132 Brazilian Journal of Physics, vol. 28, no. 2, June, 1998 Fractals and the Distribution of Galaxies y Marcelo B. Rib eiro and Alexandre Y. Miguelote Observat orio Nacional{CNPq, 21945-970,Rio de Janeiro, Brazil Received February 12, 1998 This pap er presents a review of the fractal approach for describing the large scale distribu- tion of galaxies. We start by presenting a brief, but general, intro duction to fractals, which emphasizes their empirical side and applications rather than their mathematical side. Then we discuss the standard correlation function analysis of galaxy catalogues and many obser- vational facts that brought increasing doubts ab out the reliability of this metho d, paying sp ecial attention to the standard implicit assumption of an eventual homogeneityofthe distribution of galaxies. Some new statistical concepts for analysing this distribution are presented. Without the implicit assumption of homogeneity they bring supp ort to the hy- p othesis that the distribution of galaxies do es form a fractal system. The Pietronero-Wertz's single fractal hierarchical mo del is presented and discussed, together with the implications of this new approach for understanding galaxy clustering. I. Intro duction and isotropy of the matter distribution. This scenario is still thoughtby many to b e the b est theoretical frame- work capable of explaining the large scale matter dis- The goal of mo dern cosmology is to nd the large tribution and spacetime structure of the Universe. scale matter distribution and spacetime structure of the The view outlined ab ove, which has b ecome the or- Universe from astronomical observations. It dates back tho dox homogeneous universe view, has, never b een from the early days of observational cosmology the re- able to fully overcome some of its ob jections. In partic- alization that to achieve this aim it is essential that ular, many researchers felt in the past, and others still an accurate empirical description of galaxy clustering feel to day, that the relativistic derived idea of an even- b e derived from the systematic observations of distant tual homogenization of the observed matter distribution galaxies. As time has passed, this realization has b e- of the Universe is awed, since, in their view, the empir- come a program, which in the last decade or so to ok ical evidence collected from the systematic observation a great impulse forward due to improvements in as- of distant cosmological sources also supp orts the claim tronomical data acquisition techniques and data ana- that the universal distribution of matter will not even- lysis. As a result an enormous amount of data ab out tually homogenize. Therefore, the critical voice claims the observable universe was accumulated in the form of that the large-scale distribution of matter in the Uni- the nowwell-known redshift surveys. Some widely ac- verse is intrinsically inhomogeneously distributed, from cepted conclusions drawn from these data created a cer- the smallest to the largest observed scales and, p erhaps, tain con dence in many researchers that such an accu- inde nitely. rate description of the distribution of galaxies was just ab out to b e achieved. However, those conclusions are Despite this, it is a historical fact that the inho- mainly based on a standard statistical analysis derived mogeneous view has never b een as develop ed as the from a scenario provided by the standard Friedman- ortho dox view, and p erhaps the ma jor cause for this nian cosmological mo dels, which assume homogeneity situation was the lackofworkable mo dels supp orting Present address: Dept. F sica Matem atica, Instituto de F sica{UFRJ, Caixa Postal 68528, Ilha do Fund~ao, Rio de Janeiro, RJ, 21945-970, Brazil; E-mail: [email protected]. y Former participant of the PIBIC program at Observat orio Nacional{CNPq. Present address: Engenharia de Sistemas e Com- putac~ao, Coppe-UFRJ, Ilha do Fund~ao, Rio de Janeiro, Brazil; E-mail: [email protected]. Marcelo B. Rib eiro and Alexandre Y. Miguelote 133 may b e useful for readers not familiar with their ba- this inhomogeneous claim. There has b een, however, sic ideas and metho ds. Therefore, we shall not present one ma jor exception, in the form of a hierarchical cos- an extensive, let alone comprehensive, discussion of the mological mo del advanced byWertz 1970, 1971, al- sub ject, which can b e found in Mandelbrot 1983, though, for reasons that will b e explained b elow, it has Feder 1988, Takayasu 1990 and Peitgen, Jurgens unfortunately remained largely ignored so far. and Saup e 1992, if the reader is more interested in Nevertheless, by the mid 1980's those ob jections the intuitive notions asso ciated