[Paleopsych] Science: Multiscaling properties of large-scale structure in the universe

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Multiscaling properties of large-scale structure in the universe
Martinez, V J,  Paredes, S,  Borgani, S,  Coles, P. Science. Washington: 
Science. Washington: Sep 1, 1995. Vol. 269, Iss. 5228; pg. 1245, 3 pgs

[This is a terrifically important article. I had stated in my book that 
evidence for a created universe would be that the large-scale structure of the 
universe would *not* look like the result of random fractal processes. Benoit 
Mandelbrot, in _The Fractal Geometry of Nature_, had a computer-generated 
picture of large-scale clustering, but he did not fit it to any data. The 
article below shows the result of getting at the parameters of such a process.

[The fit was good. Had the authors not been able to do so, one of my three most 
cherished hypotheses, that of non-creation, would have been put into serious 
jeopardy.

[It would similarly have been put into jeopardy if the size distribution of 
species in a genus did not fit the outcome of a birth and death process, 
whereby new species come into being as a result of splitting at a certain 
random rate and go extinct at another random rate. That the two rates can be 
measured and get a good fit with the data was shown long ago by Yule, G. Udny 
1924. "A Mathematical Theory of Evolution, Based upon the Conclusions of Dr. 
J.C. Willis, F.R.S." _Philosophical Transactions of the Royal Society, London_, 
ser. B, 213: 21-87.

[Anyone who cherishes a hypothesis should try hard and honestly to think of 
what it would take for him to abandon his hypothesis. "Convincing proof" is 
cheating, since he gets to decide what will convince him!

[My other two are that the co-evolution of genes and culture is crucial to 
understanding the evolution of human societies and that our root basic concepts 
(cost in economics, infinity in mathematics--I need to make a list) are at 
bottom intuitive KNOW-HOW concepts and cannot be nailed down into precise, 
Aristotelian KNOW-THAT genus-species definitions.]

Abstract (Document Summary)

The large-scale distribution of galaxies and galaxy clusters in the universe 
can be described in the mathematical language of multifractal sets. A 
particularly significant aspect of this description is that it furnishes a 
natural explanation for the observed differences in clustering properties of 
objects of different density in terms of multiscaling.

-------------

One of the key problems in modern cosmology is understanding how the spatial 
clustering of objects such as galaxies and galaxy clusters can provide clues 
about the evolution of primordial density inhomogeneities under the action of 
gravitational instability. The traditional tool for quantifying the spatial 
correlations of cosmological objects is the two-point correlation function 
xi(r), defined in terms of the probability delta-P of finding a pointlike 
object of a given type, such as a galaxy, in a small volume delta-V at a 
distance r from a given object of the same type

delta-P = n[1 + xi(r)]delta-V (1)

where n is the mean number density of objects. The two-point correlation 
function for galaxies, xi sub gg (r), is well fitted by a power law in the 
range r = 0.1h sup -1 to 10h sub -1 Mpc (1, 2): xi sub gg (r) = (r/r sub 0 ) 
sup -gamma , with an exponent gamma == 1.8 +/- 0.1 and a correlation length r 
sub O == (5 +/- 1)h sup -1 Mpc (h is the Hubble constant in units of 100 km s 
sup -1 Mpc sup -1 .) Analyses of samples of galaxy clusters, however, have 
yielded power-law fits to the cluster-cluster correlation function, xi sub gg 
(r), of a form similar to that of xi sub gg (r) but with exponents (3-10) 
varying in the range gamma = 1.6 to 2.6 and correlation lengths from 13h sup -1 
to 30h sup -1 Mpc, with a strong dependence of r sub 0 on the richness class of 
the clusters selected (11, 12)

Szalay and Schramm (13) noted that the clustering correlation lengths r sub 0,i 
for different classes of objects i can be described in terms of a unified 
scheme in which r sub 0,i scales with mean separation: (equation omitted) where 
n sub i is the mean number density of the objects. If the two-point correlation 
functions of different classes of objects all have a power-law shape with 
almost the same exponent gamma == 1.8, then xi(r) can be expressed in a 
universal dimensionless form (14, 15)

