Fractals and the Distribution of Galaxies

132 Brazilian Journal of Physics, vol. 28, no. 2, June, 1998

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].

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

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

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

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.

study gures, sets and functions. In the past sets or

Falconer 1990 o ers the following p oint of view on

functions that are not suciently smo oth or regular

this matter.

tended to b e ignored as \pathological" or considered as

\The de nition of a `fractal' should b e re-

mathematical \monsters", and not worthy of study,be-

garded in the same way as the biologis t regards

ing regarded only as individual curiosities. Nowadays,

the de nition of `life'. There is no hard and fast

de nition, but just a list of prop erties charac-

there is a realization that a lot can and is worth b eing

teristic of a living thing, such as the abilityto

said ab out non-smo oth sets. Besides, irregular and bro-

repro duce or to move or to exist to some extent

ken sets provide a much b etter representation of many

indep endently of the environment. Most living

things have most of the characteristics on the

phenomena than do the gures of classical geometry.

list, though there are living ob jects that are

exceptions to each of them. In the same way,

it seems b est to regard a fractal as a set that

has prop erties such as those listed b elow, rather

I I.2 The Hausdor Dimension

than to lo ok for a precise de nition which will

almost certainly exclude some interesting cases.

... When we refer to a set F as a fractal,

therefore, we will typically have the following

in mind.

An imp ortant step in the understanding of fractal

1. F has a ne structure, i.e., detail on arbi-

dimensions is for one to b e intro duced to the Hausdor -

trarily small scales.

Besicovitch dimension often known simply as Haus-

2. F is to o irregular to b e describ ed in tradi-

tional geometrical language, b oth lo cally

dor dimension. It was rst intro duced by F. Haus-

and globally.

dor in 1919 and develop ed later in the 1930's byA.S.

3. Often F has some form of self-similari ty,

Besicovitch and his students. It can take non-integer

p erhaps approximate or statistical.

values and was found to coincide with many other def-

4. Usually, the `fractal dimension' of F de-

ned in some way is greater than its

initions.

top ological dimension.

In obtaining the dimension that b ears his name,

5. In most cases of interest F is de ned in a

very simple way, p erhaps recursively."

Hausdor used the idea of de ning measures using cov-

ers of sets rst prop osed by C. Carath eo dory in 1914.

It is clear that Falconer's \de nition" of fractals in-

Here in this article, however, we shall avoid a formal

cludes Mandelbrot's lo ose tentative one quoted ab ove.

mathematical demonstration of the Hausdor measure,

Moreover, Falconer keeps op en the p ossibility that a

and keep the discussion in intuitive terms. We shall

given fractal shap e can b e characterized by more than

therefore o er an il lustration of the Hausdor measure

one de nition of fractal dimension, and they do not

whose nal result is the same as achieved by the formal

necessarily need to coincide with each other, although

pro of found, for instance, in Falconer 1990.

they have in common the prop erty of b eing able to take

The basic question to answer is: howdowe mea- fractional values. Therefore, an imp ortant asp ect of

sure the \size" of a set F of p oints in space? A simple studying a fractal structure once it is characterized as

manner of measuring the length of curves, the area of suchby,say, at least b eing recognized as self-similar in

surfaces or the volume of ob jects is to divide the space some way is the choice of a de nition for fractal dimen-

into small cub es of diameter as shown in Figure 1. sion that b est applies to, or is derived from, the case

See B elair and Dubuc 1991.

Marcelo B. Rib eiro and Alexandre Y. Miguelote 135

the sum of the volumes of the cub es needed to cover

the surface:

3 1

V = N A :

This volume vanishes as ! 0, as exp ected. Now, can

we asso ciate a length with a surface? Formally we can

simply take the measure

L = N ! A ;

which diverges for ! 0. This is a reasonable result

since it is imp ossible to cover a surface with a nite

Figure 1. Measuring the \size" of curves Feder 1988.

numb er of line segments. We conclude that the only

useful measure of a set of p oints de ned by a surface in

Small spheres of diameter could have b een used

three-dimensional space is the area.

instead. Then the curve can b e measured by nding

the number N of line segments of length needed

to cover the line. Obviously for an ordinary curvewe

have N =L = . The length of the curve is given by

L : L = N

In the limit ! 0, the measure L b ecomes asymptoti-

cally equal to the length of the curve and is indep endent

of .

In a similar waywe can asso ciate an area with the

set of p oints de ning a curveby obtaining the number

of disks or squares needed to cover the curve. In the

case of squares where each one has an area of , the

Figure 2. Measuring the \size" of a surface Feder 1988.

numb er of squares N gives an asso ciated area

2 1

There are curves, however, that twist so badly that

A = N L :

their length is in nite, but they are such that they

In a similar fashion the volume V asso ciated with the

ll the plane they have the generic name of Peano

line is given by

. Also there are surfaces that fold so wildly curves

that they ll the space. We can discuss such strange

3 2

V = N L :

sets of p oints if we generalize the measure of size just

Now, for ordinary curves b oth A and V tend to zero

discussed.

as vanishes, and the only interesting measure is the

So far, in order to give a measure of the size of a

length of the curve.

set F of p oints, in space wehave taken a test function

Let us consider now a set of p oints that de ne a

h = d { a line, a square, a disk, a ball or a

surface as illustrated in Figure 2. The normal measure

cub e { and havecovered the set to form the measure

is the area A, and so wehave

H F = h . For lines, squares and cub es wehave

the geometrical factor d=1. In general we nd

0 2

A : A = N

that, as ! 0, the measure H F is either in nite

Here we nd that for an ordinary surface the number or zero dep ending on the choice of d, the dimension of

of squares needed to cover it is N =A = in limit the measure. The Hausdor dimension D of the set F

of vanishing , where A is the area of the surface. We is the critical dimension for which the measure H F

may asso ciate a volume with the surface by forming jumps from in nity to zero see Figure 3:

See x2.3.1

136 Brazilian Journal of Physics, vol. 28, no. 2, June, 1998

0; d> D;

d d

H F = d = dN 1

1; d< D:

The quantity H F is called the d-measure of the I I.3 Other Fractal Dimensions and Some Ex-

amples of Fractals

set and its value for d = D is often nite, but may

b e zero or in nity. It is the p osition of the jump in

H as function of d that is imp ortant. Note that this

The illustration of the Hausdor dimension shown

de nition means the Hausdor dimension D is a lo cal

previously maybevery well from a mathematical p oint

prop erty in the sense that it measures prop erties of sets

of view, but it is hard to get an intuition ab out the

of p oints in the limit of a vanishing diameter of size

fractal dimension from it. Moreover, we do not havea

of the test function used to cover the set. It also follows

clear picture of what this fractional value of dimension

that the D may dep end on p osition. There are several

means. In order to try to answer these questions let us

p oints and details to b e considered, but they are not

see di erent de nitions of fractal dimension and some

imp ortant for us here see Falconer 1990.

examples.

The familiar cases are D = 1 for lines, D = 2 for

planes and surfaces and D = 3 for spheres and other -

I I.3.1 Similarity Dimension and Peano Curves

nite volumes. There are innumerable sets, however, for

which the Hausdor dimension is noninteger and is said

to b e fractal. In other words, b ecause the jump of the

The fractals we discuss may b e considered to b e sets

measure H F can happ en at noninteger values of d,

of p oints emb edded in space. This space has the usual

when H F is calculated for irregular and broken sets

top ological dimension whichwe are used to, and from

the value D where the jump actually o ccurs is usually

aphysicist's p oint of view it coincides with degrees of

noninteger.

freedom de ned by the numb er of indep endentvari-

ables. So the lo cation of a p oint on a line is determined

by one real numb er and a set of two indep endent real

numb ers is needed to de ne a plane. If we de ne dimen-

sion by degrees of freedom in this way,we can consider

a d-dimensional space for any non-negativeinteger d.

In fact, in mechanics it is conventional to consider the

motion of m particles in 3 dimensions as b eing the mo-

tion of one particle in a 6m-dimensional space if we take

each particle's lo cation and momentum as indep endent.

The dimension de ned by degrees of freedom seems

very natural, but more than 100 years ago it was found

to contain a serious aw. In 1890 Giusepp e Peano de-

scrib ed a curve that folds so wildly that it \ lls" the

plane. We shall describ e b elow what we mean by plane-

l ling curves and howwe can generate them, but let us

Figure 3. Graph of H F against d for a set F . The Haus-

keep this meaning in a qualitative and intuitive form

dor dimension is the critical value of d at which the jump

for the moment. The imp ortant p oint is that those from 1 to 0 o ccurs Falconer 1990.

Marcelo B. Rib eiro and Alexandre Y. Miguelote 137

Peano curves can b e drawn with a single stroke and

they tend to cover the plane uniformly, b eing able not

only to avoid self-intersection but also self-contact in

some cases. Since the lo cation of a p oint on the Peano

curve, like a p ointinany curve, can b e characterized

with one real numb er, we b ecome able to describ e the

p osition of any p oint on a plane with only one real num-

b er. Hence the degree of freedom, or the dimension, of

this plane b ecomes 1, which contradicts the empirical

value 2. This pro cess of natural parameterization pro-

duced by the Peano \curves" was called Peano motions

or plane- l ling motions by Mandelbrot 1983, p. 58.

