Interacting dark matter and q-deformed dark energy non-minimally coupled to gravity

In this paper, we propose a new approach to study the dark sector of the universe by considering the dark energy as emerging a q-deformed bosonic scalar field which is not only interacting with the dark matter, but also non-minimally coupled to gravity, in the framework of standard Einsteinian gravity. In order to analyze the dynamic of the system, we first give the quantum field theoretical description of the q-deformed scalar field dark energy, then construct the action and the dynamical structure of these interacting and non-minimally coupled dark sector. As a second issue, we perform the phase space analysis of the model to check the reliability of our proposal by searching the stable attractor solutions implying the late-time accelerating expansion phase of the universe.


Introduction
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 that, Supernova Ia [1,2], large scale structure [3,4], the baryon acoustic oscillations [5] and cosmic microwave background radiation [6][7][8][9]. According to the standard model of cosmological data 70% of the content of the universe consists of dark energy. Moreover, the remaining 25% of the content is an unknown form of matter having a mass but in non-baryonic form that is called as dark matter and the other 5% of the energy content of the universe belongs to ordinary baryonic matter [10]. While the dark energy spread all over the intergalactic media of the universe and produces a gravitational repulsion by its negative pressure to drive the accelerating expansion of the universe, the dark matter is distributed over the inner galactic media inhomogeneously and it contributes to the total gravitational attraction of the galactic structure and fixes the estimated motion of galaxies and galactic rotation curves [11,12].
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 [13][14][15][16][17][18][19][20][21][22]. On the other hand, there are also models in which the dark energy non-minimally couples to gravity in order to provide quantum corrections and renormalizability of the scalar 2 field in the curved spacetime. Also the crossing of the dark energy from the quintessence phase to phantom phase, known as the Quintom scenario, can be possible in the models that the dark energy interacts with the gravity. If the dark energy minimally couples to gravity, the equation of state parameter of the dark energy cannot cross the cosmological constant in the Friedmann-Robertson-Walker (FRW) geometry; therefore it is possible to emerge the Quintom scenario in the model that the dark energy non-minimally couples to gravity [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].
The constitution of the dark energy can be alternatively the cosmological constant  with a constant energy density filling the space homogeneously [38][39][40][41]. As the varying energy density dark energy models, instead of the cosmological constant, quintessence, phantom and tachyon fields can be considered. However, all these different dark energy models are same in terms of the non-deformed field constituting the dark energy. There is no reason to prevent us assuming the dark energy as a deformed scalar field, having a negative pressure, too, as required the dark energy has to be. Therefore, we propose that the dark energy considered in this study is formed of the deformed scalar field whose field equations are defined by the deformed oscillator algebras.
The quantum algebra and quantum group structure were firstly introduced by Kulish, Reshetikhin, Sklyanin, Takhtajan and Faddeev [42][43][44][45] during the investigations of integrable systems in quantum field theory and statistical mechanics. Quantum groups and deformed boson algebras are closely related terms. It is known that the deformation of the standard boson algebra is first proposed by Arik-Coon [46]. Later on, Macfarlane and Biedenharn have realized the deformation of boson algebra in a different manner from Arik-Coon [47,48]. The relation between quantum groups and the deformed oscillator algebras can be constructed obviously with this study by expressing the deformed boson operators in terms of the ) 2 ( q su Lie algebra operators. Therefore, the construction of the relation between quantum groups and deformed algebras leads the deformed algebras of great interest with many different applications. The deformed version of Bardeen-Cooper-Schrieffer (BCS) many-body formalism in nuclear force, deformed creation and annihilation operators are used to study the quantum occupation probabilities [49]. As another study, in Nambu-Jona-Lasinio (NJL) model the deformed fermion operators are used instead of standard fermion operators and this leads an increase in the NJL four-fermion coupling force and the quark condensation related to the dynamical mass [50]. The statistical mechanical studies of the deformed boson and fermion systems have been familiar in recent years [51][52][53][54][55][56][57][58][59][60][61]. Moreover, the investigations on the internal structure of composite particles involve the deformed fermions or bosons as the building block of the composite structures [62,63]. There are also applications of the deformed particles in black hole physics [64][65][66][67]. The range of the deformed boson and fermion applications diverse from atomic-molecular physics to solid state physics in a widespread manner [68][69][70][71][72][73].
The ideas on considering the dark energy as the deformed scalar field have become common in the literature [74][75][76][77]. In this study, we then take into account the deformed bosons as the scalar field dark energy interacting with the dark matter and also non-minimally coupled to gravity. In order to confirm our proposal that the dark energy can be considered as a deformed scalar field, we firstly introduce the dynamics of the interacting and non-minimally coupled dark energy, dark matter and gravity model in a spatially-flat FRW background, then perform the phase space analysis to check whether it will provide the late-time stable attractor solutions implying the accelerated expansion phase of the universe.

