THE CONCEPT OF FRACTAL COSMOS: II. MODERN COSMOLOGY · THE CONCEPT OF FRACTAL COSMOS: II. MODERN...

Serb. Astron. J. � 165 (2002), 45 – 65 UDC 524.83Review paper

THE CONCEPT OF FRACTAL COSMOS: II. MODERN COSMOLOGY

P. V. Grujic

Institute of Physics, P.O. Box 57, 11080 Belgrade, Yugoslavia

(Received: April 24, 2002)

SUMMARY: Development of the concept of fractal cosmos after Anaxagoras hasbeen followed up to the present. It is shown how the concept reappeared in theearly Renaissance as a vague idea and subsequently took up a concrete formulationat the beginning of the 20-eth century. The modern cosmology state of affairshas been considered in view of the fractal paradigm and the current disputes andcontroversies discussed. It is argued that the concept of the hierarchical cosmos isstill alive and might become an essential ingredient within the modern view of theuniverse.

1. INTRODUCTION

In the previous article (Grujic 2001, to be re-ferred to as I) we examined Anaxagoras’ worldviewand compared his hierarchical cosmos with that ofDemocritus. We analyzed a number of possible in-terpretations of his cosmology and argued that hisideas could be put in terms of the modern conceptof fractal structuring of the material objects.

The idea of hierarchical structuring of the ma-terial world is an elaborate concept of a simple gen-eral principle that has underlined almost all cosmolo-gies in the ancient world, not only European one, theassertion that microcosmos is equivalent to macro-cosmos. This postulate, on its part, stems from theprinciple of economy, that has been best formulatedby William Occam (Occam’s razor). In practical so-cial sphere all ages have witnessed various realiza-tions of this ”equivalence principle”, which has beenbest epitomized by temples conceived as miniaturesof the entire cosmos, as the most conspicuous case ofthe gothic cathedrals shows.

In the next chapter we shall see how a numberof European thinkers took up the idea in the mostprimitive form. Then, in the following chapters we

consider the fractal concept within the last centurycosmology. Next, we present some modern conceptsof the hierarchical universe, and discuss some aspectsof the observational evidence and its possible inter-pretations, as appears the subject of the current con-troversies. Finally, we discuss the fractal paradigmfrom the epistemological point of view and outlinefuture possible developments of the subject.

2. RENAISSANCE AND POST-RENA-ISSANCE EUROPE

Anaxagoras had no direct following, either inAntiquity, or in Medieval Europe. But a line ofhis thought continued via Aristotelian and Platonictradition as a concept of hierarchical world of real-ity (Copleston 1976). The first prominent represen-tative of this concept was Nicolaus of Cusanus (b.1401), who held that in each particular object thewhole universe is reflected. The universe exists inevery finite thing, as contracte, and cosmos consistsof a multitude of single entities, each related to theother and to the whole in such way that one canspeak of ”unity in a multitude” (Copleston 1976).

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In particular, the human is microcosmos, and com-prises in itself material and intellectual reality, andhence appears as a small universe, or world. Gior-dano Bruno (1548-1600) conceived the universe asan infinite reality, with a multitude of (equivalent)worlds, in both Democritus and democratic manners.But at the same time he adopted Cusanus’ idea ofmaterial accidental minimum constituents, which hedubbed monas, as counterparts of mathematical uni-ties. This idea was further developed by FranciscusMercurius van Helmont (1618-1699) in Belgium, whoconceived the material world as a collection of mon-adas, which may form complex structures.

All these doctrines were mixed with religiousand sometimes mystic ingredients and could be con-sidered a part of a speculative philosophy. The crownof this monadic approach was the doctrine of God-fried W. Leibniz (1646-1716). Though he met vanHelmont, he presumably developed his doctrine ofmonadas independently. Leibniz’s concept of monadappears neither simple nor clear. As Hartmann no-ticed (Hartmann 1946) the construct is full of appar-ent contradictions, many of which arise due to an in-herent mind versus matter dichotomy. It seems thatLeibniz incorporated both Abderian atomistic pointof view and Anaxagoras’ world picture, as can beseen from the following passages from Monadologie(Leibniz, 1914; we retain the original transcription).

3. ...Et ces monades sont les veritableatomes de la nature et en un mot leselements des choses....

65. ..., parce que chaque portion de lamatiere n’est pas seulement divisible al’infini, comme les anciens ont reconnu,mais encore sous-divisee actuellementsans fin chaque partie, dont chacune aquelque meuvement propre: autrementil serait impossible, que chaque portiondela matiere put exprimer l’univers ...

This dichotomy is resolved, however, if one ob-serves that the atomic aspect stems from the Leibnizinterpretation of monad as a spiritual entity, whichreflects the totality of the world, in the informa-tional sense. The hierarchical aspect is based, onthe other hand, on a biological ansatz, reflecting theproperties of the living bodies, as revealed by then re-cently discovered microscope. This was exactly whatAnaxagoras inferred (see I), without a microscope.On a wider scale, however, one might argue thatLeibniz’s monadic affinities stem from his generaladherence to continuum and belief that Natura nonfacit saltus. Indeed, he used to play with geometricstructures that fill in the space as much as possi-ble, like that Leibniz packing of a circle, as Mandel-brot (1983) dubbed it. He even ventured to meditatethat the exponent k in

(d/dx

)kF (x), in his newly

developed calculus, need not necessarily assume in-teger values only, as he mentioned it in a letter tol’Hospital (see, e.g. Mandelbrot 1983).

3. NEWTONIAN ERA

The idea that monad is coupled to the entirecosmos, albeit in a spiritual sense, may be regardedas a particular aspect of a concept of universal inter-action between (material) cosmic objects, or at leastmutual correlations. The physical basis of this ideawas the Newton’s concept of the universal gravita-tion (Newton 1687), which Leibniz, understandably,failed to invoke. But the law of the universal attrac-tion was not compatible with the assumption of aninfinite universe, as advocated by Giordano Bruno,for instance. Newton accepted the same hypothesis,but was warned by a number of people, like a youngtheologian Richard Bently in 1692, who raised thequestion of the stability of the stellar systems. Thiswas a precursor to the later conundrum known asSeeliger-Neuman’s paradox. Another objection wasput by a physician and antiquarian William Stuck-ley (1720) concerning the luminosity of the night sky(which should appear to our eyes like the Milky Way,he argued). The latter question was subsequentlydiscussed by Newton, Stuckley and Halley (duringthe breakfast at Newton’s home) as the latter re-ported at the Royal Society: ”Another ArgumentI have heard of urged that if the number of FixedStars were more than finite, the whole superficies oftheir apparent Sphere would be luminous” (see, e.g.,Redhead 1998). The question seems to be raised byother people even before that occasion and might betraced as far back as to Kepler (see, e. g., Mandel-brot 1983). This was the beginning of the puzzleknown today as Olbers’ paradox (see, e.g. Martinov1965), though some authors prefer the term BlazingSky Effect.

We see, hence, that the concept of an infiniteworld is not that simple as might have appeared tohis proponents. Obviously, from the point of viewof those who were concerned with the stability andluminosity of the apparent cosmos, a paradigm ofa static, infinite and homogeneous solution (model)would not do. Newton himself became aware of theseproblems and planned to tackle them in a new editionof Principia (which he never accomplished), playing,for instance, with the stars of various ”magnitudes”,an idea along the hierarchical model line of thought.The important thing to note is, nevertheless, thatwith Newton a qualitatively new approach to the cos-mology as such has been adopted, that based on as-tronomical observations and mathematical analysis.This was a direct outcome of Galileo’s new method-ology, which was a quantitative analysis, as opposedto the speculative and qualitative scholastic consid-erations in his time. The newly introduced force ofuniversal attraction made the universe a physical sys-tem, instead of a mere collection of celestial bodies.But, the same interaction, being both universal andessentially attractive, raised serious question as tothe stability of the universe. Coupled with the as-sumption of an infinite universe, both in spatial and

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temporal sense, this new aspect of the physical real-ity made the cosmology a science, but not an easyone.

Kant’s picture of the universe (Kant 1925,1968) was essentially Abderian one. Kant boldly in-ferred from the available observational evidence thatthe so-called nebulae were extragalactic systems, likeour own Galaxy. He also considered the conspicuouslack of homogeneity in the observable universe (Kant1755, 1925, 1968):

”This part of my theory which givesit its greatest charm ... consists of thefollowing ideas ... It is ... natural ...to regard [the nebulous] stars as being... systems of many stars ... [They] arejust universes and, so to speak, MilkyWays ... It might further be conjecturedthat these higher universes are not with-out relation to one another, and that bythis mutual relationship they constituteagain a still more immense system ...which perhaps, like the former, is yetagain but one member in a new com-bination of numbers! We see the firstmembers of a progressive relationship ofworlds and systems; and the first part ofthis infinite progression enables us al-ready to recognize what must be conjec-tured of the whole. There is no end butan abyss ... without bound.”

Kant adopted, also the idea of a changing uni-verse, with cosmoses arising and disappearing, in arepetitive manner (1755, see Kant 1925), thus pro-moting Anaxagoras’ and Democritus’ ideas of theplurality of worlds. Another important contributionof Kant to the subject was the first, albeit qualita-tive, conception of the mechanism of the creation ofour Solar system, which meant Anaxagoras’cosmogo-ny (we use the term cosmogony in a wider sense, re-ferring to the universe, see I) put into more concreteand realistic physical terms. When Laplace put for-ward, independently, his cosmogonical model (1797,see Laplace 1925), based on the atomistic hypothesis,the revival of the Presocratic cosmology was almostcomplete.

