Stochastic self-similar and fractal universe

Abstract

The structures formation of the Universe appears as if it were a classically self-similar random process at all astrophysical scales. An agreement is demonstrated for the present hypotheses of segregation with a size of astrophysical structures by using a comparison between quantum quantities and astrophysical ones. We present the observed segregated Universe as the result of a fundamental self-similar law, which generalizes the Compton wavelength relation. It appears that the Universe has a memory of its quantum origin as suggested by R. Penrose with respect to quasi-crystal. A more accurate analysis shows that the present theory can be extended from the astrophysical to the nuclear scale by using generalized (stochastically) self-similar random process. This transition is connected to the relevant presence of the electromagnetic and nuclear interactions inside the matter. In this sense, the presented rule is correct from a subatomic scale to an astrophysical one. We discuss the near full agreement at organic cell scale and human scale too. Consequently the Universe, with its structures at all scales (atomic nucleus, organic cell, human, planet, solar system, galaxy, clusters of galaxy, super clusters of galaxy), could have a fundamental quantum reason. In conclusion, we analyze the spatial dimensions of the objects in the Universe as well as space-time dimensions. The result is that it seems we live in an El Naschie's E-infinity Cantorian space-time; so we must seriously start considering fractal geometry as the geometry of nature, a type of arena where the laws of physics appear at each scale in a self-similar way as advocated long ago by the Swedish school of astrophysics.

Dynamical systems on El Naschie’s ε(∞) Cantorian space–time

“…In the previous papers, the authors consider the compatibility of a Stochastic Self-Similar, Fractal Universe with the observation and the consequences of this model. In particular, it was demonstrated that the observed segregated Universe is the result of a fundamental selfsimilar law, which generalizes the Compton wavelength relation, R 2 is the Golden Mean value [1]. This expression agree with the Golden Mean and with the gross law of Fibonacci and Lucas [2,3].…”

“…If jAj À1 ¼ ð ffiffi ffi 5 p À 1Þ=2 and N = n À 1, where À 1 6 n 6 1, then C F = e (n) is the E-infinity Cantorian space. Mohamed El Naschie in [15] showed the relationship between the Cantor space C and e (1) . As He reports: ''the relationship comes from the cardinality problem of a Borel set in polish spaces Thus we call a subset of a topological space a Cantor set if it is homeomorphic to the Cantor space''.…”

“…Following El Nashie [5], let us start from PlankÕs fundamental energy quantization relation 1 As we will see in the next paragraph S (1) is the action to consider the Cantorian material support instead of a classical continuum; consequently to be more precise we have to write d(S + S (1) ) = 0.…”

“…Particularly, the model is related to Penrose tiling and thus to e (1) theory (Cantorian space-time theory) as proposed by El Naschie [8,9] as well as with Connes Noncommutative Geometry [10]. In [1] the authors presented a descriptive model of segregated universe, then considered a dynamical model to explain the results and to give the evolution of the structures [11]. In [12] a waveguiding and mirroring effects are considered with respect to the large scale structure of the Universe.…”

Accelerated expansion in a stochastic self-similar fractal universe

“…Iovane et al [15], using a stochastic self-similar and fractal universe model, estimated the value of γ as 0.5. In [1] the virial theorem and a scaling law are applied to a large-scale astrophysical structure to obtain a time-dependent gravitational constant, G(t, N) given by 2…”