^{3}.

In this paper, we propose a new approach to study the dark sector of the universe by considering the dark energy as an emerging

The dark energy is accepted as the effect of causing the late-time accelerated expansion of universe which is experienced by the astrophysical observations such as Supernova Ia [

Miscellaneous dark models have been proposed to explain a better mechanism for the accelerated expansion of the universe. These models include interactions between dark energy, dark matter, and the gravitational field. The coupling between dark energy and dark matter seems possible due to the equivalence of order of the magnitudes in the present time [

The constitution of the dark energy can be alternatively the cosmological constant

The quantum algebra and quantum group structure were firstly introduced by Kulish et al. [

The ideas on considering the dark energy as the deformed scalar field have become common in the literature [

The field equations of the scalar field dark energy are considered to be defined by the

Early Universe scenarios can be well understood by studying the quantum field theory in curved spacetime. The behavior of the classical scalar field near the initial singularity can be translated to the quantum field regime by constructing the coherent states in quantum mechanics for any mode of the scalar field. It is now impossible to determine the quantum state of the scalar field near the initial singularity by an observer, at the present universe. In order to overcome the undeterministic nature, Hawking proposes to take the random superposition of all possible states in that spacetime. It has been realized by Berger with taking random superposition of coherent states. Also the particle creation in an expanding universe with a nonquantized gravitational metric has been investigated by Parker. It has been stated by Goodison and Toms that if the field quanta obey the Bose or Fermi statistics, when considering the evolution of the scalar field in an expanding universe, then the particle creation does not occur in the vacuum state. Their result gives signification to the possibility of the existence of the deformed statistics in coherent or squeezed states in the Early Universe [

Motivated by this significant possibility, we propose that the dark energy consists of a

Now the Friedmann equations will be derived for our interacting dark matter and nonminimally coupled

Now we will define the Dirac-Born-Infeld type action integral of the interacting dark matter and

Now the phase-space analysis for our interacting dark matter and nonminimally coupled

The cosmological properties of the proposed

Critical points and existence conditions.

Label | | | | | | Existence |
---|---|---|---|---|---|---|

| 0 | | 0 | | | |

| 0 | | 0 | | | |

| 0 | | | | | |

| 0 | | | | | |

Now we should find

We need to obtain the four sets of eigenvalues and investigate the sign of the real parts of eigenvalues, so that we can determine the type and stability of critical points. If all the real parts of the eigenvalues are negative, the critical point is said to be stable. The physical meaning of the stable critical point is a stable attractor; namely, the Universe keeps its state forever in this state and thus it can attract the universe at a late time. Here an accelerated expansion phase occurs because

Eigenvalues and stability of critical points.

Critical points | Eigenvalues | | | Stability | ||
---|---|---|---|---|---|---|

| −3.0000 | | | Stable point for | ||

Saddle point for | ||||||

| ||||||

| −1.0642 | −1.5576 | −5.5000 | 0.1000 | 1.0000 | Stable point for |

−1.0193 | −1.0193 | −7.2507 | 1.0000 | 1.0000 | ||

−0.8407 | −0.8407 | −7.7519 | 2.0000 | 1.0000 | ||

−0.7080 | −0.7080 | −8.0701 | 3.0000 | 1.0000 | ||

−0.6014 | −0.6014 | −8.3107 | 4.0000 | 1.0000 | ||

−0.5121 | −0.5121 | −8.5060 | 5.0000 | 1.0000 | ||

−0.4353 | −0.4353 | −8.6709 | 6.0000 | 1.0000 | ||

−0.3680 | −0.3680 | −8.8136 | 7.0000 | 1.0000 | ||

−0.3082 | −0.3082 | −8.9395 | 8.0000 | 1.0000 | ||

−0.2544 | −0.2544 | −9.0520 | 9.0000 | 1.0000 | ||

−0.2055 | −0.2055 | −9.1535 | 10.0000 | 1.0000 |

Now we will study the cosmological behavior of each critical point by considering the attractor solutions in scalar field cosmology [

Two-dimensional projections of the phase-space trajectories for stability condition

Two-dimensional projections of the phase-space trajectories for stability condition

Two-dimensional projections of the phase-space trajectories for stability conditions

Two-dimensional projections of the phase-space trajectories for stability conditions

All the plots in Figures

Three-dimensional plots of the phase-space trajectories for the critical points

We know that the deformed dark energy model is a confirmed model since it reproduces the same stability behaviors, critical points, and perturbation matrices with the standard dark energy model, but the auxiliary variables of deformed and standard models are not the same. The relation between deformed and standard dark energy can be represented regarding auxiliary variable equations in (

Behavior of the auxiliary variables

In this study, we propose that the dark energy is formed of the negative-pressure

These figures show that, by using the convenient parameters of the model according to the existence and stability conditions and the present day values, we can obtain the stable attractors as

The

The consistency of the proposed

The author declares that there is no conflict of interests regarding the publication of this paper.

^{∗}observations: determination of cosmological parameters

_{q}(2) and a q-analogue of the boson operators

_{q}(2) fermionic system

_{q}(1,1) description of vibrational molecular spectra