CHAOS 5541 No. of Pages 5 ARTICLE IN PRESS UNCORRECTED PROOF

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1 1 Chaos, Solitons and Fractals xxx (2007) xxx xxx 2 Hierarchical Cantor set in the large scale structure 3 with torus geometry 4 R. Murdzek 5 Physics Department, Al. I. Cuza University, Blvd. Carol I, Nr. 11, Iassy , Romania 6 Accepted 2 January Abstract 9 The formation of large scale structures is considered within a model with string on toroidal space-time. Firstly, the 10 space-time geometry is presented. In this geometry, the Universe is represented by a string describing a torus surface. 11 Thereafter, the large scale structure of the Universe is derived from the string oscillations. The results are in agreement 12 with the cellular structure of the large scale distribution and with the theory of a Cantorian space-time. 13 Ó 2007 Elsevier Ltd. All rights reserved Introduction 16 The recently completed galaxy surveys as such the Two Degree Field Galaxy Redshift Survey [1] have shown that, at 17 the largest scales, the distribution of galaxies exhibits structures that are not satisfactorily explained within the frame- 18 work of the standard cosmological model. The properties of the large scale structure are well characterized using the 19 fractal geometry language. Following this idea, it has been shown that the fractal dimension of the large scale galaxy 20 distribution is close to two and that there are no evidences for transition to homogeneity. A basic assumption of the 21 standard cosmological model is that, on some large enough scales, the universe presents a homogeneous distribution 22 of matter. But the last three decades proved that the observational data are consistent with Charlier s conception of 23 a hierarchical universe [2] and in agreement with El Naschie s E-infinity theory [3] rather than with the standard model 24 description. 25 In this respect, De Vaucouleurs [4] showed that, within wide limits, the available data satisfied a mass distribution 26 law MðrÞ /r 1;3. Since then, the issue has continued to take importance with the growth of the data. In 1995, Baryshev 27 et al. [5], showed that the scale of the largest inhomogeneities is given by the sizes of the sample s boundaries. In the last 28 two decades, several redshift surveys, such as those performed by Giovanelli and Haynes [6], De Lapparent et al. [7], 29 Broadhurst et al. [8], Da Costa et al. [9] and Vettolani et al. [10] have been emphasized massive structures such as sheets, 30 filaments, clusters and voids, and have shown that large structures are common features of the observable universe. The 31 pencil beam survey data gathered in ESO Slice Project has been analyzed by Pietronero and Sylos Labini [11]. Within 32 their work it has been shown that the distribution of galaxies confirms the law MðrÞ /r 2. The same result of fractal 33 distribution of dimension (D ¼ 2 0:2) has been found in the analysis performed by Guzzo et al. [12] and Moore address: rmurdzek@yahoo.com /$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi: /j.chaos

2 2 R. Murdzek / Chaos, Solitons and Fractals xxx (2007) xxx xxx 34 et al. [13]. Recently, it has been shown that this distribution looks more like a cellular structure among all scales rather 35 than a distribution which presents a cut-off to homogeneity [14]. 36 But such hierarchical structures are found in the Cantorian space-time. The Cantorian space-time [15] contains an 37 infinite number of sets E(i), where the index i means the topological dimension of the smooth space into which the 38 fractal set is contained. A representative example is the Cantor set, Eð0Þ, a fractal which is contained into a space of 39 topological dimension one, but whose Hausdorff dimension is equal to Similarly, the fractal dimension of the dis- 40 tribution of galaxies inside the smoothed space-time of integer dimension differs from the topological dimension of the 41 space into which is being contained. 42 Another problem of the standard cosmological model is the well-known singularity problem. One way to avoid it 43 but to keep the positive results of the expanding models is to consider a cyclic universe, without singularities, as it is 44 described within the ekpyrotic model [16]. 45 In this work, we shall show that the fractal structure of the large scale distribution of matter can be derived from a 46 brane-world scenario with toroidal space-time: in Section 2, we shall briefly discuss the geometry of the background 47 space-time. We also shall show that such a space-time supports the formation of standing waves. Then, our three- 48 dimensional space is represented by an oscillating closed string embedded into a higher dimensional space. In Section 49 3, we shall make the connection between the structure of the oscillating loop and the Cantor set. Some conclusions are 50 also discussed A string on a torus 52 In what follows we shall consider that: 53 By averaging, the Universe can be described as a smooth four-dimensional brane, embedded into an N-dimensional 54 space, as it is described in brane-world scenarios [16,17]. 55 The four-dimensional space-time is globally flat, isotropic and homogeneous, in respect with the Cosmological 56 Principle. 57 The Universe exhibits a cyclic behavior by following a succession of expanding and contracting stages. This state- 58 ment implies that our space-time is closed. 59 The formation of cosmic structures reflects the behavior of the space-time. 60 As the large scale distribution of galaxies proves, the fact that the Universe looks three-dimensional is due to the 61 smoothing (averaging) of the included structures In order to visualize such a space-time we shall consider the geometry of a torus. In this model our three-dimensional 64 space is represented as the great circle of the torus which evolves on torus surface. It means that from this point of view 65 the three-dimensional space is represented as a closed loop moving on the torus. 66 Toroidal geometry is presented in the literature through the work of Arfken, Moon and Spencer, Morse and Fesh- 67 bach [18]. 68 Using the notations from Fig. 1, we have for the torus surface the Riemannian metric [19] Fig. 1. Notations used for the torus parameters and coordinates.

