Newsgroups: bit.listserv.frac-l
Date:         Wed, 26 Aug 1992 23:32:56 MDT
From:         Milan PALUS <PCPMP@CSEARN.BITNET>
              26 Aug 1992 12:23:51 EDT from <serge@SLOTH.ASTRO.UNC.EDU>

>I am interested in astrophysical applications of fractals ...

This could be of some interest:



PHYSICS REPORTS
Volume 213, No. 6, published May 16, 1992

The fractal structure of the universe,
   P.H. Coleman and L. Pietronero

We re-evaluate the experimental data concerning the large scale
structure of the universe utilizing the theoretical methods of
modern statistical mechanics. This allows us to test the implicit
assumptions of analyticity and homogeneity present in the usual
analysis. The results are quite surprising: the previous methods
of analysis are mathematically inconsistent for all the cases
that can be explicitly tested.
In view of the untested homogeneity assumption, these methods
give rise to a finite correlation length even in systems that
do not possess one (i.e. those with long range power law
correlations). In particular, we demonstrate that the famous
galaxy correlation length {\it r}$_{0}$ \approx {\it 5h}$^{-1}$
Mpc is a spurious result due to these inconsistencies. In fact
it is related to the cut-off due to the finiteness of the sample
and not to the correlation properties of galaxies. We can
therefore predict that the value of {\it r}$_{0}$ will increase
with sample depth when deeper complete catalogs are available.
The reanalysis of galaxy and cluster catalogs without any
assumption shows long range (fractal) correlations up to the
sample limits. This explains the previous contradiction between
the correlation analysis and the observation of large scale
structures like voids and superclusters. The correlation length,
in fact, cannot be much smaller than the size of the largest
structures one can observe.
We also consider the angular projection of systems with long
range correlations and explain why the angular distributions
appear to be more homogeneous than the three dimensional ones.
These studies are then generalized by an analysis of the whole
distribution of visible matter (by including an estimate of the
galaxy masses) which shows well defined multifractal properties.
This allows one to clarify various other problems such as the so-
called "luminosity segregation" phenomenon and the "richness-
clustering" relation.
In addition the multifractal behavior implies that the upper cut-
off of the luminosity function scales with the size of the sample
with a characteristic exponent which we determine. This fact has
direct implications for galaxy number counts and for several
other problems. One may even conjecture in this respect, that
quasars may actually represent the continuation of this scaling
behavior to much larger scales.
In summary a new picture appears which is fundamentally different
from the usual one but actually much simpler: the distribution
of visible matter in the universe appears to be fractal and
multifractal up to the present observational limits (200 Mpc;
about one twentieth of the Hubble radius of the entire universe)
without any evidence for homogenization. This implies that the
nature of the distribution is strongly non-analytic and no simple
average density can be defined. These results have profound
consequences for our knowledge of the properties of the universe
as well as for the theoretical framework that one should use to
describe it.

 1. Introduction                                           313
 2. Usual analysis of galaxy correlations                  318
 3. Correlation analysis without assumptions               325
 4. New analysis of the galaxy distribution                335
 5. Properties of fractal structures                       344
 6. Multifractal properties of the distribution of
    visible matter                                         351
 7. Correlations of clusters                               365
 8. Correlation properties of projections                  369
 9. Pencil beam surveys and negative dimensions            379
10. Theoretical implications                               382
11. General picture - conclusions                          387

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Regards,

Milan


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* Dr. Milan PALUS                |    tel.: +42 2 8574661     *
* Lab for Appl Math & Bioeng     |    tel.: +42 2 8559972     *
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