# Couplingq-Deformed Dark Energy to Dark Matter

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

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## Abstract

We propose a novel coupled dark energy model which is assumed to occur as aq-deformed scalar field and investigate whether it will provide an expanding universe phase. We consider the q-deformed dark energy as coupled to dark matter inhomogeneities. We perform the phase-space analysis of the model by numerical methods and find the late-time accelerated attractor solutions. The attractor solutions imply that the coupled q-deformed dark energy model is consistent with the conventional dark energy models satisfying an acceleration phase of universe. At the end, we compare the cosmological parameters of deformed and standard dark energy models and interpret the implications.


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## Body

## 1. Introduction

The standard model of cosmology states that approximately 5% of the energy content of the universe belongs to ordinary baryonic matter of standard model of particle physics. The other 95% of the energy content of the universe is made of the dark sector. Particularly, 25% of the content is an unknown form of matter having a mass but in nonbaryonic form that is called dark matter. The remaining 70% of the content consists of an unknown form of energy named dark energy. On the other hand, it is known that the universe is experiencing accelerating expansion by astrophysical observations, Supernova Ia, large-scale structure, baryon acoustic oscillations, and cosmic microwave background radiation. The dark energy is assumed to be responsible for the late-time accelerated expansion of the universe. The dark energy component of the universe is not clustered but spreads all over the universe and it generates gravitational repulsion due to its negative pressure for driving the acceleration of the expansion of the universe [1–10].In order to explain the viable mechanism for the accelerated expansion of the universe, cosmologists have proposed various dynamical models for dark energy and dark matter, which include possible interactions between dark energy, dark matter, and other fields, such as gravitation. Particularly, the coupling between dark energy and dark matter is proposed since the energy densities of two dark components are of the same order of magnitude today [11–18].Since there are a great number of candidates for the constitution of the dark energy, such as cosmological constant, quintessence, phantom, and tachyon fields, different interactions have been proposed between these constituents, dark matter, and gravitational field in the framework of general relativity [19–29]. However, the corresponding dynamical analyses of the interactions between different dark energy models and the dark matter and the gravity have been studied in the framework of teleparallel gravity which uses the torsion tensor instead of the curvature tensor of general relativity [30–39].The main motivation of this study comes from the recent studies in the literature [40–45] which involves the deformation of the standard scalar field equations representing the dark energy. In this study, we propose a novel dark energy model as a q-deformed scalar field interacting with the dark matter. Since the dark energy is a negative-pressure scalar field, this scalar field can be considered as a q-deformed scalar field. The q-deformed scalar field is in fact a q-deformed boson model, and the statistical mechanical studies of q-deformed boson models have shown that the pressure of the deformed bosons is generally negative. Not only do the different deformed boson models have a negative pressure, but also the different deformed fermion models can take negative pressure values [46–50]. Here, we consider the q-deformed bosons and propose that the scalar field which is produced by these deformed bosons constitutes the dark energy in the universe. We also investigate the dynamics of the coupling q-deformed dark energy and dark matter inhomogeneities in the Friedmann-Robertson-Walker (FRW) space-time. In order to confirm our proposal, we perform the phase-space analysis of the model whether the late-time stable attractor solutions exist or not, since the stable attractor solutions imply the accelerating expansion of the universe. We finally compare the cosmological parameters of the q-deformed and standard dark energy model and interpret the implications of the comparison.


## 2. Dynamics of the Model: Couplingq-Deformed Dark Energy to Dark Matter

In our model, the dark energy consists of the scalar field whose field equations are defined by theq-deformed boson fields. Since the idea of q-deformation to the single particle quantum mechanics is a previous establishment in the literature [51–53], it has been natural to construct a q-deformed quantum field theory [54–56]. While the bosonic counterpart of the deformed particle fields corresponds to the deformed scalar field, the fermionic one corresponds to the deformed vector field. Here, we take into account the q-deformed boson field as the q-deformed scalar field which constitutes the dark energy under consideration. The q-deformed dark energy couples to dark matter inhomogeneities in our model. Now, we begin with defining the q-deformed dark energy in the FRW geometry.Quantum field theory in curved space-time is important in the understanding of the scenario in the Early