R. Murdzek / Chaos, Solitons and Fractals xxx (2007) xxx xxx 3 70 ds 2 ¼ ðbþacos mþ 2 d/ 2 þ a 2 dm 2 : 71 After a short session in Maple, from this line element we obtain the connections

3 R. Murdzek / Chaos, Solitons and Fractals xxx (2007) xxx xxx 3 70 ds 2 ¼ ðbþacos mþ 2 d/ 2 þ a 2 dm 2 : 71 After a short session in Maple, from this line element we obtain the connections coefficients. Then, we have C 1 12 ¼ a sin m ðb þ a cos mþ 73 C 2 ðb þ a cos mþ sin m 11 ¼ a 74 The geodesic equations are then derived and d 2 dt 2 /ðtþ d 2 dt mðtþ 2 2a sinðmþ d /ðtþ d mðtþ dt dt ¼ 0 b þ a cosðmþ ðb þ a cosðmþþ sinðmþ d /ðtþ 2 dt ¼ 0: a 82 Eqs. (3) and (4) as well as the visualization of the geodesics (Fig. 2) show that on the toroidal space-time the null- 83 geodesics are closed curves, which means that the formation of standing waves is allowed on the torus surface String oscillations and fractal space-time 85 In our model, the space-time is represented by the closed loop which describes the torus surface. We aim to have a 86 description of the string oscillations which ignores the details of the wiggles. Then, the most simple solution is 88 sðx; tþ ¼A sinðk n xþ cosðx n tþ; where x ¼ðbþacos mþu: ð5þ 89 Perturbations will appear because of the oscillations, in the effective density of the string. We compute the amplitude 90 of these perturbations as functions of x, starting from the energy distribution along the string. 91 If we require a length L of string to form a closed loop, from the periodicity condition it follows that 93 sðx; tþ ¼sðx þ L=2; tþ: ð6þ 94 We can compute the energy distribution of the oscillations along the string by averaging over a period and summing 95 over all oscillations modes. Then, we have V ðxþ ¼ A02 X 1 1 nxp L n 2 sin2 ; 97 L ð7þ n¼0 98 where A 0 ¼ const, and n an odd integer. 99 Let T 0 be the tension in the string in the absence of oscillations. Then, the effective tension along the string is then given through 103 ov ðxþ T eff ¼ T 0 : ð8þ ox 104 We must note here that the product of the effective density l eff and effective tension remains constant along the string 105 [20] ð1þ ð2þ ð3þ ð4þ Fig. 2. Geodesic on the torus surface: the torus is geodesically complete.

4 R. Murdzek / Chaos, Solitons and Fractals xxx (2007) xxx xxx 106 108 T eff l eff ¼ l 2 ; 109 where l ¼ const. 110 From Eqs. (8) and (9), we have 112 l eff ðxþ ¼l 2 1 T 0 ov ðxþ ox Fig. 4.

4 4 R. Murdzek / Chaos, Solitons and Fractals xxx (2007) xxx xxx T eff l eff ¼ l 2 ; 109 where l ¼ const. 110 From Eqs. (8) and (9), we have 112 l eff ðxþ ¼l 2 1 T 0 ov ðxþ ox Fig. 4. Cantor set D cantor ¼ 0:6309 (a) and Cantor cube D c ¼ 1:893 (b). : ð10þ 113 Fig. 3 illustrates the effective density of the string as a function of the position x on the string. The picture is that of a 114 cellular universe where clusters form by accumulating the matter in the regions of maximum densities on the string. 115 These regions correspond to the nodes which are formed by the standing waves. Because of the fact that every harmonic 116 participates in the distribution of the energy on the string, we will have different scales of clustering which correspond to 117 different energies. The distribution of the nodes on the string is perfectly analogous to Cantor set, meaning that the 118 fractal dimension of this distribution is equal to fractal dimension of the Cantor set which is D cantor ¼ 0:6309. The frac- 119 tal dimension of the distribution in the three-dimensional space will be that of a Cantor cube (Fig. 4), D c ¼ 1:893. This 120 result is concordant with the fractal dimension of the galaxy distribution. ð9þ Concluding remarks 122 Using the torus geometry, we have obtained a new representation of the Universe, in which the stationary waves on 123 the torus surface determine the appearance of the structures. These structures perfectly correspond to the Cantorian

