[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
[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
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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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
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.]
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