Rugiero Boscovich. In an attempt to recon-cile Newton’s and Leibnitz’s ideological backgroundsconcerning the nature of space and time and theperennial questions regarding the divisibility of mat-ter Boscovich (1711-1787) discussed the latter in histreatise from 1758, 1763 Theoria Philosophiae Nat-uralis (Boscovich 1922). Boscovich’s solution to theproblem of an infinite divisibility of matter is purelygeometrical one. Making use of his primary con-struct of material points, he defines particles of thefirst, second, etc order. Thus, we read (III.395):

A given mass, however small, dis-tributed over a given space, however la-rge, so that there remains no small and

empty space larger than a given, no ma-tter how much small space, without anyparticle of that mass.... We understandthat this small mass is divided into asmany particles and that each of themis placed in a small volume. They canfurther be divided at wish, so that newparts of each particle cover wall of thissmall volume... ”

Although no direct reference is made concern-ing Kant hypothesis expounded above (Kant’s workappeared 3 years before Boscovich’s), it resemblesthe former, at least formally. As for Boscovich’s so-lution of the problem of maximum filling the spacewith a finite amount of matter, will reappear in thecosmological work of Fournier, as we shall see lateron.

XIX century witnessed no significant advanceconcerning our inference into the structure and dy-namics of cosmos. The principal concern was to re-solve the paradoxes related to an infinite world as-sumption, as raised by Newton’s contemporaries. In1826 Heinrich Olbers (1758-1840) made the luminos-ity paradox even more astounding, by noting that thesky should be as luminous as the Sun surface. Hisargument runs like this. Since in an infinite universeour line of sight should encounter a star wherever welook at the sky, the latter should shine like a sur-face of any star, and like our Sun, for that matter.Moreover, if one integrates the total electromagneticradiation coming from all stars in the universe, oneobtains an infinitely bright sky. Of course, when oneaccounts for the screening effect, the net brightnessis reduced to a finite value (Martinov 1965).

Equally disturbing was the gravitational para-dox, known as Seeliger-Neuman paradox, as first for-mulated by Seeliger in Astr. Nachr. No 3273 (1895).If a celestial object, like a star, is surrounded byan innumerable like objects in an infinitely extendedcosmos, the net gravitational force is undeterminedboth in direction and magnitude, being of the form∞−∞, and may, therefore assume any value withinan interval (0,∞). One might be tempted to cir-cumvent this inconvenience by assuming a sphericalsymmetric distribution of surrounding masses, butthis would violate the assumption of an homogeneouscosmos, which in its turn is just one of aspects of thegeneral Copernican principle. One notes that therewould be no screening effect in the case of gravita-tional interaction, since any object is transparent tothe force of universal attraction. This difference withrespect to the luminosity paradox stems from the es-sentially different nature of the physical quantitiesinvolved. The light (electromagnetic radiation) is adynamic physical field, which propagates through theempty space with a finite velocity c, and interactswith matter in various ways, including absorption,reflection, etc. Gravitational field is a static quan-tity, ever and everywhere present, not propagatingand thus never absorbed by the matter of any formand nature. (We disregard for the moment the hy-pothetical existence of gravitational waves, predictedby General Relativity).

It is with these dilemmas that the cosmologyentered the twentieth century.

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4. EARLY TWENTIETH CENTURY

4.1 Some theoretical preliminaries

Since the appearance of Einstein’s General Re-lativity in 1915, cosmology gained the ground for be-coming a scientific theory. Further development ofcosmology relied on two premises. First, there ex-ists a fundamental theory, Einstein’s or otherwise,which can provide a mathematical tool to formulatea selfconsistent picture of our universe. Second, for-mal solutions and cosmological models should sat-isfy a number of fundamental postulates. The latterare the so-called cosmological principles (see, e. g.Narlikar 1979, Barrow and Tipler 1986). The first(strong) cosmological principle asserts that the uni-verse at every instant of the (global) cosmic time ap-pears isotropic and homogeneous. These postulatesare of a geometric nature and are satisfied, separatelyor both, by a number of cosmological paradigms.The second of these appears more general, for if amedium is homogeneous, it is isotropic too, but theopposite does not hold. If only the first of the two issatisfied, a number of Newtonian models can be con-structed (Thatcher 1982). In particular, Lemaitre’smodel belongs to this class, too (Narlikar 1979). The-se models constitute a class of monocentric models,with a singular, space point singled out as a centreof the universe. From this point, universe looks thesame in all directions. If this property of the universeis satisfied for any arbitrary point (or observer), theuniverse is homogeneous, and the second principleis satisfied, the postulate of homogeneity. Finally,the so-called Strong cosmological principle requiresthat the universe remains the same (identical to it-self), regardless of the flow of (cosmic) time. Thistime homogeneity principle is usually called The per-fect cosmological principle, for it comprises all threesymmetries - rotational and translational in spaceand time.

What is the role of these principles? Theirepistemological status appears at least twofold.First, since they invoke a particular symmetry withinan abstract space, they simplify the mathematicalproblem of solving corresponding equations. Buttheir meaning does not exhaust itself by these techni-cal benefits. They carry, albeit implicitly, strong mo-tivations from outside of the purely scientific sphere,e. g. esthetic, even religious. To some cosmologists,these principles play a role of a cosmological dogma,and carry thus a flavour of faith (see, e.g. Ribeiro andVideira 1998, on dogmatism in cosmology). Consid-ering that an observational evidence may be vagueand indecisive, as the case with observational cos-mology often is, these prejudices may play crucialrole in interpreting the data and are usually behindthe ensuing disputes, as ideological background.

4.2 The observational evidence

Until the advent of large telescopes the cos-mology was to a large extent a speculative subject,though the Kant-Laplace hypothesis could be con-

sidered scientific, albeit qualitative, approach. De-spite Giordano Bruno’s and Kant’s concepts of animmense, if not infinite, universe, cosmos consisted ofthe visible (naked eye) sky, the largest stellar systembeing our Milky Way. Stars remained the elemen-tary constituents of the cosmos, though a numberof sky objects of obscure nature, like nebulae couldbe discerned on the night sky (but see I for Kant’sinference on the matter). In the absence of a reli-able method to determine distances of objects seenon the sky, the cosmos remained within the realm ofour Galaxy. But even within this limited picture, thesimilarity principle was operative, with a clear anal-ogy between our planetary system and the galacticdisk. Both systems possess an axial symmetry, point-ing toward the same physical mechanism of formingthese rotational structures. It was this analogy thatinspired Lambert (see Charlier 1922) to conceive theworld as a generalized planetary system.

The breakthrough was made first in 1912 byHenrietta Leavitt (1868-1921) (see, e. g. Aczel 1999,for a popular account), who established a relationbetween Cepheid variable average apparent bright-ness and its period of apparent magnitude variation(period-luminosity relation). This enabled her to de-fine a standard astronomical means for estimatingdistance from systems containing Cepheids. In 1917Vesto Slipher from Lowell observatory published apaper where he showed that the spiral nebulae werereceding from us with an immense velocity, accord-ing to his measurements of Doppler red-shift of theseobjects, which he considered to be intragalactic. Butit will take another 12 years for Hubble to establishthat these nebulae were extragalactic replicas of ourown Galaxy and that they recede with the speed pro-portional to their distances from us (Hubble’s law,see e.g. Peebles 1993 for a more detailed account ofthe matter). These findings will have a dramatic ef-fect on our picture of the universe, as we shall seelater on. But before proceeding along these lines, wefirst turn to the early attempts to reconcile the ideaof an infinite universe with observational evidence,as required by two cosmological paradoxes. And wethus turn to the concept of fractal cosmos.

4.3 Charlier’s fractal model

The first hint at the possible hierarchical cos-mos was made by Fournier d’Albe (1907), who de-vised a curious geometrical structure obeying theso-called octahedral principle (see, e.g. Mandelbrot1983). His cosmos had the property that the con-tent of matter within each sphere was proportionalto its radius. This condition was sufficient to protectthe universe against both cosmic paradoxes. (Such amodel yields the fractal dimension D = 1, i.e. one-dimensional cosmos, see later).

Following an idea due to Kant’s contempo-rary Lambert, Charlier conceived the fractal uni-verse with galaxies as elementary constituents (Char-lier 1908, 1922), each containing N1 stars (see, e.g.Ribeiro 1994 for a more detailed account of the con-tributors to the concept of a fractal cosmos in the

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last century). He then defines a cluster of galax-ies G1 as a generalized galaxy, with N2 constituentmembers, then a cluster of clusters G2 with N2 mem-bers, etc. Then to each of consecutive constituentsGi radius Ri, i = 0, 1, 2, ... is ascribed, as well as thecorresponding masses Mi. Assuming, for simplicity,a spherical shape of all (sub)units Gi Charlier takesthat Ni units fill up Gi in such a manner that aneffective radius ρi is defined as

ρi =Ri

N1/3i

(1)

so that an average distance between two nearest ne-ighbours in Gi is 2ρi. One has for the masses thefollowing relation (M0 is the mass of a star)

Mi = NiMi−1 = NiMi−1Ni−2 · · · N2N1M0 (2)

With these definitions Charlier shows thatboth paradoxes disappear.

Seeliger’s paradox. The aim is to demonstratethat the net Newtonian force on a system Gi is finite.If one assumes the most unfavourable situation of Gisituated at the edge of Gi+1, the net gravitationalforce on Gi will be the sum of all forces from thecosmic matter

FN =∞∑

i=1

Mi

R2i

(3)

From the requirement that the sum converges (finitevalue) one has from (2) and (3)

Ri

Ri−1>

√Ni (4)

If this condition is met, the overall force exerted byall cosmic gravitational sources on a single subunitwill remain finite.

Olbers’ paradox. If the luminosity of a galaxyGi at a distance ρi from the observer (at Earth, moreprecisely at the Sun) is hi and the total luminosity ofGi (counted from its centre) is Li, and if one assumesthat Gi is situated at the centre of Gi+1, the totalelectromagnetic energy influx at Earth would be

L =∞∑

i=1

Li (5)

The number of Gi in Gi+1 within the sphereof radius r is then

ni =r3

ρ(6)

and the apparent luminosity is

hapi = hi

ρ2i

r2(7)

The total apparent luminosity stemming fromall Gi is (after integrating over all layers from 0 toRi)

Li = 3hiN1/3i (8)

The next step now is to find out the relation-ship between hi and hi+1. It is interesting to notehere that in his original derivation (Charlier 1908)Charlier made a mistake and the following treatmentwas due to Seeliger, who sent to Charlier his deriva-tion in a letter (Franz Selety derived the same rela-tion independently, and communicated it to Charlier,after the paper by the latter was submitted).