In order to see howPeano curves ll the plane, let

us intro duce rst a new de nition of dimension based

on similarity.Ifwe divide an unit line segmentinto N

parts see Figure 4 we get at the end of the pro cess each

part of the segment scaled by a factor =1=N , which

means N =1.Ifwe divide an unit square into N

Figure 5. Construction of the von Ko ch curve F .At each

stage, the middle third of eachinterval is replaced by the

1=2

similar parts, each one is scaled by a factor =1=N

other two sides of an equilateral triangle Falconer 1990.

if N = 4 the square is scaled by half the side length;

Let us see some examples of calculations of the

so N =1.Now if an unit cub e is divided in N parts,

1=3

similarity dimension of sets. Figure 5 shows the con-

each scaled by =1=N again if N = 8 the cub e

struction of the von Ko ch curve, and any of its seg-

is scaled by half the side length, wehave N =1.

ments of unit length is comp osed of 4 sub-segments

Note that the exp onents of corresp ond to the space

each of which is scaled down by a factor 1/3 from

dimensions in each case. Generalising this discussion

its parent. Therefore, its similarity dimension is

wemaysay that for an ob ject of N parts, each scaled

1:26. This non-integer dimension, D = log 4 = log 3

down from the whole by a ratio , the relation N =1

greater than one but less than two, re ects the prop-

de nes the similarity dimension D of the set as

erties of the curve. It somehow lls more space than a

log N

simple line D = 1, but less than a Euclidean area of

D = : 2

log 1=

the plane D = 2. The Figure 5 also shows that the

von Ko ch curve has a nite structure which is re ected

The Hausdor dimension describ ed previously can b e

in irregularities at all scales; nonetheless, this intricate

seen as a generalization of this similarity dimension.

structure stems from a basically simple construction.

Unfortunately, similarity dimension is only meaningful

Whilst it is reasonable to call it a curve, it is to o irreg-

for a small class of strictly self-similar sets.

ular to have tangents in the classical sense. A simple

calculation on the von Ko ch curve shows that E is of

length ; letting k tend to in nity implies that F

has in nite length. On the other hand, F o ccupies zero

area in the plane, so neither length nor area provides a

very useful description of the size of F .

After this discussion we start to have a b etter idea

of what those fractal dimensions mean. Roughly, a di-

mension provides a description of howmuch space a set

lls. It is a measure of the prominence of the irregu- Figure 4. Dividing a segment, a square and a cub e

larities of a set when viewed at very small scales. We Takayasu 1990.

138 Brazilian Journal of Physics, vol. 28, no. 2, June, 1998

may regard the dimension as an index of complexity. Peano curve are inevitable from a logical and intuitive

We can therefore exp ect that a shap e with a high di- p oint of view, and after an in nite numb er of itera-

mension will b e more complicated than another shap e tions we e ectively haveaway of mapping a part of

with a lower dimension. a plane by means of a top ologically one-dimensional

curve: given some patch of the plane, there is a curve

which meets every p oint in that patch. The set b ecomes

everywhere dense. The Peano curve is p erfectly self-

similar, which is shown very clearly in the gure. The

generator used to construct the Peano curve is not the

only p ossible one, and in fact there are many di erent

ways of constructing such plane- lling curves using dif-

ferent generators, although they are generically known

as Peano curves. Mandelbrot 1983 shows many dif-

ferent and b eautiful examples of di erent generators of

Peano curves.

I I.3.2 Coastlines

Figure 6. Construction of Peano's plane- lli ng curve with

initiator and generator. In each step the segment is divided

The Hausdor and similarity dimensions de ned so

in N = 9 parts, each scaled down by =1=3. This means

far provide de nitions of fractal dimension for pure frac-

the similarity dimension is D =2.For reasons of clarity

tals, that is, classical fractal sets in a mathematically

the corners where the curve self-contacts have b een slightly

idealized way. Although some of these classical fractals

rounded Peitgen, Jurgens and Saup e 1992.

can b e used to mo del physical structures, what is nec-

A nice and mathematically fundamental example

essary now is to discuss real fractal shap es which are

of this prop erty are the plane- lling curves mentioned

encountered in natural phenomena. Hence, we need to

ab ove, whichwe are now in p osition to discuss in a more

apply as far as p ossible the mathematical concepts and

quantitative manner. There are many di erentways of

to ols develop ed so far in the study of real fractal struc-

constructing these plane- lling curves and here we shall

tures, and when the mathematical to ols are found to

take the original generator discussed byPeano. Fig-

b e inadequate or insucient, we need to develop new

ure 6 shows the construction of the curve and we can

ones to tackle the problem under study. If the existing

see that the generator self-contacts. The gure only

to ols prove inadequate, it is very likely that a sp eci c

shows the rst two stages, but it is obvious that after a

de nition of fractal dimension appropriate to the prob-

few iterations the pattern b ecomes so complex that pa-

lem under consideration will havetobeintro duced. Let

p er, p encil and our hands turn out to b e rather clumsy

us see next two examples where we actually encounter

and inadequate to ols to draw the gure, b eing unable

such situations. The rst is the classical problem of

to pro duce anything with much ner detail than that.

the length of coastlines, where a practical application

Therefore, some sort of computer graphics is then nec-

of the equation 1 gives the necessary to ols to charac-

essary to obtain a detailed visualization of such curves.

terize the shap e.

The imp ortant asp ect of this construction is that the

The length of coastlines is the classical fractal prob-

segment is divided in N = 9 parts, each scaled down by

lem that was characterized as such and solved by Man-

=1=3, which means the similarity dimension of the

delbrot. It provides a very interesting example of a

Peano curveisD =2.

fractal shap e in nature and how our intuitive notion

of recti able curves is rather slipp ery. What we shall So we see that this curve has similarity dimension

attempt to show next is a brief and straightforward pre- equals to the square, although it is a line whichdoes

sentation of the problem and its solution, as nothing re- not self-intersect. The removal of one single p oint cuts

places the fascinating discussion made by Mandelbrot the curveintwo pieces which means that its top olog-

himself Mandelbrot 1983. ical dimension is one. The self-contact p oints of the

Marcelo B. Rib eiro and Alexandre Y. Miguelote 139

and try to measure the length of the coast by using an

yardstick of length along the coastline of the map,

and count the numb er of steps N needed to cover it

entirely.Ifwecho ose a large yardstick then wewould

not have to b other ab out even the deep est fjords, and

can estimate the length to b e L = N . However,

someb o dy could raise ob jections to this measurement

based on the unanswered questions ab ove, and we could

try a smaller yardstick. This time the large fjords of

Figure 7 would contribute to the measured length, but

the southeastern coast would still b e taken relatively

easily in few measurements of the yardstick. Neverthe-

less, a serious discussion would demand more detailed

maps, which in consequence reveal more details of the

coast, meaning that a smaller yardstick is then made

necessary. Clearly there is no end to this line of investi-

gation and the problem b ecomes somewhat ridiculous.

The coastline is as long as wewant to make it. It is a

nonrecti able curve and, therefore, length is an inad-

equate concept to compare di erent coastlines as this

measurement is not ob jective, that is, it dep ends on the

yardstickchosen.

Coasts, which are obviously self-similar on nature,

can, nevertheless, b e characterized if we use a di erent

metho d based on a practical use of equation 1. In

Figure 7. The coast of the southern part of Norway. The

gure was traced from an atlas and digitized at ab out

Figure 7 the coast of Norway has b een covered with

1800 1200 pixels. The square grid indicated has spacing

a set of squares with edge length , with the unit of

of 50 km Feder 1988.

length taken to equal the edge of the frame. Count-

Let us start with the Figure 7 showing the southern

ing the numb er of squares needed to cover the coastline

part of Norway. The question in this case is: how long is

gives the number N . If we pro ceed and nd N for

the coast of Norway? On the scale of the map the deep

smaller values of we are able to plot a graph of N

fjords on the western coast show up clearly. The details

versus , for di erent grid sizes. Now it follows from

of the southern part are more dicult to resolve, and if

equation 1 that asymptotically in the limit of small

one sails there one nds ro cks, islands, bays, faults and

the following equation is valid:

gorges that lo ok much the same but do not showup

even in detailed maps. This fact is absolutely intuitive

N / : 3

and anyone who has walked along a b each and lo oked at

So the fractal dimension D of the coastline can b e deter- the same b each in the map afterwards can testify it. So

mined by nding the slop e of log N plotted as a func- b efore answering the question of how long is the coast

tion of log . The resulting plot for the coastline shown under analysis wehave to decide on whether the coast

in gure is presented in Figure 8, with D 1:52, and of the islands should b e included. And what ab out the

the same metho d pro duces D 1:31 for the coast rivers? Where do es the fjord stop b eing a fjord and

of Britain. These twovalues agree with the intuition b ecomes a river? Sometimes this has an easy answer

already asso ciated with fractal dimensions in the sense and sometimes not. And what ab out the tides? Are we

that the more irregular coast, the Norwegian in this discussing the length of the coast at low or high tide?

case, should have a higher value for D than the British Despite these initial problems wemay press ahead

140 Brazilian Journal of Physics, vol. 28, no. 2, June, 1998

coast. By means of equation 3 we can obtain the systems in general havea characteristic lower length

expression for the length of coastlines as scale, we need to take that into consideration in our

physical applications of fractals. For instance, the prob-

L / ;

lem of the length of coastlines necessarily involves a

lower cuto in its analysis as b elow certain scale, say,

which shows its explicit dep endence on the yardstick

at the molecular level, we are no longer talking ab out

chosen. The dimension D in equation 3 is determined

coastlines. For the same reason we sometimes haveto

by counting the number of boxes needed to cover the set

assume upp er cuto s to the fractal structures we are

as a function of the b ox size. It is called box counting

analysing. This highlights once more an imp ortant as-

dimension or box dimension.

p ect of application of fractals to real physical phenom-

ena: each problem must b e carefully analysed not only

to lo ok for the appropriate fractal dimension or di-

mensions as wemayhave more than one de ned in the

problem, but also to see to what extent the fractal

hyp othesis is valid to the case in study.

