Cascade birth of universes in multidimensional spaces
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ARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
Cascade Birth of Universes in Multidimensional Spaces S. G. Rubin Moscow Engineering Physics Institute, Kashirskoe sh. 31, Moscow, 115409 Russia e-mail: [email protected] Received October 30, 2007
Abstract—The formation mechanism of universes with distinctly different properties is considered within the framework of pure gravity in a space of D > 4 dimensions. The emergence of the Planck scale and its relationship to the inflaton mass are discussed. PACS numbers: 04.50.-h, 04.50.Cd, 04.50.Kd DOI: 10.1134/S1063776108040109
1. INTRODUCTION The dynamics of our Universe is well described by a modern theory containing 30–40 parameters. The number of these parameters, whose values are determined experimentally, is too large for the theory to be considered final. In addition, it is well known that the range of admissible parameters must be extremely narrow (fine tuning of parameters) for the birth and existence of such complex structures as our Universe, which is difficult to explain. Extensive literature is devoted to a discussion of this problem [1]. One way of solving the problem is based on the assumption about the multiplicity of universes with different properties [2–4]. Rich possibilities for justifying this assumption are contained in the idea of multidimensionality of our space itself. The number of extra dimensions has long been a subject for debate. For example, the Kaluza– Klein model originally contained one extra dimension. At present, infinite-dimensional spaces [5] and even variable-dimensional spaces [6] are being discussed. In this paper, the concept of superspace is extended to a set of superspaces with different, unlimited (above) numbers of dimensions. Based on the introduced extended superspace, we suggest the formation mechanism of universes with distinctly different properties and the emergence mechanism of the Planck scale. The probability of the quantum transitions that produce lowerdimensional subspaces is discussed. Let us define the superspace D = (MD; gij ) as a set of metrics gij in space MD to within diffeomorphisms. On a space-like section Σ, let us introduce a metric hij (for details, see the book [7] and the review [8]) and define the space of all Riemannian (D – 1) metrics: Riem ( Σ ) = { h ij ( x ) x ∈ Σ }.
The amplitude of the transition from one arbitrarily chosen section Σin with the corresponding metric hin to another section Σf with a metric hf is hf
A f , in = 〈 h f , Σ f h in, Σ in〉 =
∫ Dg exp [ iS ( g ) ].
(1)
h in
In what follows, we use the units = c = 1. The topologies of the sections Σin and Σf can be different. We will be concerned with the quantum transitions in which the topology of the hypersurface Σf is a direct product of the subspaces, MD – 1 – d ⊗ Md. The space Md is assumed to be compact. Below, we will explore the question of what class of geometries on the hypersurface Σf can initiate classical dynamics. The entire analysis is performed within the framework of nonlinear gravity in a space of D > 4 dimens
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