Noncommutativity, Saez-Ballester theory and kinetic inflation

TL;DR

This paper develops a noncommutative extension of the Saez-Ballester cosmological theory, demonstrating that noncommutativity can induce inflationary behavior and late-time acceleration, resembling Starobinsky inflation, through numerical analysis.

Contribution

It introduces a noncommutative deformation in the Saez-Ballester theory and explores its cosmological implications, including inflation and acceleration, using the Hamiltonian formalism.

Findings

Noncommutativity induces an inflationary phase with a graceful exit.

Late-time universe exhibits a zero acceleration epoch.

Model resembles Starobinsky inflation at the field equation level.

Abstract

This paper presents a noncommutative (NC) version of an extended S\'{a}ez-Ballester (SB) theory. Concretely, considering the spatially flat Friedmann-Lema\^{\i}tre-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.