Time-dependent toroidal compactification proposals and the Bianchi type II model: Classical and quantum solutions

Abstract

In this work 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 without the contributions of fluxes as first approximation. This approach is applied to anisotropic cosmological Bianchi type II model for which we study the classical coupling of the anisotropic scale factors with the two real scalar moduli produced by the compactification process. Also, we present some solutions to the corresponding Wheeler-DeWitt (WDW) equation in the context of Standard Quantum Cosmology and we claim that these quantum solution are generic in the moduli scalar field for all Bianchi Class A models. Also we gives the relation to these solutions for asymptotic behavior to large argument in the corresponding quantum solution in the gravitational variables and is compared with the Bohm's solutions, finding that this corresponds to lowest-order WKB approximation.

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

“…We can observe that the quantum solution in the Ω sector is similar to the corresponding FRW cosmological model, found in different schemes [12,22,23,32,43]. Also, similar analysis based on a Bianchi type II model can be found in recent published paper by two authors of this work [51].…”

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

“…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.…”