One starts from the luminosity h2 of G1 at thedistance ρ2

h2 = N1h1ρ21

ρ22

(9)

which gives a general relationship

hi = hi−1Ni−1

ρ2i−1

ρ2i

(10)

From (8), (9) and (10) one obtains

Li

Li−1= Ni

R2i−1

R2i

(11)

The sum (5) to remain finite one has

Ni

R2i−1

R2i

< 1 (12)

which is equivalent to (4). Thus, one arrives atthe remarkable property of Charlier’s universe thatit solves both Seeliger-Neuman’s and Olbers’ para-doxes.

This result might have been anticipated on thegeneral grounds, considering that both gravitationalforce and apparent luminosity follow the same fall-off behaviour having regard to the distance from theobserver. The latter property, in its turn, stems fromthe fact that the gravitational force can be expressedin terms of lines of force, whereas the electromag-netic radiation may be represented by an emission ofparticles (photons), and both emanations obey theEuclidian geometry.

Charlier showed also that the same conclusionholds if the assumption of the central galaxy nestingis relaxed. He also derived the relationship betweenthe distances between members of each galaxy

ρi

ρi−1>

√Ni−1Ni (13)

What happens if a body (star, for instance)falls from the galaxy of the next higher order intoa galaxy, with a zero initial velocity? Charlier con-sidered that problem, too and found the relationshipbetween final velocities attained

vi

vi−1< N

1/4i (14)

Finally, Charlier derived another remarkableproperty of the galaxies (conceived as subunits with-in his hierarchical scheme), namely that all subunitshave the same period of (Keplerian) motion

T =

√3πδ

G(15)

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where G is the gravitational (Newton’s) constant andδ the mean density of a galaxy. Hence, each galaxy ischaracterized by its own unique period. This prop-erty, which appears a specific realization of Poinca-re’s cycle theorem (see, e.g. Pars 1965), makes thewhole fractal model even more appealing.

The most remarkable property of Charlier’suniverse is that the overall (mean) density of his hier-archical cosmos is zero. It is this feature that standsbehind, albeit implicitly, the above resistance to thegravitational and luminosity paradoxes. This featurewill prove significant for further development of themodern cosmology, as we shall see below.

Before we proceed further, two points mustbe stressed here. First, at the time Charlier con-trived his hierarchical model no evidence for the frac-tal structure was available. Second, Charlier did notaddress the question as to in which way the struc-ture he proposed might have formed. Both questionsturned out tricky ones, up to the present time.

4.4 The expanding universe

The year 1922, when Charlier’s paper appe-ared, witnessed several remarkable events, whichproved to be crucial for the further development ofscience. While working with Sommerfeld, WernerHeisenberg conceived semiclassical (in modern par-lance) models of Hydrogen and Helium, with half-integer quantum numbers (see, e.g. Briggs 1999).The models reproduced the experimental data on theZeeman effect and the ground-state Helium energyremarkably well, but Heisenberg gave up publishinghis results after a fierce Bohr’s opposition to the ideaof non-integer quantum numbers. This marked theend of the pursuit for atomic models based on theconcept of the electron trajectory, and led eventuallyto Heisenberg’s discovery of the Matrix (quantum)mechanics. In the same year Einstein was awardedNobel prize for his theoretical explanation of thephoto-effect. The idea behind the theory was themuch disputed concept of a quantum of (electromag-netic) energy, later to be dubbed photon, that was asort of resurrection of Newton’s corpuscular conceptof light. Encouraged by these events, two years laterDe Broglie published his formula that related the mo-mentum of a microscopic particle and an associatedwavelength, which turned out instrumental for devel-oping the Wave mechanics by Schrodinger. (Heisen-berg’s non-integers will be rediscovered in modernsemiclassical theory as the so-called Maslov’s index,see, e.g. Percival 1977).

Russian mathematician Friedmann publishedin 1922 his dynamical solution of Einstein’s equationof General Relativity, where he demonstrated thatthe universe could have started from a singular point-like state of infinite density and then expand. Thisdynamical model proved not only realistic, as thesubsequent observations corroborated, but solved athe same time several problems related to Einstein’s(and Newton’s, for that matter) static cosmologicalmodel. We enumerate them here.

Stability problem. As is known from the sci-ence of mechanics, no stable structure could be main-

tained in a static system where constituents interactby attractive and/or repulsive forces. It holds forall attractive force, like Newton’s gravitational in-teraction, and for Coulombic systems, like a plasmaof ions and electrons. To overcome this shortcom-ing of his static model, Einstein introduced the con-cept of mixed interaction, combining the attractiveforce with a hypothetical long-distance repulsive in-teraction. As is well known this remedy, which isessentially the concept of a molecular force, was in-troduced via so-called cosmological constant Λ (see,e.g. Collins et al 1989), as appearing in the Einsteinequation

Rµν − 12gµνR +

Λc2

gµν =−8πGN

c4Tµν (16)

where Rµν is the so-called Ricci tensor, R is Ricciscalar, gµν is the metric tensor and Tµν is the energy-momentum tensor, to save the phenomena. Gener-ally, one may establish a dynamical equilibrium ina system with monotonous forces, like Newton andCoulomb ones, either by rotation, or expansion ofthe system. While the first approach appears legit-imate for microscopic systems, like atoms (see, e.g.Grujic 1999), the concept of a rotating cosmos raisesa number of conceptual difficulties, which do not ap-pear, however, unsurmontable (see, e.g. Reboucasand de Lima 1981, for the theoretical aspects, andBirch 1982, for the observational evidence).

Much simpler solution to the stability of cos-mos is the expansion hypothesis, which asserts thatall galaxies, as the principal cosmic constituents (bui-lding blocs), recede from each other. Formally, thisrepresents a homothetic transformation, which ma-kes all distances in the universe increase in time.This assumption removes the stability problem bytour de force, since the possibility of two bodies, e.g.,to fall into each other is simply excluded by makingthem separating apart all the time. The overall sys-tem appears stable being put into a collective mo-tion, which by itself prevents gravitational collapse.But this assumption has a weak points too, as weshall see now.

Thus, the concept of an expanding universeproved to be fatal for all static models, includingCharlier’s one, just as the concept of microphysicswithout classical trajectories turned out lethal to theclassical models of atomic and subatomic systems.

The overall expansion can not be a universalprocess in our universe. Galactic and subgalactic sys-tems do not obey this rule, as we know. Whateverthe mechanism of galaxy formation is invoked, theoverall rotation appears a dominant feature of thecollective dynamics. Of course, when one goes to thelower physical levels, other forces enter the game,like chemical, Coulombic, etc, and the problem ofstability takes on various forms as one goes from onesystem to another, or from one physical level to theother. Hence, when talking about the universal ex-pansion, one must keep in mind that it refers to aparticular (possibly undetermined) cosmic domain,starting from galaxies to a possibly upper level ofthe cosmological reality. In particular, it has been

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shown that atomic dimensions have not been notice-ably changed since the early cosmic era, when atomsformed.

The problem of stability of (sub)galactic struc-tures in the context of a particular cosmological mo-del appears as an important aspect of the cosmologyin the most general term, but we shall not dwell onit here.

As it often turns out, an idea in natural sci-ence has a precursor in an accidental mathematicalresult. Modern cosmology is no exception. The firstexpanding ”universe” was treated by de Sitter, asearly as 1917, a year after Einstein set up his Gen-eral Relativity equation (16) without Λ). Cosmology,like physics, for example, is an exact science, and thelatter is an art of approximations, i.e. of inexactness.De Sitter found the solution of Einstein’s equation byreducing the content of matter (and radiation alto-gether) to zero. By doing so, he found the solutionfor the scaling parameter S of the form

S(t) = eHt (17)where H is Hubble (time independent) constant,which can be defined by (Narlikar 1977)

H2 =12Λ (18)

Note that the zero-content universe is not no−universe. In particular, the empty space is not anEuclidean one, but is endowed with the structure(Minkowsky space). Though trivial at the first gla-nce, de Sitter’s model proved very important fromthe heuristic point of view. First, it gave an idea ofa dynamic cosmos and second, the later paradigmof an inflatory universe (see, e.g. Collins et al 1989)was inspired directly by de Sitter’s solution. Whatis of a particular importance to us here is that, aswe mentioned above, Charlier’s cosmos had a zero-average-density property, too.

Cosmological paradoxes. When writing his fa-mous paper in 1922 Charlier was well aware of the(preliminary) observational evidence for galaxies re-ceding from us (in fact, his own observatory at Lundwas much engaged in this sort of astrophysical inves-tigations, as Charlier himself mentioned). He seemedto feel that that ”small cloud” on the otherwisebright ”fractal sky” might be announcing a storm.The latter did materialize in solving automaticallyboth paradoxes, within the expanding cosmos para-digm. We first consider the luminosity paradox.

The explanation runs in two ways. The firstrefers to the finite age of cosmos, which is implied bythe one-way collective motion (eventually as a partof a cyclic dynamics, at the worst). According to(17) the estimated age would be t0 = 1

2H . Accord-ingly, the amount of the electromagnetic (or of anyother type) energy that can reach us must be finite,since only a finite portion of the (observable) cosmosis an available source of light. The second type ofexplanation relies on the red shift as Doppler effect

(see, however, Voracek 1985, 1986, for other inter-pretations). From Hubble law

z ≡ ∆λ

λ=

Hr

c(19)

where r is the distance from a galaxy, the net fluxfrom a spherical layer through a unit area plane is(Martinov 1963)

φ = 2πNLdr (20

where N is the number of stars in a unit volume,and L the average luminosity of a star. If one as-sumes that the star spectre is Planckian one, afterintegrating over distance and frequency

Φ =2h

c2

∫ ∞

0

dr

∫ ∞

0

ν3γ4

ehνkT γ − 1

dν, γ = 1 +Hr

c(21)

where h is Planck constant and T temperature, onearrives at the finite flux at any point in the space

Φ =2πcNL

3H(22)

We note that the same conclusion is reached if a rig-orous relativistic expression for the red shift

z =(1 + v

c )1/2

(1 − vc )1/2

(23)

where v is the velocity of the source, is used. Wenote, also, that more realistic upper integral bound-aries in (21) (instead of ∞) would further reduce thevalue of Φ.

