Fractal analysis of the large-scale stellar mass distribution in the Sloan Digital Sky Survey
aa r X i v : . [ a s t r o - ph . C O ] J u l Prepared for submission to JCAP
Fractal analysis of the large-scalestellar mass distribution in the SloanDigital Sky Survey
Jos´e Gaite
Applied Physics Dept., ETSIAE, Univ. Polit´ecnica de Madrid, E-28040 Madrid, SpainE-mail: [email protected]
Abstract.
A novel fractal analysis of the cosmic web structure is carried out, employing theSloan Digital Sky Survey, data release 7. We consider the large-scale stellar mass distribution,unlike other analyses, and determine its multifractal geometry, which is compared with thegeometry of the cosmic web generated by cosmological N -body simulations. We find a goodconcordance, the common features being: (i) a minimum singularity strength α min = 1, whichcorresponds to the edge of diverging gravitational energy and differs from the adhesion modelprediction; (ii) a “supercluster set” of relatively high dimension where the mass concentrates;and (iii) a non-lacunar structure, like the one generated by the adhesion model. Keywords: cosmic web, galaxy clusters, redshift surveys, superclusters
Contents N -body simulations 9 v for our volume-limited samples 134.4 Coarsening the volume-limited samples 144.5 Multifractal spectrum 14 Introduction
The large scale structure of the universe can be described as a “cosmic web,” with character-istic though irregular geometric features that extend over lengths of tens of megaparsecs. Onlarger scales, the isotropy and homogeneity of the universe gradually manifest themselves,in accord with the Cosmological Principle. The cosmic web structure can be generated bymodels of the cosmic gravitational dynamics, namely, the Zeldovich approximation and theadhesion model [1, 2] and has been found in galaxy surveys [3, 4] and cosmological N -bodysimulations [5–7]. The structure actually consists of a web of filaments and sheets of mul-tiple sizes, which represent the patterns of gravitational clustering of matter. This type ofgeometric structure, with features on ever decreasing scales, belongs in the domain of frac-tal geometry. Of course, the geometric features of the real cosmic-web have sizes that arebounded below by some scale determined by non-gravitational dynamics (and above by thehomogeneity scale).Fractal models of the universe arose from the idea of a hierarchy of galaxy clusters thatcontinues indefinitely towards the largest scales [8], an idea that originated a debate aboutthe scale of transition to homogeneity [9–14]. Many fractal analyses of galaxy clusteringhave been motivated or influenced by this debate, but the fractal analysis of the large-scalestructure of the universe is interesting in its own right. Early fractal analyses and, specifically,multifractal analyses comprise analyses of the distribution of galaxies [15–17] and also ofthe distribution of dark matter in cosmological N -body simulations [18–20]. Recent N -bodysimulations have better resolution and their analysis reveals new fractal aspects of the cosmicweb [21–26]. However, the combined studies of galaxy surveys and N -body simulations havenot led to a full description of the fractal geometry of the cosmic web and, in particular, to adefinite relation between the geometries of the distributions of galaxies and of dark matter.For example, it is not clear that the much studied power-law dependence of the galaxy-galaxycorrelation function coincides with an analogous dependence of the correlation function ofthe dark matter distribution. It is not even clear that these two correlation functions can bedirectly related, because individual galaxies are not like dark matter particles.Galaxies are visible because of their baryonic content, and the combined dynamics ofcold dark matter and baryonic gas has also been the object of simulations of large-scale struc-ture formation. The comparison between the fractal features of the baryonic gas and darkmatter distributions in the result of one of these simulations shows that they are essentiallyequal [24]. This suggests that a direct comparison of the fractal features of the distributionsof observed visible mass and of simulated dark matter should show good concordance.The study of galaxy clustering by means of correlation functions considers galaxies asequivalent point-like particles [10], much like the particles of dark matter or gas of cosmo-logical N -body simulations. Of course, real galaxies are not point like and their spatialextensions are considerable, and even larger than their visible components (even consideringonly the baryonic part). Nor are galaxies equivalent to one another. Galaxy catalogs provideus with their locations as point-like particles and also with some characteristics that actuallydistinguish them but normally do not provide us with their masses. The neglect of galaxymasses in the analysis of the large-scale structure is equivalent to assigning the same massto all galaxies, which is a questionable approximation. Pietronero [15] already noticed thebroad range of known galaxy masses and argued that it makes the properties of the spatialmass distribution substantially more complex, to such extent that a multifractal analysis isnecessary, instead of the calculation of correlations of galaxy positions. However, in absenceof galaxy masses in the catalogs, the multifractal analysis that has been usually performed– 2 –nly considers the number density of galaxies [11, 14]. A notable exception is the early workof Pietronero and collaborators in which they calculated the masses of galaxies from the ob-served luminosities by assuming a simple mass-luminosity relation (a power law) and thencecarried out a proper multifractal analysis [9, 12].That work of Pietronero et al as well as the contemporary multifractal analyses ofother researchers that did not take galaxy masses into account were limited by the galaxycatalogs then available. Fortunately, we have now available better catalogs, which contain, inparticular, good estimates of stellar masses of galaxies, obtained with sophisticated methods[27, 28]. These masses can be used to achieve a more realistic description of the distributionof visible mass. In fact, a quantitative comparison between the statistical properties of thedistributions of matter in galaxy surveys and of gas or dark matter in N -body simulationsrequires us to take galaxy masses into account. In contrast, the spatial extensions of galaxiesare hardly relevant for the study of the large-scale structure.We analyze in this work the galaxy distribution in the Sloan Digital Sky Survey, datarelease 7 (SDSS-DR7), employing the New York University Value-Added Galaxy Catalog(NYU-VAGC) [29] and taking into account the stellar mass content of galaxies. We restrictourselves to the statistical and geometric properties of the cosmic web that can be determinedby a multifractal analysis (for a morphological analysis of the supercluster-void network inSDSS-DR7, see Ref. [30]). Previous studies of the distribution of SDSS galaxies in redshiftspace are mainly concerned with the problem of the transition to homogeneity, namely, thetransition from middle-scale power-law correlations to very-large-scale uniformity [31–34].These studies do not consider the galaxy masses and look for homogeneity in the numberdensity of galaxies. The scale of homogeneity will feature in our multifractal analysis, asa relevant parameter, but we focus on the properties of the multifractal spectrum and itscomparison with the multifractal spectrum found in N -body simulations of the Lambda colddark matter (LCDM) model, with or without gas. At any rate, we will try to compare ourresults with previous results, especially, with the results of Verevkin et al [33] and Chac´on-Cardona et al [34], who also study the SDSS-DR7. We also compare our analysis withmultifractal analyses of older catalogs.To summarize this work, we first describe the SDSS data employed and the definition ofvolume-limited samples (Sect. 2). Next, we describe the details of our method of multifractalanalysis, including examples of its application to N -body simulations (Sect. 3). The resultsof the analysis of three volume-limited samples of the SDSS, especially, the stellar-massdistribution multifractal spectrum, are contained in Sect. 4, including a comparison with theresults of cosmological N -body simulations. Finally, we present our conclusions and discussthem in Sect. 5. The Sloan Digital Sky Survey in its seventh data release [35] covers one quarter of the skyand has information about galaxies, quasars and stars. The galaxy data have been improvedand included in the The New York U. Value-Added Galaxy Catalog (NYU-VAGC) by theresearch group of M.R. Blanton and D.W. Hogg [29]. Two cuts in apparent magnitude aremade in the Petrosian r spectral band, located at 6165 ˚A: the upper cut, which has to bepresent in every survey, indicates the faintest objects that can be detected, while the lowercut is made to prevent contamination by very bright objects. Following Ref. [30], we takethe lower cut at apparent magnitude m r = 12 . . < m r < . z ≃ .
