The Concept of Fractal Cosmos: Iii. Present State

Serb. Astron. J. } 182 (2011), 1 - 16 UDC 524.88 DOI: 10.2298/SAJ1182001G Invited review

THE CONCEPT OF FRACTAL COSMOS: III. PRESENT STATE

P. V. Gruji´c

Institute of Physics, P.O. Box 57, 11080 Belgrade, Serbia E–mail: [email protected]

(Received: November 2, 2010; Accepted: November 2, 2010)

SUMMARY: This is the sequel to the previous accounts on the rise and development of the concept of fractal cosmos, up to year 2001 (Gruji´c 2001, 2002). Here we give an overview of the present-day state of art, with the emphasis on the latest developments and controversies concerning the model of hierarchical universe. We describe both the theoretical advances and the latest empirical evidence concerning the observation of the large-scale structure of the observable universe. Finally we address a number of epistemological points, putting the fractal paradigm into a broader cosmological frame. Key words. Cosmology: miscelaneous

1. PROLOGUE pictures do not appear as clear cut patterns, and the same holds for the entire Cosmos as well. In the previous reviews of the fractal cosmol- Fractals appear ubiquitous in Nature, Arts ogy we have shown that the concept of hierarchical and Science. To numerous well known examples structuring was deeply rooted into mind of philoso- (see, Mandelbrot 1983) some new cases may be phers and scientists, from the Pre-Socratics to the mentioned, like molecular biology (DNA) (Zuo-Bing modern time (Grujić 2001, 2002). It appears somewhat ironic that it was the modern advance on the 2003), linguistic (Chinese letters) (Johnston 2008), subject of fractality that has made us appreciate the paintings (Taylor et al. 1999), (Mureika 2005), optics old achievements of our ancestors. With the risk of (Berry et al. 2001), field theory (Casimir effect) (Fu- assigning our concepts to the past times (Whiggish naro 2009), superconductivity (see, e.g. Cartlidge syndrome) it was only after developing more precise 2010), solid state physics (see, e.g. Pacey 2010), ideas of the fractality that we are able to recognize science of material (Farr 2009), to mention some of the old concepts akin to that. It concerns partic- them only. There is no clear connection at a fun- ularly Anaxagoras’ homoeomerias, which perplexed damental level that would make these cases result- his contemporaries. Another great contribution to ing from an underlying principle or fundamental law, the concept the hierarchical structure of the world but the diversity of cases and their proliferating num- was due to Kant, whose ideas on the subject passed ber points toward something deeper than the mere almost unnoticed, overshadowed by his cosmogony phenomena. Cosmos turns out the largest object (Kant 1968). Einstein himself was aware of the hi- possibly endowed with fractal, or at least hierarchi- erarchical cosmos, but dismissed the idea on some cal structure, with intriguing properties, which pose general grounds (Einstein 1922). The latter, how- a number of questions, both at the ontological and ever, should not disturb the ”fractal community”, epistemological levels. Before we go on, it is fair to since we know the first cosmologist of the previous notice that in many of the quoted instances fractal century used to reject many other models (like those 1 P. V. GRUJIC´ due to Friedman and Lemaitre), not to mention his not yet been realized (see, e.g. Baryshev 1999, 2005 own cosmological constant Λ. The most significant and Baryshev and Teerikorpi 2002) for a more pop- contribution to the modern paradigm of hierarchi- ular account). The case of parallels in geometry cal cosmos was due to Charlier in 1922, who put the was ”solved”, or resolved by abandoning the Eu- model on the solid astrophysical grounds (Charlier clidian paradigm of ”flat space” (fourth postulate 1922). about right angles) and allowing for other possi- Equally great contribution to the very con- bilities, which resulted in constructing hyperbolic cept came from mathematicians from the last cen- and spherical geometries of Lobachevski and Rie- tury. In fact, the first mentioning of fractal concept mann, respectively. Formally, hierarchical cosmos could be found with Plato, (Plato, Republic, Book was designed as a response to Eleatic challenge (see, VI,509; see, e.g. Jaeger 1973). We first mention e.g. Kirk et al. 1983) by Anaxagoras, who solved Felix Hausdorff, who, at the very beginning of 20th the problem of filling an infinite space with a finite century, introduced a new kind of dimension, now amount of matter (see Grujić 2001 and references called Hausdorff’s dimension and which Mandelbrot used for the very definition of fractal objects (see, therein). There is no appealing need for a hierarchical e.g. Grujić 2002). Cosmos, as there was no need for non-Euclidian ge- During the latest decade considerable advance ometries. Yet it is a traditional wisdom that every was made with regard to the theoretical cosmol- possibility should be examined and no cosmological ogy and astrophysical evidence of the cosmos at model, even paradigm, should be a priori rejected. large. Though not epoch making discoveries may Further, as the observational technic is advancing, be claimed, a number of important results have been this need may appear soon enough that the fractal achieved in laying down fundamental background of (or some similar) paradigm becomes compelling. the hierarchical structure in general and fractal cosmology in particular. On the observational side the main event was discovery of the accelerating uni- 2.1. Preliminary remarks verse, which has provoked some of the radical amend- ments to the prevailing Standard Theory of the cos- Any attempt to conceive the Universe as a mic dynamics. Cosmos, that is to ascribe to the totality of our phys- On the empirical side three principal goals ical environment a particular structure, is based in- have been strived for: (i) the upper observational evitably on a number of premises, upon which one limit Rup beyond which a homogeneous distribution tries to contrive a particular cosmological model. of the cosmic matter, mainly galaxies, may be ac- These premises are called, in this particular case, cepted; (ii) the nature of the inhomogeneous distri- cosmological principles or cosmological postulates. bution particularly in view of the possible hierarchi- These principles are then used as guiding rules for cal structure; (iii) the accurate value of the fractal di- elaborating fine details of the model under construc- mension D for R R . As for the theoretical side, tion (see, e.g. Grujic 2007). Their role appears at f ≤ up least twofold. First, cosmological principles identify a number of monographs has been published which the basic ideological status of the cosmologists and have advanced our understanding of the study of cos- second, they introduce simplifications into the formal mic structure in general. We mention first the mono- (mathematical) approach, enabling one to produce fi- graph (Gabrielli et al. 2005) which presents a thor- nal formal solutions of the structure one is searching ough approach concerning the nature of the statisti- for. cal methods used in detecting and interpreting the The essential simplification in contriving Cos- cosmic matter distribution and the books by Nottale, mos is to ascribe a particular symmetry to the Uni- (Nottale 1993, 2011) which lay down the foundations verse. The choice is dictated by a number of premises of the fractal structure of space and time at the very one adopts, intentionally or implicitly. In an ab- fundamental level. We shall discuss these and some stract sense these symmetries may be divided into other important contributions to the subject later two broad classes according to the attributes: homo- on. In the next Section we discus a number of gen- geneous and inhomogeneous systems. eral theoretical aspects, and in Section 3 a heuristic parallel between a specific case of the atomic dynamics and the Standard Cosmological Model (SCM) is 2.1.1. The concept of homogeneity made. In Section 4 theoretical advances have been presented and in Section 5 modeling fractal universe The notion of cosmic homogeneity appears and the relevant observational evidence of the cosmic subtle and deserves some further elaborations. Be- structure is analyzed and discussed. Section 6 is ded- fore talking about cosmic structuring, one must first icated to considering some epistemological questions specify the scale of the space (physical or other- and concluding remarks are given. wise). We know that the universe is not, and can not be homogeneous on the microscopic, mesoscopic 2. THEORETICAL CONSIDERATIONS and macroscopic scales, for it is the inhomogeneities on these scales which make the nontrivial structur- The case of the fractal cosmology resembles ing, including that of life, possible. On the truly much the story of parallels in geometry: though con- cosmic scale, however, the notion of homogeneity be- ceived even in Antiquity, it has been running in par- comes vague, if not problematic (as we shall see later allel with the prevailing (”standard”) paradigm since on). The issue resembles much that of the theory of then, with the prospects of ” crossing” that have gaseous matter phase, which was initially developed 2 THE CONCEPT OF FRACTAL COSMOS: III. PRESENT STATE for the simple quasi-homogeneous state, close to the (ii) Space homogeneous thermodynamical equilibrium. Only after this sim- Majority of cosmological models belong to this ple case was understood, the theory of inhomoge- class including the above-mentioned Einstein’s static neous gases, far from the thermodynamical equilib- cosmological model and Lemaitre’s model (see Chap. rium, was developed. With this analogy in mind, one 4.1). may consider the Universe as a cold gas, a collection of galaxies as the principal elementary constituents, (iii) Space-time homogeneous playing the role of atoms. We shall, in the following, The original de Sitter’s model implied ho- distinguish three principal cases: mogeneity within the abstract space-time, but as Lemaitre found out later, its spatial part turned (i) Quasi-homogeneous systems out inhomogeneous (see, e.g. Nussbaumer and Bieri If one can define an average inter-galactic mu- 2009). tual distance, dqh, on the scale with characteristic length an order of magnitude larger than dqh, and (iv) Space and time homogeneous we partition the entire cosmic space into cells of such Einstein’s first cosmological model satisfied dimensions, one may speak of a homogeneous distri- the so-called Perfect Cosmological Principle, with bution if, on the average, each cell contains approxi- the universe remaining the same and homogeneous mately the same number of galaxies. all the time. (ii) Semi-homogeneous systems (v) Scale homogeneous (invariant) If one can determine the length λ so that for This is the case we shall be interested in here. scales d λ an inhomogeneous system becomes ho- It implies that the matter distribution in the cos- mogeneous,≥ one may speak of a semi-homogeneous mic space is such that a series of scales may me structure. Strictly speaking, this definition appears defined with (approximately) the same structure at trivial, in the sense of the case (i) above, but, as we each level. If we have various discernable levels, shall see in the following in the cosmological struc- but different structuring at different scales, one deals turing, it has a real meaning. with a hierarchical model. If the same structuring is repeating at each scale, one speaks of a fractal model. (iii) Essentially inhomogeneous systems In the reality situation appears somewhat more com- If there is no finite λ in case (ii), i.e. the sys- plicated, and more complex models, like multi-fractal tem is inhomogeneous at all scales, we speak of an es- ones, are invoked to describe the actual state of af- sential inhomogeneity. It is exactly the case we shall fair. be considering here with regard to the cosmological models and observations. Another definition of 2.1.2. The concept of symmetry an essentially inhomogeneous system is that nowhere within the system the average density can be determined. Symmetry refers primarily to the geometric properties of real or abstract systems, but the no- Generally, these definitions need not necessar- tion has its counterpart in the dynamic, better to say, ily refer to the real space or physical systems, but kinematic sector. Symmetry operations, like transla- rather apply to the formal, geometrical or mathe- tion, rotation etc. do occur in real systems, though matical in general, aspects of the real systems. It is not in all of them. It was a great formal achievement in this abstract sense that the hierarchical structure, of Emmy Noether to link the translation in space which may be based on the so-called scaling symme- with the conservation of impulse, and time homo- try, can be conceived as scale homogeneity. As elab- geneity with the energy conservation (e.g. Goldstein orated elsewhere (cf e.g. Grujic 2007) all present-day 1981). cosmological models, or better, paradigms, may be Scaling or the so-called self-similarity transfor- divided into the following classes, according to the mations are abounding in nature. Growing of living symmetry properties: creatures, both plant and animal, is a good exam- ple of changing size, while (approximate by) keeping (i) Time homogeneous the same form. Stalactites and stalagmites are good The universe remains the same all the time, examples in the nonliving world. After astronomical whatever its structure may be. Einstein’s first cos- discovery of the expanding universe in late twenties mological model, the static universe, belongs to this class of time-invariable universe. The original so- of the last century (e.g. Nussbaumer and Bieri 2009), called steady-state models (Narlikar 1977) described we witness that the entire cosmos undergoes global scaling towards ever larger dimensions. As we shall a time-invariant universe, locally and globally alike. see later on, the hierarchical model has its dynamical With the discovery of the universal expansion (see, extension, just as Friedmann and Lemaitre found for e.g. Nussbaumer and Bieri 2009 for a detailed ac- Einstein’s original static cosmos. But which physi- count) the original model was modified so as to ac- cal consequence stems from the scaling symmetry? count for the global change in time but retaining the We shall address this issue later on. Here we shall local constant matter density. So one may speak of address a parallel between the Coulombic and self- a semi-stationary cosmos. gravitating systems (see, e.g. Grujic 1993).

