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

Authors: Emre Dil
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

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


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## Body

## 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 ω=1 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 constantΛ 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 suq(2) 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 theq-deformed boson fields in our model. Constructing a q-deformed quantum field theory after the idea of q-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 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 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 aq-deformed 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 [79](1)ϕqx=∫d3k2π3/212wk1/2aqkeikx+aq∗ke-ikx.The following commutation relations for the deformed annihilation operator aq(k) and creations operator aq∗(k) in q-bosonic Fock space are given by [45](2)aqkaq∗k′-q2aq∗k′aqk=δk-k′,aqkaqk′-q2aqk′aqk=0,where q is a real deformation parameter in interval 0<q<∞ and [N^(k)]=aq∗(k)aq(k) is the deformed number operator of kth mode whose eigenvalue spectrum is given as(3)Nk=1-q2Nk1-q2.Here N^(k)=as∗(k)as(k) is the standard nondeformed number operator. By using (2) in (1), we can obtain the commutation relations and planewave expansion of the q-deformed scalar field ϕq(x), as follows:(4)ϕqxϕq∗x′-q2ϕq∗x′ϕqx=iΔx-x′,where(5)Δx-x′=-12π3∫d3kwksin⁡wkx-x0.The metric of the spatially flat FRW spacetime in which the q-oscillator algebra represents the q-deformed scalar field dark energy is defined by(6)ds2=-dt2+a2tdr2+r2dθ2+r2sin2⁡θdϕ2,and for a FRW metric(7)wk2=g∑iki2a2+m,where g=det⁡gμν. Also the relation between deformed and standard annihilation operators aq and as [85] is given as(8)aq=asN^N^,which is used to obtain the relation between deformed and standard bosonic scalar fields by using (3) in (8) and (1):(9)ϕq=ϕ1-q2N^1-q2N^.Here we have used the Hermiticity of the number operator N^.Now the Friedmann equations will be derived for our interacting dark matter and nonminimally coupledq-deformed dark energy model in a FRW spacetime by using the scale factor a(t) 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](10)Uμ=1,0,0,0,and the energy-momentum tensor follows as(11)Tμν=ρ+pUμUν+pgμν=ρ00000gijp0.A more suitable form can be obtained by raising one, such that(12)Tνμ=diag⁡-ρ,p,p,p.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(13)ρtot=ρq+ρm,ptot=pq+pm,where ρq and pq are the energy density and the pressure of the q-deformed dark energy and ρm and pm 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 p=ωρ which relates the pressure and the energy density and ω is called the equation of state parameter. We then express the total equation of state parameter, such that(14)ωtot=ptotρtot=ωqΩq+ωmΩm,where Ωq=ρq/ρtot and Ωm=ρm/ρtot are the density parameters for the q-deformed dark energy and the dark matter, respectively. Then the total density parameter is defined as(15)Ωtot=Ωq+Ωm=κ2ρtot3H2=1.We now turn to Einstein’s equations of the form Rμν=κ2(Tμν-(1/2)gμνT). 