with fractals and their to ok a new vigour with the arrival of a new metho d for applications, or in Barnsley 1988 and Falconer 1990 describing galaxy clustering based on ideas of a radi- if the interest is more mathematical. The literature on cally new geometrical p ersp ective for the description of this sub ject is currently growing at a b ewildering pace irregular patterns in nature: the fractal geometry. and those b o oks represent just a small selection that In this review weintend to show the basic ideas b e- can b e used for di erent purp oses when dealing with hind this new approach for the galaxy clustering prob- fractals. This section consists mainly of a summary lem. We will not present the ortho dox traditional view of a background material basically selected from these since it can b e easily found, for instance, in Peebles sources. The discussion starts on the mathematical as- 1980, 1993 and Davis 1997. Therefore, we shall con- p ects asso ciated with fractals, but gradually there is a centrate in the challenging voice based on a new view- growing emphasis on applications. p oint ab out the statistical characterization of galaxy clustering, whose results go against many traditional I I.1 On the \De nition"ofFractals exp ectations, and whichkeep op en the p ossibility that the universe never b ecomes observationally homoge- The name fractal was intro duced by Benoit B. Man- neous. The basic pap ers where this fractal view for delbrot to characterize geometrical gures which are the distribution of galaxies can b e found are relatively not smo oth or regular. By adopting the saying in Latin recent. Most of what will b e presented here is based 1 that nomen est numen, he decided to \exert the right on Pietronero 1987, Pietronero, Montuori and Sy- of naming newly op ened or newly settled territory land- los Labini 1997, and on the comprehensive reviews marks." Thus, he \coined fractal from the Latin ad- by Coleman and Pietronero 1992, and Sylos Labini, jective fractus. The corresp onding Latin verb frangere Montuori and Pietronero 1998. means `to break:' to create irregular fragments. It is The plan of the pap er is as follows. In section 2 we therefore sensible ... that, in addition to `fragmented' present a brief, but general, intro duction to fractals, as in fraction or refraction, fractus should also mean which emphasizes their empirical side and applications, `irregular', b oth meanings b eing preserved in fragment but without neglecting their basic mathematical con- " Mandelbrot 1983, p. 4. cepts. Section 3 brie y presents the basic current ana- In attempting to de ne fractals mathematically, lysis of the large scale distribution of galaxies, its dif- Mandelbrot 1983 o ered the following: \A fractal is culties and, nally, Pietronero-Wertz's single fractal by de nition a set for which the Hausdor -Besicovitch hierarchical mo del that prop oses an alternative p oint dimension strictly exceeds the top ological dimension". of view for describing and analysing this distribution, as We shall discuss later the Hausdor dimension, but the well as some of the consequences of such an approach. imp ortant p oint here is that Mandelbrot himself has The pap er nishes with a discussion on some asp ects of since retreated from this original tentative de nition the current controversy ab out the fractal approach for as it proved to b e unsatisfactory, in the sense that it describing the distribution of galaxies. excluded some sets which ought to b e regarded as frac- tals. In a private communication to J. Feder, Man- II. Fractals delbrot prop osed instead the following lo ose tentative de nition: \a fractal is a shap e made of parts similar to This section intro duces a minimum background ma- the whole in some way" Feder 1988, p. 11. Even so, terial on fractals necessary in this pap er and which 1 To name is to know. 134 Brazilian Journal of Physics, vol. 28, no. 2, June, 1998 in study. As the current trend app ears to indicate that in the scho ol on \Fractal Geometry and Analysis" that this absence of a strict de nition for fractals will pre- to ok place in Montreal in 1989, and was attended by 2 vail, the word fractal can b e, and in fact is, even among one of us MBR , Mandelbrot declined to discuss the sp ecialists, used as a generic noun, and sometimes as an problem of de nition by arguing that any one would adjective. b e restrictive and, p erhaps, it would b e b est to con- sider fractals as a collection of techniques and metho ds Fractal geometry has b een considered a revolution applicable in the study of these irregular, broken and in the waywe are able to mathematically represent and self-similar geometrical patterns.