4. We choosefor our initial sample the following redshift limits: z > . z < .
1, because we do not need the sample to be deep. Indeed,SDSS galaxies with z > . z > . As in any redshift survey with limits in apparent magnitude, the mean number densityof SDSS-DR7 galaxies decreases with redshift. It is necessary to construct volume-limited subsamples of the full sample to correct this radial selection effect [12, 14]. Unlike in Refs. [31–34], in which volume-limited subsamples of galaxies are defined by their ranges of absolutemagnitudes, here they are defined by redshift ranges, which determine the correspondingabsolute magnitude ranges. The cuts in absolute magnitude are given by the expression ofthe absolute magnitude in terms of apparent magnitude and redshift [10]: M abs = m r − R ( z ) − − K ( z ) , (2.1)where m r is the apparent magnitude in the Petrosian r band, R is the luminosity distance inMpc, and K ( z ) is the k-correction for the SDSS r band. In Refs. [31, 33, 34], two differentapproximations for the calculation of the k-correction have been employed. Here we employthe approximation of Chilingarian et al [37], which is appropriate in our case.Since we construct volume-limited (VL) samples by specifying their ranges of redshiftand the redshifts are associated to individual galaxies, it is easy to analyze how a VL samplechanges when R ( z ) or K ( z ) change, for example, after a change of the cosmological model. Inthis regard, we assume a standard LCDM cosmology, with Ω m = 0 .
3, Ω Λ = 0 .
7, but we havechecked that changes in the parameters within reasonable ranges do not alter the results.For the Hubble constant, we may think of the choice h = 1, but our VL samples, in terms ofranges of z , are independent of h , because their construction only involves ratios of pairs ofvalues of R ( z ). In consonance, distances will be expressed in Mpc/ h .Before deciding the VL samples to employ, it is convenient to explain our choice ofcoordinates, with the angular selection, and also consider the requirements of our multifractalanalysis. The selection of our VL samples is described in Sect. 4.1. Let us consider first the angular coordinates for fixed radial distance. The most convenientangular coordinate systems are orthogonal systems such that they preserve the area, thatis to say, such that the element of area is just the product of the line elements along thetwo coordinates, like in Cartesian coordinates. This type of coordinates is common in geog-raphy [38] and have been employed for multifractal analysis of N -body simulated halos inRef. [26]. Equal area coordinates can be defined in terms of angular spherical coordinates:one coordinate is just the longitude and the other is the sine of the latitude. The equatorialcoordinates, namely, right ascension α and declination δ , are indeed spherical coordinatesand could be used for this purpose. However, the SDSS imaging camera scans the sky in strips along particular great circles, so that the appropriate coordinate system, called the survey coordinate system in Ref. [39], is a different system of angular spherical coordinates,– 4 – - - Figure 1 . Our system of equal-area angular coordinates sl and f and the selected rectangle[ − . , . × [0 . , . < z < with poles at α = 95 ◦ , δ = 0 ◦ , and α = 275 ◦ , δ = 0 ◦ . The latitude and longitude measuredfrom these poles are called λ and η , respectively. In Ref. [39], the origin of η is set to the point α = 185 ◦ , δ = 32 . ◦ but we have set it to α = 185 ◦ , δ = 0 ◦ , that is to say, the middle pointof the semicircle δ = 0 ◦ , α ∈ [95 ◦ , ◦ ]. Therefore, our equal-area coordinates are sl = sin λ and f = η + 32 . ◦ ( π/ ◦ ) (in radians). They are related to α and δ by the transformations sl = sin( α − π/ δ, tan f = tan δ/ cos( α − π/ α and δ must be expressed in radians).With this choice of coordinates, we can form a regular region, namely, a rectangle, suchthat it covers most of the SDSS main area. The rectangle is the product of the intervals − . < sl < . − . < f < . sl and f , namely, Ω = 1 . · .
222 = 1 .
899 steradians. This rectangle is displayed in Fig. 1, where the fractions of area do represent fractions ofsolid angle and hence of galaxy number, because sl and f are equal-area coordinates.For the radial coordinate r , the natural definition is the comoving distance that corre-sponds to the assumed values of the cosmological parameters. The relation between r andthe luminosity distance in Eq. (2.1) is R = (1 + z ) r . Let us assume that some mass is distributed in a region of space. The multifractal analysisof this distribution can be carried out in two different ways: either by using a lattice of cells(boxes) covering the region (a method called by Falconer [40] “coarse multifractal analysis”)or by using point-centered spheres, where the points span the support of the distribution. The “rectangle” so defined is not a spherical rectangle , because the two opposite sides with constant sl ,that is to say, constant latitude λ , are not great circles and are therefore curved in the intrinsic geometry ofthe spherical surface. – 5 –arte [41] compares both methods. For a distribution of equal-mass particles, the calculationof the two-point correlation function is equivalent to the calculation of the point-centeredsecond multifractal moment. In fact, the calculation of point-centered statistical moments,or just the second moment, in the form of conditional density, has been the main methodof fractal analysis of the SDSS galaxy distribution [31–34]. However, a lattice multifractalanalysis is more practical to cope with a large amount of data and avoids the problem ofchoosing a maximum radius for point-centered spheres. In the analysis of distributions ofequal-mass particles, this method boils down to an elaboration of counts-in-cells statistics;but it is also practical for unequal-mass particles. We explain it now.Given the lattice, fractional statistical moments are defined as M q = X i (cid:16) m i M (cid:17) q , (3.1)where the index i runs over the set of non-empty cells, m i is the total mass of the particlesin the cell i , while M = P i m i is the total mass of all the particles, and q ∈ R . Somedistinguished integral moments are: M , which is just the number of non-empty cells; M ,normalized to one; and M , which is related to the two-point correlation function.In a regular distribution, with a well-defined density everywhere, if we take a sufficientlyfine mesh, then the mass contained in any cell is proportional to the cell volume v . Therefore, M q ∼ v q − . This does not apply to singular distributions. But the singularities of adistribution can be such that the q -moments are non-trivial power laws of v in the v → τ ( q ) = 3 lim v → log M q log v , (3.2)provided that the limit exists (for every q ). Such a distribution is called multifractal. Fora regular distribution, τ ( q ) = 3( q − v ,that is to say, in a range of negative values of log v (a range of small scales). In fact, theexponent is normally defined as the slope of the graph of log M q versus log v , and its valueis found by numerically fitting that slope.The standard lattice in multifractal analysis is the Euclidean rectangular and evencubical lattice [40, 41]. In fact, a cubical lattice is perfectly adapted to the analysis of N -bodysimulations. However, volume-limited galaxy samples are