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 μν=00 and μν=ij, as follows:(16)-3a¨a=κ22ρ+3p,(17)a¨a+2a˙a2=κ22ρ-p,respectively. Here dot also represents the derivative with respect to cosmic time t. Using (16) and (17) gives the Friedmann equations for the FRW metric as(18)H2=κ23ρq+ρm,H˙=-κ22ρq+pq+ρm+pm,where H=a˙/a 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(19)ρ˙q+3Hρq+pq=-Q,(20)ρ˙m+3Hρm+pm=Q,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 andq-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 Tμν for the q-deformed dark energy and the dark matter in order to get the energy density ρ and pressure p of these dark objects explicitly. Then the action is given as(21)S=∫d4x-gR2κ2-12gμν∂μϕq∂νϕq-V-ξfϕqR+Lm,where ξ is a dimensionless coupling constant between q-deformed dark energy and the gravity, so ξf(ϕq)R denotes the explicit nonminimal coupling between energy and the gravity. Also Lq=-(1/2)gμν∂μϕq∂νϕq-V-ξf(ϕq)R and Lm 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 [89]:(22)Tμνq=-2∂Lq∂gμν+gμνLq=∂μϕq∂νϕq+2ξfϕq∂R∂gμν-12gμνgαβ∂αϕq∂βϕq+2V-gμνξfϕqR.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 δR=Rμνδgμν+gμνδRμν. This leads us to finding the variation of the contraction of the Riemann tensor identity, as follows: δRμν=δRμρνρ=∇ρ(δΓνμρ)-∇ν(δΓρμρ). Here ∇μ represents the covariant derivative and Γνμρ represents the Christoffel connection. By using the metric compatibility and the tensor nature of δΓνμρ, we finally obtain(23)δRδgμν=Rμν+gμν□-∇μ∇ν,where □=gαβ∇α∇β is the covariant d’Alembertian. Using (23) in (22) gives(24)Tμνq=∂μϕq∂νϕq-12gμνgαβ∂αϕq∂βϕq-gμνV+2ξRμν-12gμνRfϕq+2ξ□fϕq-2ξ∇μ∇νfϕq.Then the μν=0,0 component of the energy-momentum tensor leads to the energy density ρq:(25)ρq=T00q=12ϕ˙q2+V+6ξH2fϕq+6ξHf′ϕqϕ˙q,where prime refers to derivative with respect to the field ϕq and we use □=-∂02-3H∂0, because of the homogeneity and the isotropy for ϕq in space. Also R00=-3a¨/a and R=6[a¨/a+a˙2/a2] is used for the FRW geometry. The μν=i,i components of Tμνq also give the pressure pq as(26)pq=giiTiiq=12ϕ˙q2-V-2ξ2H˙fϕq+3H2fϕq+f′′ϕqϕ˙q2+f′ϕqϕ¨q+2Hf′ϕqϕ˙q,where we use (∇i∇i)f(ϕq)=(∂i∇i-Γiiλ∇λ)f(ϕq)=-Γ110∂0f(ϕq) with Γ110=a˙a for the FRW spacetime. We can now obtain the equation of motion for the q-deformed dark energy by inserting (25) and (26) into the evolution equation (19), such that(27)ϕ¨q+3Hϕ˙q+∂V∂ϕq+ξRf′ϕq=-Qϕ˙q.The usual assumption in the literature is to consider the coupling function as f(ϕq)=ϕq2/2 [90] and the potential as V=V0e-κλϕq [91–93]. 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 (9) as ϕq=Δ(q)ϕ in the equations (25)–(27) and obtain(28)ρq=12Δ2ϕ˙2+e-κλΔϕ+3ξH2Δ2ϕ2+6ξHΔ2ϕϕ˙+12Δ˙2ϕ2+ΔΔ˙ϕϕ˙+6ξHΔΔ˙ϕ2,(29)pq=12Δ2ϕ˙2-e-κλΔϕ-2ξΔ2H˙ϕ2+32H2ϕ2+ϕ˙2+ϕϕ¨+2Hϕϕ˙+1-8ξΔΔ˙ϕϕ˙+12-2ξΔ2ϕ2-2ξΔΔ¨ϕ2-4ξHϕϕ˙ΔΔ˙ϕ2,(30)Δϕ¨+3ΔHϕ˙-κλe-κλΔϕ+ξΔRϕ˙+2Δ˙ϕ˙+Δ¨ϕ+3HΔ˙ϕ=-βκρm.Here we consider that the particles in each mode can vary by creation or annihilation in time for Δ=(1-q2N)/(1-q2)N; therefore its time derivatives are nonvanishing. On the other hand, the common interaction current in the literature Q=βκρmϕ˙q is used here [17].Now the phase-space analysis for our interacting dark matter and nonminimally coupledq-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 proposedq-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 as(31)x=κΔϕ˙6H=Δxs,y=κe-κλΔϕ3H=e-κλϕΔ-1ys,z=κΔ˙ϕ6H,zs=0u=κΔϕ=Δus,where xs, ys, zs, and us are the standard form of the auxiliary variables in q→1 