3 P. V. GRUJIC´

3. SMALL-ENERGY SYSTEMS 3.1. Lemaitre’s universe

In 1927 Belgium cosmologist, reverend George One defines dilatation transformations by Lemaitre, found remarkable solutions to the Einstein General Relativity cosmological equation, not being r θr, (1) → aware of the previous results due to Alexander Fried- which expand or shrink the physical system. We mann (see, e.g. Nussbaumer and Bieri 2009). These call homogeneous functions those with the properties dynamical solutions described the universe which ex- (Landau and Lifshitz 1976) pands or shrinks, depending on the initial conditions with the original Einstein’s static solution as a spe- V (θr) = θλV (r), (2) cial case. But Lemaitre went further than Friedman, contriving a physical model of a dynamical universe. where the real numbers θ and λ are the The expansion begins from a state of enormous, but scaling parameter and degree of homogeneity re- finite, density, and almost infinitely small dimen- spectively. A physical system whose potential func- sion. This primordial state he called cosmic atom, tion has the properties as in Eq. (2), scales under inspired by the remarkable contemporary advance these homothetic transformations: of the atomic physics, due to the application of the the time as newly invented Quantum Mechanics. By invoking the quantum mechanical nature of this initial state t θ3/2t, (3) → universe Lemaitre proposed what we could consider the energy as the forerunner of the modern Big Bang cosmological paradigm. By undergoing an explosion, constituents E E/θ . (4) → of the primordial atom flew away, making the uni- A mechanical classical system, whose pairwise verse expand, as we observe it today. One remark- inter-constituent interactions are described by: able feature of this model is that it recovers space homogeneity (a cornerstone of the majority of cosmo- a logical models)- every point in the (physical) space V = ij , (5) i,j rλ appears equivalent and an observer at any point sees ij other constituents fly away from him. has the potential function In the following we need to note that these constituents are galaxies. Further, according to the present-day theoretical elaborations and obser- V = X Vij. (6) vational evidence, our universe seems to possess zero i

