The Fractal Geometry of the Cosmic Web and Its Formation

Abstract

The cosmic web structure is studied with the concepts and methods of fractal geometry, employing the adhesion model of cosmological dynamics as a basic reference. The structures of matter clusters and cosmic voids in cosmological N-body simulations or the Sloan Digital Sky Survey are elucidated by means of multifractal geometry. A non-lacunar multifractal geometry can encompass three fundamental descriptions of the cosmic structure, namely, the web structure, hierarchical clustering, and halo distributions. Furthermore, it explains our present knowledge of cosmic voids. In this way, a unified theory of the large-scale structure of the universe seems to emerge. The multifractal spectrum that we obtain significantly differs from the one of the adhesion model and conforms better to the laws of gravity. The formation of the cosmic web is best modeled as a type of turbulent dynamics, generalizing the known methods of Burgers turbulence.The geometry of the cosmic web has been observed in galaxy surveys [12][13][14] and cosmological N-body simulations [15][16][17]. In addition, the fractal geometry of the large scale structure has been also studied in the distribution of galaxies [18][19][20] and in cosmological N-body simulations [21][22][23]. Recent N-body simulations have very good resolution and they clearly show both the morphological and self-similar aspects of the cosmic web. However, while there seems to be a consensus about the reality and importance of this type of structure, a comprehensive geometrical model is still missing. In particular, although our present knowledge supports a multifractal model, its spectrum of dimensions is only partially known, as will be discussed in this paper.We must also consider halo models [24], as models of large scale structure that are constructed from statistical rather than geometrical principles. In the basic halo model, the matter distribution is separated into a distribution within "halos", with given density profiles, and a distribution of halo centers in space. Halo models have gained popularity as models suitable for analyzing the results of cosmological N-body simulations and can even be employed for designing simulations [25]. Halo models are also employed for the study of the small scale problems of the Lambda-cold-dark-matter (LCDM) cosmology [26]. Remarkably, the basic halo model, with some modifications, can be combined with fractal models [27][28][29][30][31]. At any rate, halo models have questionable aspects and a careful analysis of the mass distribution within halos shows that it is too influenced by discreteness effects intrinsic to N-body simulations [31,32].We review in this paper a relevant portion of the efforts to understand the cosmic structure, with a bias towards methods of fractal geometry, as regards the description of the structure, and towards nonlinear methods of the theory of turbulence, as regards the formation of the structure. So this review does not have as broad a scope as, for example, Sahni and Coles' review of nonlinear gravitational ...

Scale Symmetry in the Universe

“…To be precise, despite common prejudices, generic mass distributions are strictly singular, that is to say, the mass density is not well defined and is in fact either zero or infinity at every point [36]. The singularities give rise to a spectrum of dimensions, as indeed happens in the mass distributions that appear in cosmology [35].…”

“…In the adhesion model of structure formation, the collapse initially leads to matter sheets (two-dimensional singularities), then to filaments (one-dimensional singularities), and finally to point-like singularities [33,34]. The web structure so formed is a particular example of multifractal mass distribution and is indeed a rough model of the actual cosmic structure, but it is not a very accurate model [35]. The problem is that the formation of matter filaments and especially point-like singularities, in Newtonian gravitation, can only occur after the dissipation of an infinite amount of gravitational energy, so that the process requires a fully relativistic treatment (Box 32.3, [29]), beyond the scope of the adhesion model.…”

“…On the other hand, multifractals are the natural generalization of simple hierarchical mass distributions, namely simple fractal sets, also referred to as monofractal distributions. Therefore, we are led to study the geometry of multifractal mass distributions and the proper way to characterize them [35]. Examples of monofractal, multifractal and adhesion model web structures are shown in Figure 1.…”

“…16, [38]). However, in the adhesion model, the mass distribution becomes singular and multifractal as a consequence of the scale invariant initial conditions [35,45].…”

“…We proceed to study how to generate the fractal geometry of the large scale structure, especially combining the Vlasov dynamics and the Poisson-Boltzmann-Emden equation [20] with the Zel'dovich approximation and the adhesion model of the early stage of structure formation [33,34]. The fractal geometry of the web structure that arises in the cosmological evolution according to the adhesion model is thoroughly studied in [35]. In this paper, we take a further step in the attempt to describe analytically the fractal geometry of the universe, by means of simple models and taking advantage of the power of the scale symmetry.…”