limit. We now write the density parameters for the dark matter and q-deformed scalar field dark energy in the autonomous system by using (28) with (36):(32)Ωm=κ2ρm3H2,(33)Ωq=κ2ρq3H2=x2+y2+ξu2+26ξxu+z2+2xz+26ξzu.Then the total density parameter reads(34)Ωtot=κ2ρtot3H2=x2+y2+ξu2+26ξxu+z2+2xz+26ξzu+Ωm=1.We should also obtain the κ2pq/3H2 in the autonomous form to write the equation of state parameters, such that(35)κ2pq3H2=1-4ξx2-y2+23ξ+4ξ2su2+8ξ2-ξu2+263ξxu+1-4ξz2+21-4ξxz+263ξzu+2ξβuΩm-2ξλy2u,where s=-H˙/H2. Using (33) and (35), we find the equation of state parameter for the dark energy as(36)ωq=pqρq=1-4ξx2-y2+23ξ+4ξ2su2+8ξ2-ξu2+263ξxu+1-4ξz2+21-4ξxz+263ξzu+2ξβuΩm-2ξλy2ux2+y2+ξu2+26ξxu+z2+2xz+26ξzu-1.Also from (33) and (36), the total equation of state parameter can be obtained as(37)ωtot=ωqΩq+ωmΩm=1-4ξx2-y2+23ξ+4ξ2su2+8ξ2-ξu2+263ξxu+1-4ξz2+21-4ξxz+263ξzu+2ξβuΩm-2ξλy2u+γ-1Ωm,where γ=1+ωm is defined to be the barotropic index. We need to give the junk parameter s in the autonomous form, such that(38)s=-H˙H2=321+ωtot=321+1-4ξx2-y2+23ξ+4ξ2su2+8ξ2-ξu2+263ξxu+1-4ξz2+21-4ξxz+263ξzu+2ξβuΩm-2ξλy2u+γ-1Ωm.Pulling s from the right-hand side of (38) to the left-hand side gives(39)s=1+1-4ξx2-y2+8ξ2-ξu2+263ξxu+1-4ξz2+21-4ξxz+263ξzu+2ξβuΩm-2ξλy2u+γ-1Ωm23-23ξu2-4ξ2u2-1.While s is a junk parameter alone, it gains physical meaning in the deceleration parameter qD, such that(40)qD=-1+s=-1+1+1-4ξx2-y2+8ξ2-ξu2+263ξxu+1-4ξz2+21-4ξxz+263ξzu+2ξβuΩm-2ξλy2u+γ-1Ωm23-23ξu2-4ξ2u2-1.Now 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 N=ln⁡a. For any quantity F, this derivative has the relation with the time derivative as F˙=H(dF/dN)=HF′. Then we will obtain X′=f(X), where X is the column vector including the auxiliary variables and f(X) is the column vector of the autonomous equations. We then find the critical points Xc of X, by setting X′=0. We then expand X′=f(X) around X=Xc+U, where U is the column vector of perturbations of the auxiliary variables, such as δx, δy, δz, and δu for each constituent in our model. Thus, we expand the perturbation equations up to the first order for each critical point as U′=MU, where M is the matrix of perturbation equations. The eigenvalues of perturbation matrix M determine the type and stability of each critical point [99–108]. Then the autonomous form of the cosmological system is(41)x′=-3x-3z+sx-z′+sz+6ξsu-26ξu+62λy2-62βΩm,(42)y′=sy-62λyx-62λyz,(43)z′=-3x-3z+sx-x′+sz+6ξsu-26ξu+62λy2-62βΩm,(44)u′=6x+6z.Here (41) and (43) 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, y, and u form a complete orthonormal set for the phase-space. Therefore, we set (41) and (43) in a single autonomous equation as(45)x′+z′=-3x-3z+sx+sz+6ξsu-26ξu+62λy2-62βΩm.The autonomous equation system (42), (44), and (45) represents three invariant submanifolds +z=0, y=0, and u=0 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 q→1 limit in terms of the standard auxiliary variables, such as(46)xs′=-3xs+ssxs+6ξssus-26ξus+62λys2-62βΩm,ys′=ssys-62λysxs,us′=6xs.Here 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 Ωtot=1 in (34) and also by assuming ωtot=-1 and qd=-1 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 xs, ys, and us instead of x+z, y, and u, in the q→1 standard dark energy model limit.Table 1
Critical points and existence conditions.
