arXiv:0910.2178v1 [astro-ph.CO] 12 Oct 2009
Statistical Properties of the Spatial Distribution of
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:
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
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 . 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  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.
• 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 . We chose the method of
conditional density in spheres for the reasons stated by Vasil’ev . 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
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
For magnitude-limited catalogs the radial distribution N(z) is approximated by the following
empirical formula (see, e.g., [4, 5]):
N(z) = Azγexp
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
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:
2Terms “density” and “concentration” are synonyms in this sense, since the concentration is the density of
point sources with the unit mass.
0 0.05 0.1 0.15
0.2 0.25 0.3
0.02 0.04 0.06
Fig. 9. Radial distribution (top) and the curve of relative fluctuations (bottom) for the 6dF catalog.
The first domain ∆z = zm/75.
• the empirical formula (2), which is often used to approximate radial distributions of ob-
jects in magnitude-limited catalogs, yields equally adequate minimum root-mean-square
approximation for both uniform and fractal distributions with dimensions exceeding 2.0.
At smaller dimensions the scatter becomes too large and the formula is inapplicable.
We found the fractal dimension to correlate with the deviation of the true radial dis-
tribution from the approximating formula, and not with the parameters of the best-fit
Our analysis of real catalogs yielded the following results:
• the data of the 2dF catalog imply a fractal dimension of 2.20 ± 0.25 in the interval from
2 to 20 Mpc. No reliable conclusions can be made on larger scales about the dimension
and scale of irregularities due to the intrinsic biases of the method. Deeper surveys and
surveys with better sky coverage are needed for this task.
Because of its incompleteness, the 6dF catalog can not yet be used to derive a reliable
estimate for the fractal dimension;
• An analysis of radial distributions revealed the significant irregularities both in the 2dF
and 6dF catalogs. Deviations from smooth distribution exceed 7σpand their scale lengths
amount to 70 Mpc. The scale length and magnitude of irregularities correlate rather well
with fractal-dimension estimates in the 2.1–2.4 interval.
I am sincerely grateful to Yu. V. Baryshev for formulating the problem, for assistance and
constant attention to this work, and to V. P. Reshetnikov for his useful advices and assistance
in preparing the paper. This work was supported by the Russian Foundation for Basic Research
(grant no. 09-02-00143).
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