Universe. Quantum mechanically, constructing the coherent states for any mode of the scalar field translates the behavior of the classical scalar field around the initial singularity into quantum field regime. At the present universe, the quantum mechanical state of the scalar field around the initial singularity cannot be determined by an observer. Therefore, Hawking states that this indeterministic nature can be described by taking the random superposition of all possible states in that space-time. Berger has realized this by taking the superposition of coherent states randomly. Parker has studied the particle creation in the expanding universe with a nonquantized gravitational metric. When the evolution of the scalar field is considered in an expanding universe, Goodison and Toms stated that if the field quanta obey the Bose or Fermi statistics, then the particle creation does not occur in the vacuum state. Therefore, the scalar field dark energy must be described in terms of the deformed bosons or fermions in the coherent states or squeezed state [56–62]. Also, the q-deformed dark energy is a generalization of the standard scalar field dark energy. The free parameter q makes it possible to obtain a desired dark energy model with a preferred interaction or coupling by setting up the deformation parameter to the suitable value.This motivates us to describe the dark energy as aq-deformed scalar field interacting with the dark matter. We can give the Dirac-Born-Infeld type action of the model as S=κ2∫(lg+lϕq+lm+lr)d4x, where κ2=8πG, lg=-(1/2κ)gR, and lm and lr are the gravitational, dark matter, and radiation Lagrangian densities. Then, the deformed dark energy Lagrangian density is given for (+,-,-,-) metric signature as [63](1)lϕq=g12gμν∇μϕq∇νϕq-m2ϕq2,where g=-det⁡gμν and ϕq is the q-deformed scalar field operator for the dark energy and ∇μ is the covariant derivative which is in fact the ordinary partial derivative ∂μ for the scalar field. From the variation of the q-deformed scalar field Lagrangian density with respect to the deformed field, we obtain the deformed Klein-Gordon equation:(2)∂lϕq∂ϕq-∂μ∂lϕq∂∂μϕq=0,∂μ∂μϕq+m2ϕq=0.Also, we get the energy-momentum tensor of the scalar field dark energy from the variation of Lagrangian lϕq with respect to the metric tensor, such that(3)Tμνϕq=2g∂lϕq∂gμν-gμνlϕq=∂μϕq∂νϕq-12gμνgαβ∂αϕq∂βϕq-m2ϕq2,from which the timelike and spacelike parts of Tμνϕq read as follows:(4)T00ϕq=12ϕ˙q2-12gii∂iϕq2+12m2ϕq2,Tiiϕq=-12giiϕ˙q2+12∂iϕq2-12giigjj∂jϕq2+12giim2ϕq2,where i,j=1,2,3 represent the spacelike components.In order to calculate the deformed energy density and the pressure functions of theq-deformed energy from the energy-momentum components (4), we need to consider the quantum field theoretical description of the deformed scalar field in a FRW background with the metric(5)ds2=dt2-a2tdx2+dy2+dz2.Then, the canonically quantized q-deformed scalar field ϕq is introduced in terms of the Fourier expansion [57](6)ϕq=∫d3kaqkfk+aq+kfk∗,where aq(k) and aq+(k) are theq-deformed boson annihilation and creation operators for the quanta of theq-deformed scalar field dark energy in the kth mode. k denotes the spatial wave vector and obeys the relativistic energy conservation law ω2=-giik2+m2. Here, fk is a set of orthonormal mode solutions of the deformed Klein-Gordon equation, such that(7)fkxμ=exp⁡ikμxμ2π32ω,where kμ=(ω,-k) is the four-momentum vector and satisfies the relations(8)∫d3xfkfk′∗=δ3-k+k′2ω,∫d3xfkfk′=e2iωtδ3-k-k′2ω,∫d3xfk∗fk′=δ3k-k′2ω,∫d3xfk∗fk′∗=e-2iωtδ3k+k′2ω.We can give the spatial average of theq-deformed scalar field energy-momentum tensor Tμνϕq as [59](9)T-μνϕq=∫d3xTμνϕq.By using (9), the spatial average of the timelike energy-momentum tensor component is obtained as(10)T-00ϕq=∫d3x12ϕ˙q2-12gii∂iϕq2+12m2ϕq2.This can be determined from (6)–(8) term by term, as follows:(11)12m2∫d3xϕq2=12m2∫d3x∬d3kd3k′aqkfk+aq+kfk∗aqk′fk′+aq+k′fk′∗=12∫d3km22ωaqkaq-ke2iωt+aq+kaqk+aqkaq+k+aq+kaq+-ke-2iωt,(12)12∫d3xϕ˙q2=12∫d3kω2-aqkaq-ke2iωt+aq+kaqk+aqkaq+k-aq+kaq+-ke-2iωt,(13)12gii∫d3x∂iϕq2=12∫d3kgiik22ωaqkaq-ke2iωt+aq+kaqk+aqkaq+k+aq+kaq+-ke-2iωt.Combining (11), (12), and (13) gives(14)T-00ϕq=12∫d3kωaqkaq+k+aq+kaqk,where ω2=m2-k2/a2(t) for the FRW space-time. Correspondingly, the average of the spacelike energy-momentum tensor component can be determined, such that(15)T-iiϕq=∫d3x-12giiϕ˙q2+12∂iϕq2-12giigjj∂jϕq2+12giim2ϕq2=-12∫d3kgiiω2-aqkaq-ke2iωt+aq+kaqk+aqkaq+k-aq+kaq+-ke-2iωt+∫d3kki22ωaqkaq-ke2iωt+aq+kaqk+aqkaq+k+aq+kaq+-ke-2iωt+12∫d3kgiigjjk22ωaqkaq-ke2iωt+aq+kaqk+aqkaq+k+aq+kaq+-ke-2iωt+12∫d3kgiim22ωaqkaq-ke2iωt+aq+kaqk+aqkaq+k+aq+kaq+-ke-2iωt.From the identities aq(-k)e2iωt=aq+(k) and aq+(-k)e-2iωt=aq(k), (15) turns out to be(16)T-iiϕq=12∫d3k1ω2ki2-a2tω2aqkaq+k+aq+kaqk,for the FRW geometry, and ki is the ith spatial component of the wave vector.Theq-deformation of a quantum field theory is constructed from the standard algebra satisfied by the annihilation and creation operators introduced in the canonical quantization of the field. The deformation of a standard boson algebra satisfied by the annihilation and creation operators of a bosonic quantum field theory was firstly realized by Arik-Coon [51], and then Macfarlane and Biedenharn [52, 53] independently realized the deformation of boson algebra different from Arik-Coon. Hence, theq-deformed bosonic quantum field theory of the scalar field dark energy is constructed by theq-deformed algebra of the operators aq(k) and aq+(k) in (6), such that(17)aqkaq+k′-q2aq+k′aqk=δ3k-k′,(18)aqkaqk′-q2aqk′aqk=0,(19)Nk=aq+kaqk.Here, q is a real deformation parameter, and [Nk] is the deformed number operator whose eigenvalue spectrum is given as [51](20)nk=1-q2nk1-q2,where nk is the eigenvalue of the standard number operator Nk.The corresponding vector spaces of the annihilation and creation operators for theq-deformed scalar field dark energy are theq-deformed Fock space state vectors, which give information about the number of particles in the corresponding state. Theq-deformed bosonic annihilation and creation operators aq(k) and aq+(k) act on the Fock states nk as follows:(21)aqknk=nknk-1,aq+knk=nk+1nk+1,Nknk=aq+kaqknk=nknk.By taking the quantum expectation values of the spatial averages of energy-momentum tensor with respect to the Fock basis nk, we obtain the energy density and the pressure of theq-deformed dark energy. Using ρϕq=T00 and pϕq=-Tii for the energy density and pressure of theq-deformed scalar field dark energy, we obtain(22)∫ρϕqd3x=nkT-00ϕqnk=∫d3kω2nkaqkaq+k+aq+kaqknk=1+q2nk+1∫d3kϕωϕ2,whereq-deformed boson algebra in (17) is used in the second line. Because theq-deformed boson algebra in (17)–(19) transforms to be the standard boson algebra and [nk]=nk in the q→1 limit, the energy density ρϕq of theq-deformed dark energy transforms into the energy density ρϕ of the standard dark energy as(23)∫ρϕd3x=2nk+1∫d3kϕωϕ2.Hence, the energy density ρϕq of theq-deformed dark energy can be written in terms of the energy density ρϕ of the standard dark energy by(24)ρϕq=1+q2nk+12nk+1ρϕ=Δqnkρϕ.Accordingly, the pressure of theq-deformed scalar field dark energy can be written from (6) as(25)∫pϕqd3x=nk-T-iiϕqnk=∫d3k12ω2k2a2t-ω2nkaqkaq+k+aq+kaqknk=1+q2nk+1∫d3k12ω2k2a2t-ω2,where giiki2=(1/a2(t))(k12+k22+k32)=k2/a2(t) is used. In the q→1 limit, theq-deformed pressure pϕq of dark energy transforms to the standard pressure pϕ of the dark energy, such that(26)∫pϕd3x=2nk+1∫d3k12ω2k2a2t-ω2.Consequently, theq-deformed pressure pϕq of dark energy can be obtained in terms of the standard pressure pϕ of the dark energy; thus, (27)pϕq=1+q2nk+12nk+1pϕ=Δqnkpϕ.Also, the commutation relations and plane-wave expansion of theq-deformed scalar field ϕq(x) are given by using (17)–(19) in (6), as follows:(28)ϕqxϕq+x′-q2ϕq+x′ϕqx=iΔx-x′,where(29)Δx-x′=-12π3∫d3kwksin⁡wkx-x0.On the other hand, the deformed and standard annihilation operators, aq and as, are written as [62](30)aq=asNkNk.From this, we can express the deformed bosonic scalar fields in terms of the standard one by using (20) in (30) and (6):(31)ϕq=1-q2nk1-q2nkϕ=Δqϕ,where we use the Hermiticity of the number operator N.Now, we will derive the Friedmann equations for our couplingq-deformed dark energy to dark matter model with a radiation field in FRW space-time by using the scale factor a(t) in Einstein’s equations. We can achieve this by relating the scale factor with the energy-momentum tensor of the objects in the considered universe model. It is a common fact to consider energy and matter as a perfect fluid, which will naturally be generalized to dark energy and matter. An isotropic fluid in one coordinate frame gives an isotropic metric in another frame which coincides with the first frame. This means that the fluid is at rest in commoving coordinates. Then, the four-velocity vector is given as [63](32)Uμ=1,0,0,0,while the energy-momentum tensor reads(33)Tμν=ρ+pUμUν+pgμν=ρ00000gijp0.Raising one index gives a more suitable form(34)Tνμ=diag⁡-ρ,p,p,p.For a model of universe described by Dirac-Born-Infeld type action and consisting of more than one form of energy momentum, we have totally three types of energy density and pressure, such that(35)ρtot=ρϕq+ρm+ρr,(36)ptot=pϕq+pr,where the pressure of the dark matter pm is explicitly zero in the total pressure ptot (36). From the conservation of equation for the zero component ∇μT0μ=0, one obtains ρ∝a-3(1+w). Here, w is the parameter of the equation of state p=wρ which relates the pressure and the energy density of the cosmological fluid component under consideration. Therefore, pressure is zero for the matter component and wm=0, but for the radiation component wr=1/3 due to the vanishing trace of the energy-momentum tensor of the electromagnetic field. We then express the total equation of state parameter as(37)wtot=ptotρtot=wϕqΩϕq+wrΩr.While the equations of state parameters are given as wϕq=pϕq/ρϕq and wr=pr/ρr=1/3, the density parameters are defined by Ωϕq=ρϕq/ρtot, Ωr=ρr/ρtot for theq-deformed dark energy and the radiation fields, respectively. Since the pressure of the dark matter is pm=0, then the equation of state parameter is wm=pm/ρm=0 and the density parameter is Ωm=ρm/ρtot for the dark matter field having no contribution to wtot (37), but contributing to the total density parameter, such that(38)Ωtot=Ωϕq+Ωm+Ωr=κ2ρtot3H2=1.We now turn to Einstein’s equations of the formRμν=κ2(Tμν-1/2gμνT). Then, by using the components of the Ricci tensor for FRW space-time (5) and the energy-momentum tensor in (34), we rewrite Einstein’s equations, for μν=00 and μν=ij:(39)-3a¨a=κ22ρ+3p,(40)a¨a+2a˙a2=κ22ρ-p,respectively. Here, the dot also represents the derivative with respect to cosmic time t. Using (39) and (40) gives the Friedmann equations for FRW metric as(41)H2=κ23ρϕq+ρm+ρr,H˙=-κ22ρϕq+pϕq+ρm+ρr+pr,where H=a˙/a is the Hubble parameter and ρr=3pr. From the conservation of energy, we can obtain the continuity equations forq-deformed dark energy, dark matter, and