(Equation 2 omitted)

where beta = 0.2 to 0.3 (14, 15). This relation is remarkably well fitted by 
optical clusters, x-ray clusters, groups of galaxies, quasi-stellar objects, 
and radio galaxies, and although there are still significant uncertainties in 
the power-law fits for clusters, the general trend of increasing r sub 0 with 
richness seems to be well established observationally and is reproduced in 
numerical simulations of cluster clustering (16, 17). On the other hand, 
optical galaxies and IRAS galaxies (those first observed with the Infrared 
Astronomical Satellite), the objects for which most data are available, do not 
appear to fit into this scheme (14, 15, 18), because they are characterized by 
larger values of beta == 1.1 (15). This could be because small-scale (<5h sup 
-1 Mpc) galaxy clustering is principally determined by nonlinear gravitational 
effects and is therefore enhanced with respect to the weakly nonlinear 
clustering displayed by clusters on large scales (>20h sup -1 Mpc). In this 
report we shall show how these observational trends--in particular, the 
apparent difference in clustering behavior between clusters of galaxies and 
galaxies themselves--can be explained in terms of the multiscaling phenomenon, 
which is associated with the application of density thresholds to multifractal 
sets.

Scaling is said to occur in a geometrical pattern whenever some quantity 
describing the spatial distribution has a power-law dependence on scale. For 
example, fractal models of coastlines have a length L that depends on the 
resolution d used to measure it according to L(d) oc d sup 1-D for some 
noninteger D: in this case, D is the Hausdorff dimension of the coastline 
structure. In contrast to simple fractals like this, which are described by a 
single scaling dimension (D), multifractal sets involve a spectrum of scaling 
indices: the density around different points is characterized by different 
(local) fractal dimensions. A particularly important signature of multifractal 
scaling is the fact that moments of the distribution of differing order q scale 
in a manner described by different dimensions D sub q (for a simple fractal, D 
sub q = D for all q). Such objects have proven extremely useful in describing a 
variety of nonlinear phenomena in turbulence, chaotic dynamics, and disordered 
systems (19), and there is now considerable evidence that galaxy clustering is 
intrinsically multifractal in character, perhaps connected with the supposed 
self-similarity of gravitational evolution (20).

In the context of galaxy clustering, the important exponent is the correlation 
dimension D sub 2 , which is defined in terms of scaling of the correlation 
integral C(r) over a distance s

(Equation 3 omitted)

where A is a constant. The index D sub 2 is a clean and easy to interpret 
measure of clustering strength: the larger the value of D sub 2 . the weaker 
the large-scale clustering. Note that power-law scaling of C(r) implies 
power-law scaling of xi(r) only if xi(r) >>1. If this is the case, then Eq. 3 
yields D sub 2 == 3 - gamma. However, this equality is not expected to hold in 
the range of scales where the correlation function is of order unity. 
Consequently, if the correlation integral behaves as a power-law when xi(r) == 
1, the function xi(r) itself does not. By differentiating Eq. 3 with respect to 
r and putting r = r sub 0 , we obtain

(Equation 4 omitted)

which furnishes a useful estimator of r sub 0 .

Multiscaling (21) is the general term given to scaling behavior in which the 
characteristic exponent (in this case D sub 2 ) is a slowly varying function of 
scale or of the threshold density used to select objects of different richness 
from an underlying distribution. This form of scaling is a general consequence 
of applying a density cutoff to a multifractal set. Regions of higher density 
in multifractal sets have smaller values of the scaling indices. If, for 
example, such a set is "censored" by removing all of the points where the local 
density is less than some given value epsilon, then the correlation dimension 
of the surviving set will be smaller than that of the uncensored set: D sub 2 
(epsilon) < D sub 2 .