Wannierian threshold law (Dimitrijevic et al. 1994). tem (compound state) occurs without central (priv- Finally, if the effective central charge assumes a neg- ileged) body making the entire system more demo- ative value, the interaction with the outgoing elec- cratic than in the atomic case. On the other hand, on trons becomes repulsive and another threshold law an intermediate global scale, ”local” deviations from follows. The latter case resembles the cosmological the overall Hubble flow may provide situation similar model with the dark energy, which pushes galaxies to that of Wannier’s model. We may imagine places from each other, causing the overall acceleration of like the centre of the Great Attractor, (with overall the expanding universe as has already been observed infall of the cosmic matter towards the attractor cen- (see, e.g. Schneider 2006). tre), from which radical deviations from the global red-shift we experience today may be observed, in- 3.4. Microcosmos versus megacosmos cluding the possibility of a systematic blue shift. In a civilization at the technological level similar to ours We have dwelled over the parallel microcosmos from the beginning of 20th century the overall out- - megacosmos so as to emphasize the unique posi- look of the surrounding cosmos may be verry dif- tion cosmology has among all other physical systems ferent from our present-day image of the Universe. studies from the methodological and epistemological We shall return to this local deviation effect when point of view. The central dogma of any cosmologi- considering the possible mechanism for forming hier- cal model is the Copernican principle, which excludes archical structures. any special position in the cosmic space (Weak cos- Before we go on, let us note that time evo- mological principle). The universe expands, not from lution of the atomic and cosmic systems differ es- a single point, but from any point. The only vari- sentially. In the microscopic world we prepare the able is the so-called scale parameter S(t), distance initial state of our entire system (impinging particle between neighbouring galaxies. The effect of galaxy and target), with the compound, presumably chaotic clustering, opposing the overall Hubble flow (global system being an intermediate phase in the overall expansion), competes with the effect of the mutu- evolution. On the other hand the cosmological ”ini- ally receding cosmic constituents. Hence, the overall tial state” (like Lemaitre’s ”cosmic atom”) appears uniformity in the cosmic matter distribution, as as- ”put by hand”, at least within the Standard model sumed by majority of the cosmological models, can (the Big Bang hypothesis). True, new models which be neither universal nor holding all the time. account for the ”time before the Big Bang” do ap- In the case of atomic small-energy systems, pear, like those within the string theory (Gasperini as discussed above, recombinations among the con- and Veneziano 1993), or various cyclic universe mod- stituents in the final channels compete with the to- els, but we shall not dwell on it here (see, e.g. Ellis tal disintegration process and thus determine prob- ability for multiple-escape. This recombination may 1999 for a broader discussion). mean that electrons, for instance, fall back to the central ion (or nucleus), or, in the presence of 4. MODELING FRACTAL STRUCTURES positrons, for instance, formation of positroniums. Central ions play the role of cosmic attractors. Evi- dently, the interplay of the Newtonian universal grav- In conceiving space and time Newton postu- itational attraction and the general expansion of the lated strong division between them and the matter, Universe determines in the final account the dynam- which he inserted into an otherwise empty space, as a ical structure of the Cosmos. The global expansion universal receptacle of the inert content of the Uni- serves as the stabilizing force, keeping the galaxies verse. It is not easy to discern the epistemological away from each other. In the atomic case it is the aspect of such a postulate from the physical ontol- inter-particle repulsion which contributes to the final ogy it implied. On the other hand, Leibnitz gave escape to infinity (fragmentation process). In fact, preference to matter, which played more fundamen- the more realistic comparison would not be between tal role than with Newton. To Leibnitz the only the Universe and an atomic fragmentation process, observable were the relations between matter con- but between the latter and local instabilities. The stituents of the Universe with notions like absolute latter can result in a massive fall into ”great attrac- space loosing meaning. Finally, Einstein reformu- tors”. This phenomenon raises the question of the lated the issue by going back to the space and time real meaning of the concept of the universal expan- as real observable but subordinated to matter which sion (the expanding Universe). determines space-time properties. In the following Wannier’s model for the small-energy system we shall see how the modern cosmological consider- fragmentation applies equally to the self-gravitating ations deal this issue with various initial premises few-body systems, as shown in Grujic and Simonovic giving rise to the observable cosmic features. (1990), which brings the methodologies of the few- body system dynamics even more closely to each 4.1. Matter, space and time other. Both radii of the symmetrically outgoing Three most fundamental physical entities play atomic particles (electrons or positrons, Wannier the crucial role in conceiving the cosmological struc- 1953, for instance) and the scale factor S(t) in the tures in a way which may differ significantly from Einstein-de Sitter model (see, e.g. Peebles 1993, the laboratory/local circumstances. Two principal Collins et al. 1989), grow in time as t2/3. The schools of thought may be discerned in the European expansion of the overall intermingled transient sys- culture. The first one may be ascribed to Democritus tem of the incoming body and the target subsys- (the Abderian school) which postulated the empty 6 THE CONCEPT OF FRACTAL COSMOS: III. PRESENT STATE space (void) filled with atoms. Aristotle rejected the same level are identical, and at different levels pos- notion of void (kenon) and gave primacy to the mat- sess the same shape. Passing from one object to ter, which is the only reality and the rest just derived another at the same level by a number of symmetry quantities. As stressed by Roscoe (2008) Democri- transformations leaves the objects unchanged (simi- tus’ line was followed by Newton whereas Leibniz, larity transformations). Passing from an object at a Berkley and Einstein gave preference to Aristotle’s particular level to another at different level changes relationist approach. the size but not the shape of the object - selfsimilar These outlooks gave rise to two distinct, transformation, or scaling transformation. though interrelated approaches: (i) studying the If similarity transformations are approximate structure of space-time irrespective of the matter, ones, we speak of quasi-fractals (hierarchical struc- (ii) space-time as a secondary attribute of the inert tures are quasi-fractals in this sense). If a parameter matter. But the first approach can hardly be consid- which characterizes a fractal at different levels (like ered independent of the matter distribution, if not the fractal dimension) changes from level to level, ontologically then epistemologically. It is from our we speak of the multifractal structure (multifrac- observations of the matter properties that we con- tals). Some fractals have an infinite number of levels, ceive the space-time attributes, as we shall see later whereas some possess a finite number of them. In on. In particular, the eventual fractality of the space- fact, only mathematically constructed fractals may time continuum appears designed after the similar have infinitely many levels in principle whereas in properties of the matter distribution, regardless of nature one finds objects with a few levels. The class the possible mutual influence. of so-called strange attractors belongs to the former 4.1.1. Functions and curves whereas the snowflakes to the latter case. Of course, quasifractals abound in nature, from coastal lines, Dynamics of the inert particles systems is de- bronchial systems, plants etc. It is this approximate scribed, at least classically, by functions of mass, fractality which one expects to observe in cosmology, space and time. Functions are integrable quanti- as we shall see later on. ties, though these integrals may differ according to Exact fractals appear rare both in nature and the specific cases. Continuous functions (curves) ap- mathematics but are convenient objects to define the most important property of fractality. We shall con- pear integrable by ordinary integrals of the calculus, sider briefly some of them on the so-called fractal whereas discontinuous functions, which can not be curves. represented by curves, need a more general definition of integration, like the Lebesgue’s one. Derivatives 4.1.3. Fractal curves may be derived from the corresponding integrals, but These mathematical objects serve the best to not all functions appear differentiable, as the case of illustrate the essence of fractality. One of the most discontinuous ones illustrates. When integrals are used examples is the so-called Koch curve (see, e.g. defined, then one may define the first, second etc Figure 3.3 in Roscoe 2008), which may be used to derivatives in the manner of the standard Newton- Leibnitz calculus but Leibnitz found that a fractal construct snowflakes, for instance. One can not ex- derivative may be defined instead of natural ones hibit, of course, any fractal object fully, but just illus- (see, e.g. Atanackovic et al. 2008). We note here trate the iterative procedure for constructing a frac- that the fractal derivatives now find a broad space tal curve, for instance. Any fractal curve has the of application in various fields, including astronomy. topological dimension DT = 1, for there is always a Further, not even all continuous curves can be dif- way to map it onto a ”normal curve”, which is one- ferentiated as the case of the so-called fractal curves dimensional. In fact, one of definitions of fractality shows. It is this situation that introduces need for a is that a fractal system has fractal dimension greater more general definition of the derivative, as we shall than its topological dimension, DF > DT . This defi- see later on. nition is not quite correct since there are exceptions but may be used as a quick guide when estimating 4.1.2. What are fractals? the nature of an object. Definitions of fractal objects are numerous and The most important parameter of a fractal ob- often vague. The original definition by Benoit Man- ject is its fractal dimension. In the ”normal” physical delbrot states that objects, curves, functions, or sets space the length between any two points on a frac- are fractal when: ”their form is extremely irregular tal curve is infinite. The number of segments rises and/or fragmented at all scales” (Roscoe 2008). This as one increases the resolution ǫ (”zoom”), becoming definition appears broad enough to encompass many infinite in the limit ǫ 0. If 0 is the cardinal number of a countable set,→ then ℵin the case of a fractal particular situations, but not much helpful in pro- ℵ viding precise idea what fractal structure is. Here object = 2 0 . we enumerate some classes of irregular system which Deepℵ er structure as revealed by ”zooming in” can be classified as fractal. the curve (or any other fractal object) contains ”in- (i) Hierarchical structures. If we may define a visible” parts of the curve, as if the latter has been number of physical or otherwise scales, with each of embedded into a higher-dimensional space. It makes, them comprising similar objects of the same size, one therefore, sense to define a fractal dimension, which speaks of the hierarchical structures. Army, state ad- is, by definition, higher than the topological one. If ministration, brain are examples of such structuring. within the given segment of a fractal curve we have p (ii) Selfsimilar structures. These are often (rectilinear) parts, each of length q, then the fractal called fractals in a less strict sense. Objects at the dimension is given by 7 P. V. GRUJIC´

