Advances in High Energy Physics

Volume 2016 (2016), Article ID 7380372, 17 pages

http://dx.doi.org/10.1155/2016/7380372

## Interacting Dark Matter and -Deformed Dark Energy Nonminimally Coupled to Gravity

Department of Physics, Sinop University, Korucuk, 57000 Sinop, Turkey

Received 5 October 2016; Revised 10 November 2016; Accepted 17 November 2016

Academic Editor: Sergei D. Odintsov

Copyright © 2016 Emre Dil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

In this paper, we propose a new approach to study the dark sector of the universe by considering the dark energy as an emerging -deformed bosonic scalar field which is not only interacting with the dark matter, but also nonminimally 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 -deformed scalar field dark energy and then construct the action and the dynamical structure of this interacting and nonminimally 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.

#### 1. 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 as Supernova Ia [1, 2], large-scale structure [3, 4], the baryon acoustic oscillations [5], and cosmic microwave background radiation [6–9]. 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 nonbaryonic form that is called 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 [13–22]. On the other hand, there are also models in which the dark energy nonminimally couples to gravity in order to provide quantum corrections and renormalizability of the scalar 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 where 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 boundary in the Friedmann-Robertson-Walker (FRW) geometry; therefore it is possible to emerge the Quintom scenario in the model where the dark energy nonminimally couples to gravity [23–37].

The constitution of the dark energy can be alternatively the cosmological constant with a constant energy density filling the space homogeneously [38–41]. 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 the same in terms of the nondeformed field constituting the dark energy. There is no reason to prevent us from assuming that the dark energy is a deformed scalar field, having a negative pressure, too, as expected from the dark energy. 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 et al. [42–44], 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 [45]. Later on, Macfarlane and Biedenharn have realized the deformation of boson algebra in a different manner from Arik-Coon [46, 47]. 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 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 [48]. As another study, in Nambu-Jona-Lasinio (NJL) model, the deformed fermion operators are used instead of standard fermion operators and this leads to an increase in the NJL four-fermion coupling force and the quark condensation related to the dynamical mass [49]. The statistical mechanical studies of the deformed boson and fermion systems have been familiar in recent years [50–60]. Moreover, the investigations on the internal structure of composite particles involve the deformed fermions or bosons as the building block of the composite structures [61, 62]. There are also applications of the deformed particles in black hole physics [63–66]. The range of the deformed boson and fermion applications diverses from atomic-molecular physics to solid state physics in a widespread manner [67–72].

The ideas on considering the dark energy as the deformed scalar field have become common in the literature [73–76]. In this study, we then take into account the deformed bosons as the scalar field dark energy interacting with the dark matter and also nonminimally 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 nonminimally coupled dark energy, dark matter, and gravity model in a spatially flat FRW background and 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.

#### 2. Dynamics of the Model

The field equations of the scalar field dark energy are considered to be defined by the -deformed boson fields in our model. Constructing a -deformed quantum field theory after the idea of -deformation of the single particle quantum mechanics [45–47] has naturally been nonsurprising [77–79]. 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 -deformed bosonic scalar field as the -deformed dark energy under consideration. In our model, the -deformed dark energy interacts with the dark matter and also nonminimally couples to gravity.

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 [79–84].

Motivated by this significant possibility, we propose that the dark energy consists of a -deformed scalar field whose particles obey the -deformed algebras. Therefore, we now define the -deformed scalar field constructing the dark energy in our model. The field operator of the -deformed scalar field dark energy can be given as [79]The following commutation relations for the deformed annihilation operator and creations operator in -bosonic Fock space are given by [45]where is a real deformation parameter in interval and is the deformed number operator of th mode whose eigenvalue spectrum is given asHere is the standard nondeformed number operator. By using (2) in (1), we can obtain the commutation relations and planewave expansion of the -deformed scalar field , as follows:whereThe metric of the spatially flat FRW spacetime in which the -oscillator algebra represents the -deformed scalar field dark energy is defined byand for a FRW metricwhere . Also the relation between deformed and standard annihilation operators and [85] is given aswhich is used to obtain the relation between deformed and standard bosonic scalar fields by using (3) in (8) and (1):Here we have used the Hermiticity of the number operator .