This kind of thresholding is reminiscent of the idea of "biasing" (22), in 
which regions of high primordial density are identified with observable 
objects. Our philosophy is, however, quite different from this in that we apply 
our thresholding to the nonlinear density field modeled as a multifractal, 
rather than the initial (Gaussian) density perturbations. In this respect, our 
approach improves considerably on the standard approach to biasing.

To demonstrate the applicability of this description to a realistic example, we 
first analyzed a series of numerical simulations of the distribution of rich 
clusters (17). In these simulations, clusters are identified as the highest 
peaks in the evolved density field. Consequently, cluster populations 
characterized by progressively larger mean separations d sub 1 are selected by 
applying progressively higher density thresholds. A plane projection of the 
three-dimensional density field of one of these simulations is shown in Fig. 1 
for a so-called CHDM model, in which 30% of the critical density of the 
universe is provided by one flavor of massive neutrinos and the remainder is 
made up mainly of cold relic particles (23). (Fig. 1 omitted) This model has 
been shown to be reasonably successful at accounting for observed large-scale 
structure. The morphology corresponds to that of a multifractal distribution, 
rather than a simple fractal. Figure 2 shows the correlation integral results 
for such simulated universes. (Figure 2 omitted)

Multiscaling behavior is clearly present in these simulations: D sub 2 varies 
with the characteristic interparticle distance of each sample d sub i . We have 
also analyzed the variation of the correlation length r sub 0 as a function of 
d sub i calculated by means of Eq. 4. Results are given in Table 1 and 
correspond to fitting C(r) over the scale range 10h sup -1 to 50h sup -1 Mpc. 
(Table 1 omitted) As expected on the basis of our multiscaling hypothesis, 
richer clusters (that is, with larger d sub i ) generate less steep C(r) and 
larger r sub 0 . The remarkably small errors in the fitting parameters 
(especially on D sub 2 ) show that there is an excellent power-law fit to C(r) 
over the entire range of scales considered.

A similar qualitative behavior is manifested by the observed distribution of 
cosmic objects. If galaxies and galaxy systems with increasing richness are 
considered to be selected by applying a density threshold in the mass 
distribution, the multiscaling argument implies that the corresponding values 
of the correlation dimension D sub 2 must decrease with increasing density. We 
show here the results of a correlation integral analysis of different galaxy 
and cluster samples. For galaxies we use the Center for Astrophysics (CfA) 
sample (2), the Perseus-Pisces sample (24), and the QDOT (Queen Mary, Durham, 
Oxford, and Toronto)-IRAS redshift survey (25). The cluster samples used are 
the Abell and ACO (Abell-Corwin-Olowin) catalogs (26), the Edinburgh-Durham 
Cluster Catalog (EDCC) redshift survey (8), the ROSAT x-ray-selected cluster 
sample (10), and the APM (Automated Plate Measuring) cluster catalog (9). We 
have performed the calculation of C(r) directly on the CfA and QDOT galaxy 
surveys and Abell and ACO catalogs; in the other cases we just have integrated 
the published values of xi(r). Although this latter procedure is 
straightforward, the numerical integration does tend to exaggerate errors: 
direct calculations of C(r) are generally better.