4.3. Scaling Matter DF = log p/log q. (9) Space and time are abstract concepts con- The length of the segment after n iterations in conceived by removing (abstracting) matter from the structing the curve is physical reality. They may be considered ontologically as more fundamental than (inert) matter but, L = L (p/q)n. (10) n 0 by the very abstracting procedure, they come after We note that fractality is essentially discrete prop- matter (epistemological aspect). It was this obser- erty and one needs to ”zoom” 3n times to jump to the vation which lead Roscoe (2002, 2008) to invert the n-th level. Fractal curves are not differentiable, not procedure when examining the structure of space and being sufficiently smooth for the ordinary concept time at large. of differentiation. In a sense, they resemble parti- Distribution of galaxies (and any other ob- cle trajectories in a real space according to Nelson’s ject in general) and their clusters etc. on the cos- stochastic quantum mechanics (Nelson 1966, 1985), mic scale is usually estimated by counting galaxies with the difference that the latter appear differen- within a sphere of radius R with the observer at the tiable almost everywhere. Nelson trajectories are centre. The number of encompassing galaxies, taken essentially stochastic but one may define stochastic as point-like elementary constituents, is given by: fractality too so that the similarities, at least at the D heuristic level, do exist. NR = AR . (13) Fractal dimension of a curve, for instance, may depend on the position measured by a parameter If D = 3 we have a homogeneous distribution. along the curve. Thus we have a variable fractal di- For D = DF < 3 we talk of fractal structure. Gen- mension instead of constant one. Equally, different erally, we have DF = 0 - point distribution, DF = 1 resolution levels may have different fractal dimen- - linear distribution, DF = 2 - surface distribution sion in which case one speaks about multifractality. and DF = 3 -volumetric, or space filling distribution These elaborations, however, move us away from the (see, e.g. Mureika 2007). simple geometrical attributes. As we shall see later Number of objects within a sphere of radius on, one often tries to find way out of difficulties in R is given by: recognizing fractal cosmic distribution by resorting to more general notions, like variable or nonunique NR = f(R) . (14) parameters. If Eq. (14) is a monotonous function, then we may write the inverse relation 4.2. The Theory of Scaling Relativity of Spacetime R = g(N) . (15)