We note that in order to replace a static New-tonian model, as Charlier’s one, the model of anexpanding universe had to invoke the ”most heavyartillery” of the century, the quantum physics andRelativity theory. Finally, we mention here the wellknown fact that an expanding universe is not a uni-que cosmological paradigm, though it has been wi-dely accepted and considered a standard model. Forthe alternative paradigm, Steady state hypothesis,see, e.g. Narlikar (1974) (see Arp et al 1990, fora more recent account of the relevant observationalevidence).

5. THE FRACTAL PARADIGM

In a preceding section we enumerated somecosmological principles that govern a number of cos-mological paradigms. Where the fractal, self-similarcosmos stands in this context? In particular, is afractal cosmic structure endowed with isotropy andhomogeneity? It possesses the so-called local isotro-py, that is all cosmic points are equivalent concern-ing the isotropy (no special cosmic points), but thesepoints are not uniformly distributed (see, e. g. Saar1988, Einasto et al 1988, Sylos Labini et al 1998).Hence, fractal cosmos is not homogeneous.

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5.1 Some cosmogonic remarks

As we mentioned above, Charlier did not ad-dress the problem of structure formation, while set-ting up his hierarchical model, as an a priori concept.The question arises, however, whether his model con-tradicts the concept of ever expanding universe. Weaddress this question briefly, before reviewing thepresent day state of affairs.

It is a common evidence that the cosmos in oursurrounding is neither homogeneous nor uniformlyexpanding. These facts are not independent, of cou-rse, and can be used to explain the present state ofthe cosmic inhomogeneous distribution of matter, inparticular of galaxies. We know that not all galaxiesare receding from us (as the case with Andromedashows). Moreover, a large scale cosmic contractiontowards Virgo cluster (the Great Attractor) convin-ces us that perturbations with regard to the overallexpansion are possible and may be of considerablemagnitude. All these phenomena are connected withthe problem of the (gravitational) stability of the cos-mic matter, or with the concept of gravitational col-lapse. It is this instability that might be responsiblefor the large-scale structures, as observed today, andwhich might be operative in forming a fractal dis-tribution, at least up to certain cosmic scales (butsee later). We shall now consider this phenomenonwithin the modern cosmological perspective.

5.2 Fractal or nonfractal, the question is now

Hence, is the universe fractal or not? As wehave seen, the observed nonuniform distribution ofgalaxies (clusters, superclusters, etc) does not con-tradict the principal assumption of the majority ofmodern cosmological models (at least those based onGeneral relativity) that the universe is homogeneousand isotropic (cosmological principles). The ques-tion is now could the two concepts, uniform universeand fractal cosmos, be reconciled. To put it differ-ently, the question is, if a hierarchical structure doesappear, up to which cosmic scale it persists. Thenext question is if an eventual fractal distribution isobserved, could it be described by a simple fractalmodel, or one must contrive a multifractal cosmos(see I)?

A proper approach to the fractal issue shouldencompass at least the following points (questions):

(i) If at least a part of the universe has a(multi)fractal structure, what would be the mech-anism of its hierarchical formation?

(ii) How should this (multi)fractal structurebe conceived from the point of view of an external(metaphysical, see later) observer?

(iii) What would our observational evidencelook like that should corroborate such a particularform of the observable universe?

5.2.1 Fractal formation

As we saw in I, Anaxagoras was content withthe solution of the type Noυς απo µεχανες (Deus ex

machina). We need, of course, a more convincing ex-planation. It must be said rightaway here that thereis no a general model for fractal formation in nature(or any other structure, for that matter), and eachparticular physical system requires a special mech-anism, if conceived at all (see, e.g. Gouyet 1996).The situation is different in the purely mathemati-cal domain, of course, as we showed in I (see, e.g.Ribeiro 1994). In the case of mathematical physics(e.g. nonlinear dynamics) fractals do appear, but asphenomenological objects, that has, however, a re-stricted heuristic value (see, e.g. Handke 1994).

One can distinguish two possible scenarios forstructuring the universe. One proceeds from a uni-que structureless universe, which then falls apart in aspecified manner. This ”scenario from above” com-prises more than unique initial state, like a uniformuniverse, a unique (cosmic) superstring, etc. Theother scenario ”from below” assumes the existenceof initial small agglomerations, which in due timemerge to form larger and larger clusters. Both sce-narios are relevant to the issue of the fractal cosmos(see, e.g. Hogan 1980).

Without claiming to be exhuastive on the sub-ject, we quote a number of possible mechanisms offorming a hierarchical cosmic structures. Within the”from below” scenario, one possible way to concen-trate cosmic matter would be based on the gravita-tional instability, that might force objects like galax-ies (conceived in general manner, as Charlier did) tomake agglomerations, as transient units on the wayof gravitational collapse. The peculiar velocities ofcelestial systems, which divert from the overall Hub-ble flow, might be considered as evidence for such atendency. If the fractal paradigm is accepted, thispicture appears to be in conformity with the generalidea of the cosmological collapse, within the conceptof a closed universe, followed by Big Bang (cyclicmodels).

Self-gravity as a structuring force

Fractal structure arising from self-gravity mayform in various cosmic media, from the interstellarmatter to galaxy clustering. Methods developed arebased on the renormalization group, used extensivelyin many areas, like the quantum field theory, sta-tistical physics, condense matter etc. If the systemwith an hamiltonian H possesses scale invariance,represented by an operator R, a series of successivehamiltonians are generated by

Hn+1 = RHn (24)All hamiltonians have the same structure, but differ-ent values for the parameters they include. WithinZel’dovich model, which belongs to the top-downscenario, the collapse of matter starts from largescale, and subsequently one-dimensional (filaments)and two-dimensional (sheets) structures are formed(Zel’dovich 1978; for other approaches see, e.g. Co-mbes 1999 and references therein).

In a recent series of papers a statistical me-thod, exploiting the scale-invariance of the gravita-tional force, has been developed, within down-top

52


scenario (see, e.g. Vega et al 1998, Combes 1999),which comprises both interstellar matter and galax-ies. The latter arise from the collapse of smallerstructures, and the structures at higher levels areformed, in a non-linear manner, with various effects,like the turbulence, self-criticality etc accounted for.This thermodynamic approach appears general eno-ugh to promise a unifying picture of fractal structur-ing phenomena.

Strings and the cosmic structuring process

If the former mechanism is ascribed to thelater phase of the universe evolution, the cosmic st-rings hypothesis pertains to the cosmogonic era. Ac-cording to some authors, the cosmic strings, whichare not to be confused with the strings from thequantum field theory, but are defined as topologi-cal defects of the early cosmic space-time manifold(see, e.g. Collins et al 1989), might be instrumen-tal to the galaxies formation. Two different mecha-nisms are envisaged within the context. First, mat-ter was accumulated around the early formed cos-mic strings, giving rise to the today observed galaxydistribution. Another possibility is that the strings,endowed with angular momenta, were torn apart bythe centrifugal forces, and the process proceeding ina number of consecutive phases. At the first stagethe string broke into large pieces, from which super-clusters arose. Then at consecutive stages the piecesbroke themselves into smaller parts, from which clus-ters and galaxies formed.

Presumably, two classes of the cosmic strings,as primordial topologically stable objects, formed atthe time of phase transition in the very early uni-verse, are defined, the open and closed (loop) strings.Both may be considered candidates for precursors ofthe observed cosmic structures. Here we mention ahybrid model, which starts with open strings thatbreak and form string loops, which themselves breakinto smaller loops, etc, then the matter is graduallyaccreted around them, and the Abell clusters areinvoked as an observational support for the model(Turok and Brandenberger 1985). Here, we shall ex-pose in some detail the model suggested by Tassie(1986) in a number of papers, which appears par-ticularly interesting to the subject, as we shall seebelow.

The model proceeds from an observation thatthere is a general relation between the angular mo-mentum J and mass M (see Tassie 1986, and refer-ences therein)

Ja = κMβ (25)

with κ as a universal constant, estimated as

κ ≈ 4 · 102 GN

c(26)

where GN is Newton’s gravitational constant, andc velocity of light. As for β, for a large class ofcelestial objects the estimate is β ≈ 2. However,a more stringent analysis of the observational datashows that β varies from class to class, within an in-terval (1 2

3 , 1 34 ), whereas Carrasco et al (1982) found

for the overall classes β = 1.94± 0.09. The explana-tion was that these regularities appear remnants ofprimordial rotational motions of the cosmic strings,conceived as rigid rotators. As for their origin theidea due to Kibble (1985) was that, in analogy withmagnetic vortices in superconductors, these vortexstrings arose after a sort of phase transition at theearly stage of the Universe. Even more seducinganalogy is that with superfluids (like that of He4),which resembles much the ancient Pelasgian mythicpicture of the world creation from Chaos, with Eu-rinome and Ophion (Graves 1963; also, Grujic 1996,unpublished). The beauty of the string hypothesisis that it can be related to the quantum field world,where one finds a number of relations similar to thosementioned above. Thus, in the case of Regge trajec-tories, one has for the hadrons

Jh = κhM2 + J(0), κh = h/(GeV/c2)2, (27)

The constant κh differs much from its counterpartin cosmology, but the essence of the model remains.Since the superstring theory claims that it can ex-plain the host of microworld phenomena and canbe considered to be the basis for an ultimate the-ory of matter, like GUT etc (see, e.g. Collins et al1989), this formal similarity gains in attractiveness.It lends support, also, to a number of bold hypoth-esis, like that of the self-similar cosmos due to Old-ershaw (1989), who argues that the whole materialworld, from the elementary particles to the entireuniverse, has been designed after a fractal pattern.(See, also, Oldershaw 2001)

The actual decay mechanism for the primor-dial strings is not a matter of consensus among thosewho hold on to this model. Some argue that the frag-ments stemming from a unique, primordial cosmicstring, should be massless, and therefore not candi-dates for the subsequent structuring of the universe.Other authors consider that finite mass fragmentscould have been the outcome of this disintegration,forming a successive hierarchical structures. Withineach of class of objects (that is, at a particular hier-archical level) a number of processes might have con-tributed to the observed variations of the constantsκ and β in (25), like the so-called tidal interactions,collisions, etc.