defined in spherical sectors, whichmakes such lattices inadequate, because they lead to a loss of data. It is preferable to definea rectangular lattice in the coordinates adapted to spherical sectors that have been definedin Sect. 2.2 (which is not a rectangular lattice in Euclidean space). In addition, we requirethat the cells have identical volume. This can be achieved by dividing the angular-coordinaterectangle, given by the intervals of sl and f , into equal area sub-rectangles (like in Ref. [26]),and by also splitting the range of r into intervals with constant ∆( r ) = ( r + ∆ r ) − r .Such a lattice is not unique and we will further require that the resulting cells are reasonablyregular, with aspect ratios not very different from one (Sect. 4.4).Besides the moment exponents τ ( q ), a multifractal is also characterized by its local dimensions. The local dimension α at the point x is the exponent of mass growth from that A continuous mass distribution with a well-defined density is said to be absolutely continuous . Althoughthis property may seem natural, the standard methods of randomly generating continuous mass distributionsproduce strictly singular distributions, namely, distributions with no positive finite density anywhere [42]. – 6 –oint outwards, that is to say, m ( x , r ) ∼ r α ( x ) , where m ( x , r ) is the mass in a ball or box oflinear size r centered on x . The local dimension measures the “strength” of the singularity:the smaller is α , the more divergent is the density at x and the stronger is the singularity.Actually, singularities correspond to α <
3, that is to say, to a divergent density, whereaspoints with α > α constitutes a fractal set with a (Hausdorff) dimension that depends on α ,denoted by f ( α ). In terms of τ ( q ), the spectrum of local dimensions is given by α ( q ) = τ ′ ( q ) , q ∈ R , (3.3)and the spectrum of fractal dimensions f ( α ) is given by the Legendre transform f ( α ) = q α − τ ( q ) . (3.4)Standard self-similar multifractals have a typical spectrum of fractal dimensions that spansan interval [ α min , α max ], is concave (from below), and fulfills f ( α ) ≤ α [40, 41]. Furthermore,the equality f ( α ) = α is reached at one point, with α < q = 1 in Eq. (3.4)[notice that Eq. (3.2) gives τ (1) = 0]. The corresponding set of singularities contains thebulk of the mass and is called the “mass concentrate.”As a complement to the multifractal spectrum f ( α ), it is useful to define the spectrumof R´enyi dimensions D q = τ ( q ) q − , (3.5)because they have an information-theoretic meaning [41, 43]. Indeed, they express the power-law behavior of the R´enyi q -entropies of the coarse distribution in the limit of vanishingcoarse-graining volume, v →
0. The dimension of the mass concentrate α = f ( α ) = D isassociated to the ordinary entropy and is also called the entropy dimension. D = − τ (0) isthe box-counting dimension of the support of the distribution (let us recall that the supportof a mass distribution is the smallest closed set that contains all the mass [40]). D alsocoincides with the maximum value of f ( α ). D = τ (2) is the correlation dimension. Fora regular distribution, τ ( q ) = 3( q −
1) and D q = α = f ( α ) = 3. For a uniform fractal (a unifractal or monofractal ), α , f ( α ) and D q are also constant but D q = α = f ( α ) <
3. Ingeneral, D q is a non-increasing function of q .As said above, the convergence to the limit in Eq. (3.2) must take place in a range ofsmall values of v . Naturally, v must be small in comparison to the homogeneity volume v ,which is the smallest volume such that the mass fluctuations in it are small and approximatelyGaussian (assuming that homogeneity holds on sufficiently large scales). For cell sizes v closeto v or larger, the fluctuations tend to vanish and M q ≈ v q − . This relation is an asymptoticequality, provided that the total sample volume is normalized to one. On account of it, wedefine, for a given cell size v , the coarse exponent as τ ( q ) = 3 log( M q /v q − )log( v/v ) . (3.6)In this fraction, both the numerator and denominator vanish when v approaches v frombelow (the former approximately and the latter exactly). Their quotient tends to τ ( q ) =3( q − τ for regular distributions. The coarse exponent (3.6) dependson both v and v but must become independent of v when v ≪ v , provided that the limit v → v must not be ignored, because the rate of– 7 –onvergence to the limit does depend on the value of v . In particular, if one sets v to one,namely, the total sample volume, and this volume greatly exceeds the homogeneity volume,the coarse exponents can be so inaccurate that no convergence can be observed and it is notpossible to speak of a scaling limit. In other words, if v is not set correctly, the availablerange of v may not be long enough for us to obtain reliable values of the functions τ ( q ) and f ( α ). We discuss the choice of v in Sect. 4.2.When the cell volume v reaches v , each cell can be considered as an independentrealization of the stochastic process that generates the cosmic web structure. The multifractalspectrum f ( α ) measures the probability of finding mass concentrations of strength α in arealization. This probability is estimated, in a lattice with small v , as the number of cellswith strength α divided by the number of non-empty cells, approximately, v − f ( α ) / /v − D / = v [ − f ( α )+ D ] / . This probability is maximal and close to one when f ( α ) takes its maximumvalue D , because most non-empty cells have the corresponding value of α . On the contrary,the probability is minimal for α min or α max , because they normally occur only once. If thesample only occupies the homogeneity volume ( v = 1), then we have just one realization and f ( α min ) = f ( α max ) = 0, so that the cells with α min or α max occur just once. A larger sample( v <
1) contains more than one realization of the stochastic process. If α min and α max still occur only once, then there is less than one cell with α min or α max per realization and f ( α min ) = f ( α max ) <
0; that is to say, there are negative fractal dimensions . This anomalyhas been discussed by Mandelbrot [44]. It is linked to the use of coarse multifractal analysis,to the extent that any dependence on v must disappear as v →
0. Indeed, when f ( α ) < α diminishes as v →
0. In the limit, any set of singularitiesof strength α with f ( α ) < almost surely empty. Therefore, we can discard this part ofthe spectrum. In coarse multifractal analysis, the computation of the coarse exponents (3.6) is subject toerrors that increase for small cell volume v , with the consequent limitation of the availablescale range. The computation of M q by Eq. (3.1) is subject to errors because the mass m i in each cell is uncertain. This uncertainty is due to the uncertainty in the positions andextensions of galaxies combined with the uncertainty in the galaxy masses, namely, in theavailable stellar mass estimates of the galaxies. Although the latter type of error does notexist if one assumes that the galaxy masses are equal, this assumption is intrinsically muchmore erroneous. Adequate methods of error estimation are explained in Sects. 3.2 and 4.5,but it is convenient to consider before some generalities.The major cause of limitation in the available scale range for a sample lies in the discretenature of the sample: if there are many galaxies in a cell, the statistical uncertainties inpositions or masses tend to compensate each other, whereas the uncertainty is largest in thecells with the smallest number of particles. The effect on the coarse exponent τ ( q ) dependson the value of q . The form of M q in Eq. (3.1) shows that, for q >