In his comprehensive treatise on the theory With this starting positions Roscoe (2002) of scale transformation Nottale applied the methods posed the questions (see also Roscoe 2008): developed by him and others (Nottale 2011) to many Is it possible to associate a globally inertial branches of natural science. We shall here mention space and time with a non-trivial global matter distri- some of results relevant to our fractal model of cos- bution and, if it is, what are the fundamental proper- mos. The central point of the scale transformation ties of this distribution? In the context of the simple approach is development of the basic operation of the model analyzed, he finds definitive answers to these differential calculus. As mentioned before, standard questions so that: A globally inertial space/time can analysis does not apply to fractal object (curves or be associated with a non-trivial global distribution otherwise). of matter: If we start with a curvilinear coordinate (x, ǫ) and introduce an infinitesimal dilation ǫ ′ 2 Lǫ = ǫ(1+dρ), then the derivative with respect to−the→ M = m0(R/R0) + 2pm0m1/d0(R/R0), (16) scale parameter (we omit the standard coordinate x) is written as where m0, m1, d0, R0 are undetermined constants. Hence, the answer to the second question is: This global distribution is necessarily fractal ′ (ǫ ) = (ǫ + ǫdρ) = (1 + Ddρ) (ǫ), (11) with D = 2. Time is likewise derived from the dis- L L e L placements of the particles dr, defining the time interval: where the dilation operator is defined as dt2 =< dr dr > /v2 . (17) ∂ ∂ | 0 D = ǫ = , (12) e ∂ǫ ∂lnǫ Here, parameter v0 is not a particle velocity but a universal conversion constant linking the time and which appears as an analogue to the standard in- space quantities. This universe consists of photon- finitesimal calculus. like mass particles with constant (essentially nonzero) velocity and resembles the Bose-Einstein en- semble. The author concludes that: 8 THE CONCEPT OF FRACTAL COSMOS: III. PRESENT STATE