5 R. Murdzek / Chaos, Solitons and Fractals xxx (2007) xxx xxx space-time. The fractal dimension given by the distribution of the nodes on the string and is in good concordance to the 125 fractal dimension of the galaxy distribution. 126 References 127 [1] also at: astro-ph/ [2] Charlier CVL. Wei Eine Unendliche Welt Aufgebaut Sein Kann. Astronomioch Fysik 1908;4:1; 129 Charlier CVL. How an infinite world may be built up. Arkiv Mat Astron Physik 1922;16:1 35; 130 Charlier CVL. Publ Astron Soc Pacific 1924;37: [3] El Naschie MS. A review of E infinity theory and the mass spectrum of high energy particle physics. Chaos Soliton Fract ;19: [4] De Vaucouleurs G. The case for a hierarchical cosmology. Science 1970;167: [5] Baryshev YuV, Sylos Labini F, Montuori M, Pietronero L. Facts and ideas in modern cosmology. Vistas Astron 1995;38: [6] Giovanelli R, Haynes MP. A 21 cm survey of the Pisces Perseus supercluster. 1: the declination zone to 33.5 degrees. Astron 136 J 1985;90: ; 137 Giovanelli R, Haynes MP, Chincarini GL. Morphological segregation in the Pisces Perseus supercluster. Astrophys J ;300: [7] De Lapparent V, Geller MJ, Huchra JP. The mean density and the two-point correlation function for the CfA1 redshift survey 140 slices. Astrophys J 1988;332: [8] Broadhurst TJ, Ellis RS, Koo DC, Szalay AS. Large scale distribution of galaxies at the galactic poles. Nature 1990;343: [9] Da Costa MJ et al. A complete southern sky redshift survey. Astrophys J 1994;424:L [10] Vettolani G, et al. Preliminary results from the ESO Slice Project (ESP). Proc of Schloss Rindberg Workshop: Studying the 144 Universe with Cluster of Galaxies. 145 [11] Pietronero L, Sylos Labini F. Comments on the CfA2 analysis, preprint; [12] Guzzo L, Iovino A, Chincarini G, Giovanelli R, Haynes MP. Scale invariant clustering in the large scale distribution of galaxies. 147 Astrophys J Lett 1992;382:L [13] Moore B, Frenk CS, Efstathiou G, Saunders W. The clustering of IRAS galaxies. Mon Not R Astron Soc 1994;269: [14] Einasto J. The structure of the universe on 100 Mpc scales preprint: arxiv.org/abs/astro-ph/ [15] El Naschie MS. Elementary prerequisites for E-infinity theory. Chaos Soliton Fract 2006;30: ; 151 El Naschie MS. Intermediate prerequisites for E-infinity theory. Chaos Soliton Fract 2006;30:622 8; 152 El Naschie MS. Advanced prerequisites for E-infinity theory. Chaos Soliton Fract 2006;30: [16] Khoury J, Ovrut BA, Steinhardt PJ, Turok N. Phys Rev D 2001;64; 154 Steinhardt Paul J, Turok Neil Available from: hep-th/ [17] Randall L, Sundrum R. Phys Rev Lett 1999;83:3370. Available from: hep-th/ [18] Arfken G. In: Mathematical method for physicists. Orlando, FL: Academic Press; p ; 157 Moon P, Spencer DE. Field theory handbook, including coordinate systems, differential equations, and their solutions. New York: 158 Springer-Verlag; p ; 159 Morse PM, Feshbach H. Methods of theoretical physics, part I. New York: McGraw-Hill; p [19] Murdzek R. The torus universe. Romanian J Phys 2006;51(9 10); 161 Murdzek R. The geometry of the torus universe. Int J Mod Phys 2007;16(3): [20] Vilenkin A. Phys. Rev. D 1990;41:

THE GEOMETRY OF THE TORUS UNIVERSE

International Journal of Modern Physics D Vol. 16, No. 4 (2007) 681 686 c World Scientific Publishing Company THE GEOMETRY OF THE TORUS UNIVERSE R. MURDZEK Physics Department, Al. I. Cuza University, Iassy,