A&A 397, 401-403 (2003)
DOI: 10.1051/0004-6361:20021528
R. Thieberger
Physics Department, Ben Gurion University, Beer Sheva, 84105, Israel
Received 30 April 2002 / Accepted 5 June 2002
Abstract
This work is a continuation of a previous publication Thieberger
and Spiegel concerning the use of two dimensional catalogues
in unraveling the large scale distribution of galaxies. Since the
data in two-dimensional catalogues are more plentiful than those in
three dimensions, the luminosity method can be helpful in using
two dimensional data to unravel the nature of the three dimensional
distribution. In this work we analyse the Cfa catalogue and
demonstrate that the results indicate a distribution of two dimensions
in the nearer range and three dimensions in the farther range.
The possibility of a different regime for very short distances is
discussed.
Key words: galaxies: general - cosmology: large-scale structure of Universe
Standard cosmology is based on the assumption that the universe is spatially homogeneous, at least on scales sufficiently large to justify its approximation by a Friedmann-Lemaître-Robertson-Walker (FLRW) model. The high isotropy measured in the cosmic microwave background radiation (CMBR) is usually taken as strong evidence in support of this hypothesis.
The structures seen in galaxy catalogues - groups, clusters and superclusters, distributed along voids, filaments and walls - are not viewed as contradicting this Principle, as the common opinion is that the scales on which the universe is assumed to be homogeneous are much larger than those subtended by these structures. However, the consensus on a homogeneous feature of structures, even on very large scales, has never been complete (see e.g. Pietronero 1987).
In more recent times, three-dimensional galaxy catalogues have been made available, they support the findings of approximate self-similarity in the large-scale distribution of galaxies within prescribed scale intervals (Provenzale 1991). Since the data in two-dimensional catalogues are more plentiful than those in three dimensions and are likely to remain so for some time (Cress et al. 1996), it seems worthwhile to discuss further the ways in which the two-dimensional data may be used to unravel the nature of the three-dimensional distribution. The basic issue in analyzing a two-dimensional catalogue goes back to work on stellar statistics (Trumpler & Weaver 1953). The significant point in the stellar studies was the effect of the luminosity distribution of the observed objects on the projected distributions.
The general outline of this issue, which is the main concern of our present attempt has been given previously (Thieberger & Spiegel 2002). Just as in the abovementioned publication, we shall leave out of the account some other features of the galaxy case that remain to be considered in detail, namely source evolution and cosmological effects (Spedalere & Schucking 1980; Ribeiro 1995; Celerier & Thieberger 2000).
In this study we limit ourselves to a three dimensional catalogue so as to be able to evaluate the results. Therefore we chose the cfa catalogue, which has been widely studied, and therefore where we can perform all necessary comparisons so as to understand more precisely our approach.
One of our motivations for the present study comes from our interest in trying to distinguish the different scaling regimes that may exist in the galaxy distribution, though on different length scales. It has been suggested (Murante et al. 1998) that three scaling regimes may be discerned in the distribution of galaxies: on the smallest scales the results are consistent with a distribution of density singularities; in the intermediate range, there is scaling behaviour suggestive of flat structures such as Zeldovitch (1970) favored; on the largest scales, the data indicate a homogeneous distribution of galaxies with nonfractal behavior. This categorization is based on the results of an analysis of three-dimensional catalogues. However, it has to be admitted that those data may as yet not be adequate to clearly decide such issues and that is why we wish to consider whether we may reliably use two-dimensional catalogues for immediate needs. Because of statistical problems, we find it at the moment difficult to study the close region behaviour, therefore we will limit ourselves in this study to the two other regions namely, the two dimensional region and the three dimensional one.
Before describing our calculations, we devote the two next sections to a short description of the correlation integral method and to a brief outline of the influence of luminosity effects by imposing a Schechter luminosity function on the objects of study and attempt to see how this influences the distribution of the projected objects, or rather of their apparent magnitudes. Then we devote a section to a model catalogue so as to help us understand the results of the observational data.
Calculating dimensions for galaxies enters into difficulty
for the cases when the distribution is fractal (Coleman &
Pietronero 1992). A robust characterization of the structures
of point sets is given by the correlation integral (Grassberger
& Procaccia 1983), defined as:
We may interpret C2(r) as
where
is the
average number of galaxies within a distance r of a typical galaxy in
the set. As r goes to zero, C2 should go to zero and, for general
distributions, we express this conclusion as
.
The exponent D2 is called the correlation dimension
and it is necessarily
3 for an embedding space of dimension three.
When D2 is not an integer, the distribution is called fractal
(Mandelbrot 1982).
In Eq. (1) we have a double summation. In certain cases, for
example when we have a pencil beam, this inner region has to be
very small and so sometimes one has to use just the
distances from a single point. This of course results in a strong
deterioration of the statistics. For comparison, we will perform
such calculations and we will call it the "pencil beam method"
(the dimension obtained will be denoted by
,
comparing it
to the usual "Grassberger Procaccia method" (with dimension
).
In considering the distribution on the celestial sphere of galaxies or of radio sources the role of the intrinsic luminosities of the observed objects must be allowed for. Such effects appear in N(>f), the number of galaxies detected whose flux density exceeds a given value, f (Peebles 1993). As is done in Thieberger and Spiegel, we assume that the luminosity function of the objects in our study is independent of position and of time.