Let us see a sp eci c example. In order to apply

the ideas of the previous sections, we can replace a

mathematical line by a linear chain of \molecules" or

monomers. Figure shows the replacement of a line by

Figure 8. The numberof`boxes' of size needed to cover

achain of monomers, a two-dimensional set of p oints

the coastline in the previous gure as a function of . The

by a planar collection of monomers, and a volume bya

straight line is a t of N / to the observations.

The fractal dimension is D 1:52 Feder 1988.

packing of spheres. Let us call the radius R the small-

est length scale of the structure under study. In this

The b ox counting dimension prop oses a system-

case R will b e the radius of the monomers in gure .

atic measurement which applies to any structure in

The numb er of monomers in a chain of length L =2R

the plane, and can b e readily adapted for structures

in the space. It is p erhaps the most commonly used

N = :

metho d of calculating dimensions and its dominance

lies in the easy and automatic computability provided

For a group of monomers in a circular disk wehave the

by the metho d, as it is straightforward to countboxes

prop ortionality

and maintain statistics allowing dimension calculation.

N / :

The program can b e carried out for shap es with and

without self-similarity and, moreover, the ob jects may

For the three-dimensional close packing of spherical

be emb edded in higher dimensional spaces.

monomers into a spherical region of radius R, the num-

b er of monomers is

I I.3.3 Cluster Dimension

N / :

Let us see now the application of fractal ideas to the

problem of aggregation of ne particles, such as those of

By generalising these relations wemaysay that the

so ot, where an appropriate fractal dimension has to b e

numb er of particles and the cluster size measured by

intro duced. This kind of application of fractal concepts

the smallest sphere of radius R containing the cluster

to real physical systems is under vigorous development

is given by

at present, inasmuch as it can b e used to study such

systems even in a lab oratory, where they can b e grown,

; 4 N /

or in computer simulations.

where D is the dimension of the distribution that may The Hausdor dimension D in equation 1 requires

b e non-integer, that is, fractal. Equation 4 is a the size of the covering sets to vanish, but as physical

Marcelo B. Rib eiro and Alexandre Y. Miguelote 141

number-radius relation and D is the cluster fractal di-

mension of the aggregation. The cluster fractal dimen-

sion is a measure of how the cluster lls the space it

o ccupies.

Figure 10. Random cluster containing 50,000 particles ob-

tained from a two-dimensional di usion-li mi ted aggregation Figure 9. Simple aggregation of spherical monomers Feder

pro cess DLA with D =1:71 Feder 1988.

1988.

Before closing this section, it seems appropriate to

discuss something ab out the limitations of the fractal

A fractal cluster has the prop ertyofhaving a de-

geometry.For this purp ose we shall quote a paragraph

creasing average density as the cluster size increases, in

from Peitgen, Jurgens and Saup e 1992, p. 244 which

away describ ed by the exp onent in the numb er radius

b eautifully expresses this p oint.

relation. The average density will have the form

\The concept of fractal dimension has in-

D E

R R R ; 5

spired scientists to a host of interesting new

work and fascinating sp eculations. Indeed, for

a while it seemed as if the fractal dimensions

where E is the Euclidean dimension of the space where

would allow us to discover a new order in the

the cluster is placed. Therefore, a cluster is not neces-

world of complex phenomena and structures.

This hop e, however, has b een damp ened by

sarily fractal, even if it is p orous or formed at random,

some severe limitations. For one thing, there

as its densitymay b e constant. Note that the shap e

are several di erent dimensions which give dif-

of the cluster is not characterized by the cluster fractal

ferent answers. We can also imagine that a

structure is a mixture of di erent fractals, each

dimension, although D do es characterize, in a quanti-

one with a di erentvalue of b ox counting di-

tativeway, the cluster's feature of \ lling" the space.

mension. In such case the conglomerate will

have a dimension which is simply the dimension

Figure 10 showsavery much studied typ e of clus-

of the comp onents with the largest dimen-

ter, obtained by the di usion-limited aggregation pro-

sion. That means the resulting numb er can-

cess DLA. In this pro cess the cluster is started by

not b e characteristic for the mixture. What we

would really liketohave is something more like

a seed in the centre, and wandering monomers stick

a sp ectrum of numb ers which gives information

to the growing cluster when they reach it: if the ran-

ab out the distributio n of fractal dimensions in

dom walker contacts the cluster, then it is added to it

a structure. This program has, in fact, b een

carried out and runs under the theme multi-

and another walker is released at a random p osition on

fractals."

the circle. This typ e of aggregation pro cess pro duces

For reasons of simplicitywe shall not deal with multi- clusters that have a fractal dimension D = 1:71 for

fractals in this pap er. di usion in the plane. Numerical simulations show that

As a nal remark, as Pietronero 1988 has p ointed the fractal dimension is D =2:50 for clusters in three-

out, the fractal concept provides a description of these dimensional space.

142 Brazilian Journal of Physics, vol. 28, no. 2, June, 1998

I I I.1 The Standard Correlation Function Ana- irregular structures on nature, but it do es not imply

lysis the formulation of a theory for them. Indeed, Mandel-

brot has not pro duced a theory to explain how these

structures actually arise from physical laws. A study of

the interrelations b etween fractal geometry and phys-

The standard statistical analysis assumes that the

ical phenomena is what may b e called the \theory of

ob jects under discussion galaxies can b e regarded as

fractals", and forms the ob jectiveofintense activityin

p oint particles that are distributed homogeneously on a

the eld nowadays. This activity is basically divided in

suciently large scale. This means that we can mean-

two main streams. The rst tries to understand how

ingfully assign an average numb er density to the distri-

it has come ab out that many shap es in nature present

bution and, therefore, we can characterize the galaxy

fractal prop erties. Hence, the basic question to answer

distribution in terms of the extent of the departures

is: where do fractals come from? The second approach

from uniformityonvarious scales. The correlation func-

is to assume as a matter of fact the existence of fractal

tion as intro duced in this eld byP.J.E.Peebles

structures and to study their physical prop erties. This

around 25 years ago is basically the statistical to ol that

generally consists of assuming a simple fractal mo del as

p ermits the quantitative study of this departure from

a starting p oint and studying, for example, some basic

homogeneity.

physical prop erty, like the di usion on this structure.

We shall discuss in a momenthow to obtain the ex-

As examples of such studies, there are the fractal

plicit form of the two-p oint correlation function, but it

growth models, which are based on a sto chastic growth

is imp ortant to p oint out rightnowtwo essential asp ects

pro cess in which the probability is de ned through

of this metho d. First that this analysis ts very well in

Laplace equation e.g., DLA pro cess. They are con-

the standard Friedmannian cosmology which assumes

sidered the prototyp es of manyphysical phenomena

spatial homogeneity, but it do es not takeinto consider-

that generate fractal structures. As other examples, we

ation any e ect due to the curvature of the spacetime.

have the self-organized critical systems, which are such

In fact, this metho d neglects this problem altogether

that a state with critical prop erties is reached sp onta-

under the assumption that the scales under study are

neously,by means of an irreversible dynamical evolu-

relatively small, although it do es not o er an answer to

tion of a complex system e.g., sandpile mo dels. They

the question of where we need to start worrying ab out

p ose problems similar to the fractal growth pro cess, and

the curvature e ects. In other words, this analysis do es

use theoretical metho ds inspired by the latter, like the

not tell us what scales can no longer b e considered rel-

so-called \ xed scale transformation", that allows to

atively small.

deal with irreversible dynamics of these pro cess and to

Secondly that if the homogeneity assumption is not

calculate analytically the fractal dimension.

justi ed this analysis is inapplicable. Moreover, b ecause

I I I. The Fractal Hyp othesis for the Distribution

this analysis starts by assuming the homogeneity of the

of Galaxies

distribution, it does not o er any kind of test for the hy-

pothesis itself. In other words, this correlation analysis

In this section we shall discuss how the fractal con-

cannot disprove the homogeneous hypothesis.

cept can b e used to study the large scale distribution

Let us presentnow a straightforward discussion on of galaxies in the observed universe. We start with a

how this statistic can b e derived Raine 1981. If the brief summary of the standard metho ds used to study

average numb er density of galaxies isn = N=V then we this distribution. Later we will see the problems of this

1=3

from a given havetogoonaverage a distance n ortho dox analysis and the answers given to these prob-

galaxy b efore another is encountered. This means that lems when we assume that the large scale distribution

lo cal departures from uniformity can b e describ ed if of galaxies forms a self-similar fractal system. Some

we sp ecify the distance we actually go from any par- implications of the use of fractal ideas to describ e the

ticular galaxy b efore encountering another. This will distribution of galaxies, like a p ossible crossover to ho-

sometimes b e larger than average, but sometimes less. mogeneity, are also presented.

