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Is the Universe Fractal? Author(s): Vicent J. Martínez Source: Science, New Series, Vol. 284, No. 5413 (Apr. 16, 1999), pp. 445-446 Published by: American Association for the Advancement of Science Stable URL: https://www.jstor.org/stable/2898357 Accessed: 12-04-2019 01:26 UTC

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SCIENCE'S COMPASS

PERSPECTIVES: COSMOLOGY

Is the Universe Fractal? Vicent J. Martinez

O ne of the fundamental issues in modem cosmology is the question of whether the spatial distribution

of galaxies is homogeneous at a given scale. The cosmological principle, formu-

lated originally by Einstein, states that the large-scale uni-

verse is spatially homogeneous and

isotropic. It is this principle, together with Einstein's general relativity, that provides the theoretical framework on which the standard hot big bang model for the origin of the universe is based. However, the principle is an assumption and needs to be verified by observations.

The majority of astrophysicists accept the validity of the cosmological principle. Others follow the ideas envisaged by Charlier (1) and de Vaucouleurs (2) of an unbounded clustering hierarchy in which stars group into galaxies, galaxies into clusters, clusters into superclusters, and so on. This hierarchical clustering view of the universe was recently taken up by authors arguing for a self-similar or fractal distri- bution of galaxies (3, 4).

In recent years, the controversy over whether the universe is smooth on large scales or has an unbounded fractal hierar- chy has received increasing attention (5), because analyses of recent galaxy redshift surveys have reached different conclusions.

During the past two decades, catalogs of galaxies mapping the universe in three dimensions have been compiled (6). These surveys list not only the position on the ce- lestial sphere of each galaxy but also its redshift. By the Hubble law, the latter is proportional to the distance of the galaxy. Comparison of the galaxy positions in the southern slices of the Las Campanas cata- log (7) with the first slice of the Center of Astrophysics second survey (CfA2) (6) (see top figure) shows "the beginning of the end" (8): Although we can see the same structures (walls, filaments, and voids) in the Las Campanas slice as in the CfA2 catalog, we do not see similar struc- tures of larger size than those in the CfA2 sample. In a fractal pattern, the size of these structures should be larger for the deeper slice. This diagram would thus sug-

The author is in the Departament d'Astronomia i As- trofisica, Universitat de Valencia, Burjassot, 46100 Valencia, Spain. E-mail: [email protected]

gest that homogeneity is being reached at larger scales.

The most popular tool for statistical analysis of redshift galaxy surveys is the two-point correlation function, t(r) (9), which measures the clustering in excess [4(r) > 0] or in defect [4(r) < 0] compared with a Poisson distribution, for which t(r) = 0. In contrast, the correlation integral C(r) (10) measures the average number of galaxies within a sphere of radius r of any given galaxy. In a fractal set, this function is proportional to rD2, where D2 is the cor- relation dimension, one of the most com- mon "fractal" dimensions used in the liter- ature. For a uniform distribution, C(r) is proportional to the volume of the sphere, and therefore D2 = 3. If, instead of taking an average, we look at the number of neighbors included in a sphere of radius r centered on Earth, M(r), we can -define the "fractal dimension" DM as the exponent of the relation M(r) o rDM (mass-radius rela- tion). This relation is less accurate than C(r), which considers all galaxies in the sample as possible centers but has the ad- vantage that the measure of the dimension can be extended to much larger scales, be- cause the redshift surveys are typically centered at the observer on Earth.

It is established that t(r) follows a well-defined power law at small separa-

tions, t(r) = (r/ro)-l8, where the correla-

A matter of scale. The galaxy distribution for

the southern slices of the Las Campanas red- shift survey together with the first slice of the

CfA2 catalog at the Northern Hemisphere. Al-

though the depth of the Las Campanas slices is four times (in redshift) the depth of the CfA2 slice, the size of the structures is the same in both samples, contrary to what is expected for an unbounded fractal.

tion length rO 5h- Mpc (h is the Hubble constant in units of 100 km s- Mpc-1; Mpc = 3.26 x 106 light-years) is the dis- tance at which the density of galaxies is on average twice the mean number density. Given the power-law behavior of t(r), in the range where t(r) >> 1, the correlation integral provides a value of D2 = 1.2. This result, together with the fact that the corre- lation function of clusters of galaxies,

Ccjr), was originally fitted to a power law with the same exponent [4cC(r) r- 8], has led several authors (11) to model the uni- verse's large-scale structure as a bounded fractal with dimension D2 = 1.2.

