Statistical properties of the spatial distribution of galaxies

Abstract

The methods of determining the fractal dimension and irregularity scale in simulated galaxy catalogs and the application of
these methods to the data of the 2dF and 6dF catalogs are analyzed. Correlation methods are shown to be correctly applicable
to fractal structures only at the scale lengths from several average distances between the galaxies, and up to (10 − 20)%
of the radius of the largest sphere that fits completely inside the sample domain. Earlier the correlation methods were believed
to be applicable up to the entire radius of the sphere and the researchers did not take the above restriction into account
while finding the scale length corresponding to the transition to a uniform distribution. When an empirical formula is applied
for approximating the radial distributions in the samples confined by the limiting apparent magnitude, the deviation of the
true radial distribution from the approximating formula (but not the parameters of the best approximation) correlate with
fractal dimension. An analysis of the 2dF catalog yields a fractal dimension of 2.20 ± 0.25 on scale lengths from 2 to 20
Mpc, whereas no conclusive estimates can be derived by applying the conditional density method for larger scales due to the
inherent biases of the method. An analysis of the radial distributions of galaxies in the 2dF and 6dF catalogs revealed significant
irregularities on scale lengths of up to 70 Mpc. The magnitudes and sizes of these irregularities are consistent with the
fractal dimension estimate of D =2.1–2.4.

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arXiv:0910.2178v1 [astro-ph.CO] 12 Oct 2009
Statistical Properties of the Spatial Distribution of
Galaxies
N. Yu. Lovyagin1
St.-Petersburg State University, Universitetskij pr. 28, St.-Petersburg, 198504 Russia
Astrophysical Bulletin, 2009, Vol. 64, No. 3, pp. 217–228.
The original publication is available at www.springerlink.com:
http://www.springerlink.com/content/m04u0814r17065l1.
Abstract
The methods of determining the fractal dimension and irregularity scale in simulated
galaxy catalogs and the application of these methods to the data of the 2dF and 6dF cat-
alogs are analyzed. Correlation methods are shown to be correctly applicable to fractal
structures only at the scale lengths from several average distances between the galaxies,
and up to (10–20)% of the radius of the largest sphere that fits completely inside the
sample domain. Earlier the correlation methods were believed to be applicable up to the
entire radius of the sphere and the researchers did not take the above restriction into
account while finding the scale length corresponding to the transition to a uniform distri-
bution. When an empirical formula is applied for approximating the radial distributions
in the samples confined by the limiting apparent magnitude, the deviation of the true
radial distribution from the approximating formula (but not the parameters of the best
approximation) correlate with fractal dimension. An analysis of the 2dF catalog yields a
fractal dimension of 2.20±0.25 on scale lengths from 2 to 20 Mpc, whereas no conclusive
estimates can be derived by applying the conditional density method for larger scales due
to the inherent biases of the method. An analysis of the radial distributions of galaxies
in the 2dF and 6dF catalogs revealed significant irregularities on scale lengths of up to
70 Mpc. The magnitudes and sizes of these irregularities are consistent with the fractal
dimension estimate of D = 2.1–2.4
1.INTRODUCTION
The spatial distribution of galaxies bears signatures of both the initial conditions in the
early Universe and the evolution of the primordial density perturbations. An analysis of various
galaxy samples performed using the two-point correlation function showed that this function
has a power-law form ξ(r) = (r0/r)γon scale lengths ranging from 0.01 to 10 Mpc (hereafter we
adopt a Hubble constant of H0= 100km/s/Mpc) with a slope of γ = 1.77 and the parameter
r0= 5 Mpc [1]. It has long been considered that the scale of the r0parameter is the typical
irregularity scale length, and the distribution of galaxies becomes uniform starting from the
scale length of r0 = 5 Mpc. However, the discovery of structures with the scale lengths of
several tens and hundreds Mpc [2] in recent surveys has cast doubt upon this hypothesis.
In this context, the problems of applicability limits and reliability of the correlation methods
of the analysis of spatial distribution of galaxies, and finding new methods for describing large
and very large structures acquire special importance.
At present, two kinds of data on the galaxy redshifts are of great importance.
1E-mail: lovyagin@mail.com
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• The first kind are the redshift catalogs covering large areas (solid angles) of the sky, but
limited to small redshifts (up to z ? 0.5) (2dF, 6dF, SDSS, etc.). Such catalogs can be
analyzed via applying the correlation methods to determine the fractal dimension.
• The second kind is represented by the deepfield catalogs of photometric redshifts. Such
studies cover small solid angles (of the order of 1◦×1◦), but extend to much larger redshifts
z > 1 (up to 6) (COSMOS, HDF, HUDF, FDF and others). Correlation methods are
difficult to apply to such catalogs due to the small radius of the largest sphere that fits
entirely inside the small solid angle considered.