The x-ray, Abell, ACO, and APM cluster correlation integrals are all well-fit 
by a power law with exponent D sub 2 == 2.1 (Fig. 3). (Fig. 3 omitted) The EDCC 
sample yields a value of D sub 2 == 1.8 over the same scaling range. A value of 
D sub 2 == 2.5 applies to the optical galaxy catalogs (CfA and Perseus-Pisces), 
and a value of D sub 2 = 2.8 is obtained for the QDOT-IRAS galaxies. These 
values for D sub 2 are quite different from those obtained with gamma == 1.8 
for the slope of xi(r) and in the relation D sub 2 = 3 - gamma. This is 
essentially because the range of scales for which we obtain the various 
estimates of D sub 2 in Fig. 2 does not coincide with the range where xi sub gg 
(r) behaves as a power law; clusters of galaxies sample this range of scales 
particularly poorly. Because the smaller D sub 2 is, the larger is the 
departure from a uniform space-filling distribution, these results mean that 
clusters of galaxies have stronger correlations than optical galaxies, which, 
in turn, have stronger correlations than IRAS galaxies. It is natural to 
interpret these trends in terms of multiscaling of objects identified in terms 
of different richness thresholds. A self-consistent picture emerges in which 
clusters correspond to higher matter densities than typical optical galaxies, 
which are themselves located (on average) in denser environments than IRAS 
galaxies. Even the apparently anomalous behavior of EDCC is consistent with 
this trend: clusters from this sample are, on average, richer than in the other 
cluster samples, so its behavior confirms the multiscaling of clusters of 
different density seen in the simulations we have already described.

An important point to emerge from this analysis is that the most natural and 
effective way to characterize scaling properties of the clustering of objects 
of different intrinsic richness is through the correlation integral C(r) rather 
than the two-point correlation function xi(r). Although differences in the two 
descriptions are small if xi(r) >> 1, in the regime where xi == 1, no 
distribution can simultaneously display scaling of both xi(r) and C(r). The 
correlation integral description allows a wide range of empirical clustering 
data to be unified into a single coherent framework within which multiscaling 
is a natural consequence. For example, the fact that D sub 2 for IRAS galaxies 
is larger than that for optical samples indicates that IRAS galaxies are less 
correlated than optical galaxies, or in other words, that optical galaxies 
correspond to higher peaks of the density distribution.

Using Eq. 4, we can obtain clean estimates of r sub 0 for these data sets. For 
the ACO sample, we get r sub 0 == 23h sup -1 Mpc, and for the Abell sample, r 
sub 0 == 26h sup -1 Mpc in the range 10h sup -1 to 50h sup -1 Mpc, whereas for 
the APM cluster catalog, we get r sub 0 16.7 sup h-1 Mpc in the range 1 sup h-1 
to 40 sup h-1 Mpc, in agreement with the value reported by Dalton et al. 
fitting xi(r) directly to a power law (9).

What is missing at the moment from this approach is a detailed understanding of 
the way initial conditions and dynamics interact to produce the observed 
scaling properties. Nevertheless, the ability to incorporate the dependence of 
clustering strength on richness into a unified multifractal scaling paradigm 
through the multiscaling hypothesis is a considerable benefit of this approach. 
Moreover, the robustness of C(r) scaling compared to that of xi(r) strongly 
motivates the use of C(r) as a diagnostic of clustering pattern and dynamics. 
Only by the use of appropriate statistical tools such as this will the new 
generation of galaxy redshift surveys lead to a theoretical understanding of 
the origin of large-scale structure in the universe.

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19. G. Paladin and A. Vulpiani, Phys. Rep. 156, 147 (1987).

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24. L. Guzzo, A. Iovino, G. Chincarini, R. Giovanelli, M. P. Haynes, ibid. 382. 
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25. V. J. Martinez and P. Coles, bid. 437, 550 (1994).

26. S. Borgani, V. J. Martinez, M. A. Perez, R. Valdarnini, ibid. 435, 37 
(1994).

27. We thank L. Guzzo, K. Romer, and G. Dalton for making their xi(r) results 
available to us. We are also grateful to L. Moscardini and M. Plionis for 
permission to use cluster simulations done by them together with P.C. and S.B. 
The Particle Physics and Astronomy Research Council provides P.C. with an 
Advanced Research Fellowship. S.P. is supported by a fellowship of the 
Ministerio de Educacion y Ciencia. This research was partially supported by a 
European Community Human Capital and Mobility Programme network (contract ERB 
CHRX-CT93-0129) and by the project GV-2207/94 of the Generalitat Valenciana.

13 April 7995; accepted 13 July 1995. [It took almost minus 6000 years for this 
paper to get accepted.]