Label x + z y u ω t o t q D Existence A 0 1 0 - 1 - 1 λ = 0, Ωm=0 B 0 - 1 0 - 1 - 1 λ = 0, Ωm=0 C 0 ( 4 ξ / λ ) - 2 / λ + 4 / λ 2 + 1 / ξ - 2 / λ + 4 / λ 2 + 1 / ξ - 1 - 1 λ ≠ 0, Ωm=0 D 0 - ( 4 ξ / λ ) - 2 / λ + 4 / λ 2 + 1 / ξ - 2 / λ + 4 / λ 2 + 1 / ξ - 1 - 1 λ ≠ 0, Ωm=0Now we should findδs from (39), which will exist in the perturbations δx′+δz′, δy′, and δu′, such that(47)δs=21-4ξx+z+263ξu1Pδx+δz+-2y-4ξλyu1Pδy+8ξ2-ξ2u+263ξx+z+2ξβΩm-2ξλy2+43ξ+8ξ2su1Pδu,where P=2/3-(2/3)ξu2-4ξ2u2. Then the perturbations δx′+δz′, δy′ and δu′ for each phase-space coordinate in our model can be found by using the variations of (42), (44), and (45), such that(48)δx′+δz′=21-4ξx+z2+-s+3P+4ξ2u2+83ξ-8ξ26x+zu1Pδx+δz+-2x+z+6λPy-4ξλx+zyu-26ξyu-46ξ2λyu21Pδy+10ξ2-ξ2x+zu+263ξx+z2+2ξβx+zΩm-2ξλx+zy2+43ξ+8ξ2x+zsu+8ξ2-ξ26ξu2+26ξ2βuΩm-26ξ2λy2u+43ξ+8ξ26ξsu2+6ξs-2P1Pδu,δy′=21-4ξx+zy+263ξuy-62λyP1Pδx+δz+-2y2-4ξλy2u+sP-62λx+zP1Pδy+8ξ2-ξuy+263ξx+zy+2ξβyΩm-2ξλy3+43ξ+8ξ2syu1Pδu,δu′=6δx+δz.From (48), we find the 3×3 perturbation matrix M whose elements are given as(49)M11=21-4ξx+z2+-s+3P+4ξ2u2+83ξ-8ξ26x+zu1P,M12=-2x+z+6λPy-4ξλx+zyu-26ξyu-46ξ2λyu21P,M13=10ξ2-ξ2x+zu+263ξx+z2+2ξβx+zΩm-2ξλx+zy2+43ξ+8ξ2x+zsu+8ξ2-ξ26ξu2+26ξ2βuΩm-26ξ2λy2u+43ξ+8ξ26ξsu2+6ξs-2P1P,M21=21-4ξx+zy+263ξuy-62λyP1P,M22=-2y2-4ξλy2u-+sP-62λx+zP1P,M23=8ξ2-ξuy+263ξx+zy+2ξβyΩm-2ξλy3+43ξ+8ξ2syu1P,M31=6,M32=M33=0.We insert the linear perturbations (x+z)→(xc+zc)+(δx+δz), y→yc+δy, and u→uc+δu about the critical points in the autonomous system (42), (44), and (45), 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 A, B, C, and D with the corresponding existing conditions, such that(50)MA=MB=-30-26ξ0-30600,(51)MC=4ξ2uC2P-3-46λuC2ξ2P-26ξyCuCP+6yCλ-62ξ2-16ξ3uC2P-26ξ2λyC2uCP-26ξ26ξyCuC3P-6λyC2-2yC2P-4ξλyC2uCP-2ξλyC3P-2ξ-16ξ2uCyCP600,where yC=4ξ/λ(-2/λ+4/λ2+1/ξ) and uC=-2/λ+4/λ2+1/ξ,(52)MD=4ξ2uD2P-346λuD2ξ2P+26ξyDuDP-6yDλ-62ξ2-16ξ3uD2P-26ξ2λyD2uDP-26ξ6λyD2-26ξyDuD3P-2yD2P-4ξλyD2uDP-2ξλyD3P-2ξ-16ξ2uDyDP600,where yD=-4ξ/λ(-2/λ+4/λ2+1/ξ) and uD=-2/λ+4/λ2+1/ξ. Also by using xs, ys, and us instead of x+z, y, and u in the perturbation matrix elements above, we obtain the standard perturbation matrix elements in q→1 limit. Then substituting the standard critical points we again obtain the same matrices MA, MB, MC, and MD. Therefore the stability of the standard model agrees with the stability of the deformed model.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 becauseωtot=-1<-1/3. However, if the suitable conditions are satisfied, there can even exist an accelerated contraction for ωtot=-1<-1/3 value. Eigenvalues of the four M matrices and the stability conditions are represented in Table 2, for each critical point A, B, C, and D. From 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, due to the complexity of the matrices (51) and (52). The stability conditions of each critical point are listed in Table 2, according to the sign of the real part of the eigenvalues.Table 2
Eigenvalues and stability of critical points.
