Noncommutativity, Sáez–Ballester Theory and Kinetic Inflation

Abstract

This paper presents a noncommutative (NC) version of an extended Sáez–Ballester (SB) theory. Concretely, considering the spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric, we propose an appropriate dynamical deformation between the conjugate momenta and, applying the Hamiltonian formalism, obtain deformed equations of motion. In our model, the NC parameter appears linearly in the deformed Poisson bracket and the equations of the NC SB cosmology. When it goes to zero, we get the corresponding commutative counterparts. Even by restricting our attention to a particular case, where there is neither an ordinary matter nor a scalar potential, we show that the effects of the noncommutativity provide interesting results: applying numerical endeavors for very small values of the NC parameter, we show that (i) at the early times of the universe, there is an inflationary phase with a graceful exit, for which the relevant nominal condition is satisfied; (ii) for the late times, there is a zero acceleration epoch. By establishing an appropriate dynamical framework, we show that the results (i) and (ii) can be obtained for many sets of the initial conditions and the parameters of the model. Finally, we indicate that, at the level of the field equations, one may find a close resemblance between our NC model and the Starobinsky inflationary model.

Inflation and Fractional Quantum Cosmology

“…into the WDW Equation (12). Straightforward differentiation plus condition (14) (which is equivalent to WKB) (The semi-classical approximation and also the WKB wave function are important concepts in the context of quantum cosmology.…”

“…Note that in a particular case where α = 2, Equation ( 27) reduces to its standard counterpart (12). To obtain the semi-classical solution of this fractional WDW equation, we use the alternative equivalent representation of the Riesz fractional derivative (or fractional Laplacian) [60], given by…”

“…In addition to the well-known standard models of inflation, recent scenarios using non-commutative cosmological models, Generalized Uncertainty Principle (GUP) and other modified gravity models have shown that some problems with the standard models (such as graceful exit and the Hubble constant problem) can be solved, see, for instance [7][8][9][10][11][12].…”

Noncompactified Kaluza–Klein Gravity

“…In contrast to the latter, the SB theory includes the action of ordinary matter that is not coupled to the SB scalar field. Various cosmological models have been established in the context of the SB theory to study open problems in both the classical and quantum regimes [49][50][51][52][53][54]. Applying the similar reduction procedure used to construct the MBDT, without assuming the presence of the higher dimensional matter fields and imposing the cylindricity condition on the extra coordinate, it has been shown that the SB field equations associated to the bulk split into four sets of effective field equations on any hypersurface orthogonal to the extra dimension [9].…”

Shape Invariant Potentials in Supersymmetric Quantum Cosmology

“…Adler-Deser-Misner (ADM); see[50] for more details 5. In this work, we consider a framework in which the scalar field is minimally coupled to gravity, see also[51,52,53]. Instead, one can choose other interesting gravitational models where φ is non-minimally coupled, see for instance,[54,55,56,57,58].…”