As we mentioned earlier, two possible modesof galaxy formation out of strings might be con-ceived. According to one model, strings comprise buta small part of the overall cosmic matter, otherwiseuniformly distributed, and galaxies form by accre-tion of the surrounding matter around these ”cosmicseeds”. The other alternative assumes that the cos-mic matter is contained in the strings, from the verybeginning of the string decays. These distinctionsare important for the observational cosmology. For ifthe former alternative holds, the universe must havebeen more uniform in the past and by looking intodeep cosmic space, one should see more and moreeven distribution of the cosmic matter. On the otherhand, if the string structure persisted up to the re-mote past, the distant parts of the universe shouldbe more inhomogeneous, resembling string configu-rations more than the nearby cosmic objects.

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Before leaving this string paradigm, let us me-ntion that the string hypothesis goes beyond the frac-tal structure concept, and other sorts of universesmay arise from a collection of primordial strings. Thecase of one unique string appears but an extreme one,out of many other possibilities (see, e.g. Tassie 1986,and references therein). But we shall not dwell on ithere. Also, we note that all these similarities havemore heuristic significance, rather than theoreticalvalue. In a sense, these attempts to present a uni-fying picture of structuring material world resemblethe approach of some authors to quantify celestialsystems just like atomic ones, by introducing a gen-eralized Planck constant (see, e.g. Grujic 1993, andreferences therein).

6. THE FRACTAL PATTERN

Whoever designed the Cosmos, (S)He eviden-tly did not consult Alphonso X, and simple realiza-tions of a project in ideal geometrical terms, as con-sidered by Charlier, for example, are out of question.In the following, we shall adopt the convention dueto Ribeiro and Videira (1998), and designate ideal(true) cosmic objects by capital initials, like Cosmos,as metaphysical entities. Our models, which seek toapproach real structures, as tentative descriptions ofthe reality, we take simply as (the) cosmos, for ex-ample, and the same for our observational inference.Further, we shall distinguish three kinds of relationsbetween different celestial objects. First, we haveanalogies, as between binary, ternary etc stars andthe analogous binary, ternary etc galactic groups.Similarly, one observes similarity between galactic(intrinsic) rotation and the corresponding dynam-ics of galactic (super)clusters. Second, we notice,for example, a striking similarity between globular(stellar) clusters and some galactic clusters, as ex-amplified by that galactic cluster in Coma Berenices(see, e.g. Bakulin et al 1977). Finally, one mighthave a strong similarity between (adjacent) hierar-chical levels of the cosmic reality, to which a trulyself-similarity transformation may be applied. It isimportant to stress here that one should define inadvance which kind of relationship he or she is re-ferring to, when seeking, or claiming, the fractal (orany other) kind of cosmic structure.

6.1 Fractality, self-similarity, scale-invarianceand hierarchical ordering

We employ the three first terms as synonyms,though the last one conveys best the essence of themeaning behind these terms. As for the hierarchi-cal ordering, it stands somewhat apart, and we shallelucidate its relationship with the first three notions,before going on with the issue of the cosmic structure(cf, e.g. Mandelbrot 1983, Ribeiro 1994).

Hierarchy may, but need not necessarily implyscale-invariance. This is best illustrated by enumer-ating the basic elements of geometry, as defined by

Euclid, for instance - point, line, area, volume, eachbeing a part of the next one. In this context, oneshould ask what are the basic structures observed inthe surrounding universe, as a kind of building blocksof our cosmos? Starting from the elementary ”point”unit, a galaxy (0D), we have further filaments (1D),sheets (2D), and finally voluminous galactic clusters(3D). This series of the cosmic elements, which fol-lows the geometric pattern we just mentioned, shouldbe completed by the cosmic voids. Origin of the firstfour cosmic ingredients is still a matter of research,not to say controversies, while for the cosmic voidssee, e. g. Collins et al (1989). What is of importanceto us here is, first, that such a hierarchical series isnot scale-invariant, and second, a truly self-similarglobal cosmic structure may consist of all of theseelements, as ingredients of a complex (multy)fractalpattern.

6.2 Fractal properties

Strictly speaking, fractality is a wider notionthan self-similarity (see, e.g. Mandelbrot 1983). InChapter 2 of I we quoted a number of general fea-tures of fractal objects. In the following we shallneed a more detailed description of these mathe-matical objects, before proceeding with attempts toidentify them on the sky. Since these geometricalstructures are embedded into Euclidian space (justas their physical representatives are immersed intothe real physical space), we need a measure thattells us how compactly these objects fill the hostspace. One of the best quantitative indicators is the(proper) dimension ascribed to a fractal, just as onespeaks of proper length in relativistic kinematics, forinstance. One first defines the so-called Hausdorff β-dimensional outer measure of a set A, by introducingall possible coverings of A, with sets with diametersεi ≤ ε, for a given value of ε. If one designates thisfamily of coverings Γε

A, then the Hausdorff measureis (see, e.g., Mandelbrot 1983; Martinez and Jones1990, and references therein)

Hβ(A) = limε→0

(inf)ΓεA

∑i

εβi , β ≥ 0, (28)

where inf stands for infimum, the greatest lowerbound of a set of numbers. Then the Hausdorf di-mension of A, DH(A), is defined by the relations

Hβ(A) = ∞, β < DH(A), (29)

Hβ(A) = 0, β > DH(A), (30)One important property of the above defined

measure is that it is identically zero for a countablesets. Mandelbrot ’s definition of a fractal is that itis an object whose Hausdorff dimension is strictlylarger than its topological dimension. Thus, for theKoch’s curve shown in Fig. 2 in I, one finds D =1.262, instead of D = 1 (as we would expect from adecent line on a surface, or imbedded into 3D space.

If Cosmos is endowed with a hierarchical stru-cture, how can an observer from Earth notice it?

54


Generally, in the absence of an Observer, one mustresort to an indirect evidence. Before we proceedwith description of some methods employed, a pre-liminary consideration of formal and practical possi-bilities seems in order.

An (intelligent) observer living in a ND-spacecan infer by direct observations objects belongingto (ND − 1)-dimensional space. Since we live in athree-dimensional physical space (except, possibly,for the Flatlanders), we observe surfaces of (three-dimensional) objects. It is only through further pro-cessing of the (visual, dactylic, etc) data that weexperience the full voluminosity of the real worldaround us. If we are restricted to the visual infer-ence, as is the case with astronomical observation,we are left to the projection of the celestial systemsonto a surface. It is due to this sort of restrictionthat the Ancients conceived the Cosmos as fixed ona sphere, (which sphere in due course multiplied),but the surface-like picture (spherical models) dom-inated modern cosmology up to Kepler’s time. It isfrom this ”projection restriction” position that onehas to extend his experience in order to get a real-istic, many-dimensional picture of the Cosmos. Weshall enumerate the principal steps towards this goal.

(i) Moving from the projected to the cosmicdepths, by estimating distances from us to the celes-tial objects (static approach).

(ii) Leaving the picture of a static, geomet-ric structure, for a dynamic, time varying universe(kinematic description).

(iii) Accounting for the relativistic effects, ofboth Special and General Relativity, because of thefinite speed of light, which is the principal means ofobservation (relativistic picture).

The third point means one has to abandon,in principle, the Euclidian space and resort to anabstract, four-dimensional manifold. This point be-comes more and more relevant as one moves to moredistant objects, and after some (not necessarily stric-tly determined) limit, all three points become notonly relevant, but inseparably entangled. For in-stance, it is the kinematic of the universe that allowsus to estimate cosmic distances, via red shift, for ex-ample (whatever interpretation of the latter wouldbe).

The principal advances in elucidating the(phase space) structure of the cosmos have followedthis methodological distinction hronologically, as itcould have been expected. Thus, we start with thestatic universe.

The main sources of information for inferring apossible regular structuring of the universe are galac-tic catalogues (see, for instance, the seminal paper bySylos Labini et al 1998). But since one does not ex-pect an easy discernable cosmic structure, a suitablemethodology of extracting relevant features from theaccumulated data must be decided upon, prior to anydata processing. And it is here that controversiesarise.

As stressed by Ribeira (1994), there are twomethodological approaches that may be adopted.One is to start from a particular cosmological para-digm (even model) and try to recognize in the astro-

nomical records whether this can be ascribed to theobservable universe. The other approach would be toavoid any preconceived paradigm, or model, and seewhether a distinct structure emerges from the dataprocessing. These approaches could be considered asbelonging to deductive and inductive methods, re-spectively.

But which are principal cosmological para-digms available to us? The first is that based on thecosmological principles we mentioned before, that ison the assumptions of an homogeneous and isotropicuniverse. Both the Standard (Big Bang) and SteadyState theories belong to this paradigm. The otherparadigm is the concept of the fractal, hierarchicalcosmos. The question arises – are these cosmologicalpictures possible (not necessarily the only) alterna-tives, or could each be a part of a more general sit-uation. To put it in another way, could the universebe homogeneous at the very large (not necessarilyyet specified) scale, but possessing some discernablestructure at smaller scale? For we know that ourobservable part of the universe appears inhomoge-neous (what makes it termed cosmos, after all). Thisquestion points towards the old Boltzmannian ther-modynamic conundrum, why we happen to live ina structured part of the universe, which otherwiseshould be at the thermodynamic equilibrium. Butour issue is of somewhat different nature. It is notthe issue of a possible gigantic fluctuation that bringsabout self-similar cosmos, but the question -could, ifconfirmed by observational evidence, such a cosmoscoexist with an overall homogeneous universe? Thisis a tricky question, in particular considering thatthe universe appears structured as far as we (bet-ter to say - catalogues) see. We shall return to thisepistemological question later on.