Dynamics of the model
The field equations of the scalar field dark energy are considered to be defined by the qdeformed boson fields in our model. Constructing a q-deformed quantum field theory after the idea of q-deformation of the single particle quantum mechanics [46][47][48] has naturally been non-surprising [78][79][80]. The bosonic part of the deformed particle fields corresponds to the deformed scalar field and the fermionic counterpart corresponds to the deformed vector field.
In this study, we consider the q-deformed bosonic scalar field as the q-deformed dark energy under consideration. In our model, the q-deformed dark energy interacts with the dark matter and also non-minimally couples to gravity.
Early Universe scenarios can be well understood by studying the quantum field theory in curved space-time. 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 space-time. It has been realized by Berger with taking random superposition of coherent states. Also the particle creation in an expanding universe with a non-quantized gravitational metric has been investigated by Parker. It has been stated by Goodison and Tom that if the field quanta obey the Bose or Fermi statistics, when considered 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 the possibility of the existence of the deformed statistics in coherent or squeezed states in the Early Universe [80][81][82][83][84][85].
Motivated by this significant possibility, we propose that the dark energy consists of a qdeformed scalar field whose particles obey the q-deformed algebras. Therefore, we now define the q-deformed scalar field constructing the dark energy in our model. The field operator of the q-deformed scalar field dark energy can be given as [80]   The following commutation relations for the deformed annihilation operator ) (k a q and creations operator ) ( * k a q in q-bosonic Fock space are given by [46] ) where q is a real deformation parameter in interval    q 0 and is the deformed number operator of k-th mode whose eigenvalue spectrum is given as is the standard non-deformed number operator. By using (2) and (3) in (1), we can obtain the commutation relations and planewave expansion of the q-deformed , as follows where The metric of the spatially flat FRW space-time in which the q-oscillator algebra represents the q-deformed scalar field dark energy is defined by and for a FRW metric which is used to obtain the relation between deformed and standard bosonic scalar fields by using (4) in (9) and (1): Here we have used the Hermiticity of the number operator N .

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Now the Friedmann equations will be derived for our interacting dark matter and nonminimally coupled q-deformed dark energy model in a FRW space-time by using the scale factor ) (t a in Einstein's equations. In order to obtain these equations, we relate the scale factor to the energy-momentum tensor of the objects in the model under consideration. We use the fluid description of the objects in our model by considering energy and matter as a perfect fluid, which are dark energy and matter in our model. An isotropic fluid in one coordinate frame leads to an isotropic metric in another frame coinciding with the frame of the fluid. This means that the fluid is at rest in commoving coordinates. Then the four velocity of the fluid is given as [53] ) and the energy-momentum tensor follows as A more suitable form can be obtained by raising one, such that Since we have two constituents, q-deformed dark energy and the dark matter in our model, the total energy density and the pressure are given by where are the density parameters for the q-deformed dark energy and the dark matter, respectively. Then total the density parameter is defined as We now turn to Einstein's equations of the form ) ( . Then by using the components of the Ricci tensor for a FRW space-time (7) and the energy momentum tensor in (13), we rewrite the Einstein's equations, for 00   respectively. Here dot also represents the derivative with respect to cosmic time t. Using (18) and (19) gives the Friedmann equations for the FRW metric as where a a H /   is the Hubble parameter. From the conservation of energy, we can obtain the continuity equations for the q-deformed dark energy and the dark matter constituents in the form of evolution equations, such as where Q is an interaction current between the q-deformed dark energy and the dark matter which transfers the energy and momentum from the dark matter to dark energy and vice versa. Q vanishes for the models having no interaction between the dark energy and the dark matter.
Now we will define the Dirac-Born-Infeld type action integral of the interacting dark matter and q-deformed dark energy non-minimally coupled to gravity in the framework of Eisteinian general relativity [88][89][90]. After that we will obtain the energy-momentum tensor  T for the q-deformed dark energy and the dark matter in order to get the energy density  and pressure p of these dark objects explicitly. Then the action is given as where  is a dimensionless coupling constant between q-deformed dark energy and the gravity, so R f q ) (  denotes the explicit non-minimal coupling between energy and the gravity.
and m L are the Lagrangian densities of the q-deformed dark energy and the dark matter, respectively. Then the energy-momentum tensors of the dark energy constituent of our model can be calculated, as follows [91] In order to find the derivative of the Ricci scalar with respect to the metric tensor, we use the variation of the contraction of the Ricci tensor identity . This leads us to find the variation of the contraction of the Riemann tensor identity, as follows . Here   represents the covariant derivative and    does the Christoffel connection. By using the metric compatibility and the tensor nature of     , we finally obtain where g      is the covariant d'Alembertian. Using (26) in (25) gives Then the 0 , 0   component of the energy-momentum tensor leads to the energy density where prime refers to derivative with respect to the field q  and we use 2 00 3H because of the homogeneity and the isotropy for where we use ) for the FRW spacetime. We can now obtain the equation of motion for the q-deformed dark energy by inserting (28) and (29) into the evolution equation (22), such that The usual assumption in the literature is to consider the coupling function as 2 / ) ( 2 q q f    [92] and the potential as [93][94][95]. In order to find the energy density, pressure and equation of motion in terms of the deformation parameter q, we use the above coupling function and potential with the rearrangement of equation (10) as in the equations (28)-(30) and obtain, Here we consider that the particles in each mode can vary by creation or annihilation in time , therefore its time derivatives are non-vanishing. On the other hand, the common interaction current in the literature is used here [17].
Now the phase-space analysis for our interacting dark matter and non-minimally coupled qdeformed dark energy model will be performed, whether the late-time stable attractor solutions can be obtained, in order to confirm our model.