0, the errors in the largervalues of m i are more important, whereas, for q <
0, the errors in the smaller values of m i are more important. In consequence, the values of τ ( q ) for q <
0, and hence the values of f ( α ) for α > α >
3) are more difficultto establish, because voids are usually undersampled. Moreover, the uncertainty of M q for q < v . The values of M q for q > v , but their uncertainty also increases with decreasing v , as the discretization errors grow.– 8 –n general, the geometry of both clusters and voids is only discernible when they containsufficient numbers of galaxies.In the case of equal masses, the variable that rules the discretization errors is themean number density n of the sample. The discretization length n − / (length of the cubewith one particle on average) is the overall scale for the onset of discretization errors (thesmaller scales can be said to belong to the “shot-noise regime” [13]). The structure ofcosmic voids, in particular, is lost on scales smaller than the discretization length, while thestructure of clusters persists on somewhat smaller scales [22–24]. In fact, it is easy to seethat singularities of strength α are sampled down to the length scale v / − /α n − /α , where v is the homogeneity volume. This criterion is only valid for equal masses, but it could beregarded as a heuristic rule for unequal masses. For galaxies in a certain mass range, a highernumber density obviously diminishes the discretization errors, but it is not easy to comparethe errors for different mass ranges and how they affect different ranges of α . We shall seethat low-luminosity and therefore low-mass volume-limited samples are generally preferableover the full range of α (Sect. 4.5).In regard to the actual evaluation of errors, we can estimate the errors of the moments M q for a given v , in terms of the errors in positions and masses (Sects. 3.2 and 4.5), butthis does not really tell us how reliably the coarse exponents τ ( q ) approach their limit valuesfor v →
0. In fact, the approach to the v → v for larger α (smaller q ).The maximum scale range for convergence extends from the homogeneity scale v / down tothe α -dependent discretization scale written above, which gives, in the equal-mass case, thescale factor v / v / − /α n − /α = ( v n ) /α . It is indeed smaller for larger α (and tends to one for α → ∞ ). The crucial non-dimensionalvariable is v n , which is the number of particles in a homogeneity volume. This number isvery large in recent N -body simulations and guarantees good convergence even for α &
4, asshown in Sect. 3.2. Unfortunately, the situation is much worse in galaxy surveys.In summary, the estimation of errors in the coarse multifractal spectra for a givensample is less useful than the analysis of convergence of those spectra in the available scalerange. This range is limited by the growth of the magnitude of errors for diminishing v ,which restricts the maximum value of α for which convergence holds. The overall accuracyof our results relies on several consistency checks. First, the results for a given volume-limitedgalaxy sample must be self-consistent across its available scaling range, which amounts to aproof of convergence to a multifractal limit. Second, the results for different samples must beconsistent. Finally, the multifractal geometry of the stellar mass thus obtained, after passingthe preceding checks, must be consistent with the multifractal geometry of dark matter andgas derived from N -body simulations. N -body simulations The multifractal analysis of LCDM N -body simulations shows that a sufficient convergencecan be achieved for all values of local dimension α and reveals a typical multifractal spectrum,represented in [22, Fig. 5], [24, Fig. 2], or [26, Fig. 2]. For example, the analysis of the Bolshoi(= Big) simulation [26] gives the multifractal spectrum displayed in Fig. 2. Notice the goodconvergence of the coarse multifractal spectra, which correspond to the scale of 3.9 Mpc/ h (a– 9 – Α f H Α L Bolshoi
Figure 2 . The multifractal spectrum of the present time dark matter distribution in the Bolshoisimulation, as a typical multifractal spectrum of LCDM N -body simulations. This graph clearlyshows the convergence of coarse multifracta spectra. fraction 2 − of the total box length) and seven subsequently halved scales (a total factor of2 = 128). But only the larger scales can give the multifractal spectrum in the zone of voids( α >
3) and reach the maximum value f ( α ) = 3. Essentially the same multifractal spectrumof Fig. 2 is found in other LCDM N -body simulations, and, moreover, the analysis of theMare-Nostrum simulation, which includes gas, shows that the same multifractal spectrum isfound for the distribution of gas [24, Fig. 2]. The salient features of the common spectrumare the following.First of all, it has the typical concave shape that corresponds to a self-similar multifractal[40, 41]. The maximum value of f ( α ), equal to the box-counting dimension of the support ofthe distribution D , is very close to 3, which is a special value. Notice that M = v − whenthere are no empty cells in the lattice, yielding, according to Eq. (3.6), D = − τ (0) = 3.However, in the v → D = 3. Therefore, D = 3occurs either with no voids or with a sequence of voids of sizes that decrease too rapidly.Indeed, what singles out a fractal hierarchy of voids is that it fulfills the Zipf law or thatit follows the Pareto distribution [8, 23]. The analysis of LCDM N -body simulations, e.g.,the Bolshoi simulation in Fig. 2, shows that there are actually no empty cells, starting from v . v and well into the scaling range, and the voids that arise on lower scales seem tobe due to undersampling. This suggests that the cosmic web has a nonlacunar multifractalgeometry, with no totally empty voids [23]. As regards the mass concentrate, the dimension α = f ( α ), given by the point oftangency to the diagonal in the graph, seems to be about 2.4, but it cannot be determinedprecisely. The strongest singularities have α ≃
1. This value, namely, the law of mass growth m ( r ) ∝ r , corresponds to the singular isothermal sphere profile or to a filament, in the fullyisotropic or extremely anisotropic cases of mass concentrations, respectively. The strongestmass depletions have α ≃ . N -body simula-tions with the largest values of v n , that is to say, with the best mass resolution. In the This conclusion refers to Mandelbrot’s original definition of lacunarity [8] and does not mean that a non-vanishing lacunarity cannot be defined. Indeed, the concept of lacunarity has proved to be subtle and thereare various definitions. The notion of nonlacunar fractal, as a fractal set that is everywhere dense , was alsointroduced by Mandelbrot [8]. – 10 –olshoi simulation, the homogeneity scale is one eighth of the simulation box, which yieldsan overall scale range ( v n ) / = 256. This range is reduced for α > α present: one can observe in Fig. 2 convergence ofthree coarse multifractal spectra even for α ≃ .
