Time-Dependent Toroidal Compactification Proposals and the Bianchi Type I Model: Classical and Quantum Solutions

Abstract

We construct an effective four-dimensional model by compactifying a ten-dimensional theory of gravity coupled with a real scalar dilaton field on a time-dependent torus. This approach is applied to anisotropic cosmological Bianchi type I model for which we study the classical coupling of the anisotropic scale factors with the two real scalar moduli produced by the compactification process. Under this approach, we present an isotropization mechanism for the Bianchi I cosmological model through the analysis of the ratio between the anisotropic parameters and the volume of the Universe which in general keeps constant or runs into zero for late times. We also find that the presence of extra dimensions in this model can accelerate the isotropization process depending on the momenta moduli values. Finally, we present some solutions to the corresponding Wheeler-DeWitt (WDW) equation in the context of standard quantum cosmology.

Cosmologies with Scalar Fields from Higher Dimensions Applied to Bianchi Type VIh=-1 Model: Classical and Quantum Solutions

“…On the other hand, we see that the solution associated to the equation (41b) is the same for all Bianchi Class A cosmological models. It was the main result obtained in [2,3]. This can be appreciated by observing the Hamiltonian operator (30), which can be split as Ĥ(a, c, φ, σ)Ψ = Ĥg (a, c)Ψ + Ĥm (φ, σ)Ψ = 0, where Ĥg y Ĥm represents the With the last results, we obtain the general solution to the WDW equation ( 40) whose wave function Φ can be built by taking the superposition of the functions (44) and (45), that is…”

“…It was the main result obtained in [3,4]. This can be appreciated by observing the Hamiltonian operator (35), which can be split asĤ(a, c, , )Ψ =Ĥ g (a, c)Ψ + H m ( , )Ψ = 0, whereĤ g yĤ m represents the Hamiltonian for gravitational sector and the moduli fields, respectively, and the scale factors have the transformations = , = .…”

“…with σ a real parameter. By reducing the higher dimensional action (1) to four dimensions and rewrite it in the Einstein frame (for more details see [2,3]) this gives the reduced action…”

“…They work in the context of quantum cosmological models in a n-dimensional anisotropic space by introducing massless scalar fields [7]. It will be our subject as it was in our previous results [2,3] We can mention that the moduli fields will satisfy a Klein-Gordon like equation in the Einstein frame as an effective theory. This will be possible to appreciate by taking a variation of the action (4) with respect to each of the moduli fields.…”

“…The above problems have suggested to consider the presence of higher-dimensional degrees of freedom in the cosmology derived from four-dimensional effective theories. Some features of the presence of higher-dimensional effective theories is to consider an effective action with a graviton and a massless scalar field, the dilaton, describing the evolution of the universe [2,3]. On the other hand, it is well known that relativistic theories of gravity, such as general relativity or string theories, are invariant under reparametrization of time.…”