Critical points Eigenvalues ξ λ Stability Aand B −3.0000 - ( 1 / 2 ) 9 - 48 ξ - 3 / 2 ( 1 / 2 ) 9 - 48 ξ - 3 / 2   Stable point for0<ξ≤3/16 with λ,β∈R Saddle point forξ<0 with λ,β∈R  Cand D −1.0642 −1.5576 −5.5000 0.1000 1.0000 Stable point for0<ξ, λ=1 and β∈RSaddle point, if ξ<0 and λ≠1 −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 1.0000Now we will study the cosmological behavior of each critical point by considering the attractor solutions in scalar field cosmology [110]. We know that the energy density of a scalar field has a role in the determination of the evolution of Universe. Cosmological attractors provide the understanding of evolution and the factors affecting on this evolution, such that, from the dynamical conditions, the evolution of scalar field approaches a particular type of behavior without using the initial fine tuning conditions [111–121]. 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 (42), (44), and (45) we use adaptive Runge-Kutta method of 4th and 5th order, in MATLAB programming. We use the present day values for the dark matter density parameter Ωm=0.3, interaction parameter β=14.5, and 0<γ<2 values in solving the differential equation system [94, 122]. Then the solutions with the stability conditions of critical points are plotted for each pair of the solution set being the auxiliary variables x+z,y, and u.Critical Point A. This point exists for λ=0 which means that the potential V is constant. Acceleration occurs at this point because of ωtot=-1<-1/3, 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 0<ξ≤3/16 with λ,β∈R, but it is a saddle point meaning the universe evolves between different states for ξ<0. In Figure 1, we illustrate the 2-dimensional projections of 4-dimensional phase-space trajectories for the stability condition ξ=0.15 and for the present day values β=14.5, γ=1.5, and Ωm=0.3 and three auxiliary λ values. This state corresponds to a stable attractor starting from the critical point A=(0,1,0), as seen from the plots in Figure 1.Figure 1
Two-dimensional projections of the phase-space trajectories for stability conditionξ=0.15 and for present day values β=14.5, γ=1.5, and Ωm=0.3. All plots begin from the critical point A being a stable attractor.
(a) (b) (c)Critical Point B. Point B also exists for λ=0 meaning that the potential V is constant. Acceleration phase is again valid here since ωtot=-1<-1/3, but this point refers to contraction phase because y is negative here. For the stability of the point B, it is again stable for 0<ξ≤3/16 with λ,β∈R, but it is a saddle point for ξ<0. Therefore the stable attractor behavior is represented for contraction starting from the critical point B=(0,-1,0), as seen from the graphs in Figure 2. We plot phase-space trajectories for the stability condition ξ=0.15 and for the present day values β=14.5, γ=1.5, and Ωm=0.3 and three auxiliary λ values.Figure 2
Two-dimensional projections of the phase-space trajectories for stability conditionξ=0.15 and for present day values β=14.5, γ=1.5, and Ωm=0.3. All plots begin from the critical point B being a stable attractor.
(a) (b) (c)Critical Point C. Critical point C occurs for λ≠0 meaning a field dependent potential V. The cosmological behavior is again an acceleration phase since ωtot<-1/3 and an expansion phase since y is positive. Point C is stable for 0<ξ, λ=1, and β∈R and saddle point if ξ<0 and λ≠1. 2-dimensional projections of phase-space are represented in Figure 3, for the stability conditions ξ=2, λ=1 and for the present day values β=14.5, Ωm=0.3, and three auxiliary γ values in the present day value range. The stable attractor starting from the critical point C can be inferred from the plots in Figure 3.Figure 3
Two-dimensional projections of the phase-space trajectories for stability conditionsξ=2, λ=1 and for present day values β=14.5, Ωm=0.3. All plots begin from the critical point C being a stable attractor.