the radiation constituents in the form of evolution equations, such as(42)ρ˙ϕq+3Hρϕq+pϕq=-Q′,(43)ρ˙m+3Hρm=Q,(44)ρ˙r+3Hρr+pr=Q′-Q,where Q is an interaction current between theq-deformed dark energy and the dark matter which transfers the energy and momentum from the dark matter to dark energy and vice versa. Q and Q′ vanish for the models having no coupling between the dark energy and the dark matter. For the models including only the interactions between dark energy and dark matter, the interaction terms become equal Q′=Q. The case Q<0 corresponds to energy transfer from dark matter to the other two constituents, the case Q′>0 corresponds to energy transfer from dark energy to the other constituents, and the case Q′<0 corresponds to an energy loss from radiation. Here, we consider that the model only has interaction between dark energy and dark matter and Q′=Q [64].The energy densityρ and pressure p of this dark energy are rewritten explicitly from the energy-momentum tensor components (4) obtained by the Dirac-Born-Infeld type action of couplingq-deformed dark energy and dark matter, such that [65–68](45)ρϕq=T00ϕq=12ϕ˙q2+12m2ϕq2,pϕq=-Tiiϕq=12ϕ˙q2-12m2ϕq2,where the dark energy is space-independent due to the isotropy and homogeneity. Now, the equation of motion for theq-deformed dark energy can be obtained by inserting (45) into the evolution equation, such that(46)ϕ¨q+3Hϕ˙q+m2ϕq=-Qϕ˙q.In order to obtain the energy density and pressure and equation of motion in terms of the deformation parameterq, (31) and its time derivative will be used. Because the number of particles in each mode of theq-deformed scalar field varies in time by the particle creation and annihilation, the time derivative of Δ(q)is given as(47)Δ˙q=-q2nkn˙kln⁡q1-q21-q2nknk-n˙k1-q2nk21-q2nk3.Substituting (31) and (47) in (45) and (46), we obtain(48)ρϕq=12Δ2qϕ˙2+12Δ2qm2ϕ2+12Δ˙2qϕ2+ΔqΔ˙qϕϕ˙,(49)pϕq=12Δ2qϕ˙2-12Δ2qm2ϕ2+12Δ2qϕ2+ΔqΔ˙qϕϕ˙,(50)Δqϕ¨+3ΔqHϕ˙-Δqm2ϕ+2Δ˙qϕ˙+Δ¨qϕ+3HΔ˙qϕ=-βκρm.Here, we consider the commonly used interaction current asQ=βκρmϕ˙q in the literature [15], in order to obtain stationary and stable cosmological solutions in our dark model. The deformed energy density and pressure equations (24) and (27) are the same as (48) and (49), respectively. While (24) and (48) are the expression of the deformed energy density, accordingly (27) and (49) are the deformed pressure of the dark energy in terms of the deformation parameterq. The functions of the deformation parameter in (24) and (48) are(51)Δqnk≈Δq,1+q2nk+12nk+1≈1-q2nk1-q2nk,since nk values are very large, and nk is given as a function of time:(52)nk≈t3+16t2+16t-1412+16.We now perform the phase-space analysis of our couplingq-deformed dark energy to dark matter model if the late-time solutions of the universe can be obtained, in order to confirm our proposal.


## 3. Phase-Space Analysis

We investigate the cosmological properties of the proposedq-deformed dark energy model by performing the phase-space analysis. We need to transform the equations of the dynamical system into their autonomous form [26–28, 36, 37, 69–71]. The auxiliary variables are defined to be(53)xϕq=κΔϕ˙+Δ˙ϕ6H,yϕq=κe-κλΔϕ3H.We consider an exponential potential as V=V0e-κλϕq instead of the potential V=1/2m2ϕq2 in Lagrangian (1), as the usual assumption in the literature, because the power-law potential does not provide a stable attractor solution [16, 18, 72–76].We also express the density parameters for theq-deformed scalar field dark energy, dark matter, and the radiation in the autonomous system by using (35), (36), and (48) with (53):(54)Ωϕq=κ2ρϕq3H2=xϕq2+yϕq2,(55)Ωm=κ2ρm3H2,(56)Ωr=κ2ρr3H2,and then the total density parameter is given by(57)Ωtot=κ2ρtot3H2=xϕq2+yϕq2+Ωm+Ωr=1.Also, the equation of state parameter for the dark energy is written in the autonomous form by using (35) and (36) with (53):(58)wϕq=pϕqρϕq=xϕq2-yϕq2xϕq2+yϕq2.Then, the total equation of state parameter in the autonomous system from (37) and (54)–(56) and (58) is obtained as(59)wtot=xϕq2-yϕq2+Ωr3.We also define s=-H˙/H2 in the autonomous system by using (41) and (59), such that(60)s=-H˙H2=321+wtot=321+xϕq2-yϕq2+Ωr3.s is here only a jerk parameter which is used in other equations of cosmological parameters. However, the deceleration parameter qd which is not used in the equations but is not also a jerk parameter is defined as(61)qd=-1-H˙H2.Now, we convert the Friedmann equation (41), the continuity equations (43) and (44), and the equation of motion (50) into the autonomous system by using the auxiliary variables in (53)–(56) and their derivatives with respect to N=ln⁡a, for which the time derivative of any quantity F is F˙=H(dF/dN). Thus, 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 δxϕq,δyϕq,δΩm, and δΩr 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 the critical points for each critical point [77–79].With the definitions for the interaction current and the potential, the autonomous form of the cosmological system reads [80–89](62)xϕq′=-3xϕq+sxϕq+62λyϕq2-62βΩm,yϕq′=syϕq-62λyϕqxϕq,Ωm′=Ωm-3+6βxϕq+2s,Ωr′=Ωr-4+2s.In order to perform the phase-space analysis of the model, we obtain the critical points of the autonomous system in (62). We will obtain these points by equating the left hand sides of (62) to zero for stationary solutions, by using the condition Ωtot=1. After some calculations, five critical points are found as listed in Table 1 with the existence conditions.Table 1
Critical points and existence conditions.