This result is to be compared with although significant inference into dark matter dis- the distribution of galaxies in our di- tribution on small scale, such as that of galaxies, rectly observable universe, which ap- has been achieved (see e.g. Schneider 2006), little is proximates very closely perfectly iner- known about the cosmic scale distribution (see e.g. tial conditions, and which appears to be fractal with D 2 on the small- Mureika 2007 and references therein). At present, to-medium scale at≈least. If we make there are indications that dark matter is also struc- the simple assumption that the distribu- tured according to the fractal pattern with fractal dimension DDM 1.5 2.5. tion of ponderable matter traces the dis- Surely dark≈matter− must play a significant role tribution of the material vacuum then, not only within galaxies, but on large scale too (see given the extreme simplicity of the analyzed model, this latter correspondence e.g. Oldershaw 2002). Thus Carati, Cacciatori and between the model’s statements and the Galgani (2009), investigating possible influence of cosmic reality lends strong support to the remote cosmic matter on the test particle, find the idea that our intuitively experienced that the nonvanishing contribution is possible only if perceptions of physical space and time discrete and fractal matter distribution is accounted are projected out of relationships, and for (Carati et al. 2009). It should be noted here that changing relationships, between the par- many approaches to the cosmology do not assume ticles (whatever these might be) in the dark matter or/and dark energy as a necessary in- material universe in very much the way gredients, like the quantum cosmology due to Cahill described. (2007). Whatever this constituent consists of, it must This task of inferring the dimension of space be the source of gravitational force and thus interact via the matter distribution is only a part of the gen- with the rest of the inert matter. The question arises eral approach of determining the structure of space- as to the possible structure of this invisible (none- time according to the structuring of the inert matter. mitting electromagnetic radiation) matter. There is The first step was made by Einstein with his idea no a priori reason for dark matter to differ in this that it is the matter which determines the global respect from the ordinary component what would properties of space-time as Euclidean (flat) spherical imply that if the cosmos is fractal at least partly, (closed universe) or hyperbolic ones (open universe) dark matter must be fractal too. Moreover, some (see e.g. Peebles 1993, Narlikar 1977). We men- researchers argue that it was the primordial dark tion that recently, Lu and Hellaby (2007) started the matter which triggered the initial density fluctua- programme of determining numerically the metric tions which have grown to the present day observed of cosmic space within the Lemaitre-Tolman-Bondi (fractal) structure (see e.g. Francesco 2008 and ref- model (Peebles 1993) from the available cosmograph- erences therein). Anyway, considering that dark- ical data, which should enable one to test the homo- matter component dominates the matter sector, it geneity assumption rather than to postulate it. The is this hidden part of the universe which governs the task appears very ambitious indeed but more definite overall matter distribution and thus the structure of results are expected in the future. the visible cosmos (Schneider 2006). 4.4. Dark matter and dark energy 4.4.2. Dark energy - Θεoζ απø µǫχανεζ So far we were dealing with the visible cos- This phantom constituent of the universe ap- mic matter which has been estimated to comprise pears legitimate hair of the (in)famous Einstein’s only 5 percent of the total content of the universe. cosmological constant Λ. The latter was invented to Two other invisible components have been hypoth- protect the universe from the (gravitational) collapse esized, the dark (nonbarionic) matter and dark en- which followed from the General relativity (GR) fun- ergy, which share the material content by approxi- damental equation (see e.g. Peebles 1993, Narlikar mately 0.22 and 0.73 amount respectively (see e.g. 1977, Nussbaumer and Bieri 2009). This mathe- Schneider 2006, Spergel 2003). Any realistic pro- matical barrier to the inherent instability of self- posal for the cosmic structuring must account for gravitating systems was easy to put by hand but these constituents (Schneider 2006, Crittenden 2008, difficult to interpret in physical terms. Or to put Sahni 2004). Both hypothetical components are usu- it in Mark Twain’s terms, nothing easier than that - ally introduced separately or ad hoc but attempts are there exist tens of various ”physical incarnations” of made to formulate a more general theoretical frame- this cosmic entity. The most popular present day in- work from which the invisible part of the matter terpretation imagines the dark energy constituent as emerges (see e.g. Quercellini et al. 2007). Attempts a homogeneous fluid, which engenders repulsive force to couple both components are made (Zhao and Li at large distance so that the other material inert con- 2008), including the principle of holographic sector stituents repel each other at large mutual separations (Zimdahl and Pavon 2007). causing the overall cosmic expansion. The concept suffers a number of deficiencies and is still the sub- 4.4.1. Dark matter - diabolo ex machina ject of extensive speculative investigations. One of these deficiencies is the property of the dark energy Dark matter may play significant role in as- to cause the expansion of the Universe, but remain- trophysics in many respects including its influence ing unchanged itself, in particular keeping the same on structuring the overall inert matter. However, density. Likewise, it is postulated that the energy 9 P. V. GRUJIC´ causes inert matter to move without influencing the two questions are of interest here. First, is the con- energy which violates Newton’s postulate on action cept of this new constituent of our Universe compat- and reaction (see e.g. Fahr and Heyl 2007, Bean et ible with the fractal model and second, what would al. 2008). Further, all proposed candidates fail to be mechanism of coupling the fractal structure with account for the gravitational properties of the vac- the fluid which, presumably, fills the entire cosmic uum energy (Durrer and Maartens 2007, Elis et al. space. It turns that not only that the fractal struc- 2010; see also Thomas 2002). ture goes well with dark energy, but one may easily Two principal questions are posed concerning explain the observed acceleration of the global ex- the hypothetical dark energy: (i) Is it really neces- pansion. We have considered a simple Newtonian sary to invoke such an entity in order to explain the model for this matter and have shown that Char- observed cosmic acceleration and (ii) What might be lier’s model (Charlier 1922) can provide a conve- the nature of such constituent and mechanism which nient means for driving the cosmic expansion (Gru- supposedly drives the expansion. jic 2004). It relies on the fundamental interaction The answer to the first question has reached between inert matter and the surrounding fluid ac- by now an affirmative consensus (see e.g. Schneider cording to Archimedes’ hydrostatics. The former ex- 2006 but see White 2007, Ishibashi and Wald 2006 for periences anti-gravitational force due to the higher a concise discussion of possible alternatives), whereas density of the surrounding fluid. These calculations the debate as regards the physical nature of dark en- are of qualitative nature, however, and further elab- ergy still goes on (see e.g. Medved 2008, Nayak and orations are still to be made. Singh 2009). It should be mentioned, however, that a number of authors still question the cosmological origin (or significance) of the observed red shifts which 5. THE ORIGIN OF STRUCTURING are the central data or the primary source of the observational cosmology. We shall not, however, dwell on it here. The issue of forming fractal (and any other, The concept of dark energy (DE) provides the for that matter) structure has two principal aspects. means for the observed acceleration of the visible cos- First, from the epistemological point of view, propos- mos but equally puts some restrictions on the possi- ing a convincing mechanism of forming hierarchical ble processes of forming cosmic structures including (and any other) structure greatly enhances chances the fractal one. It has turned out that the presence that this structure is accepted by the cosmological of DE implies the horizon, with the proper distance community. Besides, this issue is a part of the gen- to this given by: eral question of how the structuring process began at the early stages of the universe evolution (ontological c −1 Rh = 3.67 h Gpc, (ΩΛ + Ωm = 1), aspect). Both aspects are still in the early stages of H0√1 Ωm ≈ investigation and it would be premature, if not over- − (18) ambitious, to claim any mechanism to be reliable. In where H0 is the present-day Hubble constant and the other words, we still have no standard theory of the fraction of matter sector has been taken Ωm 1/3 structure formation. What does not mean there are (see e.g. Sahni 2004, also Quercellini et al. 2007).