Now the Friedmann equations will be derived for our interacting dark matter and nonminimally coupled -deformed dark energy model in a FRW spacetime by using the scale factor 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 velocities of the fluid are given as [52]and the energy-momentum tensor follows asA more suitable form can be obtained by raising one, such thatSince we have two constituents, -deformed dark energy and the dark matter in our model, the total energy density and the pressure are given bywhere and are the energy density and the pressure of the -deformed dark energy and and are the energy density and the pressure of the dark matter, respectively. The equation of state of the energy-momentum carrying cosmological fluid component under consideration in the FRW universe is given by which relates the pressure and the energy density and is called the equation of state parameter. We then express the total equation of state parameter, such thatwhere and are the density parameters for the *-*deformed dark energy and the dark matter, respectively. Then the total density parameter is defined asWe now turn to Einstein’s equations of the form . Then, by using the components of the Ricci tensor for a FRW spacetime (6) and the energy-momentum tensor in (12), we rewrite Einstein’s equations, for and , as follows:respectively. Here dot also represents the derivative with respect to cosmic time . Using (16) and (17) gives the Friedmann equations for the FRW metric aswhere is the Hubble parameter. From the conservation of energy, we can obtain the continuity equations for the -deformed dark energy and the dark matter constituents in the form of evolution equations, such aswhere is an interaction current between the -deformed dark energy and the dark matter which transfers the energy and momentum from the dark matter to dark energy and vice versa. 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 -deformed dark energy nonminimally coupled to gravity in the framework of Eisteinian general relativity [86–88]. After that we will obtain the energy-momentum tensor for the -deformed dark energy and the dark matter in order to get the energy density and pressure of these dark objects explicitly. Then the action is given aswhere is a dimensionless coupling constant between -deformed dark energy and the gravity, so denotes the explicit nonminimal coupling between energy and the gravity. Also and are the Lagrangian densities of the -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 [89]: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 finding the variation of the contraction of the Riemann tensor identity, as follows: . Here represents the covariant derivative and represents the Christoffel connection. By using the metric compatibility and the tensor nature of , we finally obtainwhere is the covariant d’Alembertian. Using (23) in (22) givesThen the component of the energy-momentum tensor leads to the energy density :where prime refers to derivative with respect to the field and we use , because of the homogeneity and the isotropy for in space. Also and is used for the FRW geometry. The components of also give the pressure aswhere we use with for the FRW spacetime. We can now obtain the equation of motion for the -deformed dark energy by inserting (25) and (26) into the evolution equation (19), such thatThe usual assumption in the literature is to consider the coupling function as [90] and the potential as [91–93]. In order to find the energy density, pressure, and equation of motion in terms of the deformation parameter , we use the above coupling function and potential with the rearrangement of equation (9) as in the equations (25)–(27) and obtainHere we consider that the particles in each mode can vary by creation or annihilation in time for ; therefore its time derivatives are nonvanishing. 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 nonminimally coupled -deformed dark energy model will be performed, whether the late-time stable attractor solutions can be obtained, in order to confirm our model.

#### 3. Phase-Space and Stability Analysis

The cosmological properties of the proposed -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, 94–98], such aswhere , , , and are the standard form of the auxiliary variables in limit. We now write the density parameters for the dark matter and -deformed scalar field dark energy in the autonomous system by using (28) with (36):Then the total density parameter readsWe should also obtain the in the autonomous form to write the equation of state parameters, such thatwhere . Using (33) and (35), we find the equation of state parameter for the dark energy asAlso from (33) and (36), the total equation of state parameter can be obtained aswhere is defined to be the barotropic index. We need to give the junk parameter in the autonomous form, such thatPulling from the right-hand side of (38) to the left-hand side givesWhile is a junk parameter alone, it gains physical meaning in the deceleration parameter , such thatNow we convert the Friedmann equations (18), the continuity equation (20), and the equation of motion (30) into the autonomous system by using the auxiliary variables in (31) and their derivatives with respect to . For any quantity , this derivative has the relation with the time derivative as . Then we will obtain , where is the column vector including the auxiliary variables and is the column vector of the autonomous equations. We then find the critical points of , by setting . We then expand around , where is the column vector of perturbations of the auxiliary variables, such as , , , and for each constituent in our model. Thus, we expand the perturbation equations up to the first order for each critical point as , where is the matrix of perturbation equations. The eigenvalues of perturbation matrix determine the type and stability of each critical point [99–108]. Then the autonomous form of the cosmological system isHere (41) and (43) in fact give the same autonomous equations, which means that the variables and do not form an orthonormal basis in the phase-space. However, , , and form a complete orthonormal set for the phase-space. Therefore, we set (41) and (43) in a single autonomous equation asThe autonomous equation system (42), (44), and (45) represents three invariant submanifolds , , and which, by definition, cannot be intersected by any orbit. This means that there is no global attractor in the deformed dark energy cosmology [109]. 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 limit in terms of the standard auxiliary variables, such asHere we need to get the finite fixed points (critical points) of the autonomous system in (41)–(45), 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 (42), (44), and (45) to zero, by using in (34) and also by assuming and in (37) and (40), 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 , , and instead of , , and , in the standard dark energy model limit.