Label x ϕ q y ϕ q Ω m Ω r Ω ϕ q ω ϕ q ω t o t A 3 6 λ + β 2 β ( λ + β ) + 3 2 ( λ + β ) λ ( λ + β ) - 3 ( λ + β ) 2 0 β ( λ + β ) + 3 ( λ + β ) 2 - β ( λ + β ) β ( λ + β ) + 3 - β ( λ + β ) B 3 6 λ + β - 2 β ( λ + β ) + 3 2 ( λ + β ) λ ( λ + β ) - 3 ( λ + β ) 2 0 β ( λ + β ) + 3 ( λ + β ) 2 - β ( λ + β ) β ( λ + β ) + 3 - β ( λ + β )  C 6 λ 6 1 - λ 2 6 0 0 1 λ 2 3 - 1 λ 2 3 - 1  D 4 6 λ 2 3 λ 0 1 - 4 λ 2 4 λ 2 1 3 1 3  E - 1 6 β 0 1 3 β 2 1 - 1 2 β 2 1 6 β 2 1 1 3Now, we will get the perturbationsδxϕq′, δyϕq′, δΩm′, and δΩr′ for each constituent in our model by using the variations of (62), such as(63)δxϕq′=-32+92xϕq2-32yϕq2+Ωr2δxϕq+6λ-3xϕqyϕqδyϕq+62βδΩm+xϕq2δΩr,(64)δyϕq′=3xϕqyϕq-62λyϕqδxϕq+32+62λxϕq+32xϕq2-92yϕq2+Ωr2δyϕq+yϕq2δΩr,(65)δΩm′=6xϕqΩm+6βΩmδxϕq-6yϕqΩmδyϕq+6βxϕq+3xϕq2-3yϕq2+ΩrδΩm+ΩmδΩr,(66)δΩr′=6xϕqΩrδxϕq-6yϕqΩrδyϕq+-1+3xϕq2-3yϕq2+2ΩrδΩr.Thus, we obtain a4×4 perturbation matrix M whose nonzero elements are given as(67)M11=-32+92xϕq2-32yϕq2+Ωr2,M12=6λyϕq-3xϕqyϕq,M13=62β,M14=xϕq2,M21=3xϕqyϕq-62λyϕq,M22=32+62λxϕq+32xϕq2-92yϕq2+Ωr2,M24=yϕq2,M31=6xϕqΩm+6βΩm,M32=-6yϕqΩm,M33=6βxϕq+3xϕq2-3yϕq2+Ωr,M34=Ωm,M41=6xϕqΩr,M42=-6yϕqΩr,M44=-1+3xϕq2-3yϕq2+2Ωr.Then, we insert linear perturbationsxϕq→xϕq,c+δxϕq, yϕq→yϕq,c+δyϕq, Ωm→Ωm,c+δΩm, and Ωr→Ωr,c+δΩr about the critical points for three constituents in the autonomous system (62). So, we can calculate the eigenvalues of perturbation matrix M for five critical points given in Table 1, with the corresponding existing conditions.In what follows, we find and represent five perturbation matrices for each of the five critical points. We obtain five sets of eigenvalues. In order to determine the type and stability of critical points, we investigate the sign of the real parts of eigenvalues. A critical point is stable if all the real part of eigenvalues is negative. The physical meaning of the negative eigenvalue is always stable attractor; namely, if the universe is in this state, it keeps its state forever and thus it can attract the universe at a late time. There can occur accelerated expansion only forwtot<-1/3.A: (68)M=92λ+β2+3λ2λ+β-3-96λ+β+6λ2βλ+β+32λ+β-6β264λ+β96λ+β-6λ22βλ+β+32λ+β-32βλ+β+32λ+β202βλ+β+322λ+β186λ+β+6βλλ+β-3λ+β2-6λλ+β-32βλ+β+32λ+β30λλ+β-3λ+β2000-4+3λλ+βB: (69)M=92λ+β2+3λ2λ+β-396λ+β-6λ2βλ+β+32λ+β-6β264λ+β-96λ+β+6λ22βλ+β+32λ+β-32βλ+β+32λ+β20-2βλ+β+322λ+β186λ+β+6βλλ+β-3λ+β26λλ+β-32βλ+β+32λ+β30λλ+β-3λ+β2000-4+3λλ+βC: (70)M=λ2-36λ21-λ26-6β26λ120λ22-30121-λ2600λ2+λβ-30000λ2-4D: (71)M=8λ2-122-42λ2-6β226λ-2+42λ24λ2013λ004βλ+10246λ1-4λ2-123λ1-4λ201-4λ2 E: (72)M=12β2-10-6β2-126β02+λβ0000013β26β1-12β2001-12β2Eigenvalues of the fiveM matrices with the existence conditions, stability conditions, and acceleration condition are represented in Table 2, for each of the critical points A, B, C, D, and E. As seen in Table 2, the first two critical points A and B have the same eigenvalues. Here, the eigenvalues and the stability conditions of the perturbation matrices for critical points A, B, D, and E have been obtained by the numerical methods, due to the complexity of the matrices. The stability conditions of each critical point are listed in Table 2, according to the sign of the eigenvalues.Table 2
Eigenvalues and stability of critical points.