6.2.1 Static (statistical) approach

First attempts to discerns a fractal, selfsimi-lar structure from the catalogues available aimed atrecognizing scale invariance as one moves to ever re-moter galaxies, by making use of the standard statis-tical methods. It turned out, however, that the latterwere not quite appropriate for the problem at hand(Pietronero 1987). The fractal cosmological modelappears extraordinary one, for it is conceived to fulfillan extraordinary task - to distribute a finite amountof cosmic matter over an infinite universe, with a reg-ular structure (which turns out to be isotropic, butnot homogeneous).

The primitive data from the catalogues arethe angular positions and red shifts of the galaxies.Two principal statistical means for processing thesedata are (i) an estimate of the average density, (ii)the correlation function. Since it is the gravitationalinteraction that is the driving force of the cosmicstructuring, one should start from the mass densityfunction

ρ(r) =N∑i

mi δ (r − ri), (31)

with δ as Dirac’s delta function, and galaxies situatedat ri. Since catalogues provide only positions, one

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P.V. GRUJIC

passes from the mass density over to the number den-sity

n(r) =N∑i

δ(r − ri), (32)

Then one calculates the correlation function, afteraveraging over angular variable,

ζ(r) =〈n(r0)n(r0 + r)〉r0 − 〈n〉2

〈n〉2 , (33)

where the average 〈···〉 is made over all referent points(points of origin) within the given volume V . Forrandom distributions the average of product is equalto the product of averages and ζ is identically zero.The joint (conditional) probability to find an objectin the volume δV1 and another object at δV2 is

P = 〈n〉2δV1δV2[1 + ζ(r12)], (34)where r12 is the distance between two volumes. Thefunction ζ is a measure of fluctuations within the vol-ume V , and its average value is zero. If any struc-ture appears, however, it is these fluctuations whichshould reveal it, by forming a regular pattern. Thisapproach, however, assumes the existence of the av-erage density 〈n〉, as an intrinsic property, indepen-dent of V (provided the latter is large enough tomake the statistics meaningful). The problem withthe fractal structures is that they have no such a(volume independent) property. In particular, frac-tal cosmos has been designed exactly to make thedensity tend to zero, as the volume increases to in-finity.

Fig. 1. Two consecutive hierachical levels of afractal cosmological model (schematically). (FromPietronero 1987)

In order to overcome this problem Pietronero(1987) has proposed a more appropriate approach,suitable for systems where an average density neednot exist. Suppose we have a self-similar pattern, asillustrated in Fig. 1.

If one starts from a referent sphere with ra-dius r0 which contains N0 objects, then within spherewith r1 = kr0 one finds N1 = kN0 objects, etc. Gen-erally, we have the relation

rn = knr0, (35)for the n − th radius, and the number of objectswithin r would be

N(r) = ArD , A =N0

rD0

, (36)

D =logk

logk, (37)

where D is another, more practical definition of thefractal dimension. If a sample of the cosmic objectslies within a sphere of radius Rs, the average densityis

〈n〉 =N(Rs)V (Rs)

=34π

AR−γs , γ = D − 3, (38)

An early estimate of γ was γ = 1.8 (Vau-couleurs 1970; see, also, Giavalisco et al 1989), im-plying D = 1.2. Since 〈n〉 in (38) depends explic-itly of the sample radius Rs, a more appropriatequantity, the conditional density, was introduced byPietronero (Pietronero 1987)

n(r) =1

S(r)dN(r)

dr=

D

4πAr−γ , (39)

where S(r) is the area of the spherical shell withradius r. Likewise, a more appropriate correlationfunction is introduced by the same author

Γ(r) =〈n(r0)n(r0 + r)〉r0

〈n〉 , (40)

which can be written as

Γ(r) =1N

N∑i

ni(r) = n(r) =D

2πAr−γ , (41)

and depending only on the intrinsic properties of thesample, not on its dimensions. Another useful quan-tity is the volume integral

I(r) =∫ r

0

Γ(r′)r′2dr′, (42)

Before we proceed with further elaboration of thestatistical tools, we mention that another quantity,the so-called radial distribution function is often used

g(r) = 1 + ζ(r), (43)which may be represented as a power-law function

g(r) = ArD2−3, D2 ≈ 3 − γ, (44)

56


where D2 is the so-called correlation dimension. Ge-nerally one has D2 < D, except for homogeneousfractals, when D2 = D (see, e.g., Martinez and Jones1990).

In attempting to reveal a structure of Cosmos,one could expect three possibilities with respect tothe fractal model. Either Cosmos is (i) scale- invari-ant, or (ii) it is such only up to some characteris-tic distance λ0, the so-called homogeneity scale, or(iii) the fractal structure has nothing to do with theUniverse. In the mixed case of a fractal embeddedinto otherwise homogeneous Universe, one has forthe conditional number density

n(r) =D

4πAr−γ , r < λ0, (45)

n(r) = n0 =D

4πAλ−γ

0 , r ≥ λ0, (46)

In such a case one would have a simple power-law behaviour of the volume integral

I(r) = ArD , r < λ0, (47)and

I(r) = ADλD0

[1D

− 13

(r3

λ30

− 1)]

, r ≥ λ0, (48)

The advantage of using these quantities will showup when considering the observational evidence, aswe shall see later on. The length-scale λ0 is relatedto a typical dimension of the largest voids observed.Another important quantity is the so-called corre-lation length rc, which separates the regions wherethere exist correlations of the density fluctuations(with respect to the average density), from the re-gion where these correlations are absent. Obviously,the existence of λ0 implies rc, otherwise the latter ismeaningless. Finally, we mention a statistical prop-erty described by the correlation length r0, definedby putting ζ = 1. Since it turns out that r0 dependslinearly on the sample radius Rs, it has no physicalmeaning in the cosmological studies.

6.2.2 Mass distribution and multifractality

If one accounts for the galactic masses then amore general fractal pattern may be expected. Onefirst introduces a normalized distribution that canbe turned into probability by dividing the mass dis-tribution (31) by the total mass within the sample(Pietronero 1987, Sylos Labini et al 1998)

µ(r) =1

MTρ(r), MT =

N∑i

mi, (49)

If the total volume is divided into boxes of linear sizel, one defines the function

µi(ε) =∫

i−th box

µ(r)dr, (50)

for ith box, where ε = l/L and 0 < µi < 1. Onedefines then the box-counting fractal dimension as

limε→0

µi(ε) ∼ εα(x), (51)

where x is the box position. In the case of a simplefractal one recovers the fractal dimension α(x) = D,but generally α(x) may fluctuate considerably. Ifwe have a number of boxes with the same measurescaling with α, these form a subset with dimensionf(α). For a simple fractal f(α) = α = D. From f(α)one can estimate αmin and αmax that correspond tothe largest clusters and voids respectively.

Self-similarity and fractality

The concept of self-similarity is wider than thefractal morphology. Consider, for example, a func-tion V (r), which may represent a spherically sym-metric potential. If this function appears in a power-law form, λ rκ, it is homogeneous with respect to thesimilarity transformations

V (θr) = θκV (r), (52)

where the real number κ is the order of the scalingtransformation, which multiplies the function by aconstant factor, but leaves the shape intact. Gener-ally, if a physical system has the potential functionof the pair-additive form

V ([rij ]) =N∑

i<j

cijVij(rij), (53)

where [rij ] stands for the set of all coordinates of thesystem constituents, it will be scale-invariant if allpair terms have the same order of scaling κ. Clearly,a system of gravitating bodies possesses this self-similarity property (as every Coulombic system does,see, e.g. Grujic 1993). These so-called homogeneoussystems, when subjected to the full kinematic scaletransformations exhibit a number of important prop-erties (see, e.g. Landau and Lifshitz 1976), whichmake them a special class of physical systems.Among all systems with interaction function of theform (53) two distinguish themselves further, thosewith κ = −1, 2. The first exponent defines New-tonian and Coulombic (long- range interaction) sys-tems, the second the harmonic potential system (likethe harmonic oscillator). For these two potentials,every (classical) orbit is closed. It is due to thisparticular property of the gravitating systems thatCharlier found that all bodies within a cluster haveclosed orbits, as we mentioned above. Moreover, asa generalization of Kepler laws, they all possess thesame period.

This scaling property was noticed by Laplace(1925), who wrote in his System of the World (see,e.g. Mandelbrot 1983)

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”One of [the] remarkable properties[of Newtonian attraction] is, that if thedimension of all the bodies in the uni-verse, their mutual distances and theirvelocities were to increase or diminishproportionately, they would describe cu-rves entirely similar to those which theyat present describe; so that the universereduced to the smallest imaginable spacewould always present the same appear-ance to observers. The laws of naturetherefore only permit us to observe rel-ative dimensions ... Geometers’ atte-mpts to prove Euclid’s axiom about par-allel lines have been hitherto unsuccess-ful ... The notion of .... circle does notinvolve anything which depends on itsabsolute magnitude. But if we dimin-ish its radius, we are forced do diminishalso in the same proportion its circum-ference, and the sides of all inscribedfigures. This proportionality seems tobe much more natural an axiom thanthat of Euclid. It is curious to observethis property in the result of universalgravitation”

An important consequence of these scaling-properties is that such a system has no characteristiclength (and no characteristic time-period, too). Allrelations above, like that in (47), have no character-istic lengths, and it is their exponents that matter(like D, not the prefactor A).

Lacunarity

The above statistical quantities are necessaryprerequisites for identifying eventual structuring ofthe observable part of Universe, but they are notsufficient. Namely, they describe a number of struc-tural features of a general class of physical systems,but these features need not specify cosmologicallyrelevant self-similar forms. In particular, fractal di-mension D does not determine topology of a fractal,as illustrated in Fig 2.

Fig. 2. Two deterministic (upper part) and tworandom (stochastic), (lower part) fractal structures(Mandelbrot 1983). First three fractals have the sa-me lacunarities, whereas the fourth has different one.

All fractals in Fig. 2 have the same dimen-sion D, but clearly very distinct topology. For thatreason Mandelbrot (1983) has defined the so-calledlacunarity F , by considering voids within the struc-ture, as

Nr(λ > Λ) = FΛ−D, (54)

where the lefthand side of (54) is the number of voidswith the size λ > Λ. Nr scales in the same way forboth deterministic (Cantor) sets, but the prefactor Fhas different values for two sets. As for the stochasticcases, a generalization of (54) is necessary, by intro-ducing the conditional probability P (λ)

P (λ > Λ) = FΛ−D, (55)

which gives probability that, if a box of the size εcontains points of the set, this box has a neighbour-ing void of the size λ > Λ.