Marcelo B. Rib eiro and Alexandre Y. Miguelote 143

Sp ecifying this distance in each case is equivalent of giv- if the galaxies are distributed randomly on some scale

ing the lo cations of all galaxies. This is an awkward way less than that of the sample.

of doing things and do es not solve the problem. What

Supp ose now that the galaxies were not clustered.

we require is a statistical description giving the proba-

In that case the probability dP of nding galaxies

bility of nding the nearest neighb our galaxy within a

in volumes dV and dV is just the pro duct dP dP of

1 2 1 2

certain distance.

probabilities of nding each of the galaxies, since in a

However, as the probability of nding a galaxy

random distribution the p ositions of galaxies are uncor-

closer than, say, 50 kp c to the Milky Way is zero, and

related. Now, if the galaxies were correlated wewould

within a distance greater than this value is one, this is

have a departure from the random distribution and,

clearly not the sort of probability information we are

therefore, the joint probability will di er from a simple

after; what is necessary is some sort of average. Now

pro duct. The two-point correlation function ~r ;~r is

1 2

we can think that the actual universe is a particular

by de nition a function which determines the di erence

realization of some statistical distribution of galaxies,

from a random distribution. So we put

some statistical law, and the departure from random-

ness due to clustering will b e represented by the fact [1+ ~r ;~r ] dV dV 6 dP =

1 2 1 2 12

that the average value over the statistical ensemble of

1=3

as the expression of nding a pair of galaxies in vol-

this separation is less than n .

umes dV , dV at p ositions ~r , ~r . Obviously, the

1 2 1 2

In practice, however, we do not have a statistical

assumption of randomness on suciently large scales

ensemble from which the average value can b e derived,

means that ~r ;~r must tend to zero if j~r ~r j is

1 2 1 2

so what we can do is to take a spatial average over

suciently large. In addition, the assumption of homo-

the visible universe, or as much of it as has b een cat-

geneity means that cannot dep end on the lo cation of

alogued, in place of an ensemble average. This only

the galaxy pair, but only on the distance j~r ~r j that

1 2

makes sense if the departure from homogeneity o ccurs

separates them, as the probabilitymust b e indep en-

on a scale smaller than the depth of the sample, so that

dent of the lo cation of the rst galaxy.If is p ositive

the sample will statistically re ect the prop erties of the

wehave an excess probabilityover a random distribu-

universe as a whole. In other words, we need to achieve

tion and, therefore, clustering. If is negativewehave

a fair sample of the whole universe in order to ful ll

anti-clustering. Obviously >1. The two-p oint cor-

this program, and this fair sample ought to b e homo-

relation function can b e extended to de ne n-p oint cor-

geneous, by assumption. If, for some reason, this fair

relation functions, which are functions of n 1 relative

sample is not achieved, or is not achievable, that is,

distances, but in practice computations have not b een

if the universe has no meaningful average density, this

carried out b eyond the four-p oint correlation function.

whole program breaks down.

It is common practice to replace the description

For a completely random but homogeneous distribu-

ab ove using p oint particles by a continuum description.

tion of galaxies, the probability dP of nding a galaxy

So if galaxies are thought to b e the constituent parts

in an in nitesimal volume dV is prop ortional to dV

1 1

of a uid with variable density n~r , and if the aver-

and ton , and is indep endent of p osition. So wehave

aging overavolume V is carried out over scales large

dP = dV ;

1 1

compared to the scale of clustering, wehave

where N is the total numb er of galaxies in the sample.

n~r dV =n; 7

The meaning of the probability here is an average over

the galaxy sample; the space is divided into volumes

where dV is an elementofvolume at ~r . The joint prob-

dV and we count the ratio of those cells which con-

ability of nding a galaxy in dV at ~r + ~r and in dV

1 1 2

tain a galaxy to the total numb er. The probabilityof

at ~r + ~r is given by

nding two galaxies in a cell is of order dV , and so

can b e ignored in the limit dV ! 0. It is imp ortantto

n~r + ~r n~r + ~r dV dV :

1 2 1 2

state once more that this pro cedure only makes sense N

144 Brazilian Journal of Physics, vol. 28, no. 2, June, 1998

Averaging this equation over the sample gives I I I.2 Diculties of the Standard Analysis

The rst puzzling asp ect found using the metho d

n~r + ~r n~r + ~r dV dV dV : dP =

1 2 1 2 12

N V

just describ ed is the di erence in the amplitude A of

Nowifwe compare the equation ab ove with equation

the observed correlation function 10 when measured

6 we obtain

for galaxies and clusters of galaxies. While the exp o-

nent is approximately 1.7 in b oth cases, for galaxies

~ ~

n [1 + ~ ] = nRnR + ~ dV ; 8

A ' 20 and for clusters A ' 360. Less ac-

G C

curately, sup erclusters of galaxies were found to have

where ~ = ~r ~r , R = ~r + ~r and dV is the volume

2 1 1

A ' 1000 1500 see Pietronero 1987 and refer-

elementat R.

ences therein. The correlation length was found to b e

1 1

Related to the correlation function is the so-called

r ' 5h Mp c for galaxies and r ' 25 h Mp c

0 0

power spectrum of the distribution, de ned by the

for clusters.

Fourier transform of the correlation function. It is also

These are puzzling results, b ecause as A ' 18A ,

C G

p ossible to de ne an angular correlation function which

clusters app ear to b e much more correlated than galax-

will express the probability of nding a pair of galaxies

ies, although they are themselves made of galaxies.

separated by a certain angle, and this is the function

Similarly sup erclusters will then app ear to b e more cor-

appropriate to studying catalogues of galaxies which

related than clusters. From the interpretation of r

contain only information on the p ositions of galaxies on

describ ed ab ove, the galaxy distribution b ecomes ho-

the celestial sphere, that is, to studying the pro jected

mogeneous at the distance ' 10-15 h Mp c where r

galactic distribution when the galaxy distances are not

is found to b ecome zero, while clusters and sup erclus-

available. Further details ab out these two functions can

ters are actually observed at much larger distances, in

b e found, for instance, in Raine 1981, p. 10. Finally,

fact up to the present observational limits.

for the sake of easy comparison with other works it is

useful to write equation 8 in a slightly di erent nota-

tion:

hn~r n~r + ~r i

0 0

r = 1: 9

hni

The usual interpretation of the correlation function

obtained from the data is as follows. When 1 the

system is strongly correlated and for the region when

1 the system has small correlation. From direct

calculations from catalogues it was found that at small

values of r the function r can b e characterized bya

power law Pietronero 1987; Davis et al. 1988; Geller

1989:

r Ar ; 1:7; 10

where A is a constant. This p ower law b ehaviour holds

for galaxies and clusters of galaxies. The distance r

Figure 11. The b ehaviour of r plotted against the sample

size R found by Einasto, Klypin and Saar 1986. The

at which = 1 is called the correlation length, and

di erent symb ols op en circles, crosses and triangles mean

this implies that the system b ecomes essentially homo-

the di erent directions from where the samples were taken

in the sky. Calzetti et al. 1987. Further extensions of

geneous for lengths appreciably larger than this charac-

this b ehaviour, to around 100 Mp c, are shown in Pietronero

teristic length. This also implies that there should b e

1997, g. 2.

no appreciable overdensities sup erclusters or under-

densities voids extending over distances appreciably The second diculty of the standard analysis was

larger than r . rst found by Einasto, Klypin and Saar 1986 and later 0

Marcelo B. Rib eiro and Alexandre Y. Miguelote 145

con rmed byDavis et al. 1988 see also Calzetti et al. Improvements in astronomical detection techniques, in

1987; Coleman, Pietronero and Sanders 1988; Pietro-

particular the new sensors and automation, enabled as-

nero 1997. They found that the correlation length r

tronomers to obtain a large amount of galaxy redshift

increases with the sample size. Figure 11 shows this

measurements p er night, and made it p ossible by the

dep endence clearly, and it is evident that this result

mid 1980's to map the distribution of galaxies in three

brings into question the notion of a universal galaxy

dimensions. The picture that emerged from these sur-

correlation function.

veys was far from the exp ected homogeneity: clusters of

galaxies, voids and sup erclusters app eared in all scales,

with no clear homogenization of the distribution. The

rst \slice" of the universe shown by de Lapparent,

Geller and Huchra 1986 con rmed this inhomogeneity

with very clear pictures. More striking is the compari-

son of these observed slices with a randomly generated

distribution where the lack of homogenization of the ob-

served samples is clear see Figure 12. Deep er surveys

Saunders et al. 1991 show no sign of any homogeneity

b eing achieved so far, with the same self-similar struc-

tures still b eing identi ed in the samples.

Those problems together with the p ower lawbe-

haviour of r clearly call for an explanation, and while

manyhave b een prop osed they usually deal with each

of these issues separately. As we shall see, the frac-

tal hyp othesis, on the other hand, deals with all these

problems as a whole and o ers an explanation to eachof

them within the fractal picture. While we do not intend

to claim that the fractal hyp othesis is the only p ossi-

ble explanation to these problems, whether considering

them together or separately, from now on in this pap er

we shall take the p oint of view that fractals o er an

Figure 12. The picture on top shows the observed distri-

o o

bution of galaxies in the 18 wide slice centered at 35:5 .

attractively simple description of the large scale distri-

Voids and clusters are clearly visible as well as the lackof

bution of galaxies and that the mo del o ered by them

homogenization of the sample. The largest inhomogeneiti es

are comparable with the size of the sample and, therefore,

deserves a deep, serious and unprejudiced investigation,

it is not large enough to b e considered fair. The picture

b elow shows a sample of 2483 randomly distributed p oints

either in a Newtonian or relativistic framework. From

Geller 1989.

its basis, the fractal hyp othesis in manyways represents

a radical departure from the ortho dox traditional view

The third problem of the standard analysis has to

of an observational ly homogeneous universe, whichis

do with the homogeneity assumption itself and the p os-

challenged from its very foundations in many resp ects.

sibilityofachieving a fair sample, which should not b e

confused with a homogeneous sample as the standard

I I I.3 Correlation Analysis without Assumptions

analysis usually do es. A fair sample is one in which

there exists enough p oints from where we are able to

derive some unambiguous statistical prop erties. There- Before we discuss the fractal mo del itself, it is conve-

fore homogeneity must beregardedasa property of the nient to lo ok rst at the statistical to ols where a correla-

sample and not a condition of its statistical validity. tion appropriate to a fractal distribution can b e derived.