Alternatively, one can try to fit 1 +

t(r), or the correlation integral C(r), directly to a power law. This is particularly important in ranges where t(r) >> 1 does not hold. When this was done with the CfA1 redshift survey, the value obtained for the exponent was slightly larger (12), D2 = 1.3 to 1.5. At larger scales and for the Perseus-Pisces redshift survey, Guzzo et al. (13) found a value D2 = 2.2. Since then, Pietronero and co-workers (14) have analyzed all available red- shift surveys. They found that the large-scale clustering of galaxies is well described by a fractal pattern with dimension

D2 = 2 up to scales of at least 150h-' Mpc, without a transition to homogeneity. Using the mass- radius relation, these authors ex- tend the fractal range to up to 103h-' Mpc with the same di- mension DM = 2. A transition to

Enhanced online at

www.sciencemag.org/cgi/

content/fulV284/5413/445

Stromlo-APM Las Campanas

10 | +- 8 ,, il2 = 2v ESP 10

~D, 2

+~~~~~

e

1 D=3

1 10 100

r(1r1 Mpc)

Gradual transition to smoothness. The correlation func-

tion 1 + W(r) for the Stromlo-APM, the Las Campanas, and the ESP redshift surveys. For the first and the last surveys, the calculation has been performed over volume-limited subsamples. Two straight lines have been plotted for refer- ence, corresponding to a fractal with correlation dimension

D2 = 2 and to a homogeneous distribution with D2 = 3.

www.sciencemag.org SCIENCE VOL 284 16APRIL 1999 445

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SCIENCE'S COMPASS

homogeneity would require an increasing correlation dimension with scale.

Other authors (15) have found that the distribution of galaxies in the currently

available catalogs shows evidence for a transition to homogeneity at large scales.

Scaramella et al. (16) have found that the mass-radius relation provides a dimension-

ality of DM = 3 for the European Southern Observatory (ESO) slice project redshift survey (ESP) and for the ACO (Abell-Cor- win-Olowin) catalog of galaxy clusters.

How did this controversy arise? The most important criticism made by Pietronero and co-workers of the standard analysis of the galaxy catalogs is that the estimators of the correlation function are usually based on the implicit assumption that the galaxy distribution is a homoge- neous and isotropic point process. This as-

sumption affects the way that the estima- tors are corrected for boundary effects. The estimators are all based on counting the number of neighbors at a given dis- tance. When a galaxy lies close to the boundary of the sample, the count is un- derestimated. This can be corrected in dif- ferent ways, leading to different estimators of the correlation function (17). Pietronero and co-workers propose the use of a so- called minus estimator, which omits as centers for counting neighbors at a given scale r those galaxies that are closer to the boundary than r. This solution was antici- pated by Hauser and Peebles (18) but has not been used much in cosmology, mainly because it eliminates some of the informa- tion contained in the data. Moreover, at large distances, only a small fraction of galaxies are considered as centers, increas- ing the variance of the estimator. If one wants to make full use of the data con- tained in a catalog, an edge correction has to be applied, such as the Hamilton estima- tor (19). In the lower figure, previous page I show the correlation function 1 + 4(r) calculated with this estimator for the Stromlo-APM (20) and the Las Campanas redshift surveys (21) together with that for the ESP survey (22) calculated with the standard Davis and Peebles (23) estimator. The fractal behavior at small scales disap- pears at larger distances, providing evi- dence for the transition to homogeneity.

If we are prepared to believe these re- sults, then the universe is not fractal at large scales and the validity of the cosmological principle remains plausible. But are the re- sults conclusive? The defenders of the frac- tal picture of the universe raise the follow- ing arguments against them:

1) The results could be spurious be- cause the estimator used for 4(r) could in- troduce artificial homogeneity. This is

probably the crucial point of the controver-

sy, which reflects the different methodolo- gies adopted by each side. The minus esti-

mator can only be applied up to the radius

of the largest sphere that can be enclosed

within the sample boundaries, and there- fore Pietronero and co-workers are overes- timating the scale up to which fractal cor- relations are found (24). For cluster point

processes, the difference between the mi- nus estimator and the Hamilton estimator

up to the distance where the first can be applied is negligible (17); this implies that the edge correction is not distorting the correlations, but again it is not clear whether the same is true at larger distances.

2) Although the samples analyzed in the figure are presently the best available deep redshift surveys in the optical band of the spectrum (5), several problems in their construction could affect the validity of their statistical analysis. Stromlo-APM is a sparse sample (20); only one galaxy in 20 has a measured redshift. The com- plicated boundaries of the Las Campanas

survey (7), which consists of six separated slices, each 1.50 wide, makes a reliable global statistical analysis at large scales very difficult.

Because of these problems, the strongest observational evidence support- ing the cosmological principle is not based on the redshift surveys but on the isotropy of the projected deep catalogs including the Infrared Astronomical Satellite survey (5) and radio sources (9) and on the analy- sis of the x-ray and cosmic microwave backgrounds (9, 15). Assuming the validi- ty of the principle, it is remarkable that an- gular fluctuations in the temperature of the cosmic background radiation are consis- tent with a universe in which galaxies are reasonably good tracers of mass. The ob- served scaling of the angular two-point correlation function with sample depth al- so does not fit well with the fractal picture of the universe (25). The fractal hypothesis requires that the correlation length ro must increase linearly with sample depth. In contrast, Benoist et al. (26) have demon-

strated that rO depends on the intrinsic lu- minosity of the galaxies in the sample rather than on the sample depth. The fig- ures shown here although they should be viewed with the appropriate caution also show the fingerprint of a transition from the fractal regime to large-scale homo- geneity. The scale at which 1 + 4(r) flat- tens is about the same for the three sam- ples analyzed here, strengthening the case for this interpretation.

