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.
Highlights
Metric, we propose an appropriate dynamical deformation between the conjugate momenta and, applying the Hamiltonian formalism, obtain deformed equations of motion
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
Summary
Substituting the Ricci scalar associated with the metric (1) into (2), we obtain. Considering the comoving gauge, i.e., setting N = 1, employing the Hamiltonian (4), and admitting the Poisson algebra { a, φ} = 0, { Pa , Pφ } = 0, { a, Pa } = 1 and {φ, Pφ } = 1 for the phase space coordinates { a, φ; Pa , Pφ }, we obtain:. Under the NC deformation (12), Equations (5) and (7) remain unchanged. The equations associated with the momenta, i.e., Equations (6) and (8), are deformed: 2. It is easy to show that the equations of motion associated with our NC framework are given by (17). We should note that in a particular case, where the NC parameter θ vanishes, Equations (17)–(19) reduce to their non-deformed counterparts. We restrict our attention to a specific case of a formerly constructed NC framework
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