However, both kinds of catalogs can be used to analyze the radial distribution of galaxies,
built upon a sample confined by the limiting apparent magnitude. This method not only
removes the restriction on the size of the largest sphere thereby significantly increasing the
attainable research scale lengths, but it can also be applied to all galaxies in the catalog and
not only to those in a volume-limited sample thereby increasing the number of objects studied.
An analysis of fluctuations in the radial distribution of galaxies can be used to determine both
the sizes and the amplitudes of the largest structures in the galaxy sample considered.
In this paper we analyze two methods of statistical analysis of structures—a determination
of the fractal dimension, and an analysis of radial distributions. Despite the fact that our
analysis is limited to the 2dF and 6dF catalogs, we constructed our simulated lists with two
kinds of catalogs (covering large and small solid angles on the sky).
In this paper we make use of our own software, developed to simulate three-dimensional
catalogs of galaxies and to perform statistical analysis of both real and simulated samples. It
is a C++ library of functions (so far, without a user interface). We are currently preparing
its description, which will be made available, along with the source code, at our web site. The
software covers a somewhat broader scope of problems than that described in this paper, and
will be a basis for a future package meant for comprehensive statistical analysis of the spatial
distribution of galaxies.
2.METHODS USED TO ANALYZE THE STRUCTURES
2.1. Estimating the Fractal Dimension
Fractal dimension is estimated using the method of conditional density in spheres (the total
correlation function in spheres). The definitions of the total and reduced correlation functions
and a detailed description of their properties can be found in [2]. We chose the method of
conditional density in spheres for the reasons stated by Vasil’ev [3]. He showed that this method
is, on the one hand, sufficiently fast (compared to the method of cylinders), and, on the other
hand, sufficiently accurate (the conditional density in spheres is, unlike the conditional density
in shells, less subject to fluctuations) and, moreover, it can be applied to fractal structures
(unlike the method of reduced two-point correlation function, which is built assuming uniform
distribution inside the sample).
The idea of the method consists of constructing a dependence of the number of points N(r)
inside a sphere of radius r, averaged over spheres centered on all the points of the set. Only
a portion of the set is considered, therefore the averaging should be performed only over the
spheres that fit completely inside the set. The dimension is computed by the conditional number
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density2n(r) = N(r)/(4/3πr3) in logarithmic coordinates, where the slope of the line must be
equal to the fractal dimension D minus three, because the expected behavior is n(r) ∝ rD−3.
2.2.Analysis of Radial Distributions
Radial distribution is such a dependence N(z), that
dN(z,dz) = N(z)dz,(1)
where dN is the number of galaxies with redshifts between z and z + dz. The construction of
such a distribution involves counting the number of galaxies ∆N(z,∆z) inside a spherical shell
of thickness ∆z, with midradius lying at the distance corresponding to redshift z, i.e., formula
(1) transforms into
∆N(z,∆z) = N(z)∆z.
Thus, the N(z) distribution can be built in bins with a certain chosen step in ∆z. Traditionally,
the ∆N(z,∆z) variable—the number of galaxies in shells—is plotted on the curves of radial
distribution.
For magnitude-limited catalogs the radial distribution N(z) is approximated by the following
empirical formula (see, e.g., [4, 5]):
N(z) = Azγexp
?
−
?z
zc
?α?
.(2)
Here the three parameters γ, zcand α are independent from each other and A is the normalizing
factor (the integral of radial distribution is normalized to the total number of galaxies in the
sample):
∞
?
00
N(z)dz =
∞
?
Azγexp
?
−
?z
zc
?α?
dz =Azγ+1
c
Γ?γ+1
α
α
?
= N,(3)
where N is the total number of galaxies and Γ(x) is the (complete) Euler Gamma-function.
However, it is impossible, when searching for the best approximation of the radial distribution,
to compute the A (3); due to the fluctuations we have to search for it in the interval from
A −√A to A +√A.
The approximation is performed via the least squares method, i.e., one must search for
the parameter values that minimize the sum of squared residuals. The classical least squares
method cannot be applied as the approximating function is not linear in parameters. However,
a “straightforward” minimization using the fastest (gradient) descent method is also extremely
inefficient, as the minimum is indistinct and it may take a computer several days to several
months to find it. That is why we employ the grid search method, where the grid mesh and
search domain are reduced at each successive iteration.
After finding the best-fit parameters, the domains of irregularities are identified on the curve
of relative fluctuations:
σN=Nobs− Ntheor
Ntheor
,(4)
2Terms “density” and “concentration” are synonyms in this sense, since the concentration is the density of
point sources with the unit mass.
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where
Nobs= N(zi,∆z),
Ntheor= Azγexp
?
−
?z
zc
?α?????
z=zi
.