Eigenvalues λ β  A and B −0.7516 −0.7516 −0.0065 −1.0065 4.6000 0.0100 Existing condition isβ(λ+β)>-3/2 and λ(λ+β)>3. Stable point if λ and β are the given values for the negative eigenvalues in the second column. Acceleration phase occurs if λ<2β. −0.7518 −0.7518 −0.0073 −1.0073 4.1000 0.0100 −0.7521 −0.7521 −0.0083 −1.0083 3.6000 0.0100 −0.7524 −0.7524 −0.0096 −1.0096 3.1000 0.0100 −0.8249 −0.8249 −0.2994 −1.2994 4.6000 0.5100 −0.8330 −0.8330 −0.3319 −1.3319 4.1000 0.5100 −0.8431 −0.8431 −0.3723 −1.3723 3.6000 0.5100 −0.8560 −0.8560 −0.4238 −1.4238 3.1000 0.5100 −0.8850 −0.8850 −0.5401 −1.5401 4.6000 1.0100 −0.8982 −0.8982 −0.5930 −1.5930 4.1000 1.0100 −0.9143 −0.9143 −0.6573 −1.6573 3.6000 1.0100 −0.9343 −0.9343 −0.7372 −1.7372 3.1000 1.0100 −0.9354 −0.9354 −0.7414 −1.7414 4.6000 1.5100 −0.9519 −0.9519 −0.8075 −1.8075 4.1000 1.5100 −0.9716 −0.9716 −0.8865 −1.8865 3.6000 1.5100 −0.9781 −0.9781 −0.9123 −1.9123 4.6000 2.0100 −0.9957 −0.9957 −0.9826 −1.9826 3.1000 1.5100 −0.9967 −0.9967 −0.9869 −1.9869 4.1000 2.0100 −1.0148 −1.0148 −1.0591 −2.0591 4.6000 2.5100 −1.0187 −1.0187 −1.0749 −2.0749 3.6000 2.0100 −1.0348 −1.0348 −1.1392 −2.1392 4.1000 2.5100 −1.0450 −1.0450 −1.1800 −2.1800 3.1000 2.0100 −1.0581 −1.0581 −1.2324 −2.2324 3.6000 2.5100 −1.0856 −1.0856 −1.3422 −2.3422 3.1000 2.5100  Eigenvalues  C λ 2 + λ β - 3, λ2-3,  λ2-4,  λ22-3, Existing condition isλ<6. Stable point if β<3-λ2/λ and λ<3. Acceleration phase occurs if λ<2.  Eigenvalues λ β  D −0.5000 −0.5000 1.0000 0.2000 2.5000 −0.5000 Existing condition isλ>2 and ∀β.Unstable point.Acceleration phase never occurs. −0.5000 −0.5000 1.0000 0.3333 3.0000 −0.5000 −0.5000 −0.5000 1.0000 0.4286 3.5000 −0.5000 −0.5000 −0.5000 1.0000 0.5000 4.0000 −0.5000 1.0000 −1.0000 0 0 2.0000 −0.5000 1.0000 −1.0000 2.0000 0 2.0000 0.5000 1.0000 −1.0000 4.0000 0 2.0000 1.5000 1.0000 −1.0000 −2.0000 0 2.0000 −1.5000 −0.5000 −0.5000 1.0000 0 4.0000 −1.0000 −0.5000 −0.5000 1.0000 1.0000 2.5000 0 −0.5000 −0.5000 1.0000 1.0000 3.5000 0 −0.5000 −0.5000 1.0000 1.0000 4.0000 0 −0.5000 −0.5000 1.0000 1.5000 4.0000 0.5000 −0.5000 −0.5000 1.0000 1.5714 3.5000 0.5000 −0.5000 −0.5000 1.0000 1.6667 3.0000 0.5000 −0.5000 −0.5000 1.0000 1.8000 2.5000 0.5000 −0.5000 −0.5000 1.0000 2.0000 4.0000 1.0000 −0.5000 −0.5000 1.0000 2.1429 3.5000 1.0000 −0.5000 −0.5000 1.0000 2.3333 3.0000 1.0000 −0.5000 −0.5000 1.0000 2.5000 4.0000 1.5000 −0.5000 −0.5000 1.0000 2.6000 2.5000 1.0000 −0.5000 −0.5000 1.0000 2.7143 3.5000 1.5000 −0.5000 −0.5000 1.0000 3.0000 3.0000 1.5000 −0.5000 −0.5000 1.0000 3.0000 4.0000 2.0000 −0.5000 −0.5000 1.0000 3.2857 3.5000 2.0000 −0.5000 −0.5000 1.0000 3.4000 2.5000 1.5000 −0.5000 −0.5000 1.0000 3.6667 3.0000 2.0000 −0.5000 −0.5000 1.0000 4.2000 2.5000 2.0000 −0.5000 −0.5000 1.0000 −0.1429 3.5000 −1.0000 −0.5000 −0.5000 1.0000 −0.3333 3.0000 −1.0000 −0.5000 −0.5000 1.0000 −0.5000 4.0000 −1.5000 −0.5000 −0.5000 1.0000 −0.6000 2.5000 −1.0000 −0.5000 −0.5000 1.0000 −0.7143 3.5000 −1.5000 −0.5000 −0.5000 1.0000 −1.0000 3.0000 −1.5000 −0.5000 −0.5000 1.0000 −1.0000 4.0000 −2.0000 −0.5000 −0.5000 1.0000 −1.2857 3.5000 −2.0000 −0.5000 −0.5000 1.0000 −1.4000 2.5000 −1.5000 −0.5000 −0.5000 1.0000 −1.6667 3.0000 −2.0000 −0.5000 −0.5000 1.0000 −2.2000 2.5000 −2.0000   Eigenvalues λ β  E 0.2634 −0.1317 −0.1317 2.0704 0.1000 0.7100 Existing condition isβ>1/2 and ∀λ.Unstable point.Acceleration phase never occurs. 0.2634 −0.1317 −0.1317 2.4225 0.6000 0.7100 0.2634 −0.1317 −0.1317 2.7746 1.1000 0.7100 0.2634 −0.1317 −0.1317 3.1268 1.6000 0.7100 0.2634 −0.1317 −0.1317 3.4789 2.1000 0.7100 0.2634 −0.1317 −0.1317 3.8310 2.6000 0.7100 0.2634 −0.1317 −0.1317 4.1831 3.1000 0.7100 0.2634 −0.1317 −0.1317 4.5352 3.6000 0.7100 1.0273 −0.8883 −0.1390 2.0185 0.1000 2.7100 1.0273 −0.8883 −0.1390 2.1107 0.6000 2.7100 1.0273 −0.8883 −0.1390 2.2030 1.1000 2.7100 1.0273 −0.8883 −0.1390 2.2952 1.6000 2.7100 1.0273 −0.8883 −0.1390 2.3875 2.1000 2.7100 1.0273 −0.8883 −0.1390 2.4797 2.6000 2.7100 1.0273 −0.8883 −0.1390 2.5720 3.1000 2.7100 1.0273 −0.8883 −0.1390 2.6642 3.6000 2.7100 1.0368 −0.8208 −0.2160 2.0226 0.1000 2.2100 1.0368 −0.8208 −0.2160 2.1357 0.6000 2.2100 1.0368 −0.8208 −0.2160 2.2489 1.1000 2.2100 1.0368 −0.8208 −0.2160 2.3620 1.6000 2.2100 1.0368 −0.8208 −0.2160 2.4751 2.1000 2.2100 1.0368 −0.8208 −0.2160 2.5882 2.6000 2.2100 1.0368 −0.8208 −0.2160 2.7014 3.1000 2.2100 1.0368 −0.8208 −0.2160 2.8145 3.6000 2.2100 1.0437 −0.5219 −0.5219 2.0413 0.1000 1.2100 1.0437 −0.5219 −0.5219 2.2479 0.6000 1.2100 1.0437 −0.5219 −0.5219 2.4545 1.1000 1.2100 1.0437 −0.5219 −0.5219 2.6612 1.6000 1.2100 1.0437 −0.5219 −0.5219 2.8678 2.1000 1.2100 1.0437 −0.5219 −0.5219 3.0744 2.6000 1.2100 1.0437 −0.5219 −0.5219 3.2810 3.1000 1.2100 1.0437 −0.5219 −0.5219 3.4876 3.6000 1.2100 1.0485 −0.5910 −0.4575 2.0292 0.1000 1.7100 1.0485 −0.5910 −0.4575 2.1754 0.6000 1.7100 1.0485 −0.5910 −0.4575 2.3216 1.1000 1.7100 1.0485 −0.5910 −0.4575 2.4678 1.6000 1.7100 1.0485 −0.5910 −0.4575 2.6140 2.1000 1.7100 1.0485 −0.5910 −0.4575 2.7602 2.6000 1.7100 1.0485 −0.5910 −0.4575 2.9064 3.1000 1.7100We now analyze the cosmological behavior of each critical point by noting the attractor solutions in scalar field cosmology [90]. From the theoretical cosmology studies, we know that the energy density of a scalar field has an effect on the determination of the evolution of the universe. Cosmological attractors provide the understanding of the evolution and the affecting factors on this evolution; for example, from the dynamical conditions, the scalar field evolution approaches a certain kind of behavior without initial fine tuning conditions [91–101]. Attractor behavior is known as a situation in which a collection of phase-space points evolve into a particular region and never leave from there.Critical Point A. This point exists for β(λ+β)>-3/2 and λ(λ+β)>3. Because of wtot<-1/3, acceleration occurs at this point if λ<2β and it is an expansion phase since yϕq is positive, so H is positive, too. Point A is stable, meaning that the universe keeps its further evolution, if λ and β take the values for the negative eigenvalues given in the second column of Table 2. In Figure 1, we also represent the 2-dimensional and 3-dimensional projections of 4-dimensional phase-space trajectories for β=2.51, λ=3.1, λ=3.6, and λ=4.1. This state corresponds to a stable attractor starting from the critical point A=(0.21,0.70,0.45,0), as seen from the plots in Figure 1. Also, zero value of critical point Ωr cancels the total behavior Ωr′ in (66).Figure 1