146 Brazilian Journal of Physics, vol. 28, no. 2, June, 1998

in this case the function r b ecomes spurious b ecause The metho d that is going to b e intro duced obviously

the normalization of correlated densityby the average do es not imply the existence of a fractal distribution,

density eq. 9 will dep end explicitly on R . Only in but if it exists, it is able to describ e it prop erly see

Pietronero 1987; Coleman and Pietronero 1992; Pietro- the limiting case where R will the length r

S 0 0

nero, Montuori and Sylos Labini 1997; Sylos Labini, indeed b e related to the correlation length . This

Montuori and Pietronero 1998. The appropriate ana- is in fact the case for liquids, where the two-p oint cor-

lysis of pair correlations should therefore b e p erformed relation function r was originally intro duced, as any

using metho ds that can check homogeneity or fractal reasonable sample size larger than some atomic scale

for, say,water, will contain so many molecules that its prop erties without assuming a priori either one. This

average densityisawell de ned physical prop erty. Be- is not the case of the function r , which is based on

cause water has a well de ned value for hni, the function a priori and untested assumption of homogeneity.For

r will b e the same for a glass of water and for a lake, this purp ose it is useful to start with a basic discussion

referring only to the physical prop erties of water and on the concept of correlation.

not to the size of the glass. However, by just lo oking at If the presence of an ob ject at r in uences the prob-

Figure 12 it is quite clear that this is not the case for ability of nding another ob ject at r these p oints are

the distribution of galaxies. As p ointed out by Geller said to b e correlated. So there is correlation at a dis-

1989, at least up to the present observational limits tance ~r from ~r if, on average,

the galaxy average density is not well de ned.

Gr = hn~r n~r + ~r i6= hni :

0 0

On the other hand there is no correlation if

Gr hni :

From this it is clear that non-trivial structures likevoids

or sup erclusters are made, by de nition, from corre-

lated p oints. Hence, the correlation length de nition

will only b e meaningful if it separates correlated regions

from uncorrelated ones. Figure 13 shows what would b e

the typical b ehaviour wewould exp ect from the galaxy

correlation if there is an upp er cuto to homogeneity.

Figure 13. This illustrati on represents schematically the

The p ower law decay will b e eventually followed bya

exp ected b ehaviour for the galaxy correlation densityofa

at region that corresp onds to the homogeneous dis-

correlated system with a crossover to homogeneity. The

tribution. In this case the correlation length is the

at b ehaviour of the function Gr beyond some correlation

length corresp onds to the unambiguous sign of homo-

distance at which there is a change in the correlation

geneity Coleman and Pietronero 1992.

Gr : it changes from a p ower law b ehaviour to a ho-

mogeneous regime and the average density b ecomes the

same, b eing indep endent of the p osition.

The function Gr as de ned ab ove has, however,

Actually in Figure 13 wehave a situation where the

the factor hni = N=V whichmay, in principle, b e an

sample size R is larger than .Ifwe had R < ,

S 0 S 0

explicit function of the sample's size scale. Therefore,

then the average density measured would corresp ond to

it is appropriate to de ne the conditional density r

an overdensity, di erent from the real average densityof

as,

the distribution. The precise value of this overdensity

hn~r n~r + ~r i

0 0

: 11 r

would then dep end explicitly on the sample radius, and

hni

See xI I I.5.2 for the pro of of this statement.

Marcelo B. Rib eiro and Alexandre Y. Miguelote 147

By the de nition of the average numb er densityof which gives the b ehaviour of the average densityofa

galaxies, wehave sphere of radius r centered around an o ccupied p oint

averaged over all o ccupied p oints , and the integrated

n~r n~r + ~r dV ; 12 hn~r n~r + ~ri =

0 0 0 0

conditional density

and together with hni = N=V , equation 11 b ecomes

0 0 0

I r =4 r dr ; 17 r

r = n~r n~r + ~rdV : 13

0 0

which is the numb er of galaxies of a spherical region of

radius r.

Assuming

Figure 14 shows the calculation of r , r and

n~r = ~r ~r ;

r for the CfA redshift survey, where the absence of

i=1

a homogenization of the distribution within the sample

and rememb ering that

and the absence of any kind of correlation length are

clear. This result brings strong supp ort to the hyp oth-

x y 'y dy = 'x; x= x;

esis that the large scale distribution of galaxies forms

equation 13 may b e rewritten as

indeed a fractal system.

Table 1 shows the correlation prop erties of the

r = n~r + ~r = hn~r + ~r i ; 14

galaxy distributions in terms of volume limited cata- i i

i=1

logues arising from most of the 50; 000 redshift mea-

where n~r + ~r is the conditional densityofthe ith

surements that have b een made to date. The samples

ob ject, and hn~r + ~r i the nal average that refers

are statistically rather go o d in relation to the fractal di-

to all o ccupied p oints r . This corresp onds to assigning

mension D and the conditional densityr , and their

an unit mass to all p oints o ccupied by galaxies, with

prop erties are in agreement with each other.

the ith galaxy at ~r and N b eing the total number of

While various authors consider these catalogs as not

galaxies. Equation 14 measures the average density

fair, b ecause the contradiction b etween each other, Pi-

at distance r from an o ccupied p oint, and the volume V

etronero, Montuori and Sylos Labini 1997 show that

in that equation is purely nominal and should b e such

this is due to the inappropriate metho ds of analysis.

to include all ob jects, but it do es not app ear explicitly

Figure 14 shows a densitypower law decay for many

in equation 14. r is more convenient than Gr

redshift surveys, and it is clear that wehavewell de-

for comparing di erent catalogues since the size of the

ned fractal correlations from 1 to 1000 h Mp c with

catalogue only app ears via the combination

fractal dimension D 2. This implies necessarily that

i=1

and this means that a larger sample volume will only

the value of r r = 1 will scale with sample size

0 0

increases the numb er of ob jects N . Hence a larger sam-

R , which gives the limit of the statistical validityof

ple size implies a b etter statistic.

the sample, as shown also from the sp eci c data ab out

Equations 9 and 14 are simply related by

r in table 1. The smaller value of CfA1 was due to

its limited size. At this same gure we can see the so-

r = 1: 15

called Hubble-de Vaucouleurs paradox, which is caused

hni

by the co existence b etween the Hubble law and the frac-

We can also de ne the conditional average density

tal distribution of luminous matter at the same scales

r dV ; 16 r = Pietronero, Montuori and Sylos Labini 1997.

See Coleman and Pietronero 1992 for a more detailed discussion on this sub ject with many examples of calculationsofr ,

r and r using test samples.

See xI I I.5.1 for an explanation of this e ect under a fractal p ersp ective.

See Rib eiro 1995 for a relativistic p ersp ective of this \paradox", where it is shown that this is just a result of a relativistic e ect.

148 Brazilian Journal of Physics, vol. 28, no. 2, June, 1998

Table 1. The volume limited catalogues are characterized by the following parameters: sr is the solid angle, R

1 1

h Mp c is the depth of the catalogue, R h Mp c is the radius of the largest sphere that can b e contained in

1 1

the catalogue volume, r h Mp c is the length at which r 1, D is the fractal dimension and h Mp c is

0 0 0

the eventual real crossover to homogeneity that this is actually never observed. The CfA2 and SSRS2 data are not yet

available Pietronero, Montuori and Sylos Labini 1997.

Sample R R r D

D S 0 0

CfA1 1.83 80 20 6 1:7 0:2 > 80

CfA2 1.23 130 30 10 2.0 ?

PP 0.9 130 30 10 2:0 0:1 > 130

SSRS2 1.13 150 50 15 2.0 ?

LEDA 4 300 150 45 2:1 0:2 > 150

LCRS 0.12 500 18 6 1:8 0:2 > 500

IRAS 4 80 40 4.5 2:0 0:1 15

ESP 0.006 700 10 5 1:9 0:2 > 800

Figure 15. Conditional average density of galaxies plotted

as function of distance decreasing from left to right for

Figure 14. r , r and r plotted as function of length

the following redshifts surveys: CfA1 crosses, Perseus-

scale for the CfA redshift survey. There is no indication of a

Pisces circles and LEDA squares. The solid line cor-

homogenization of the sample and b oth r and r obeya

resp onds to D = 2. The Hubble redshift-distance is also

power law, a result consistent with a fractal structure for the

shown in this graph. The dotted line corresp onds to the

distribution of galaxies. The dashed line indicates a refer-

Hubble law increasing from left to right with H =55

ence slop e of 1:7 Coleman, Pietronero and Sanders 1988.

1 1

km s Mp c . This law is constructed from: galaxies

with Cepheid-distances for cz > 0 triangles, galaxies with

Tully-Fisher B-magnitudes distances stars, galaxies with

It is imp ortant to mention that there are e ects

SNIa-distances for cz > 3000 km/s lled circles Pietro-

whichmay conceal the true statistical b ehaviour of the

nero, Montuori and Sylos Labini 1997.

samples. Those e ects may lead to the conclusion that

Figure 16 shows the b ehaviour of the density com- the sample under study is homogeneous, although such

puted from the vertex of a conic sample. Atvery small a conclusion would b e wrong, since such homogeneity

distances we are not going to nd any galaxy b ecause may app ear not as a statistical prop erty of the sample,

the total numb er is rather small. After the Voronoi's but just as an e ect of its nite size.

Marcelo B. Rib eiro and Alexandre Y. Miguelote 149

length V , which is of the order of the mean particle

separation, we b egin to have signal but this is strongly

a ected by nite size e ects. The correct scaling b e-

haviour is reached for the region r . In the in-

termediate region wehave an apparent homogeneous

distribution, but it is due to the nite size e ects Pie-

tronero, Montuori and Sylos Labini 1997; Sylos Labini,

Gabrielli, Montuori and Pietronero 1996; Sylos Labini,

Montuori and Pietronero 1998.