Orthogonal projection

As discussed previously, we observe directlyangular distributions of celestial objects, that is aprojection of three-dimensional structure onto a pla-ne. What such a projection preserves of the originalstructure embedded in three-dimensional real spaceis the size of objects. If a fractal of dimension Dis projected from d = 3 to d′ = 2 subspace, theprojected structure has dimension D′ such that (see,e.g. Sylos Labini et al 1998)

D′ = D, if D < d′ = 2;

D′ = d′, if D > d′ = 2,(56)

Thus, clouds, with D ≈ 2.5 shed a compact shadowwith D′ = 2. Likewise, a fractal cosmos with D ≥ 2will exhibit a uniform angular galaxy distribution.

6.2.3 Dynamic effects

The universe is a dynamical system. Not onlyit is subjected to the universal Hubble flow (whateverits interpretation might be), but a number of corre-lated or erratic motions are superimposed upon theoverall expansion, as observed from the co-movingreference system. If a self-similar Cosmos does ex-ist, the question arises as to its time evolution, apartfrom the overall expansion. As far as we are aware,this question has not yet been addressed fully. Weshall content ourselves here with mentioning a fewrelevant topics.

We first notice that erratic movements of ga-laxies should not contribute to forming or destroy-ing eventual self-similar (or any other, for that mat-ter) structure. As for the collective movements ofgalaxies and clusters, they might be part of globalstructuring process. Clearly, as we move from thenearby surrounding and local cluster to the deepspace, the erratic motion looses its significance andthe remote parts of the universe appear evermorestationary (though not static).

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Fig. 3. A survey sample (schematically). Inclusion of point A yields a spurious contribution to the averagedensity.

Now, one might ask the question whether the-re is a more general self-similarity, within a phase-space of the universe. Whether there is a collectiveexpansion of a part of the cosmos, similar to Hub-ble flow? Or a global infall of galaxies, or clusters,resembling the Big Crunch? After all, it is this mech-anism that gives rise to the assumed galactic blackhole formation.

Surely, a more complete dynamic picture ofthe universe will clarify the issue as to the construc-tive or destructive role of the overall cosmic expan-sion regarding cosmic structuring, including fractalone. As pointed out by Ribeiro (1994), a deeperstudy of the (nonlinear) field equations might providea clue to the puzzle of fractal pattern. From numeri-cal calculations one may infer that the systems understudies exhibit structural fragility. Moreover, it iswell known that strange attractors of dynamical non-linear systems have fractal patterns in phase space.The question arises as to the possibility that tracingthe fractal structure along the past null geodesic one

might encounter such an attractor. And what wouldbe the connection between Lyapunov exponents andfractal dimensions within this context? These arethe questions that might be waiting for answers inthe near future of cosmological studies that go be-yond the geometrical, cosmographical scope.

7. THE OBSERVATIONAL EVIDENCE

In the previous section we discussed a numberof properties that Cosmos might possess, if it is atleast partially fractal. We consider now the questionas to what would be the observational evidence ofsuch a self-similar structure. In a sense, this issue isrelated to the problem of theory of measurement, asraised in other sciences of Nature, notably in Quan-tum mechanics. In the case of cosmology, it mightbe called theory of observation. We first discuss theboundary problem.

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7.1 The boundary effects

If the self-similar structure is limited to a finalcosmic volume, as determined by the homogeneityparameter λ0, one would expect that the conditionalaverage density Γ(r) becomes constant as one passesthe region r ≈ λ0. As we shall see, the galaxy surveysdo exhibit this sort of behaviour, but it turns out tobe spurious effect, due to a number of improper pro-cedures employed (see, e.g. Sylos Labini et al 1998).One of these misleading effects is due to only partialinclusion of the spheres where an average density isto be estimated. In Fig. 3 we show schematicallythe situation one encounters while processing obser-vational data from the deep sky surveys.

As shown by Pietronero and coworkers (see,e.g. Sylos Labini et al 1998), the observed flatteringof the correlation function is due to the final sizesamples, and not a genuine effect that would pointtowards a uniform distribution of galaxies at largescale. Only spheres which lie completely within thesurvey volume, as indicated by shaded areas in Fig.3, should be included in the relevant data.

We saw earlier that the correlation parameterrc is defined by the condition ζ(rc) = 1 (see (33)).Three cases are to be considered here (Pietroneroand Sylos Labini 2000).

(i) Rs .′ λ0 < rc (fractal distribution)Then the correlation function (41) behaves as

Γ(r) ∼ rD−3. (57)If there is a crossover to homogeneity (Rs

λ0) one distinguishes two cases.(ii) The sample behaves as a fractal up to a

certain distance λ0 (that is Γ(r) behaves as a powerlaw), but becomes homogeneous at scales Rs > r λ0. If there is a lower cut-off of the fractal patternrl, then one has

Γ(r) ∼ rD−3, rl ≤ r .′ λ0, (58)

Γ(r) ≈ 〈n〉Rs , λ0 .′ r ≤ Rs, (59)

If the sample is large enough, one has(iii) λ0 .′ Rs < rc Then an average density

becomes meaningful and the (standard) correlationfunction may be used

ζ(r) =(

r

λ0

)−γ

η(r), (60)

with η(r) an oscillating function, which describesfluctuations with respect to the average density〈n〉Rs .

7.2 Global effects

These fall generally into two categories, whichwe attribute, somewhat arbitrarily, to the cosmogo-nic and relativistic effects, respectively.

7.2.1 Cosmogonic effects

If the standard, Big Bang scenario is adoptedas a realistic description of the birth of Cosmos, thenit follows that the early Universe started with ap-proximately homogeneous distribution of matter, inthe very general sense of last term. Hence, whilelooking at the very distant objects, say 1010 lightyears away, one looks into the epoch when the presentday structure has not yet been formed. Consequ-ently, deep sky surveys should reveal a homogeneousuniverse, whatever the subsequent evolution of thecosmic material content would be. (See, however,chapter 5.2.1.). Moreover, since we can not have asnapshot picture of the Universe, there is no way toinfer directly an eventual structure of the Cosmos ata constant time (Cauchy) surface.

7.2.2 Relativistic effects

These are discussed by Ribeiro (2001a,b),within the context of the so-called Fractal Debate,led between the (orthodox) supporters of the stan-dard, Friedmannian paradigm and those who arguefor a (possibly infinite) fractal matter distribution.By examining carefully the meaning and use of anumber of astrophysical constructs, like the meandensity, cosmological distances, etc, Ribeiro arguesthat the apparent conflicts between two schoolsmight not be so severe, as the case might seem. Witha full account of the relativistic effects, a numberof the observational data should be reassessed andreinterpreted. The issue at stake is the question,though not explicitly stated, whether FriedmannianCosmos could yield an inhomogeneous observationalfactography and vice versa, whether a fractal Cosmosmight look like a homogeneous system according toour observational evidence.

7.3 Fractal or nonfractal, what the sky says

We give here a short overview of the presentobservational situation concerning the galaxy distri-bution and an eventual evidence for the (multi)frac-tal pattern. For a more detailed account we direct in-terested readers to the seminal article by Sylos Labiniet al (1998).

Generally, it turns out that most of the ob-served galaxies do not belong to clusters (c. 70 %).Isolated galaxies (not in groups or clusters) are veryrare, too. As for the galaxy distribution with regardto a particular morphology of a subpopulation (Hub-ble classification, galaxy size, mass, luminosity etc),which complicates greatly an overview, we shall re-strict ourselves to the most general features of thelarge-scale distribution. We mention here only thatthe fact that the most of the giant galaxies are mainlyclustered (Luminosity segregation), unlike the dwarfgalaxies. This fact was interpreted previously as a

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consequence of the larger amplitude of the correla-tion function, but has been reinterpreted by the sup-porters of the fractal cosmos as the feature of theexponent parameter of the distribution, independentof the sample size, as discussed earlier.

An analysis of 11 catalogues available has re-vealed that there is a strong correlation between thedistance and density function, as shown in Fig. 4from Sylos Labini et al (1998).

Fig. 4. Correlation analysis of relevant samplestaken from various red-shift surveys. The referencestraight line with the slope −1, which corresponds tothe fractal dimension D = 2, is shown. In the inset,results for insufficiently large distances are shown,affording a spurious (approximately) constant den-sity (from Sylos Labini et al 1998).

As can be seen from Fig. 4, the overall pic-ture, with a wide span of distance (0.5−1000Mpc/h),with Hubble constant h in units 100 Mpc/km/s (withmeasured value h ≈ 0.65), fits well the fractal pat-tern. It turns out that all surveys are mutually con-sistent and point to the D = 2 ± 0.2 fractal dimen-sion of the observable part of the Cosmos, with aclear fractal structure within (0.5 − 150Mpc/h) re-gion. As argued by Pietronero and coworkers, theproper use of the statistical analysis removes a num-ber of inconsistencies and difficulties encountered inearlier interpretations of the available data. By elim-inating all quantities depending of the size of thesample considered, the so-called galaxy-cluster mis-match (paradox) is resolved. This consists in thediscrepancy between conclusions regarding the cor-

relation length. Namely, it turns out that when theclusters are considered only, the fractal pattern isdetected up to a particular λ0, but it differs greatlyfrom the corresponding estimate for the superclus-ter analysis, etc. Once the generalized correlationfunction Γ(r) is used, instead of the standard oneξ(r), the discrepancy disappears. This case demon-strates the fact that the analysis based on a partic-ular premise is bound to yield a result that mightdiffer greatly from another interpretation which re-laxes the premise adopted in the previous one. Thepresent estimate is that λ0 ≥ 50 Mpc/h (Pietroneroand Sylos Labini 2000).

