of the analysis it is required to use definite values for the Hubble parameter and the deceleration parameter , and we have used and .
© European Southern Observatory (ESO) 1999
Online publication: November 3, 1999