As evidence that this new statistical approach is the

appropriate metho d of analysis, we can see in Figure 17

an agreementbetween various available redshift cata-

3 1

logues in the range of distances 0:1 1 10 h Mp c.

From this we can conclude that there is no tendency

to homogeneity at this scale. In contrast to Figure 17

Figure 17. Full correlation for the various redshift cata-

wehave Figure 18 where the traditional analysis based

3 1

logues in the range of distances 0:1 1 10 h Mp c. A

on r for the same catalogues of Figure 17 shows a

reference line with a slop e 1 is also shown D 2. In the

strong con ict b etween these two analytical metho ds.

insert it is shown a conic volume. The radial density is com-

Figure 18 shows that r is unable to say without any

puted by counting all the galaxies up to a certain limit, and

by dividing for the volume of this conic sample Pietronero,

doubt if there is a homogeneous scale, and this is the

Montuori and Sylos Labini 1997.

main reason from where the concept of \unfair sample"

is generated.

Figure 18. Graph of the function r versus the distance

Figure 16. Schematic b ehaviour of the density computed

r Mp c for the same galaxy catalogues of Figure 17. Here

from the vertex. Inside the Voronoi's length V small dis-

we can see rather confusing results generated by the a pri-

tances, one nds almost no galaxies. After this length the

ori and untested assumption of homogeneity, which are not

numb er of galaxies starts to grow with a regime strongly

present in the real galaxy distribution Pietronero, Montuori

a ected by nite size uctuations, and the density can b e

and Sylos Labini 1997.

approximately roughly by a constantvalue, leading to an

apparent exp onent D 3. Finally the scaling region r

Another statistical analysis is the numb er counts

is reached Pietronero, Montuori and Sylos Labini 1997; Sy-

and angular correlations. In Figure 19 we show the los Labini, Gabrielli , Montuori and Pietronero 1996.

150 Brazilian Journal of Physics, vol. 28, no. 2, June, 1998

b ehaviour of the numb er counts versus magnitude re-

lation N

scales =0:6 0:1D 3, which means that wehave

an apparent homogeneity.However, this is due to the

nite size e ects discussed ab ove, while at larger scales

the value 0:4D 2 shows correlation prop erties of

the sample in agreement with the results obtained for

r . In addition, the fact that the exp onent0:4 holds

up to magnitudes 27 28 seems to indicate that the

3 1

fractal structure may continue up to 2 3 10 h

Mp c Pietronero, Montuori and Sylos Labini 1997.

I I I.4 Pietronero-Wertz's Single Fractal Hierar-

chical Mo del

The single fractal mo del prop osed by Pietronero

1987 is essentially an application of the cluster frac-

tal dimension to the large scale distribution of galaxies,

a straightforward analysis of the consequences of this

fractal interpretation of the galactic system plus the

Figure 19. The galaxy numb er counts in the B-band from

prop osal of new statistical to ols to analyse the cata-

several surveys. In the range 12 m 19 the counts show

logues of galaxies, derived from his strong criticisms of

an exp onent =0:6 0:1, while in the range 19 m 28

the exp onentis 0:4. The amplitude of galaxy number

the standard statistical analysis based on the two-p oint

counts for m 19 solid line is computed from the deter-

correlation function. He also studied the problem from

mination of the prefactor B of the density nr =Br

amultifractal p ersp ective. What we shall attempt to

at small scale and from the knowledge of the galaxy lumi-

do here is to present a summary of the basic results re-

nosity function. The distance is computed for a galaxy with

1 1

M = 16 and the value used for H is 75 km s Mp c

lated to the discussion ab ove. Later in a review pap er,

Pietronero, Montuori and Sylos Labini 1997.

Coleman and Pietronero 1992 extended the theory,

with esp ecial emphasis on multifractals and the angu-

lar correlation function, and added new results. Further

extensions and results of this theory were also made in

I I I.4.1 Pietronero's Single Fractal Mo del

Sylos Labini, Montuori and Pietronero 1998.

Wertz's 1970, 1971 hierarchical mo del, on the The de nition of the fractal dimension as made by

other hand, was prop osed at a time when fractal ideas Pietronero 1987 is slightly di erent than in the case

had not yet app eared. However, these ideas were more of the cluster dimension discussed previously. Figure

or less implicit in his work, and as we shall see b elow, 20 showsaschematic representation of a fractal distri-

he ended up prop osing a hierarchical mo del mathemat- bution of p oints. Clearly wehave a self-similar distri-

ically identical to Pietronero's single fractal mo del, but bution as more and more structure app ears at smaller

and smaller scales, with the structure at small scales 17 years earlier. For this reason his work deserves to

b eing similar to the one at large scales. By starting b e quoted as b eing the rst indep endent mo del where

from a p oint o ccupied by an ob ject and counting how self-similar ideas were applied in the study of the large

many ob jects are present within a volume characterized scale distribution of galaxies. The reasons whyWertz's

by a certain length scale, wehave that within a certain work was forgotten for so long lies on the shortcomings

radius d , there are N ob jects; then within d = kd of its physical implications, as it will b e shown b elow.

0 0 1 0

Marcelo B. Rib eiro and Alexandre Y. Miguelote 151

there are N = kN ob jects; in general, within which allows us, together with equation 20, to com-

1 0

pute the average density hni as b eing

d = k d 18

n 0

3 N R

wehave

= R ; =3 D: 23 hni =

V R 4

N = k N 19

n 0

This is the same typeofpower law expression obtained

ob jects. Generalizing this idea to a smo oth relation, we

several years ago bydeVaucouleurs 1970, and equa-

can de ne a generalized mass-length relation between N

tion 23 shows very clearly that the average density

and d of the typ e

is not a well de ned physical prop erty for this sort of

N d= d ; 20

fractal system b ecause it is a function of the sample

size.

where the fractal dimension

log k

D = 21

I I I.4.2 Wertz's Hierarchical Mo del

log k

dep ends only on the rescaling factors k and k , and the

prefactor is related to the lower cuto s N and d , The hierarchical mo del advanced byWertz 1970,

0 0

1971 was conceived at a time when fractal ideas had

= : 22

not yet app eared. So, in developing his mo del, Wertz

was forced to start with a more conceptual discussion

Equation 20 corresp onds to a continuum limit for the

in order to o er \a clari cation of what is meantby

discrete scaling relations.

the `unde ned notions' which are the basis of any the-

ory" Wertz 1970, p. 3. Then he stated that \a clus-

ter consists of an aggregate or gathering of elements

into a more or less well-de ned group which can b e to

some extent distinguished from its surroundings by its

greater density of elements. A hierarchical struc-

ture exists when ith order clusters are themselves

elements of an i+1th order cluster.Thus, galax-

ies zeroth order clusters are group ed into first

order cluster. First order clusters are themselves

group ed together to form second order clusters,

etc, ad in nitum" see Figure 21.

Although this sort of discussion maybevery well to

start with, it demands a precise de nition of what one

means by a cluster in order to put those ideas on a more

solid fo oting, otherwise the hierarchical structure one is

talking ab out continues to b e a somewhat vague notion.

Figure 20. Schematic illustratio n of a deterministic fractal

Wertz seemed to have realized this diculty when later

system from where a fractal dimension can b e derived. The

structure is self-similar, rep eating itself at di erent scales

he added that \to say what p ercentage of galaxies o ccur

Pietronero 1987.

in clusters is b eyond the abilities of current observations

and involves the rather arbitrary judgment of what sort

of grouping is to b e called a cluster. ... It should b e

Let us now supp ose that a sample of radius R con-

p ointed out that there is not a clear delineation b etween

tains a p ortion of the fractal structure. If we assume it

clusters and sup erclusters" p. 8.

to b e a sphere, then the sample volume is given by

Despite this initially descriptive and somewhat

R ; V R =

S S

vague discussion ab out hierarchical structure, whichis 3

152 Brazilian Journal of Physics, vol. 28, no. 2, June, 1998

basically a discussion ab out scaling in the fractal sense, and the global density was de ned as b eing

Wertz did develop some more precise notions when he

lim x; r : 25

g v

b egan to discuss sp eci c mo dels for hierarchy, and his

r !1

starting p ointwas to assume what he called the \univer-

A pure hierarchy is de ned as a mo del universe which

sal density-radius relation", that is, the de Vaucouleurs

meets the following p ostulates:

densitypower law, as a fundamental empirical fact to

i for any p ositivevalue of r in a b ounded region, the

b e taken into account in order to develop a hierarchical

volume density has a maximum;

cosmology. Then if M x; r is the total mass within a

ii the mo del is comp osed of only mass p oints with

sphere of radius r centered on the p oint x, he de ned

nite non-zero mean mass;

the volume density as b eing the average over a sphere

iii the zero global density p ostulate: \for a pure hier-

of a given volume containing M .Thus

archy the global density exists and is zero everywhere"

3M x; r

x; r ; 24 see Wertz 1970, p. 18.