≈ no attempts to formulate mechanisms which could The presence of the event horizon implies that have brought us to the present cosmic situation. there exists a ”sphere of influence” around any cos- Generally, two principal classes of approaches mic body with its proper zh. For those parts of have been discerned at present: (i) Formations of the universe with z > zh, which are causally discon- small, elementary units, which than join each other nected, signals appear unreachable from now on. In in forming larger subsystems and so on (bottom-up the case of Cold Dark Mater cosmology with nonzero approach), (ii) Formation of the bulk cosmic super- Λ (ΛCDM) and ΩΛ 2Ωm 2/3 we have zh 1.8. system, which then disintegrates into ever smaller According to N body≈ simulations≈ of the evolution≈ parts (up-down approach) (see e.g. Grujić 2002 and of LCDM universe,− it turns out that after 1011 years references therein). At present, the first (bottom-up) our universe will consist of our Milky Way and An- mechanism appears preferred (see e.g. Yoshida 2009, dromeda galaxy only (Sahni 2004). Since the growth Mureika 2007). of an accelerating universe is frozen, the evolution According to the prevailing Standard Model of mass distribution will stop after approximately (Collins et al. 1989) about 380.000 years after the 3 1010 years. Hence, formation of the hierarchi- Big Bang and the inflationary phase, after the re- cal· levels may stop after a finite elapse of the cosmic combination the Universe became transparent to the time. Of course, all these considerations refer to a electromagnetic radiation. The latter remained un- particular cosmological paradigm and there are many changed bringing the information via Cosmic Mi- other models without event horizon, even without crowave Background radiation (CMB) about the ini- DE. We mention here that event horizons are ex- tial conditions for processes for the more complex cluded within the string theory, for there they pre- structures to form. The latter process lasted mil- vent formulation of the S matrix, which is funda- − lions of years (possibly billion years) when galax- mental for the Quantum mechanics. ies and other complex constituents of the present Many elaborate and sophisticated calculations day cosmos have been formed. This period of the have been carried out describing the role of the dark energy in the form of various physical fields. To us structure formation appears the least known (Cos- 10 THE CONCEPT OF FRACTAL COSMOS: III. PRESENT STATE mic Dark Age) and is the subject of intensive inves- sumed Weak Cosmological principle (dynamic uni- tigations at present. The general picture assumes verse). first formation of stars, which then form ever larger Both principles may bee reformulated as (i) structures, like galaxies, clusters, superclusters etc. Distribution homogeneous in space and time; (ii) In fact, both approaches, the bottom-up and up- Distribution uniform in space. down imply hierarchical scheme. The question arises, If one treats the elementary constituents of therefore, whether this procedure leaves signature on the cosmic matter, galaxies, as massive points, ne- the present matter distribution. glecting other degrees of freedom, like (rotational) angular momentum, uniform distribution (even den- 5.1. Hierarchical cosmos sity) implies the cosmic isotropy: universe looks the same whatever direction one chooses for the line of 5.1.1. Why hierarchical cosmos sight. The opposite need not be true: if the universe appears isotropic, it does not follow necessarily it is This question posed above invokes, further, homogeneous. The fractal cosmos is just the case in a more general one: Why should the Universe be point. structured hierarchically, irrespective of the mode In 1930-thies Weyl suggested something that of evolution? The answer to this appears easy, in Mandelbrot dubbed later (1983) as the Conditional fact too easy, if not trivial. Nature reveals to us Cosmological Principle (Mandelbrot 1983): (at present) four fundamental forces: gravitational, (iii) Conditional Cosmological Principle - uni- electromagnetic, strong (nuclear) and weak interac- verse looks the same seen from any occupied point. tions. Of these only the first one, the gravitational The emphasize here is on ”occupied”, for Man- appears operative at the large, cosmic scale. Until delbrot proved that in the case of a fractal struc- the quantum gravity (or something similar) is con- ture cosmos appears isotropic observed by a comov- trived, gravitation remains the classical interaction, ing observer from any constituent (which precludes that is, no quantization rules, no natural units as- observers from a void), and empty observed from any cribed to gravitating systems. True, there are some unoccupied point of space. In the cosmological con- specific critical quantities, like Chandrasekhar mass text, ”occupied” means ”from a galaxy” not from (see e.g. Schneider 2006), but these do not affect the a point of a cosmic void. This property bears, be- overall cosmic structuring. Hence, from planets and sides the practical significance (ontological aspect), stars upward any quantity of mass should be treated a remarkable epistemological implications, including on equal footing. In other words, any subsystem of the anthropic principle (see e.g. Barrow and Tipler gravitating bodies may be taken as a unit of a larger 1986), but we shall not dwell on it here. system. And it is exactly the rationale for modeling hierarchical systems. By picking up a predetermined subunits of mass Mi we single out a particular i th 5.2.1. Fractal models level of a hierarchical system. How much one −may go up or down along this scheme? There is no a priori upper limit, whereas the lower one is determined As we have seen before, fractality can not be by the dominance of non-gravitational interactions reached by assuming the standard statistical tools (from planet-size bodies downwards). which make use of the notion of mass density, for in- We mention here that there have been at- stance. Generally, if a radically new structure is to be tempts to push the hierarchical scheme down to designed or revealed, an appropriate set of initial as- atomic and even subatomic levels (see e.g. Older- sumptions and principles are to be adopted (Gabrielli shaw 2005 and references therein) but not many cos- et al. 2005; see also e.g. Gaite and Dominguez 2006, mologists are convinced about it. Baryshev et al. 1995 for recent reviews). Mittal and Lohiya (2003) made an attempt to 5.2. Modeling fractal cosmos arrive at the fractal cosmic distribution starting from the Conditional Cosmological Principle and assuming from the beginning the fractal cosmic geometry. It is the standard procedure of modeling (as- Defining ”hipersurface of homogeneous fractality of tro)physical systems to set up guiding principles, dimension D”, the authors show that if from any on which one elaborates further details, which dis- (occupied) point the mass within the sphere around tinguish one model from the other. The Standard the point increases as RD, the dynamics of the cos- Model is based on the Cosmological Principles, ex- mological scale factor a(t) is described by the same plicitly or not (see e.g. Peebles 1993, Grujic 2007). equation as in the standard model with the effective We quote two of them: homogeneous density. Hence, the model should be (i) Strong cosmological principle - universe is treated on equal footing as the standard model. the same everywhere and at any (cosmic) time. In an ambitious approach to an inhomoge- (ii) Weak cosmological principle - universe is neous universe without intrinsic symmetry (making the same at each point at a fixed cosmic instant. use of the Stephani model), Pompilio and Montuory Einstein’s first cosmological model (static so- (2001) examine the case of fractal distribution with lution of GR equation) implied, albeit implicitly, the specific metric constraints. Abandoning the Coper- Strong Cosmological principle, whereas later modi- nican principle the authors find that such a universe fications due to Friedman, Lemaitre and others, as- would undergo a nonuniform expansion. Accounting 11 P. V. GRUJIC´ for the acceleration it turns out that MCB unifor- 5.3. Empirical evidence mity follows just as in the case of the standard models, like those based on the inhomogeneous metric, Though the old thesis micrososmos like Lemaitre-Tolman-Bondi (LTB). macrocosmos may not be valid, some features of mi-∼ In a recent paper Calcagni (2009) proposes croworld and universe appear common in the episte- a field-theoretic approach within fractal spacetime mological or methodological sense. In both cases di- (see also Modesto 2009 where essentially the same rect observations are difficult, if not impossible and results, obtained within the Loop Quantum Gravity an extensive use of models is required in order to approach, have been presented). The system flows get insight into the objective nature of the object to from a fixed point within a spacetime of Hausdorff di- study. The common feature of the research on both mension 2, ending in the standard four-dimensional scales is the ruling maxim: to notice means to recognize. In the case of the cosmic structure if a partic- field theory with Ds = 4 as the energy decreases. In the classical limit, such systems dissipate energy- ular structuring, different from the uniform galaxy momentum in the bulk of integer topological dimen- distribution, is to be observed, a proper methodol- sion preserving the total energy-momentum. He con- ogy is to be applied. Since in the case of large-scale siders a number of implications of the model, in- structure no quantum effects are to be expected, any cluding the cosmological ones, but the work has particular irregularity must inevitably be approxi- more heuristic than practical aims. On the con- mate. Which implies that if it exists, it is not easy to observe. trary, Grujic and Pankovic proposed an analytical The principal problem in estimating a fitting formula to account for the evolution of the three-dimensional distribution is this very three- fractal dimension, as one moves from the close vicin- dimensionality. The problem reduces mainly in esti- ity (Df = 1) to the outer space where the fractal mating the (radial) distance of remote galaxies, clus- dimension is supposed to become Df = 3 (through ters, superclusters etc. This was the central issue Df = 2) (Grujic and Pankovic 2009). of estimating the relationship between the observed According to the General Relativity (GR) re- spectral shift and the distances of the galaxies which quirements all observations concerning deep space was essential for formulating the famous Hubble law should be carried out along the past light cone. (but see e.g. Nussbaumer and Bieri 2009). The in- Ribeiro and coworkers (Abdalla et al. 2001) have ference of the real large-scale structure is searched continued their investigations of the possible out- from the existing catalogues as primary sources for comes of observing the cosmic structure within this conceiving cosmic models. It is the application of spacetime hypersurface. They find that the so-called different methods in analyzing the astronomical data apparent fractal