The next generation of wide and deep redshift surveys (SLOAN and 2df) will likely provide a more conclusive answer to the question of the large-scale structure of

the universe. In the meantime, the two

sides should agree on the statistical quanti- ties that have to be measured, the most ap- propriate estimators, the cosmological cor- rections to be applied to the data, and the

scale at which a given statistical analysis

can give meaningful results.

References and Notes 1. C. V. L. Charlier, Arkiv. Mat. Astron. Fys. 4, 1 (1908);

ibid. 16,1 (1922).

2. G. de Vaucouleurs, Astron. J. 58, 30, (1953); Science 167,1203 (1970).

3. B. B. Mandelbrot, C. R. Acad. Sci. Paris Ser. A 280,

1551 (1975); The Fractal Geometry of Nature (Freedman, San Francisco, CA, 1982).

4. L. Pietronero, Physica A 144, 257 (1987); P. H. Cole- man and L. Pietronero, Phys. Rep. 231, 311 (1992).

5. M. Davis, in Critical Dialogues in Cosmology, N. Tur-

ok, Ed. (World Scientific, Singapore, 1997), pp. 13-23; L. Pietronero, M. Montuori, F. Sylos Labini, in ibid., pp.

24-49.

6. For a recent review, see 0. Lahav, in Mapping, Mea- suring and Modeling the Universe, ASP Conference

Series, vol. 94, P. Coles, V. J. Martinez, M. J Pons-Bor-

derfa, Eds. (Astronomical Society of the Pacific, San Francisco, CA, 1996), pp. 145-155; L. Guzzo, in ibid., pp. 157-170.

7. S.A. Shectman etal., Astrophys.J.470, 172 (1996). 8. R. P. Kirshner, in Dark Matter in the Universe, S.

Bonometto, J. R. Primack, A. Provenzale, Eds. (IOS, Am- sterdam, 1996), pp. 33-48.

9. P. J. E. Peebles, The Large Scale Structure of the Uni-

verse (Princeton Univ. Press, Princeton, NJ, 1980); P. J. E. Peebles, Physical Cosmology (Princeton Univ. Press, Princeton, NJ, 1993).

10. V. J. Martinez, S. Paredes, S. Borgani, P. Coles, Science 269,1245 (1995).

11. A. S. Szalay and D. N. Schramm, Nature 314, 718 (1985); X. Luo and D. N. Schramm, Science 256, 513 (1992); S. Borgani, Phys. Rep. 251, 1 (1995).

12. P. H. Coleman, L. Pietronero, R. H. Sanders, Astron. As- trophys. 200, L32 (1988); V. J. Martinez and B. J. T. Jones, Mon. Not. R. Astron. Soc 242, 517 (1990).

13. L. Guzzo, A. lovino, G. Chincarini, R. Giovanelli, M. P.

Haynes, Astrophys. J.382, L5 (1991). 14. F. Sylos Labini, M. Montuori, L. Pietronero, Phys. Rep.

293, 61 (1998). 15. L. Guzzo, New Astron. 2, 517 (1997); A. Cappi, C.

Benoist, L. N. da Costa, S. Maurogordato, Astron. As-

trophys. 335, 779 (1998); V. J. Martinez, M. J. Pons- Borderia, R. A. Moyeed, M. J. Graham, Mon. Not. R. As-

tron. Soc 298, 1212 (1998); K. K. S.Wu, 0. Lahav, M. J. Rees, Nature 397, 225 (1999).

16. R. Scaramella et al., Astron. Astrophys. 334, 404 (1998).

17. M. J. Pons-Borderia, V. J. Martinez, D. Stoyan, H. Stoy-

an, E. Saar, Astrophys. J., in press; M. Kerscher, Astron. Astrophys. 343, 333 (1999).

18. M. G. Hauser and P. J. E. Peebles, Astrophys. J 185, 757 (1973).

19. A. J. S. Hamilton, Astrophys. J.417,19 (1993). 20. J. Loveday, S. J. Maddox, G. Efstathiou, B. A. Peterson,

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print available at http://xxx.lanl.gov/abs/astro- ph/980620 1.

26. C. Benoist, S. Maurogordato, L. N. da Costa, A. Cappi,

R. Schaeffer, Astrophys. J. 472, 452 (1996). 27. I thank L. Guzzo, J. Loveday, and D. L. Tucker for making

their 0(r) results available to me. I thank S. Borgani, A. Cappi, P. Coles, L. Guzzo, M. Kerscher, J. Peebles, M. J. Pons-Borderfa, E. Saar, V. Trimble, and K. Wu for their valuable comments and suggestions. This work was partially supported by the Spanish Direcci6n General de Ensenanza Superior project PB96-0797.

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