5. For α <
3, the convergence is still moreconvincing.Moreover, it is easy to see that each coarse multifractal spectrum has a negligible errordue to errors in particle positions. Particle coordinates are given by floating-point numberswith 23-bit mantissa (excluding the sign bit). The 13 most significant bits are preserved bythe coarse graining to the smallest cell used in the Bolshoi simulation (with 0.03 Mpc /h ),leaving a 10-bit precision inside each cell. For larger cells, the precision is higher, of course.Therefore, the relative error in position inside any cell is < − . The relative error in massof cells with many particles, which are more important for q >
0, is proportional to therelative error in positions, and, in fact, the relative error in M q , for q >
0, is practicallyequal to the relative error in positions. For q <
0, cells with few particles can be importantin the computation of M q by Eq. (3.1). However, the sum in that formula can be expressed,for equal-mass particles, as a sum over number of particles per cell, so that each summandis multiplied by the number of cells with a definite number of particles. The error mayalter significantly the number of particles in single cells with few particles but will not altersignificantly each summand. In fact, the relative error in the number of cells with a definitenumber of particles is of the order of magnitude of the relative error in position, in any case.This implies that the change in the coarse multifractal spectra induced by errors in particlepositions is inappreciable in Fig. 2.If we consider the good concordance of multifractal spectra for the Bolshoi simulationand other LCDM N -body simulations [22, 24], we can say that we have a reliable multifractalspectrum of the LCDM cosmic web. In galaxy samples, the mass resolution is much worse andthe errors are considerable, so we should not expect to obtain nearly as accurate a multifractalspectrum, and we must rather compare what we obtain to the N -body simulation multifractalspectrum. This is especially true in the zone α >
3, for which the mass resolution of galaxysamples is hardly sufficient.
Here we describe in detail how we select VL samples in a few redshift intervals, how we deter-mine the value of v , needed for Eq. (3.6), and how we construct coarse lattices appropriatefor the selected VL samples. Finally, we calculate the multifractal spectra of these samples. Previous fractal analyses of SDSS galaxies have either considered several VL samples, inranges of consecutive absolute magnitudes [31, 33, 34], or focused on a particular sample[32]. In all cases, there has been a bias towards deep VL samples and, therefore, highluminosities, in accord with their common goal of analyzing the transition to homogeneity.For just finding the multifractal geometry of the mass distribution, namely, the prop-erties of singular mass concentrations and mass depletions, the relevant length scales arenecessarily smaller. In the present multifractal analysis of the stellar mass of galaxies, itmay seem that we should favor VL samples that contained large fractions of the total stellarmass density. Indeed, such samples represent the mass concentrate set, namely, the set ofsingularities that contains the bulk of the stellar mass. However, the bulk of the stellar mass– 11 –orresponds to rather bright galaxies and, hence, to moderately deep VL samples. Unfortu-nately, if we employ these samples, we miss information about voids, which is best obtainedfrom VL samples with fainter and less massive galaxies.
Redshift r (Mpc/ h ) N V (Mpc /h ) n = N/V ρ (M ⊙ h / Mpc ) galaxy mass (M ⊙ )[0 . , . . , .
9] 1765 3 . · .
048 8 . · [5 . · , . · ][0 . , .
03] [59 . , .
4] 16557 3 . · .
052 2 . · [1 . · , . · ][0 . , .
06] [118 . , .
5] 42021 2 . · .
017 1 . · [2 . · , . · ] Table 1 . Characteristics of the three volume limited samples.
Consequently, to construct our VL samples, it is convenient to take first a range oflow-redshift galaxies, that is to say, faint galaxies, and then proceed to deeper samples. Infact, we can do with just three VL samples, chosen as follows: VLS1 with z ∈ [0 . , . z ∈ [0 . , . z ∈ [0 . , .
06] (see Table 1). The first one isactually the most useful one, because not only is it useful for the study of voids (the zone α >
3) but also, as it turns out, for the study of clusters (the zone α < α <
To implement the procedure of coarse multifractal analysis described in Sect. 3, we need tocompute the q -moments M q from the set of cell masses m i , and then compute the coarseexponents τ ( q ), according to Eq. (3.6), and do it for several lattices, with decreasing cellvolumes v . To compute the coarse exponents, we need first to calculate the homogeneityvolume v . We encounter here, of course, an old problem: the determination of the scaleof homogeneity of the universe. The SDSS data have been employed for this purpose, withvarious results [31–34]. In fact, the “scale of homogeneity” is a loose concept and, as such, isbound to be defined in different ways, which produce different results, even if applied to thesame data. A practical definition can be given in terms of the normalized second moment µ ( v ) = h ρ v i = M ( v ) /v , where ρ v is the normalized coarse-grained density and the lastequality assumes that there are no empty cells in the lattice. Homogeneity is defined by µ = 1, but this value is only approached asymptotically for large v (or as v approachesthe full sample volume). The difference µ − h ( ρ v − i measures the mass variance inthe volume v and can be used, in general, to determine the regime of galaxy clustering [13]:in the homogeneous regime, the coarse-grained density is Gaussian with a small or nearlyvanishing variance.The criterion for choosing v in Refs. [24, 26] actually was that µ ( v ) = 1 .
1, that is tosay, a variance of 10%. This criterion is simple but is as arbitrary as any other, of course,because one can as well demand smaller variances, say 5%, 1% or less, hence considerablyincreasing the value of v . Indeed, the claims that homogeneity has not been found yet ingalaxy catalogs are surely due to imposing too strict criteria for homogeneity. Withoutconsidering any specific property of the mass distribution, in addition to its having, on small In fact, what is claimed by some authors is that there are signs of inhomogeneity on very large scales[31, 33, 34]. However, certain signs of inhomogeneity that may look like structures can be observed in a massdistribution with small mass-variance, for example, in a fluid in a critical state [13]. – 12 –
10 100 1000 v Μ D (cid:144) - = - Α f H Α L SDSS VLS1
Figure 3 . Results of the analysis of sample VLS1: On the left-hand side, moment µ ( v ) ( v inMpc /h ), with a fit of the scaling part ∝ v D / − . On the right-hand side, multifractal spectrum forcoarse-graining volumes v = 2 . , . , ., . · (Mpc/ h ) . scales, large values of µ , that is to say, its being strongly non-Gaussian, one cannot proposea definite criterion and one is confined to speculating about what is a sufficiently Gaussiandistribution. However, the scaling of M q ( v ) and, in particular, of M ( v ) allows us to definea more precise scale of transition to homogeneity, namely, the scale of crossover from themultifractal scaling with non-trivial values of D q to the homogeneous scaling with D q = 3for all q . It has been shown, in some cases, that this criterion roughly agrees with the 10%mass variance criterion [24, 26]. Therefore, we also examine here the scaling of M ( v ). Tobe precise, we examine the crossover from the scaling µ ∼ v D / − on middle scales to theexact value µ = 1 on very large scales. The value of v obtained in this way is sufficient forour purpose, namely, for its use in Eq. (3.6), and we do not need to consider subtle issuesabout the concept of homogeneity (see Ref. [13]). v for our volume-limited samples The calculation of the scale of transition to homogeneity as the scale of crossover from amultifractal scaling to the homogeneous scaling in, for example, VLS1 can be seen in thelog-log plot of the corresponding µ ( v ), displayed in Fig. 3. This plot contains a fit of thescaling µ ∼ v D / − , in the interval v ∈ [3 , / h , with the result D = 1 .