Figure 21. Repro duction from Wertz 1970, p. 25 of a rough sketch cross-section of a p ortion of an N = i + 2 cluster of a

p olka dot mo del.

on di erent scales. If no such scales exist, one would With this picture in mind, Wertz states that \in

haveanindefinite hierarchy in which clusters of any mo del whichinvolves clustering, there mayormay

every size were equally represented .... At the other not app ear discrete lengths which represent clustering

Marcelo B. Rib eiro and Alexandre Y. Miguelote 153

extreme is the discrete hierarchy in which cluster equation 28 the radius of the elementary p oint mass

sizes form a discrete sp ectrum and the elements of one r , is given by

size cluster are all clusters of the next lowest size" p. ; 32 r =

23. Then in order to describ e polka dot models, that

where R is the radius of a Nth order cluster with M

N N

is, structures in a discrete hierarchy where the elements

mass, V volume, and obviously that R = r . It fol-

N 0 0

of a cluster are all of the same mass and are distributed

lows from equation 32 the relationship b etween N and

regularly in the sense of crystal lattice p oints, it b e-

r ,

comes necessary for one b e able to assign some average

r = a r ; 33

prop erties. So if N is the order of a cluster, N = i is a

where R = r .

cluster of arbitrary order gure , and at least in terms

Notice that by doing this continuous representation

of averages a cluster of mass M , diameter D and com-

i i

Wertz ended up obtaining an equation eq. 33 which

p osed of n elements, each of mass m and diameter d ,

i i i

is nothing more than exactly equation 18 of Pietro-

has a density given by

nero's single fractal mo del, although Wertz had reached

it by means of a more convoluted reasoning. Actu-

= : 26

ally, the critical hyp othesis which makes his p olka dot

mo del essentially the same as Pietronero's fractal mo del From the de nitions of discrete hierarchyitisobvious

was the assumption of regularity of the mo del b ecause, that

in this case a and n b ecome constants. Also notice

M = n m = m ; 27

i1 i1 i1 i

that this continuous representation amounts to chang-

and if the ratio of radii of clusters is

ing from discrete to an inde nite hierarchy, where in

D D

i i

the latter the characteristic length scales for clustering

a = ; 28

d D

i i1

are absent. Therefore, in this representation clusters

and voids extend to all ranges where the hierarchy

then the dilution factor is de ned as

is de ned, with their sizes extending to all scales b e-

i1 i

= > 1; 29

tween the inner and p ossible outer limits of the hierar- i

i i

chy. Hence, in this sense the continuous representation

and the thinning rate is given by

of the regular p olka dot mo del has exactly the same

sort of prop erties as the fractal mo del discussed by Pi- =n loga log =

i i i1 i

= : 30

logD =D log a

i i1 i

etronero.

From equation 27 we clearly get

A regular polka dot model is de ned as the one whose

numb er of elements p er cluster n and the ratio of the

M = n M ; 34

N 0

radii of successive clusters a are b oth constants and

which is equal to equation 19, except for the di erent

indep endentofi, that is, n and a resp ectively. Conse-

notation, and hence the de Vaucouleurs densitypower

quently, the dilution factor and the thinning rate are

law is easily obtained as

b oth constants in those mo dels,

M 3M

N 0

= = r ; 35

3 3

log n= log a

a loga =n

V 4r

N 0

= ; = : 31

n log a

where is the thinning rate

The continuous representation of the regular p olka

log n

=3 : 36

log a

dot mo del, which amounts essentially to writing the

hierarchical mo del as a continuous distribution, is ob- Notice that equations 35 and 36 are exactly equa-

tained if we consider r , the radius of spheres centered tions 23, where is now called the thinning rate. Fi-

on the origin, as a continuous variable. Then, from nally, the di erential density, called conditional density

154 Brazilian Journal of Physics, vol. 28, no. 2, June, 1998

by Pietronero, is de ned as must not b e viewed as a critique of Wertz's work, but

simply as a realization of the fact that at Wertz's time

dM r

= 1 : 37

d v

nonlinear dynamics and fractal geometry were not as

4r dr 3

develop ed as at Pietronero's time, if develop ed at all,

From the presentation ab ove it is then clear that

and therefore Wertz could not have b ene tted from

from a geometrical viewp ointWertz's continuous rep-

those ideas. Despite this it is interesting to note that

resentation of the regular p olka dot mo del is nothing

even with less data and mathematical concepts he was

more than Pietronero's single fractal mo del. However,

nevertheless able to go pretty far in discussing scaling

the two approaches may b e distinguished from each

b ehaviour in the distribution of galaxies, developing a

other by some imp ortant conceptual di erences. Ba-

mo del to describ e it in the context of Newtonian cos-

sically, as Pietronero clearly de nes the exp onentof

mology, and even suggesting some p ossible ways of in-

equation 20 as a fractal dimension, that immediately

vestigating relativistic hierarchical cosmology.

links his mo del to the theory of critical phenomena

in physics, and also to nonlinear dynamical systems,

I I I.5 Consequences of the Single Fractal Mo del

bringing a completely new p ersp ective to the study of

As stated ab ove, the fractal mo del shown in the pre- the distribution of galaxies, with p otentially new math-

vious section o ers an attractively simple explanation ematical concepts and analytical to ols to investigate

for the results obtained when analysing the distribution this problem. In addition, he strongly emphasized the

of galaxies by means of the new statistical metho ds ad- fundamental imp ortance of scaling b ehaviour in the ob-

vanced by Pietronero and collab orators. Thus, when served distribution of galaxies and the fundamental role

this statistic is applied to a fractal system some imp or- of the exp onent of the p ower law, as well as p ointing

tant results that can b e related to the observed distribu- out the appropriate mathematical to ol to describ e this

tion of galaxies are obtained. Here in this subsection we distribution, namely the fractal dimension. Finally,as

shall discuss some straightforward results arising from many fractals have a statistical nature, either in their

the fractal approach to the galaxy clustering problem. description or in their construction, or b oth, and also

But, b efore we start this discussion some imp ortant re- are inhomogeneously distributed, the statistical meth-

marks must b e made. o ds capable of dealing with fractals must also b e able

Firstly, although the fractal distribution never b e- to derivewell-de ned statistical prop erties even from

comes homogeneous, it is a statistical ly fair sample in inhomogeneous samples, fractals or not, where the av-

the sense mentioned ab ove, which is contrary to the erage densitymay not b e well-de ned. Therefore, the

traditional lines where only a homogeneous sample is fractal p ersp ective brings together a completely new set

taken to b e a fair one. of statistical to ols capable of a comprehensive reinter-

Secondly, a three dimensional galaxy distribution pretation of the conclusions drawn up on the available

gures 22a,b, which has fractal prop erties when stud- data ab out the distribution of galaxies, and without

ied in 3D, app ears relatively homogeneous at some large any need of a priori assumptions ab out homogeneity.

angular scale gure 22c, lo osing some of its irregular All that is missing in Wertz's approach, and his

characteristics when pro jected on an angular distribu- thinning rate is just another parameter in his descrip-

tion. Due to this prop erty of fractal structures, it is tion of hierarchy, without any sp ecial physical meaning

necessary great care when dealing with pro jected struc- attached to it. Therefore, in this sense his contribu-

tures, as their fractal features may b ecome hidden when tion started and remained as an isolated work, ignored

dealing with 2D data Coleman and Pietronero 1992; by most, and which could even b e viewed simply as

Pietronero, Montuori and Sylos Labini 1997. an ingenious way of mo delling Charlier's hierarchy, but

Thirdly, the galaxy distribution has b een studied nothing more.

also in terms of their mass Pietronero 1987; Coleman Nonetheless, it should b e said that this discussion

Marcelo B. Rib eiro and Alexandre Y. Miguelote 155

and Pietronero 1992 and their luminosity distribution geneity of the fractal system, and although we cannot

Pietronero, Montuori and Sylos Labini 1997, which yet rule out the p ossibility of an upp er cuto to homo-

is a full distribution and not a simple set of p oints. geneity, if the fractal system were unlimited the average

The study of these distributions requires a generaliza- density will tend to zero at increasing distances. This

tion of the fractal dimension and the use of the con- p ossibility has provoked strong reactions from some au-

cept of multifractals. In this case, wehave di erent thors as they assumed it to go against established ideas

scaling prop erties for di erent regions of the system, in cosmology and, therefore, could not accept it. Nev-

while in the fractal case only one exp onentcharacterizes ertheless, it is imp ortant to p oint out that such a tra-

the entire system. A multifractal analysis also shows a ditional view is not as sound as it seems at rst, inas-

new imp ortant relation b etween the luminosity func- much as Rib eiro 1992, 1993, 1994 showed that even

tion Schechter luminosity distribution and the space Friedmannian cosmologies do allowWertz's zero global

correlation prop erties. density p ostulate under a sp eci c relativistic interpre-

Finally, equation 23 shows the intrinsic inhomo- tation.

Figure 22. A slice of the three dimensional galaxy distributio n a and b compared with the corresp onding c angular

distribution . Note that the angular distribution app ears relatively homogeneous while the real three dimensional distribution

in space is much more irregular. This picture shows the Great Wall which extends over the entire sample at least 170 h

Mp c Pietronero, Montuori and Sylos Labini 1997.

156 Brazilian Journal of Physics, vol. 28, no. 2, June, 1998

I I I.5.1 Power Law Behaviour and Fractal Di- function r are related to the sample radius R and

mension not to the intrinsic prop erties of the system.

Equation 43 allows us to obtain a new interpre-

Pietronero 1987 de ned the conditional density

tation of the di erence in the amplitude of the corre-

from an o ccupied p oint of the fractal system as b eing

lation 42. Considering now a sample radius R for

given by

the galaxy catalogue, the amplitude of the two-p oint

correlation function is clearly given by

D 1 dN r

= r ; 38 r =

S r dr 4

A = R :

G G

where S r is the area of the spherical shell of radius r

this is the same as Wertz's di erential density. Now

In the case of clusters we then have

by means of equations 15 and 23, it is straightfor-

ward to see that

R A = :

C C

3 r

r = 1: 39

The hyp othesis of self-similarity means that it is

3 R

the exp onent of the p ower law which matters as giv-

The two equations ab ove show clearly the dep endence

ing the prop erty of the structure. The amplitude is

of r on the sample size R while this is not the case

just a rescaling factor related to the size of the sample

for r . It is clear therefore, that in a fractal system

and the lower cuto , without a direct physical mean-

the function r mixes up the physical prop erties of

ing. As an example, under a rescaling of the length

the distribution with the sample size, an e ect actually

by a factor b such that r ! r = b r , a self-

observed in analysis of catalogues, as seen ab ove.

similar function will b e rescaled as the functional rela-

Since the correlation length r is de ned as the p oint

tion f r = f b r Ab f r . This is clearly

at which r = 1, wehave

satis ed bypower laws with any exp onent. In fact, for

D D

D 0

f r =f r wehave f r =f br =b f r .