A galaxy at distance r is bright enough to be seen at flux
density, f, if its luminosity exceeds L. Using a characteristic
luminosity L*, for normalization one can express the mean
number of galaxies per steradian brighter than f by (Peebles 1993):
Lets consider our specific problem. We assume that (see
Murante et al. 1998) at short distances we have a dimension
D=2 ("Zeldovitch panckakes") and at large distances we have
homogenous distribution (i.e. D=3). In this picture we have
ignored the very close region where we probably have D=1 (the
singularity picture). Then we can modify Eq. (2),
and write:
In order to understand the results which we obtain from an
observational catalogue, we constructed a number of representative
model catalogues. We took N1 points in a box 0 to 500
(arbitrary units), and around each point on a plane we put
N2 points at distances up to PL units, in each direction
on the plane. To each point we attributed a "luminosity''
obtained according to the Schechter function distribution
(Eq. (2)). In most of the catalogues we considered
the cases were we had in our box around 20 000 points
(
minus the points which fall outside the
box limits). This value was chosen so as to be of the same
order of magnitude as the Cfa catalogue.
In this way we can examine the reliability of the results
for the real catalogue. Just for the sake of learning about the
improvement obtained for larger catalogues, we checked also a few
cases of a larger amount of points. In the following table
we compare the results to the ones obtained by the use of
correlation integrals (Grassberger & Procaccia 1983).
In the table we compare our results also to the pencil beam method.
The dimensions are denoted by
and
appropriately.
In this table we give also the results obtained by performing
the dimension analysis via the luminosity effect.
Using the
versus
f)) results, we calculate
the appropriate dimensions. The result is denoted by
in
the table.
In addition to the main purpose of the table, which is the comparison beween the different methods, for a variaty of cases, we also obtain an indication of the errors for the different models. The cases of pure two or three dimensional models (the last four lines in the table) give good agreement and when the number of points is increased (see last line) the errors even for the luminosity case start to decrease considerably. The main reason for the larger overall errors in the top five lines is the consequence of our model where some points "see'' more three dimensional points then others depending on their location on the planes. This problem, being an artifact, should be less pronounced in the true catalogues.
N1 | N2 | PL |
![]() |
![]() |
![]() |
150 | 150 | 50 | 2.3 ![]() |
2.5 ![]() |
2.5 ![]() |
300 | 150 | 50 | 2.6 ![]() |
2.6 ![]() |
2.8 ![]() |
300 | 300 | 50 | 2.6 ![]() |
2.6 ![]() |
2.6 ![]() |
6 | 4800 | 200 | 2.2 ![]() |
2.3 ![]() |
2.4 ![]() |
3700 | 6 | 50 | 2.95 ![]() |
2.9 ![]() |
3.0 ![]() |
1 | 20 k | 250 | 2.00 ![]() |
2.00 ![]() |
2.01 ![]() |
20 k | 1 | .1 | 3.00 ![]() |
2.9 ![]() |
2.98 ![]() |
80 k | 1 | .1 | 3.00 ![]() |
2.9 ![]() |
3.00 ![]() |
320 k | 1 | .1 | 3.00 ![]() |
2.94 ![]() |
2.99 ![]() |
We wish to remark here that there are problems concerning the validity of the results both for very small f and for large f. For small f the finite size of our box results in the limit being f dependent, which is an artifact. For large f only a smaller and smaller fraction of the points reach the "observer", resulting in bad statistics. So we check the average dimension over the range of f where we thought the results were reliable.
In this section we describe the dimensions that we obtain based on the three dimensional data of the Cfa Catalogue of redshifts (Huchra et al. 1995), which is a compilation from several sources. A previous analysis using the standard method, described in Sect. 2, was performed, for an older catalogue, by Provenzale et al. (1997).
We wish to compare the results of the correletion integral method and the luminosity method. The correlation integral method gives for the near galaxies D=1.9 and for the far galaxies D=2.6. The luminosity method gives for the mostly near (as explained in the previous section) D=2.1, and for the mostly far ones: D=2.8.
In these cases we do not have a way to put error bars on the
values. So to get a feel for the possible error, we attribute
to each galaxy a luminosity obtained in the same manner as in
the model catalogue, instead of using the values of the
luminosities in the real Cfa catalogue. We chose nine different
sets of random numbers. In this way we could obtain
the error on the average and obtained an error of about
.
These are encouraging results demonstrating that
although the errors for the luminosity method are somewhat larger
than the ones for the correlation integral method, they are in
a reasonable range. Of course if one has a large
two dimensional catalogue and a relatively much smaller three
dimensional catalogue (as is at the moment the case) then the
larger number of galaxies in the two dimensional catalogue will
make the errors smaller then the one obtained from analysing a
red shift catalogue.
Our calculations show that we obtain the same kind of results both for the correlation integral method and the luminosity method. We tried to estimate the possible errors on the results and it seems to us that they are somewhat larger for the luminosity method. So if we have a three dimensional catalogue most probably the correlation integral give a more reliable result, but as the two dimensional catalogues are much larger, most probably the luminosity method is more reliable.
Acknowledgements
The author wants to thank Professor E.A. Spiegel for many valuable discussions concerning this work.