(a) (b) (c)Critical Point D. This point exists for λ≠0 meaning a field dependent potential V. Acceleration phase is again valid due to ωtot<-1/3, but this point refers to a contraction phase because y is negative. Point D is also stable for 0<ξ, λ=1, and β∈R. However, it is a saddle point, while ξ<0 and λ≠1. 2-dimensional plots of phase-space trajectories are shown in Figure 4, for the stability conditions ξ=2, λ=1 and for the present day values β=14.5, Ωm=0.3 and three auxiliary γ values in the present day value range. This state again corresponds to a stable attractor starting from the point D, as seen from the plots in Figure 4.Figure 4
Two-dimensional projections of the phase-space trajectories for stability conditionsξ=2, λ=1 and for present day values β=14.5, Ωm=0.3. All plots begin from the critical point D being a stable attractor.
(a) (b) (c)All the plots in Figures1–4 have the structure of stable attractor, since each of them evolves to a single point which is in fact one of the critical points in Table 1. The three-dimensional plots of the evolution of phase-space trajectories for the stable attractors are given in Figure 5. These evolutions to the critical points are the attractor solutions of our cosmological model: interacting dark matter and q-deformed dark energy nonminimally coupled to gravity, which imply an expanding universe. On the other hand, the construction of the model in the q→1 limit reproduces the results of the phase-space analysis for the nondeformed standard dark energy case. The critical points and perturbation matrices are the same for the deformed and standard dark energy models with the equivalence of the auxiliary variables as x+z=xs, y=ys, and u=us. 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). 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, with a negative pressure.Figure 5
Three-dimensional plots of the phase-space trajectories for the critical pointsA, B, C, and D being the stable attractors.
(a) (b) (c) (d)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 (31):(53)x=1-q2N1-q2Nxs,y=exp⁡-c1-q2N1-q2N-1ys,u=1-q2N1-q2Nus,where c is a constant. From the equations (53) 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 q<1 interval for large particle number, and the decrease in the variables x, y, and u refers 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 q>1 interval for large particle number. In q→1 limit deformed variables goes to standard ones.Figure 6
Behavior of the auxiliary variablesx, y, and u with respect to the deformation parameter q and the particle number N.
(a) (b) (c) (d)


## 4. Conclusion

In this study, we propose that the dark energy is formed of the negative-pressureq-deformed scalar field whose field equation is defined by the q-deformed annihilation and creation operators satisfying the deformed boson algebra in (2), 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 q-deformed dark energy nonminimally 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 nonminimally 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 ξ, interaction parameter β, and the potential constant λ. An accelerating universe exists for all stable solutions due to ωtot<-1/3. The critical points A and B are late-time stable attractors for 0<ξ≤3/16 and λ,β∈R, with the point A referring to an expansion with a stable acceleration, while the point B refers to a contraction. However, the critical points C and D are late-time stable attractors for 0<ξ, λ=1, and β∈R, with the point C referring to an expansion with a stable acceleration, while the point D refers to a contraction. The stable attractor behavior of the model at each critical point is demonstrated in Figures 1–4. In order to solve the differential equation system (42), (44), and (45) with the critical points and plot the graphs in Figures 1–4, we use adaptive Runge-Kutta method of 4th and 5th order, in MATLAB programming. Then the solutions with the stability conditions of critical points are plotted for each pair of the solution set being the auxiliary variables in x+z, y, and u.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 asA, B, C, and D.Theq-deformed dark energy is a generalization of the standard scalar field dark energy. As seen from (9) in the q→1 limit, the behavior of the deformed energy density, pressure, and scalar field functions with respect to the standard functions all approach 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 q→1 limit and the existence of the stable attractor behavior of the accelerated expansion phase of universe for the considered interacting and nonminimally 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 (31), 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 q<1 and q>1 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 q<1 interval for large particle number. Also the value of the auxiliary variables x, y, and u increases with increasing q for the q>1 interval for large particle number. In q→1 limit all the deformed variables transform to nondeformed variables.The consistency of the proposedq-deformed scalar field dark energy model is confirmed by the results, since it gives the expected behavior of the universe. The idea of considering 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 of whether the dark 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 teleparallelism.



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*Source: 7380372-2016-12-22.xml*