44. Ofcourse, the fitting line and the consequent value of D change if we change the interval of v , but one must choose an interval that yields a good fit (with small errors). The scale ofcrossover can be taken as the value of v at the crossing of the fitting line (for the fractalscaling) and the line µ = 1 (for homogeneity), which yields v = 2000 Mpc / h . However,the corresponding value of µ is somewhat high, that is to say, hardly compatible with aGaussian distribution. So we take a larger value of v , such that the magnitude of µ − v to fit, which will lower the absolute value of the slope and therefore increase the valueof v at the crossing point (and will also increase D ). Relying on these arguments, we take,for this sample, v = 4600 Mpc / h (the cell size in a 2 × × v = 4600 (Mpc/ h ) corresponds the homogeneity length scale 4600 / Mpc/ h = 17Mpc/ h , which is in reasonable agreement with the results from N -body simulations [24, 26],although it is smaller than other values for SDSS galaxies [31–34]. These values are obtainedwith different criteria, which are probably too strict, at least, for our purpose. Nevertheless,– 13 –e find that the appropriate homogeneity scales for VLS2 and VLS3 are a little larger,reaching 25 Mpc/ h for VLS3. This value is practically equivalent to the value of 30 Mpc/ h obtained by Verevkin et al [33], also for the SDSS-DR7 (although they counter that theuniform regime on larger scales still has some sort of inhomogeneities). As explained in Sect. 3, our multifractal analysis is carried out in lattices of equal-volume cellsthat are formed by a particular Cartesian product. This product results from multiplyingan angular-coordinate lattice, with the intervals of sl and f divided into equal subintervals,by the interval of r divided into subintervals with constant increment of r . Additionally, wewant the resulting cells to be reasonable regular, with aspect ratios not very different fromone. For each sample, we must prepare a set of diminishing meshes. To achieve good aspectratios, we need to adapt the set of meshes to the particular interval of r of each sample. Forthe sake of computational simplicity, once the initial coarse mesh is chosen for a sample, wegenerate a sequence of finer meshes by using binary division of subintervals. Given the rangesof sl and f , respectively, 1 .
55 and 1 .
22, an initial 4 × r depends on thesample, but the shape of the spherical sector for a sample is only determined by the ratio ofthe upper to the lower limits of r . This ratio is almost the same for VLS2 and VLS3, andit is such that for all the three samples the length of the radial interval is smaller than thelength of the angular intervals. We find that initial coarse meshes of 4 × × × × × × Several coarse multifractal spectra of VLS1, computed with Eq. (3.6) and v = 4600 Mpc / h ,are displayed in Fig. 3. The agreement with the spectrum obtained from N -body simulations(Sect. 3.2 and Fig. 2) is very convincing in the zone α <
3, corresponding to singularitiesand therefore galaxy clusters. Indeed, the dimension of the mass concentrate, given by thepoints of tangency to the diagonal, is between 2 and 2.6 and is probably about 2.4; and thestrongest singularities also have α ≃
1. For α &
3, we have convergence of only two coarsemultifractal spectra, but they show anyhow that the maximum value of f ( α ), namely, thebox-counting dimension of the distribution’s support, is very close to 3. However, the factthat all (or almost all) cells are non-empty for just two scales, which are not well below v , isnot a fully convincing proof of non-lacunarity. Anyway, the concordance between this galaxysample and the results of N -body simulations is remarkable, because the mass resolution ofgalaxies is generally much worse and, furthermore, galaxy data are subject to considerableerrors (see below). Unfortunately, the concordance breaks down for α > α is notably larger in Fig. 3 than in Fig. 2.These results are supported by the estimation of errors in the VLS1 coarse spectra, dueto errors in galaxy positions and masses. The errors in angular and radial coordinates havedifferent origin: the uncertainty in angular positions is mainly due to the size of galaxies,whereas the uncertainty in distance is mainly due to the uncertainty in the Hubble law causedby peculiar velocities. The size of the type of galaxies in VLS1 can be calculated in terms ofgalaxy mass according to R (kpc) = 0 . M/M ⊙ ) . [45]. This yields a maximum radius <
2– 14 – Α f H Α L VLS1 variance Α f H Α L mass vs. number Figure 4 . Errors in multifractal spectra: (Left) Variance in the spectrum of VLS1 at v = 191(Mpc/ h ) due to errors in galaxy positions and masses (ten variant spectra, in dashed lines). (Right)Effect of suppressing galaxy masses in VLS1 at v = 191 ., . · (Mpc/ h ) (in dashed lines). kpc, which is negligible in comparison with the sizes of the coarse-graining cells, larger than1 Mpc/ h (the error is just a bit larger than the error in the Bolshoi simulation, Sect. 3.2).As regards radial coordinates, peculiar velocities actually destroy the Hubble flow on smallscales and, hence, the determination of distance by redshift. However, the local Hubble flowis “cold” and the dispersion of peculiar velocities is as low as 30 km/s [46]. To be on the safeside, we take a dispersion of 50 km/s. Therefore, we assume, for the error in distance, thatredshifts have normal (Gaussian) errors with dispersion σ z = 1 / z = 0 . M ± . σ Gaussian deviation interval comprises 95% of probabil-ity, we can assume for M a lognormal distribution with σ M = (ln 10) 0 . / . z and M distributions, we can generate a number of random alterna-tives to our initial sample, before the construction of the VL samples. From those alternativeinitial samples, we construct the corresponding variants of VLS1, following the same pro-cedure followed for VLS1 itself. Hence, we compute the corresponding coarse multifractalspectra. We have done so for ten alternative initial samples, focusing on the most relevantVLS1 multifractal spectrum, namely, the spectrum of VLS1 at v = 191 (Mpc/ h ) , which iswithin the scaling range yet is reasonable in the void zone ( α > α < f ( α ). The error grows for α > v = 191 and 1 . · (Mpc/ h ) , we see in the right-hand side of Fig. 4 the effect ofsuppressing mass information. Remarkably, the range of α shrinks, more at the right end,belonging to voids, than at the left end, belonging to clusters. But the effect is notable inboth cases and actually is more important if α <
3, because this is the more reliable part of– 15 – Α f H Α L SDSS VLS2 Α f H Α L SDSS VLS3
Figure 5 . Results of the analysis of samples VLS2 and VLS3. the spectrum. In fact, by suppressing mass information, we ruin the concordance with thespectrum obtained from N -body simulations in that zone. The magnitude of the effect ofsuppressing masses is undoubtedly due to the broad range of masses (see Table 1).We now proceed to the analysis of deeper VL samples, namely, VLS2 and VLS3. Itdoes not provide new information: the results for clusters are nearly the same as before butthe results for voids are definitely worse (Fig. 5). In fact, the absence of any convergenceof coarse multifractal spectra in the zone α > τ ( q )cannot be obtained for negative q because q < τ ( q ) for q < f ( α ) beyond its maximum.At any rate, the part of f ( α ) up to its maximum that Coleman and Pietronero [9] calculatedoes not agree with our results: they obtain that α min = 0 .