0 0

R ; 40 r =

S 0

Therefore, under the assumption of a single self-similar

structure, the amplitudes A and A should b e related

G C

which is dep endentonR . Therefore, the fractal sce-

nario explains the dep endence of the correlation length

R A

C C

= : 44

r with the sample size. The p oint at which r =0

A R 0

G G

Since for the galaxy and cluster catalogues wehave

R : 41 r =

R 5R , equation 44 predicts with =1:8

S 0

C G

the function r is negative and for For r > r

18;

r R it can b e approximated bya power law,

which is the value found for the discrepancy in the am-

r AR r ; r r ; 42

S 0

plitudes. Note that this is a simple evaluation of the

mismatchby a single deterministic fractal whichdoes

where the two-p oint correlation function amplitude is

not takeinto accountany sto chastic pro cess, which

given by

would b e a more realistic situation in the case of the

AR = R : 43

S S

distribution of galaxies. This discussion shows therefore

that correlations of clusters app ear to b e a continuation

Considering equation 10 obtained empirically,we

of galaxy correlations in larger scales, and the discrep-

then have a fractal dimension D 1:3 for the distri-

ancy in amplitudes simply means di erent observations

bution of galaxies in this mo del for small r .For larger

of the same system at di erent depth samples.

r the fractal dimension is D 2.

From these simple equations it is clear that with the So we see that the fractal hyp othesis explains many

exception of the exp onent , all relevant features of the of the puzzling problems so far encountered in the study

Marcelo B. Rib eiro and Alexandre Y. Miguelote 157

of the large scale distribution of galaxies. It is imp or- and, considering equation 45 we obtain

tanttosay that this is accomplished with a mo del of re-

D r

r = 1; r< ; 51

markable simplicity, showing once more the strength of

4 hni

the fractal concept of dealing with this sort of complex

problems. Therefore, from a theoretical p oint of view,

r = 1; r ; 52

4 hni

due to the simplicity of the mo del and the strength

where the dep endence of hni on R implies that the

of the concept, it b ecomes seductive to apply a fractal

p oint at which r = 0 is a function of and R .

0 S

approach to mo dern cosmology.

Then, the function r , which is still inappropriate,

b ecomes an appropriate to ol only in the limiting case

I I I.5.2 Possible Crossover to Homogeneity

where R , in the sense that hni b ecomes inde-

0 S

Until nowwehave seen that this new statistical ap-

p endentofR . Then, in this case wehave

proach shows no evidence for a homogeneous distribu-

; 53 hni =

tion of the visible matter on large scale, but we cannot

yet exclude this p ossibility,asitmay o ccur at some

scale not yet observed.

1; r< ; 54 r =

As wesawinxI I I.3 the conditional densityr =

r =0; r : 55

Gr =hni,obeysapower law for r< , and for r

0 0

it is constant.

Only in this case the length r is related to the corre-

lation length ,

By equation 38, wehave

1=D 3

r =2 : 56

0 0

r = r ; r< ; 45

For the case D 2 one has r = =2.

0 0

r =n ; r : 46

0 0

IV. Conclusions

Then at ,wehave

In this pap er wehave studied the distribution of

n = : 47

0 0

galaxies by means of a Newtonian fractal p ersp ective.

Therefore the function r has a p ower law b ehaviour

We b egan with a brief intro duction to fractals, and in x

up to and it is constant thereafter.

we discussed a more appropriated statistical analysis

The integrated conditional density 17 may b e writ-

for the large scale distribution of galaxies. We also

ten as

discussed the fractal hyp othesis itself which contrasts

I r =r ; r< ; 48

to the ortho dox traditional view of an observational ly

homogeneous universe. We nished this section show-

1 1 1

D 3D

; r ; I r =D D +

0 0 0

ing the Pietronero-Wertz's single fractal hierarchical

D 3 3

mo del for describing and analysing the large scale dis-

which is the total numb er of galaxies. The change in

tribution of galaxies and some of its consequences. In

the p ower law from D to 3 indicates the crossover to

summary the main results of xI I I are:

length .

1. the employment of the two-p oint correlation func-

If wewant to consider the function r for a system

tion r for a statistical analysis of the galaxy

with a crossover to homogeneitywehave to sp ecify our

distribution is problematic due to the a priori as-

sample radius R R > explicitly. Considering

S S 0

sumption of homogeneity. This assumption is hid-

that the density hni is the total numb er of galaxies p er

den in the average density which do es not seem to

volume unity of the sample, wehave

beawell de ned physical prop erty for the galaxy

3 D

distribution b ecause it is a function of the sam-

1+ 1 50 hni =

4 D R

ple size, a result consistent with the fractal view of

158 Brazilian Journal of Physics, vol. 28, no. 2, June, 1998

much bigger than the new correlation scale, that such a distribution. In order to solve this problem

is, when R , so that the ortho dox correla- Pietronero 1987 intro duced a new statistical ap-

0 S

tion length r is related only to the new correla- proach for the galaxy distribution, which can test

tion length , and not to the sample size R . the assumption of homogeneity;

0 S

2. the comparison b etween two di erent metho ds

of analysis shows that an inappropriate metho d

Some authors, although recognizing the problems of

leads to wrong conclusions in volume limited cata-

the standard analysis, have argued against a fractal dis-

logues, due to its nite size and shallowness. Due

tribution from a somewhat unconvincing p ersp ective.

to this they may wrongly b e considered as not

For example, Davis et al. 1988 have claimed that

fair, as shown in table . So, an apparent homo-

although their analysis con rm the dep endence of the

geneous distribution in the region V r< of

correlation length with the sample size, this e ect can-

gure , where V is the Voronoi's length and is

not b e explained by Pietronero's single fractal mo del

the inferior limit for reaching the correct scaling

b ecause they found r to b e approximately prop ortional

b ehaviour, is due to the nite size e ects, and the

to the square ro ot of the sample size, while it is linear

correct scaling is reached for the region r ;

in Pietronero's picture. Davis et al. 1988 neverthe-

3. homogeneitymust b e regarded as a prop ertyof

less have o ered only statistical explanations where it

the sample and not a condition of its statistical

is not always clear what are the hidden hyp otheses as-

validity;

sumed by them. This p oint is of esp ecial concern due

to the current widespread practice of \correcting" the

4. the fractal mo del o ers an attractively simple de-

samples in order to \improve" them for \b etter" statis-

scription of the galaxy distribution in the ob-

tics. The problem with this sort of practice is that the

served universe, and the results pro duced by the

homogeneous hyp othesis is often implicit, and this fact

new statistical approach in the form of the condi-

considerably weakens that kind of statistical explana-

tional densityr and its derived functions are

tions.

consistent with this fractal picture. Therefore the

galaxy catalogues available so far may b e con-

Another imp ortant p oint that ought to b e said again

sidered as statistically fair samples of this distri-

ab out the fractal system as de ned ab ove is its total

bution, which is contrary to the traditional lines

incompatibility with the two-p oint correlation function

where only a homogeneous sample is considered

r . The problem lies in the fact that this sort of frac-

to b e fair;

tal system do es not havea well de ned average density,

at least in b etween upp er and lower cuto s. Runi,

5. from this new statistical approachwe can see an

Song and Taraglio 1988 have, nonetheless, p ointed

agreementbetween various available redshift cat-

3 1

out that there are fractal systems in which the average

alogues in the range of 0:1 10 h Mp c, as

density do es tend to a nite and non-zero value, which

shown in gure , without any tendency to homo-

led them to supp ose that this should b e the case in cos-

geneity at this scale. These redshift catalogues

mology. Although this is a reasonable p oint to raise,

ob ey a densitypower law decay and have fractal

the problem with this argument, as we see it, is again

dimension D 2;

the a priori assumption that this must b e the case in

6. a three dimensional galaxy distribution, which

cosmology, i.e., that in cosmology the average density

has fractal prop erties when pro jected on an angu-

ought to tend to a nite non-zero limit for very large

lar distribution, app ears relatively homogeneous

distances. Obviously, b ehind this idea lies the homo-

at some large scale;

geneous hyp othesis which Runi, Song and Taraglio

7. the function r b ecomes an appropriate to ol 1988 try to incorp orate into the fractal cosmology,

only in the limiting case where the sample size is but, once again, by means of an untested argument.

See Coleman and Pietronero 1992 for criticisms of this sort of practice and some examples of these \corrections" where the

homogeneoushyp othesis is actually implicit in those pro cedures.

Marcelo B. Rib eiro and Alexandre Y. Miguelote 159

From a relativistic p ersp ective, at rst it would seem Acknowledgements

reasonable to think that this should b e the case b ecause

We are grateful to F. Sylos Labini for reading the

the standard Friedmannian cosmology is spatially ho-

original manuscript and for helpful comments. The -

mogeneous, and as this is the most p opular relativistic

nancial supp ort from FAPERJ and CNPq is, resp ec-

cosmological mo del, wewould have to incorp orate some

tively,acknowledged by MBR and AYM.

sort of homogenization or upp er cuto in the mo del,

so oner or later. However, as shown by Rib eiro 1992,

1993, 1994 this viewp ointmay b e a rather misleading

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