Phase-space and stability analysis
The cosmological properties of the proposed q-deformed dark energy model can be investigated by performing the phase-space analysis. Therefore, we first transform the equations of the dynamical system into its autonomous form by introducing the auxiliary variables [15,[96][97][98][99][100], such as where s x , s y , s z and s u are the standard form of the auxiliary variables in 1  q limit. We now write the density parameters for the dark matter and q-deformed scalar field dark energy in the autonomous system by using (31) Then the total density parameter reads, as follows We should also obtain the Also from (36) and (39) While s is a junk parameter alone, it gains physical meaning in the deceleration parameter Now we convert the Friedmann equations (20), (21), the continuity equation (23), and the equation of motion (33) into the autonomous system by using the auxiliary variables in (34) and their derivatives with respect to a N ln  Here (44) and (46) in fact give the same autonomous equations, which means that the variables x and z do not form an orthonormal basis in the phase-space. However, z x  , y and u form a complete orthonormal set for the phase-space. Therefore, we set (44) and (46) in a single autonomous equation as which, by definition, cannot be intersected by any orbit. This means that there is no global attractor in the deformed dark energy cosmology [111]. We will make finite analysis of the phase space. The finite fixed points are found by setting the derivatives of the invariant submanifolds of the auxiliary variables. We can also write these autonomous equations in s s

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Here we need to get the finite fixed points (critical points) of the autonomous system in (44)- (48), in order to perform the phase-space analysis of the model. We will obtain these points by equating the left hand sides of the equations (45), (47) and (48) to zero, by using 1   tot in (37) and also by assuming in (40) and (43), for each critical point.
After some calculations, four sets of solutions are found as the critical points which are listed in Table 1 with the existence conditions. The same critical points are also valid for the Now we should find s  from (42), which will exist in the perturbations From (53)-(55), we find the 3 3 perturbation matrix M whose elements are given as We insert the linear perturbations ) about the critical points in the autonomous system (45), (47) and (48), in order to calculate the eigenvalues of perturbation matrix M for four critical points given in Table   1, with the corresponding existing conditions. Therefore, we first give the four perturbation matrices for the critical points where where . Also by using  Table 2, for each critical points  Table 2, the first two critical points A and B have the same eigenvalues, as C and D have the same eigenvalues, too. Here the eigenvalues and the stability conditions of the perturbation matrices for critical points C and D have been obtained by the numerical methods, since the complexity of the matrices (58) and (59). The stability conditions of each critical point are listed in Table 2, according to the sign of the real part of the eigenvalues.  Now we will study the cosmological behavior of each critical point by considering the attractor solutions in scalar field cosmology [112]. We know that the energy density of a scalar field has a role on the determination of the evolution of universe. Cosmological attractors provide the understanding of evolution and the affecting factors on this evolution, such as, from the dynamical conditions that the scalar field evolution approaches a certain kind of behavior without initial fine tuning conditions [113][114][115][116][117][118][119][120][121][122][123]. We know that the attractor solutions imply a behavior in which a collection of phase-space points evolve into a particular region and never leave from there. In order to solve the differential equation system (45), (47) and (48)  Acceleration occurs at this point because of 3 / 1 1     tot  , and it is an expansion phase since y is positive, so H is positive, too. Point A is stable meaning that universe keeps its further evolution, for 16 , but it is a saddle point meaning the universe evolves between different states for 0   . In Figure 1 values. This state corresponds to a stable attractor starting from the critical point ) 0 , as seen from the plots in Figure 1.
, but it is a saddle point for 0   . Therefore it is represented that the stable attractor behavior for contraction starting from the critical point , as seen from the graphs in Figure 2. We plot phase-space trajectories for the    interacting dark matter and q-deformed dark energy non-minimally coupled to gravity, which imply an expanding universe. On the other hand, the construction of the model in the 1  q limit reproduces the results of the phase space analysis for the non-deformed standard dark energy case. The critical points, perturbation matrices are same for the deformed and standard dark energy models with the equivalence of the auxiliary variables as s x z x   , s y y  and s u u  . Therefore, it is confirmed that the dark energy in our model can be defined in terms of the q-deformed scalar fields obeying the q-deformed boson algebra in (2) and (3). According to the stable attractor behaviors, it makes sense to consider the dark energy as a scalar field defined by the q-deformed scalar field, since the negative pressure of q-deformed boson field, as dark energy field.
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 same. The relation between deformed and standard dark energy can be represented regarding to auxiliary variable equations in (34).
where c is a constant. From the equations (60) we now illustrate the behavior of the deformed and standard dark energy auxiliary variables with respect to the deformation parameter q in Figure 6. We infer from the figure that the value of the deformed x , y and u decreases with decreasing q for the 1  q interval for large particle number, and the decrease in the variables x , y and u refer to the decrease in deformed energy density. Also, we conclude that the value of the auxiliary variables x , y and u increases with increasing q for the 1  q interval for large particle number. In 1  q limit deformed variables goes to standard ones. 32 FIGURE 6: Behavior of the auxiliary variables x , y and u with respect to the deformation parameter q and the particle number N .