65 and that the maximum valueof f is equal to 1.5, in contrast with our values α min = 1 and maximum of f equal to 3. Themultifractal analyses of galactic catalogs by other authors were based on the galaxy numberdensity but obtained results similar to Coleman and Pietronero’s, that is to say, obtainedthat α min is smaller than one and that the maximum of f is quite smaller than three. Theuse of better data and the reasonable agreement with the results of N -body simulations inour analysis make it more reliable.Let us compare our multifractal analysis with the recent analysis of the SDSS-DR7by Chac´on-Cardona et al [34]. This analysis is made in terms of R´enyi dimensions D q ,which provide equivalent information to the multifractal spectrum f ( α ), from a mathematicalviewpoint (Sect. 3). However, Chac´on-Cardona et al select deep VL samples, consider adistribution of dark matter halos associated to SSDS-DR7 galaxies instead of the galaxiesthemselves, and do not take into account the galaxy masses. Moreover, their definition of D q as a derivative with respect to scale is different from the standard definition of fractaldimension adopted here, in terms of the limit for vanishing scale [40, 41]. In consequence,it is difficult to compare directly the present analysis with the analysis by Chac´on-Cardona et al [34]. Naturally, the first problem is to compare their results for D q with the presentresults for f ( α ). A few partial comparisons are simple to make: for example, the value ofthe maximum of f ( α ) must be equal to D . However, while we have found, with confidence,that the maximum of f ( α ) is 3, no definite value of D can be deduced from [34, Fig. 4].At any rate, we believe that the knowledge of the multifractal spectrum f ( α ) of the largescale stellar-mass distribution is more directly useful than the knowledge of R´enyi dimensions,in general, because it allows us to deduce several consequences, which we discuss next.– 16 – Conclusions and discussion
We have calculated the multifractal spectrum of the large scale stellar-mass distribution,employing the SDSS-DR7, in an effort to determine the geometry of the baryonic cosmic weband see how it relates to the geometry of the dark matter cosmic web. Of course, the stellarmass is only a fraction of the total baryonic mass and, furthermore, the SDSS data do notcontain all the stellar mass, because of the cuts in apparent magnitudes of galaxies (and theadditional cuts in VL samples). Nevertheless, the information obtained is representative. Adifferent type of information on the distribution of baryonic mass is provided by N -bodysimulations containing baryonic gas. From one of them, we can conclude that the fractalgeometry of the distributions of gas and dark matter is the same. Assuming this identity, thequestion that we are addressing is whether or not the fractal geometry of the distribution ofthe visible stellar mass coincides with that already known common geometry.Unfortunately, even the rich SDSS data are insufficient for fully determining the mul-tifractal geometry of the stellar mass distribution. At any rate, we can assert the overallconsistency of the stellar mass multifractal spectrum, namely, the internal consistency of ourmultifractal analysis of the SDSS data and, furthermore, its consistency with the multifractalanalysis of LCDM N -body simulations. While the internal consistency is clear in the caseof mass concentrations (clusters), it is however questionable in the case of mass depletions(voids), since we only have convergence, at best, of two coarse multifractal spectra (the min-imum number to speak of convergence) and only for one sample. Therefore, we discuss thecluster ( α <
3) and void ( α >
3) cases separately, beginning with the former.The value α min = 1, common to our analysis of the SDSS-DR7 and N -body simulations,corresponds to the singular isothermal sphere profile, the standard profile of the outskirtsof individual galaxies, and also corresponds to a filament, a basic element of the cosmicweb. More in general, it is a natural lower limit, because the gravitational potential divergesat a point on which mass concentrates with α <
1. Such a mass concentration would notonly have a divergent mass density, which may not be physically forbidden, but would alsoinvolve a divergent (negative) gravitational energy, which is certainly forbidden. Of course,the gravitational energy would not actually become infinite, because the cosmic-web massdistribution is not valid down to infinitely small scales and the scaling law m ( r ) ∼ r α mustchange as r →
0, but mass concentrations with α < α < f ( α ) < α < f ( α ) < N -body simulations [24]. However, mass concentrations with f ( α ) < v →
0, as explained in Sect. 3, and this is indeed what we observe.In this regard, let us notice that the adhesion model predicts that knots are widespreadin the cosmic-web, and these knots are points with finite mass, that is to say, singularitiesof maximum strength, α = 0. They do not seem to be present in the real cosmic web. Thereason for this discrepancy is, of course, that the Zeldovich approximation can only describegravitational dynamics on the larger scales and is unable to describe gravitational collapsewith strong energy dissipation.Singularities with α close to one are very significant energy-wise, but the total massthat they contain is insignificant. The bulk of the mass concentrates on a set of singularitieswith dimension α = f ( α ) = D ≃ .
4. This is a remarkably high value, especially, when wecompare it with the results of older multifractal analyses of the galaxy distribution (with orwithout galaxy masses), which obtain a maximum of f ( α ) that is about 2: but the dimension– 17 –f the mass concentrate must be lower than the maximum dimension, of course. As regardsthe cosmic-web morphology, one could be tempted to conclude that such a high dimensionof the mass concentrate favors cosmic sheets over filaments. However, such conclusion wouldnot be warranted at all, because the fractal dimension does not directly give information onmorphology and one should employ instead the topological dimension and measures of texture [8]. Regarding void regions, formed by points with α >
3, the main conclusion is that themaximum value of f ( α ), which gives the (box-counting) dimension of the support of thedistribution, is very close to 3 (and is at α ≃ . N -body simulations, it can be concluded with confidence that thegeometry of the dark matter or the baryonic gas is non-lacunar, while the present analysisof the stellar mass distribution is not as conclusive. Voids can be perceived in the galaxydistribution (e.g., in the right-hand side of Fig. 1) but they are, presumably, an effect ofundersampling. This effect can combine with the existence of regions with very low baryonicdensity and therefore very few stars. It is to be remarked that the cosmic web generatedby the adhesion model, although somewhat different from the real cosmic web, is also anexample of the peculiar geometry of non-lacunar fractals [23].For strong mass depletions, with α >
4, the result of our analysis of the SDSS differsfrom the result of analyses of N -body simulations. The latter shows a rather sharp declineof the dimension of point sets with α > D = 1 .
44, obtained by fitting the scaling µ ∼ v D / − in VLS1, with other values of the correlation dimension. Sylos Labini et al [31] andVerevkin et al [33] obtain D = 2 (or higher) from the SDSS data, employing the DR4 andDR7, respectively. However, smaller values, obtained from various samples, appear in theliterature [14]. We must caution that the often calculated and discussed correlation dimensionis always the one that corresponds to the galaxy position correlation function, whereas ourvalue of D corresponds instead to the correlation function of the stellar mass distribution. Acknowledgments
I thank C.A. Chac´on-Cardona for correspondence and for the SDSS-DR7 file.