We can thus interpret any fluctuation exceeding the Poisson noise level of σN > 3σp, as a
structure, where3σp= 1/√Ntheor, because in a fractal distribution the characteristic fluctuation
is increased by σξ, which can be computed based on the value of the two-point correlation
function ξ(r):
1
V2
VV
σ2
ξ=
?
dfV1
?
dfV2ξ(|r1− r2|),
where V is the volume of the set [6, 7, 8].
3.CATALOGS USED
3.1.The 2dF Catalog
The 2dF catalog 2dF [9], or, more precisely, its 2dFGRS subsample, which includes the data
on the redshifts of galaxies, contains a total of 245591 objects, of which about 220 thousand
have sufficiently accurately measured redshifts. The magnitude limits in the J-band, corrected
for the Galactic extinction, are 14.0 < mJ< 19.45. Most of the galaxies have redshifts z < 0.3.
The catalog is available at http://magnum.anu.edu.au/~TDFgg.
The galaxies of the catalog concentrate in the sky in two continuous strips extending along
the right ascension, and in randomly scattered small areas. About 140 thousand galaxies are
located in the Southern strip, and about 70 thousand galaxies, in the Northern strip.
3.2. The 6dF Catalog
The 6dFGS catalog is an all-sky spectroscopic survey at Galactic latitudes |b| > 10◦[10,
11, 12]. Observations began in 2003 and were made using a multichannel spectrograph (they
have not yet been completed at the time of writing this paper). The catalog is available
at http://www-wfau.roe.ac.uk/6dFGS. In this paper we use the second data release of the
catalog, which contains 83014 galaxies with known equatorial coordinates. Of these, 71627
objects have sufficiently reliably determined redshifts. The survey has been completed in three
sky areas. In this paper we use a sample of galaxies with known R-band magnitudes.
4. SIMULATED GALAXY CATALOGS
To test the reliability and accuracy, and to identify the applicability limits of the methods,
they must be applied to simulated catalogs. To this end, we generate catalogs that simulate not
only the spatial distribution of galaxies (uniform and fractal), but also the distribution of their
absolute magnitudes (i.e., the luminosity function of galaxies). Such catalogs can be subjected
to both the correlation analysis (determination of the fractal dimension) in a volume-limited
3Here we use Ntheorand not Nobs, because the latter may be equal to zero.
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sample in a large solid angle, and to the analysis of the radial distribution in a magnitude-
limited sample either in a large or in a small solid angle.
Moreover, we use the MersenneTwister pseudorandom number generator to generate random
quantities (space positions and absolute magnitudes of galaxies). This generator, unlike the
standard linear congruent generator, produces far less correlated numbers and it is considered
suitable for the use of Monte-Carlo method [13].
In this paper we analyze a fractal model of the real distribution of galaxies parametrized by
the fractal dimension and the parameters of the luminosity function. This model describes the
power-law nature of the observed correlations of the distribution of galaxies in real catalogs.
4.1.Spatial Distribution of Galaxies
We use three models of the spatial distribution of galaxies.
Uniform distribution. The coordinates of each point of the set are generated as three random
numbers uniformly distributed in the [0,1] interval (and hence the entire set is contained
in the [0,1] × [0,1] × [0,1] cube).
Cantor dust (more precisely, its generalization to the three-dimensional case). The zero gen-
eration of this set coincides with the [0,1] × [0,1] × [0,1] cube. Each edge of the cube
is then subdivided into m equal parts, i.e., the entire cube is subdivided into m3identi-
cal subcubes, and for each such subcube the probability p of its “survival” in the next
generation is defined. The next generation consists of the set of “surviving” subcubes,
and the algorithm is then reiterated for each such subcube. The final set is the limit
obtained as the number of the generation becomes infinite: in each generation the edge
of the cube becomes shorter by a factor of m and tends to 0 as
contract to points. In case of a real distribution the process should be terminated at a
certain generation n. A point is chosen inside each of the subcubes “surviving” in the
last generation. The coordinates of this point are random numbers uniformly distributed
along the projections of the edges of the subcube onto the coordinate axes.
1
mn, , i.e., the subcubes
The theoretical dimension of such a set is known to be given by the formula
D = logm(pm3).
In our case we use the given dimension D to compute the probability p = mD−3.
Gaussian random walk and its generalization with the possibility of generating sets of 2 ?
D ? 3. dimension. The first point coincides with the coordinate origin (0,0,0). In
the classical case each successive point is obtained from the previous point by adding
to its every coordinate a normally distributed random number with zero mean and unit
variance.
The generalization that we propose here for the first time consists of the following: at
each stage we generate two points instead of one with a certain probability w. A more
accurate description of the algorithm uses the term “generation”. The zero generation
coincides with the coordinate origin (0,0,0). Every next generation is obtained from the
previous generation in accordance with the following rule: for each point of the previous
generation one or two points of the new generation are generated, like in the classical
case, by adding normally distributed random numbers to the coordinates of the previous
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