Conclusion
In this study, we propose that the dark energy is formed of the negative-pressure q-deformed scalar field whose field equation is defined by the q-deformed annihilation and creation operators satisfying the deformed boson algebra in (2) and (3), since it is known that the dark energy has a negative pressure -like the deformed bosons -acting as a gravitational repulsion to drive the accelerated expansion of universe. We consider an interacting dark matter and qdeformed dark energy non-minimally coupled to the gravity in the framework of Einsteinian gravity in order to confirm our proposal. Then we investigate the dynamics of the model and phase-space analysis whether it will give stable attractor solutions meaning indirectly an accelerating expansion phase of universe. Therefore, we construct the action integral of the interacting dark matter and q-deformed dark energy non-minimally coupled to gravity model in order to study its dynamics. With this the Hubble parameter and Friedmann equations of the model are obtained in the spatially-flat FRW geometry. Later on, we find the energy density and pressure with the evolution equations for the q-deformed dark energy and dark matter from the variation of the action and the Lagrangian of the model. After that we translate these dynamical equations into the autonomous form by introducing the suitable auxiliary variables, in order to perform the phase-space analysis of the model. Then the critical points of autonomous system are obtained by setting each autonomous equation to zero and four perturbation matrices are obtained for each critical point by constructing the perturbation equations. We then determine the eigenvalues of four perturbation matrices to examine the stability of the critical points. We also calculate some important cosmological parameters, such as the total equation of state parameter and the deceleration parameter to check whether the critical points satisfy an accelerating universe. We obtain four stable attractors for the model depending on the coupling parameter  , interaction parameter  and the potential constant  . An accelerating universe exists for all stable solutions due to  Figures 1-4. In order to solve the differential equation system (45), (47) and (48)  The q-deformed dark energy is a generalization of the standard scalar field dark energy. As seen from (10) in the 1  q limit, the behavior of the deformed energy density, pressure and scalar field functions with respect to the standard functions, they all approach to the standard corresponding function values. Consequently, q deformation of the scalar field dark energy gives a self-consistent model due to the existence of standard case parameters of the dark energy in the 1  q limit and the existence of the stable attractor behavior of the accelerated expansion phase of universe for the considered interacting and non-minimally coupled dark energy and dark matter model. Although the deformed dark energy model is confirmed through reproducing the same stability behaviors, critical points and perturbation matrices with the standard dark energy model, the auxiliary variables of deformed and standard models are of course different. By using the auxiliary variable equations in (34), we find the relation between deformed and standard dark energy variables. From these equations, we represent the behavior of the deformed and standard dark energy auxiliary variables with respect to the deformation parameter for 1  q and 1  q intervals in Figure 6. Then, the value of the deformed x , y and u or equivalently deformed energy density decreases with decreasing q for the 1  q interval for large particle number. Also, value of the auxiliary variables x , y and u increases with increasing q for the 1  q interval for large particle number. In 1  q limit all the deformed variables goes to non-deformed variables.
The consistency of the proposed q-deformed scalar field dark energy model is confirmed by the results, since it gives the expected behavior of the universe. The idea to consider the dark energy as a q-deformed scalar field is a very recent approach. There are more deformed particle algebras in the literature which can be considered as other and maybe more suitable candidates for the dark energy. As a further study for the confirmation whether the dark 36 energy can be considered as a general deformed scalar field, the other interactions and couplings between deformed dark energy models, dark matter and gravity can be investigated in the general relativity framework or in the framework of other modified gravity theories, such as teleparallel.

Conflict of Interest
The author declares that there is no conflict of interest regarding the publication of this paper.