Two- and three-dimensional projections of the phase-space trajectories forβ=2.51, λ=3.1, λ=3.6, and λ=4.1. All plots begin from the critical point A=(0.21,0.70,0.45,0) being a stable attractor.Critical Point B. Point B also exists for β(λ+β)>-3/2 and λ(λ+β)>3. Acceleration phase is again valid here if λ<2β leading wtot<-1/3, but this point refers to contraction phase because yϕq is negative here. Stability of the point B is again satisfied for λ and β values given in the second column of Table 2. Therefore, the stable attractor behavior is represented starting from the critical point B=(0.21,-0.70,0.45,0) for β=2.51, λ=3.1, λ=3.6, and λ=4.1 values, in Figure 2. The zero value of critical point Ωr again cancels the total behavior Ωr′ in (66).Figure 2
Two- and three-dimensional projections of the phase-space trajectories forβ=2.51, λ=3.1, λ=3.6, and λ=4.1. All plots begin from the critical point B=(0.21,-0.70,0.45,0) being a stable attractor.Critical Point C. Critical point C occurs for all values of β, while λ<6. The cosmological behavior is again an acceleration phase that occurs if λ<2 providing wtot<-1/3 and an expansion phase since yϕq is positive. Point C is stable if β<(3-λ2)/λ and λ<3. Two-dimensional projection of phase-space is represented in Figure 3, for β=1.5, λ=0.001, λ=0.5, and λ=1.1. The stable attractor starting from the critical pointC = (0.48, 0.87, 0, 0) can be inferred from Figure 3. We again find zero plots containing zero values Ωm and Ωr, since they cancel the total behaviors Ωm′ and Ωr′ in (65) and (66).Figure 3
Two-dimensional projections of the phase-space trajectories forβ=1.5, λ=0.001, λ=0.5, and λ=1.1. All plots begin from the critical point C=(0.48,0.87,0,0) being a stable attractor.Critical Point D. This point exists for any values of β, while λ>2. Acceleration phase never occurs due to wtot=1/3. Point D is always unstable for any values of β and λ. This state corresponds to an unstable saddle point starting from the point D=(0.54,0.38,0,0.55) for β=1.5, λ=3, λ=4, and λ=6, as seen from the plots in Figure 4. Zero plots containing the axis Ωm lead to the cancellation of the total behavior Ωm′ in (65), since Ωm=0, so they are not represented in Figure 4.Figure 4
Two- and three-dimensional projections of the phase-space trajectories forβ=1.5, λ=3, λ=4, and λ=6. All plots begin from the critical point D=(0.54,0.38,0,0.55) being an unstable solution.Critical Point E. This point exists for any values of λ, while β>1/2. Acceleration phase never occurs due to wtot=1/3. Point E is always unstable for any values of β and λ. This state corresponds to an unstable saddle point starting from the point =(-0.24,0,0.11,0.82) for  λ=1, β=1.7, β=2.6, and β=3.5, as seen from the plots in Figure 5. Zero plots containing the axis yϕq lead to the cancellation of the total behavior yϕq′ in (64), since yϕq=0, so they are not represented in Figure 5.Figure 5
Two- and three-dimensional projections of the phase-space trajectories forλ=1, β=1.7, β=2.6, and β=3.5. All plots begin from the critical point E=(-0.24,0,0.11,0.82) being an unstable solution.All the plots in Figures1–3 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. These evolutions to the critical points are the attractor solutions in couplingq-deformed dark energy to dark matter cosmology of our model, which imply an expanding universe. Therefore, we confirm that the dark energy in our model can be defined in terms of theq-deformed scalar fields obeying theq-deformed boson algebra in (17)–(20). According to the stable attractor behaviors, it makes sense to consider the dark energy as a scalar field defined by theq-deformed scalar field, due to the negative pressure ofq-deformed boson field, as dark energy field.Finally, we can investigate the relation betweenq-deformed and standard dark energy density, pressure, and scalar field equations in (24), (27), and (31). We illustrate the behavior ofq-deformed energy density and pressure in terms of the standard ones with respect to the total number of particles and the deformation parameterq in Figures 6 and 7, respectively. We observe that for a large particle number theq-deformed energy density and pressure function decrease with the decrease in deformation parameterq. On the contrary, if the particle number is small, the deformed energy density and pressure increase with the decrease in deformation parameter. Note that when the deformation parameter decreases from 1, this increases the deformation of the model, since the deformation vanishes by approaching 1. The deformation parameter significantly affects the value of the deformed energy density and pressure. In the q→1 limit, deformed energy density and pressure function became identical to the standard values, as expected.Figure 6
q-deformed energy density for various values of n and q, in terms of standard energy density.Figure 7