The most recent analysis of the CfA2-southredshift survey data confirms the fractal structure atleast up to 20 Mpc/h (Joyce et al 1999a; see, also,Joyce et al 1999b). The fact that the observed fluc-tuations around the average counts of galaxies are ofthe same magnitude as the counts themselves arguesin favour of the fractal pattern at all scales available(Gabrielli and Sylos Labini 2001).

In fact, a more scrutinized analysis revealsthat the data follow a multifractal distribution ratherthan a clear fractal pattern (Martinez and Jones1990; see, also, Sylos Labini et al 1998). These au-thors argued, however, that structures on scales lessthan 5 h−1 Mpc are more complex than the homoge-neous fractal structure, and the Housdorff dimensionfound is sheet-like D ≈ 2.1 ± 0.1).

In the most recent analysis Bak and Chen(2001) argue that the luminous matter in the uni-verse is distributed according to a multifractal pat-tern up to a certain distance, and then a crossoverto homogeneity is predicted. They applied a ”forest-fire” model, developed previously for the reaction-diffusion process in turbulent systems, and definedan apparent dimension D(�), which depends on thescale �. According to their findings, this generalizeddimension rises linearly with logarithm of the scaleand at λ0 ≈ 300 Mpc a crossover to the homogeneousdistribution, characterized by D = 3, is predicted, asshown in Fig. 5.

Fig. 5. Scale dependent dimension D(�), derivedfrom three different samples (Bak and Chen 2001).

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The authors conclude that:

The geometry of the luminous set isnot fractal when viewed over the entirerange of scales, since there is no self-similarity for different scales. It is nothomogeneous either. The scale depen-dent dimension has a clear geometricalinterpretation: At small distances, theUniverse is zero-dimensional and point-like. Indeed, energy dissipation takesplace on individual pointlike objects,such as stars and galaxies. At distancesof the order of 1 Mpc the dimension isunity, indicating a filamentary, string-like structure; when viewed at largerscales it gradually becomes 2-dimensi-onal wall like, and finally at correlationlength, ξ, it becomes uniform.

Hence, the issue about the global structureof the Universe is far from being settled. The cur-rent debate moves around the questions, as put byPietronero and Sylos Labini (2000), of (i) the properstatistical methods employed in analyzing the obser-vational data, of (ii) the implications of the observedfractal structure up to a certain scale λ0 and (iii)what would be the reliable estimate of this homo-geneity scale? The exact value of the fractal dimen-sion D is still uncertain, as well as the role of thecosmic dark matter, as discussed by various authors(see, e.g. Sylos Labini 2000). The question of thefractal structure genesis is still lacking a reliable ap-proach, though methods developed in the statisticalphysics, such as self-organized criticality, have beeninvoked in order to understand the evolution of theself-gravitating systems (see, e.g. Sylos Labini andPietronero 2001).

The search for the fractal properties of the ob-servable universe continues, both observationally andtheoretically. In a recent paper by Gaite and Man-rubia (2002) scaling of the cosmic voids has beenexamined within the model of random fractals, withnonconclusive results. The concept of a fractal uni-verse has been also tested against the more exoticcosmological paradigms, such as Linde’s selfrepro-ducing, eternal, stochastic inflation (see, e.g. Wini-tzki 2001). On a more epistemological than technicalbasis an interesting parallel with the anthropologi-cal aspect of the fractal paradigm has been madeby Zabierowski 1988), who compares the latter withBoltzmannian solutions to the cosmic time arrow co-nundrum of the time.

8. SUMMARY AND CONCLUSIONS

We have shown how the concept of a fractalcosmos was gradually developed from Renaissanceto the present time. It occupied some of the mostpowerful minds in Europe at the time, from Leibniz,Kant, to Laplace, but these ideas were formulated insuch terms that one could hardly speak about defi-nite cosmological models. The latter arose at the be-

ginning of 20-ieth century, first paralleling the rulingrelativistic cosmological paradigms, and then start-ing interfering with them. In the last period of thecosmological studies this self-similarity pattern begincompeting with the standard relativistic paradigm,and the issuing controversy may be put in the fol-lowing terms.

(i) Can the Cosmos be conceived as a fractal,self-similar structure, without violating the standardCosmological principles, to which almost all contem-porary models adhere?

(ii) What should be observational evidencethat could decide which paradigm is realized?

(iii) Could both paradigms coexist, or put inthe following terms - are these two paradigms justdifferent aspects of the unique reality, or are theyrival, if not antagonistic alternatives?

These are the current dilemmas, that run par-allel with the growing observational evidence thatcosmos is far from being smooth and homogeneous,within our present observational limits. We haveseen how much the a priori concepts have influencednot only the interpretations of the existing data, butthe very gathering of the latter. As pointed outby Pietronero and others, a particular prejudice willyield its own observational evidence. That the obser-vation is as much recognizing a preconceived modelas discovering a new fact is not bound to cosmologyonly, and is just a part of a more general epistemo-logical issue. Relaxing any principle, or cosmologicalpostulate, would surely make the cosmology more ascience about Nature, and less a theoretical game.

The fractal concept does not explain how thisstructure has arisen, but the cosmogony appears aweak point of any other approach, though the stan-dard Hot Big Bang paradigm claims a scientific sta-tus. As pointed out by many authors (Alfven 1976,Lurcat 1978, Disney 2000) the reliable observationalevidence is still too weak to support farfatching theo-retical speculations (but see Cirkovic 2002). Not onlythe current evidence based on the cosmic sources ofthe electromagnetic radiation is still insufficient toensure a convincing choice between the current com-peting alternatives, but other hypothetical entities,like the dark matter in various forms (e.g. Grujic2002) enter the game. The question whether thisnew cosmic matter/energy component, if detected,would alter radically the current models, includingthe fractal paradigm, is surely the next topic to beconsidered seriously by cosmologists (see, e.g. SylosLabini et al 1998).

The concept of a hierarchical, self-similar cos-mic structuring possesses a number of remarkableproperties that make it an ingenious solution of theprincipal conundrum in cosmology, namely how toconceive an infinite world different from the triv-ial, Abderian extensive model. This problem hasbeen tackled successfully within the Einsteinian rel-ativistic paradigm, with space-time coupled intrinsi-cally with the content of the universe. The fractalparadigm answers the same question in an essentiallydifferent, but equally selfconsistent way. The conceptof selfsimilarity tackles the cosmological puzzle of aninfinite universe in a similar way as Cantor’s set

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theory deals with infinities within the number the-ory. The matter distribution in the universe appearsdiluted in a sense exponentially, just as the field ex-pansion occurs exponentially fast within the infla-tory paradigm. That the Copernican principle is pre-served within the paradigm of scale-invariant cosmosappears a remarkable property of the fractal modeltoo.

Going back to the Presocratic era and to Ana-xagoras’ concept of the selfsimilar cosmos, two thingsare to be observed. First, though one might specu-late about the outward self-similar cosmic pattern,that is from the mesocosmos to megacosmos, Ana-xagoras was primarily concerned with the structur-ing towards the infinitely small. Modern cosmologi-cal thoughts are mainly oriented towards larger andlarger cosmic dimensions, possibly to infinite ones(but see Oldershaw 1989). The principal reasonsfor this are the quantum mechanical restrictions, im-posed to the ultimate dimensions in the microcosmicworld. It turns out that the competition betweenAbderian and Klazomenian solutions has finished indraw. The atomic paradigm has set a barrier on theroad towards infinitely small, whereas the selfsimilarpattern has enabled us to conceive an infinite uni-verse with a limited content of matter.

There appears a remarkable aspect of the his-tory of the fractality in nature, that the principalcontributors to the hierarchical cosmos failed to re-fer to their predecessors. Leibniz did not mentionAnaxagoras, Kant neither Anaxagora nor Leibniz,etc. It would be surely an interesting issue to pur-sue an answer as to the question whether it was adeliberate neglect or just the matter of ignorance.In the latter case one might argue that the conceptof selfsimilarity appears an inherent content of ourmind, and hence less one of the speculative outcomesof our endeavors to conceive the World in its total-ity. As for Anaxagoras it seems that his contributionto the concept of the selfsimilar Cosmos has been al-most totally forgotten, with rare exceptions (see, e.g.Markov 1990).

The modern cosmology has extended its do-main well beyond the borders that might be consid-ered purely scientific. It includes at the very frontline such disciplines like philosophy (see, e.g. El-lis 1999), religion, esthetic, political ideology (see,e.g. Naddaf 1998, for the case of Anaximander; see,also, Cirkovic 2002), etc. In a sense, modern cosmol-ogy turns out equally parascience at its outermostborders, as it was during the whole period of Euro-pean culture development, from the archaic Greece,to the ”Pre-Big-Bang” theoretical speculations to-day. There is nothing wrong with this, of course, aslong as one is aware of these distinctions and doesnot proclaim speculative thought a scientific theory.The role of the fractal paradigm, besides its posi-tive contribution to the cosmology as such, is just toput more weight on the physical, observational, andepistemological aspects of our search for a definitepicture of the Universe, i.e. to the classical cosmol-ogy in general.

Acknowledgments – I am grateful to Dr Milan Cirko-vic for drawing my attention to a number of relevantpapers and for valuable discussions on the subject.I am much indebted to many authors for supplyingtheir papers, in particular to Professors Sylos Labini,L. Pietronero and M. Ribeiro, for sending me theirrecent publications. The work has been partly sup-ported by MSTD of Serbia.

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KONCEPT FRAKTALNOG KOSMOSA: II. MODERNA KOSMOLOGIJA

P. V. Gruji�

Institut za fiziku. P. F. 57, 11080 Beograd, Jugoslavija

UDK 524.83Pregledni qlanak

Razvoj koncepta fraktalnog kosmosa po-sle Anaksagore pra�en je do danaxǌeg vre-mena. Pokazano je kako se koncept pojavio po-novo u ranoj Renesansi kao maglovita ideja,da bi poqetkom 20. veka dobio konkretnu for-mulaciju. Staǌe moderne kosmologije razma-

trano je s’ taqke gledixta fraktalne paradi-gme i diskutovane su teku�e kontroverze i po-lemike. Pokazano je da je koncept hijerarhi-jskog kosmosa jox uvek �iv i da mo�e da budebitan elemenat moderne slike svemira.

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