conjecture is valid for the perturbed which lead to various interpretations and thus to dif- Einstein-de Sitter cosmology (which assumes that ferent models cosmologist choose to ascribe to the ob- the pressure is much smaller than the density (cosmic servable cosmos. These may be classified into three dust) and space curvature and cosmological constant categories. (i) Those who (often vehemently) deny are zero (Peebles 1993, p. 101) where the dust distri- the fractal (and any hierarchical) structuring, (ii) bution has a scaling behaviour compatible with the Those who claim they observe fractal structure, at power-law profile of the density-distance correlation, least up to a certain cosmic distance, if not up to as observed in the galaxy catalogues. the entire observable universe, (iii) Sceptics, who do not deny hierarchical pattern but do not accept a 5.2.2. Methodological remarks clear-cut fractal ansatz, allowing for a possible multifractal case instead. We shall quote some of those inferences and interpretations. As pointed out in (Gabrielli et al. 2005) sta- Teerikorpi et al. (1998) examined the radial tistical analysis of inhomogeneous systems, particu- space distribution of KLUN-galaxies up to 200 Mpc larly those suspicious to be fractal, must abandon the and found the data compatible with Df = 2.3 and standard approach which deals with quantities like Df = 2.0. The authors conclude that if Df 2 be- the average density of the system. Equally, simula- yond 200 Mpc the position of our Galaxy w≈ould be tion of forming hierarchical structure from a homoge- close to the average in the Universe. Baryshev and neous substance proceed via the statistical methods Bukhamastova (2004), making use of the two-point appropriate to the nonstandard analysis. Pietronero conditional column density, estimated from samples et al. (2001) carried out simulation of forming struc- LEDA and EDR SDSS Df = 2.1 0.2 for cylinder ture of selfgravitating system of particles, making use lengths of 200 Mpc. ± of the function: Searching for super-large structure in deep galaxy surveys Nabokov and Baryshev (2010) car- ∗ N(< r) 3B − ried out extensive, multi-colour deep survey of galax- Γ (r) = 3h i = rD 3, (19) 4r3π 4π ies. With photometric redshift accuracy 0.03(1 + z) they found that the distribution in deep surveys like as the conditional average density within the sphere HUDF and FDF is consistent with the existence of of radius r, where N is the number of galaxies within super-large structures of luminous matter up to 2000 the sphere and B depends on the lower cut-off. They Mpc. found the system evolution to proceed towards ever Many cosmologists, starting with de Sitter increasing particle concentration into larger clusters, (1916, 1917) considered the possibility that the ob- with average density decreasing correspondingly. served cosmological redshift might be of gravitati- 12 THE CONCEPT OF FRACTAL COSMOS: III. PRESENT STATE onal origin. Baryshev (2008) examined the conse- Equally, fractal systems cast their projections quences on the redshift-distance relation for a num- onto a plane, as clouds do with their shadows on the ber of matter distributions. In the case of a homo- surface of Earth. If the fractal dimension of a cloud 2 geneous Universe this relation should be zgrav r , is Df 2 this shadow is compact, without sunny ∼ ≥ whereas for the fractal distribution with Df = 2 one regions inside. Further, it has been shown that for has zgrav r. The author argues that the field grav- Df = 2 the linear z d relation holds (see e.g. Bary- ity fractal∼model could explain satisfactory all three shev 1994 and references− therein). observed phenomena: the cosmic background radia- At a more fundamental level the fractal di- tion, the fractal large scale structure and the linear mension Df = 2 appears of the most significant Hubble law, starting with an evolution of an initially importance. It is not well know that the famous homogeneous cold gas. Feynman’s line integral formulation of Quantum Me- In their extensive study of DR5 Sloan Digital chanics was inspired by Dirac’s work (1933) (see, e.g. Sky Survey, Thieberger and Celerier (2008) exam- Park 1979). Since it was published in a less known ined the sample of 20.000 galaxies extracted from the Soviet journal (Dirac was an eager supporter of the catalogue of 332.876 galaxies. With the choice H0 = new communist regime in Russia, as many western −1 −1 70 kms Mpc and maximum distance dmax = 160 intellectuals were at the time, in particular following Mpc, the authors carried out correlation dimension his close friendship with Piotr Kapitza), his idea was calculations which allow for determining the tran- ignored by the scientific community, until Feynman sition scale to eventual homogeneous distribution. realized its value, as he himself used to acknowledge. Their analysis reveals an increase of the correlation Elaborating the role of paths in the phase-space, dimension up to D2 = 3 around 70 Mpc. Neverthe- Feynman found that these paths are continuous non- less, the authors expect data from larger catalogues, differentiable curves (fractals) and that the most sig- when available, to provide more reliable estimate. nificant contribution to the part integral comes from In Francesco et al. (2009) an analysis was those paths with Df = 2 (Nottale 2011). In view carried out of the latest Sloan Digital Sky Survey that line-integral formulation of Quantum Mechan- up to 30 Mpc/h, and large amplitude density fluc- ics appears the closest to the classical physics, this tuations are found. The two-point correlations ap- feature of the Feynman formulation gains remarkable pear self-averaging, providing the fractal dimension significance. In fact as Nottale has demonstrated D = 2.1 0.1. The authors find that at larger scales f ± (Nottale 2011) that Schrödinger equation can be de- density fluctuations appear too large in magnitude rived by scaling transformation, concept of fractality and too extended in space to be self-averaging in- with Df = 2 turns out of a particular importance. side the considered volume. The authors argue that We notice here that many theoretical work- these inhomogeneities are compatible with the frac- ers have found Df = 2 fractal dimension natu- tal correlations but not with the standard theoretical ral choice regardless of their models and initial ap- models for the scales lower than 100 Mpc/h. proaches (Calcagni 2009). Antal et al. (2009) carried out statistical anal- On the observational side, all examinations of ysis of the fluctuations in samples of the Sloan Digital the catalogues available which have discerned hier- Sky Survey Data Release 7 making use of the condi- archical distribution argue for D = 2 dimension tional galaxy density around each galaxy. They find f that the distribution appears clearly distinguishable (Sylos Labini et al. 2009, Gaite and Dominguez from a homogeneous spatial galaxy distribution. 2006, but see Thieberger and Celerier 2008). That Cosmic voids may have fractal distribution the Universe appears more homogeneous as the scale with respect to their dimension. According to some is enlarged speaks in favour of the bottom-up mod- surveys this distribution implies D = 2 (Gaite and els, which assume an initial homogeneous distribu- f tion with gravitational force forming mass concen- Dominguez 2006), but the available data are still in- trations. conclusive. It appears indicative that the actual fractal di- 5.3.1. 2 be or not 2 be mension turns out an integer, one unit smaller than the topological dimension of the Universe. It remains The search for actual value of Df has a specific to be seen if this fact has a deeper meaning. heuristic motivation, besides the practical interest. Theoretically fractal dimension of the galactic large- 6. EPILOGUE scale distribution lies within the interval (0, 3) As we have seen above, theoretical considerations single Observational evidence suggests that our uni- out D = 2 as of a particular significance. One of the verse displays structuring at various scales. At the f lowest level we have stellar systems in the form of most significant attributes of the proposed hierarchi- galaxy, as our own Milky Way. At a larger scale cal universe by Charlier (1922) was the resolution structural units like voids, filaments and sheets can of the cosmic paradoxes, Olbers’ (luminosity) (Over- be discerned in the sky. The question of a regular duin and Wesson 2008) and the Neumann-Seeliger structure on even higher level has been puzzling as- (gravitational) paradox. Newton was aware of the tronomers and cosmologists from the beginning of latter within his concept of an infinite universe where last century. The hierarchical model has been con- every (gravitating) body was to be subjected to pos- sidered as the best candidate for the cosmic regular sibly infinite gravitational force. Charlier found that structure and a lot of theoretical investigations have both paradoxes disappear if D 2. been devoted to this cosmological paradigm. On the f ≤ 13 P. V. GRUJIC´ other hand, the current observational evidence can REFERENCES not yet decide between the standard assumption of a uniform galaxy distribution and any other particu- Abdalla, E., Mohayee, R. and Ribeiro, M.: 2001, lar structuring, like the fractal paradigm. The latter Fractals, Vol. 9, No. 4 451. has a particular intrinsic value not only within cos- Antal, T., Sylos Labini, F., Vasilyev, N. L. and mological considerations as it appears ubiquitous in Baryshev, Y. V.,2009, oai:arXiv.org:0909.1507 nature. [pdf] - 187163 The issue of the hierarchical structuring ap- Atanacković, T. 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KONCEPT FRAKTALNOG KOSMOSA: III. SADAXƫE STAƫE

P. V. Gruji´c

Institute of Physics, P.O. Box 57, 11080 Belgrade, Serbia E–mail: [email protected]

UDK 524.88 Pregledni rad po pozivu

Ovo je nastavak serije o nastanku i empirijsku evidenciju koja se tiqe strukture razvoju koncepta fraktalnog kosmosa, do 2001. na velikoj skali opserviranog svemira. Naj- godine. Ovde dajemo pregled sadaxƬeg staƬa, zad razmoriemo i neka od epistemoloxkih sa naglaskom na posledƬi razvoj i kontro- pitaƬa, stavƩajui fraktalnu pardigmu u verze oko modela hijerarhijskog svemira. xiri kosmoloxki kontekst. Opisaemo teorijske pomake kao i danaxƬu