References [1] Ya.B. Zeldovich,
Gravitational instability: An approximate theory for large densityperturbations, Astron. & Astrophys. (1970) 84–89.[2] S.N. Gurbatov, A.I. Saichev and S.F. Shandarin, Large-scale structure of the Universe. TheZeldovich approximation and the adhesion model, Phys. Usp. (2012) 223–249.[3] J. Einasto, M. J˜oeveer and E. Saar, Structure of superclusters and supercluster formation,MNRAS (1980) 353–375.[4] M.J. Geller and J.P. Huchra,
Mapping the Universe, Science (1989) 897–903. – 18 –
5] A.A. Klypin and S.F. Shandarin,
Three-dimensional numerical model of the formation oflarge-scale structure in the Universe, MNRAS (1983) 891–907.[6] D. H. Weinberg and J. E. Gunn,
Large-scale Structure and the Adhesion Approximation,MNRAS (1990) 260–286.[7] L. Kofman, D. Pogosyan, S.F. Shandarin and A.L. Melott,
Coherent structures in the universeand the adhesion model, The Astrophysical Journal (1992) 437–449.[8] B.B. Mandelbrot,
The fractal geometry of nature (rev. ed. of:
Fractals , 1977), W.H. Freemanand Company (1983).[9] P.H. Coleman and L. Pietronero,
The fractal structure of the Universe, Phys. Rep. (1992)311–389.[10] P.J.E. Peebles,
Principles of Physical Cosmology , Princeton University Press (1993).[11] S. Borgani,
Scaling in the Universe, Phys. Rep. (1995) 1–152.[12] F. Sylos Labini, M. Montuori and L. Pietronero,
Scale invariance of galaxy clustering, Phys.Rep. (1998) 61–226.[13] J. Gaite, A. Dom´ınguez and J. P´erez-Mercader,
The fractal distribution of galaxies and thetransition to homogeneity, Astrophys. J. (1999) L5–L8.[14] B.J. Jones, V. Mart´ınez, E. Saar and V. Trimble,
Scaling laws in the distribution of galaxies,Rev. Mod. Phys. (2004) 1211–1266.[15] L. Pietronero, The fractal structure of the universe: Correlations of galaxies and clusters andthe average mass density, Physica A (1987) 257–284.[16] B.J. Jones, V. Mart´ınez, E. Saar and J. Einasto,
Multifractal description of the large-scalestructure of the universe, Astrophys. J. (1988) L1–L5.[17] R. Balian and R. Schaeffer,
Galaxies: Fractal dimensions, counts in cells, and correlations,Astrophys. J. (1988) L43–L46.[18] R. Valdarnini, S. Borgani and A. Provenzale,
Multifractal properties of cosmological N-bodysimulations, Astrophys. J. (1992) 422–441.[19] S. Colombi, F.R. Bouchet and R. Schaeffer,
Multifractal analysis of a cold dark matteruniverse, Astron. & Astrophys. (1992) 1.[20] G. Yepes, R. Dom´ınguez-Tenreiro and H.P.M. Couchman,
The scaling analysis as a tool tocompare N-body simulations with observations — Application to a low-bias cold dark mattermodel, Astrophys. J. (1992) 40–48.[21] J. Gaite,
The fractal distribution of haloes, Europhys. Lett. (2005) 332–338.[22] J. Gaite, Halos and voids in a multifractal model of cosmic structure, Astrophys. J. (2007)11–24.[23] J. Gaite,
Statistics and geometry of cosmic voids, JCAP (2009) 004.[24] J. Gaite, Fractal analysis of the dark matter and gas distributions in the Mare-Nostrumuniverse, JCAP (2010) 006.[25] C.A. Chac´on-Cardona and R.A. Casas-Miranda, Millennium simulation dark matter haloes:multifractal and lacunarity analysis and the transition to homogeneity, MNRAS (2012)2613–2624.[26] J. Gaite,
Smooth halos in the cosmic web, JCAP (2015) 020.[27] G. Kauffmann et al , Stellar Masses and Star Formation Histories for Galaxies from theSloan Digital Sky Survey, MNRAS (2003) 33–53.[28] M.R. Blanton and S. Roweis.
K-Corrections and Filter Transformations in the Ultraviolet,Optical, and Near-Infrared, AJ (2007) 734–754. – 19 –
29] M.R. Blanton et al , New York University Value-Added Galaxy Catalog: A Galaxy CatalogBased on New Public Surveys, AJ (2005) 2562–2578.NYU Value-Added Galaxy Catalog web page.[30] E. Tago et al , Groups of galaxies in the SDSS Data Release 7. Flux- and volume-limitedsamples, Astron. Astrophys. (2010) A102.[31] F Sylos Labini, NL Vasilyev and YV Baryshev,
Power law correlations in galaxy distributionand finite volume effects from the Sloan Digital Sky Survey Data Release Four, Astron.Astrophys. (2007) 23–33.[32] P. Sarkar, J. Yadav, B Pandey and S. Bharadwaj,
The scale of homogeneity of the galaxydistribution in SDSS DR6, MNRAS (2009) L128–31.[33] A.O. Verevkin, Y.L. Bukhmastova and Y.V. Baryshev,
The non-uniform distribution ofgalaxies from data of the SDSS DR7 survey, Astron Rep (2011) 324–40.[34] C.A. Chac´on-Cardona, R.A. Casas-Miranda, J.C. Mu˜noz-Cuartas, Multi-fractal analysis andlacunarity spectrum of the dark matter haloes in the SDSS-DR7, Chaos, Solitons and Fractals (2016) 22–33.[35] K.N. Abazajian et al , The Seventh Data Release of the Sloan Digital Sky Survey, ApJS (2009) 543–558.[36] M.R. Blanton et al,
The Luminosity Function of Galaxies in SDSS Commissioning Data, AJ (2001) 2358–2380.[37] I. Chilingarian, A.-L. Melchior and I. Zolotukhin,
Analytical approximations of K-corrections inoptical and near-infrared bands, MNRAS (2010) 1409.[38] E.W. Weisstein,
Cylindrical Equal-Area Projection, from MathWorld — A Wolfram WebResource [ http://mathworld.wolfram.com/CylindricalEqual-AreaProjection.html ].[39] C. Stoughton et al , Sloan Digital Sky Survey: Early Data Release, The Astronomical Journal (2002) 485–548.[40] K. Falconer,
Fractal Geometry (Second Edition) , John Wiley and Sons, Chichester, UK, (2003),Chapter 17.[41] D. Harte,
Multifractals. Theory and applications , Chapman & Hall/CRC, Boca Raton, Fda.(2001).[42] M. Monticino,
How to Construct a Random Probability Measure, International StatisticalReview (2001) 153–167.[43] A. R´enyi, Calcul des probabilit´es , Dunod, Paris (1966).[44] B.B. Mandelbrot,
Negative fractal dimensions and multifractals, Physica
A 163 (1990)306–315.[45] S. Shen et al , The size distribution of galaxies in the Sloan Digital Sky Survey, MNRAS (2003) 978–994.[46] I.D. Karachentsev, O.G. Kashibadze, D.I. Makarov and R.B. Tully,
The Hubble flow around theLocal Group, MNRAS (2009) 1265–1274.(2009) 1265–1274.