q-deformed pressure for various values of  n and q, in terms of standard pressure.In Figure8, we represent theq-deformed scalar field behavior in terms of the standard one. It is observed that while the deformation parameter q→1,q-deformed scalar field becomes identical to the standard one. However, it asymptotically approaches lower values, whileq decreases with large number of particles. Since the square of a quantum mechanical field means the probability density,q-deformed probability density decreases when the deformation increases, and in the q→0 limit it approaches zero.Figure 8
q-deformed scalar field for various values of n and q, in terms of standard scalar field.Also, since the dark matter pressure is taken to be zero,ωm≈0 and ωtot≈ωϕq. For the stable accelerated expansion condition ωtot≈ωϕq<-1/3, solutions require the scalar field dark energy pressure to be negative pϕq<0. From the relation pϕq=Δq(nk)pϕ in (27), we finally represent the effect of q on the deformed dark energy pressure pϕq, namely, on the accelerated expansion behavior in Figure 9. From the figure, we deduce that, for any values of q, the deformed dark energy shows the accelerated expansion behavior with the negative deformed dark energy pressure.Figure 9
Effect ofq on the accelerated expansion behavior with negative dark energy pressure.


## 4. Conclusion

Since it is known that the dark energy has a negative pressure acting as gravitational repulsion to drive the accelerated expansion of the universe, we are motivated to propose that the dark energy consists of negative-pressureq-deformed scalar field whose field equation is defined by theq annihilation and creation operators satisfying theq-deformed boson algebra in (17)–(20). In order to confirm our proposal, we considerq-deformed dark energy coupling to the dark matter inhomogeneities and then investigate the dynamics of the model. Later on, we perform the phase-space analysis, whether it will give stable attractor solutions or not, which refers to the accelerating expansion phase of the universe. Therefore, the action integral of couplingq-deformed dark energy model is set up to study its dynamics, and the Hubble parameter and Friedmann equations of the model are obtained in spatially flat FRW geometry. Later on, we find the energy density and pressure values with the evolution equations forq-deformed dark energy, dark matter, and the radiation fields 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 suitable auxiliary variables, in order to perform the phase-space analysis of the model. Then, the critical points of the autonomous system are obtained by setting each autonomous equation to zero and four perturbation matrices can be written for each critical point by constructing the perturbation equations. We determine the eigenvalues of four perturbation matrices to examine the stability of critical points. There are also some important calculated 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 β. An accelerating universe exists for all stable solutions due to wtot<-1/3. The critical points A and B are late-time stable attractors for the given λ and β values for the negative eigenvalues in the second column of Table 2. The point A refers to expansion, while the point B refers to contraction with stable acceleration for λ<2β. However, the critical point C is late-time stable attractor for β<(3-λ2)/λ and λ<3 with expansion. The stable attractor behavior of the model at each critical point is demonstrated in Figures 1–3. In order to solve the differential equation system (62) with the critical points and plot the graphs in Figures 1–5, 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 sets being the auxiliary variables in (53), (55), and (56).These figures represent the notion that, by choosing the suitable parameters of the model, we obtain the stable and unstable attractors asA, B, C, D, and E, depending on the existence conditions of critical points A, B, C, D, and E. Also, the suitable parameters with the stability conditions give the stable accelerated behavior for A, B, and C attractor models.Theq-deformed dark energy is a generalization of the standard scalar field dark energy. As seen from the behavior of the deformed energy density, pressure, and scalar field functions with respect to the standard functions, in the q→1 limit, they all approach the standard corresponding function values. However, in the q→0 limit, the deformed energy density and the pressure functions decrease to smaller values of the standard energy density and the pressure function values, respectively. This implies that the energy momentum of the scalar field decreases when the deformation becomes more apparent, sinceq reaches 1 which gives the nondeformed state. Also, when q→0 for largen values, the deformed scalar field approaches zero value meaning a decrease in the probability density of the scalar field. This state is expected to represent an energy-momentum decrease leading to a decrease in the probability of finding the particles of the field. 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 the universe for the considered coupling dark energy and dark matter model.The results confirm that the proposedq-deformed scalar field dark energy model is consistent since it gives the expected behavior of the universe. The idea to consider the dark energy as aq-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 on the purpose of confirming whether the dark energy can be considered as a general deformed scalar field, the other couplings between dark energy and dark matter and also in the other framework of gravity, such as teleparallel or maybe f(R) gravity, can be investigated.



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