Cosmic acceleration in non-canonical scalar field model - An interacting scenario
aa r X i v : . [ g r- q c ] F e b Cosmic acceleration in non-canonical scalar field model:an interacting scenario
Sudipta Das and Abdulla Al Mamon Department of Physics, Visva-Bharati,Santiniketan- 731235, India.
Abstract
In this paper we have studied the dynamics of accelerating scenario within the frameworkof scalar field models possessing a non-canonical kinetic term. In this toy model, the scalarfield is allowed to interact with the dark matter component through a source term. Wehave assumed a specific form for the coupling term and then have studied the dynamics ofthe scalar field having a constant equation of state parameter. We have also carried outthe dynamical system study of such interacting non-canonical scalar field models for powerlaw potentials for some physically relevant specific values of the model parameters. It hasbeen found that the only for two particular stable fixed points of the system, an acceleratingsolution is possible and the universe will settle down to a ΛCDM universe in future and thusthere is no future singularity in this model.
Keywords: Dark energy, Non-canonical scalar field, interaction;
Recent cosmological observations [1, 2] strongly suggest that the present cosmic acceleration isdriven by the mysterious dark energy which has large negative pressure i.e., long range anti-gravity properties. Several dark energy models have been constructed to explain this late timeacceleration of the universe using a scalar field, namely, quintessence [3, 4, 5, 6, 7], having differentforms of equation of state p = wρ for the dark energy [8, 9]. A number of other models also existin the literature in which the action is modified in various ways. To name a few, there are non-minimally coupled scalar field models [10, 11, 12, 13, 14, 15], f ( R )-gravity models [16, 17, 18, 19],non-canonical scalar field models [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], k -essencemodels [35, 36, 37] and many more. All these proposals offer quite satisfactory description of thedark energy properties in some sense or the other although each of them have its own demerit.As none of these models can be considered as superior to others, the search is on for a suitabledark energy candidate.Most of the dark energy models consider the field to be non-interacting. As nothing specific is E-mail:[email protected] E-mail : [email protected] known about the nature of dark energy, an interacting scenario may be useful and may providea more general scenario. Recently, interacting dark energy models have gained interest and thereare several works in which investigations are carried out considering an interaction between thedifferent components of the universe [38, 39, 40, 41, 42, 43]. It has been found that an interactingscenario can provide solution to a number of cosmological problems in both canonical scalar fieldmodels [44] as well as non-minimally coupled scalar field models such as Brans-Dicke scalar field[45, 46]. Motivated by these facts, in this paper we try to build up a viable dark energy modelusing a non-canonical scalar field in an interacting scenario.In general, the Lagrangian for a scalar field model can be represented as [47] L = f ( φ ) F ( X ) − V ( φ ) (1)where X = ˙ φ for a spatially homogeneous scalar field. Equation (1) infact includes all thepopular single scalar field models. It describes k -essence when V ( φ ) = 0 and quintessence when f ( φ ) = constant and F ( X ) = X . In this paper, we consider another class of models called Gen-eral Non-Canonical Scalar Field Model with its Lagrangian given by L = F ( X ) − V ( φ ). Thistype of non-canonical scalar field models were proposed by Fang et al. [48], where they couldobtain cosmological solutions for different forms of F ( X ). Recently Unnikrishnan et al. [49] haveproposed an inflationary model for the universe using a non-canonical scalar field.Following [48] and [49], we consider a non-canonical scalar field cosmological model in theframework of a homogeneous, isotropic and spatially flat FRW space-time. We also consider thatthe non-canonical scalar field is interacting and we introduce a source term through which thescalar field interacts with the matter field. Under this scenario, we derive the field equations andtry to obtain exact solutions for various cosmological parameters. A brief description of the paperis as follows: In section 2, we have derived the field equations and the conservation equations forthe non-canonical scalar field Lagrangian. We have shown that with a constant equation of stateparameter for this non-canonical scalar field, this interacting model can provide an acceleratedexpansion phase of the universe preceded by a decelerating phase. We could obtain cosmologicalsolutions for various parameters of the model. In section 3, we perform the dynamical systemstudy for this interacting non-canonical scalar field model of the universe with simple power lawpotentials. The last section contains some concluding remarks. Let us consider the action S = Z √− gdx (cid:20) R L ( φ, X ) (cid:21) + S m (2)(We have chosen the unit where 8 πG = c = 1.)where R is the Ricci scalar, S m represents the action of the background matter, the Lagrangiandensity L ( φ, X ) is an arbitrary function of the scalar field φ , which is a function of time only andits kinetic term X given by X = ∂ µ φ∂ µ φ .Variation of the action (2) with respect to the metric g µν gives the Einstein field equations as G µν = h T φµν + T mµν i (3)where the energy-momentum tensor of the scalar field φ is given by T φµν = ∂ L ∂X ∂ µ φ∂ ν φ − g µν L . (4) T mµν represents the energy-momentum tensor of the matter component which is modelled as anidealized perfect fluid, and is given by T mµν = ( ρ m + p m ) u µ u ν + pg µν (5)where ρ m and p m are the energy density and pressure of the matter components of the universerespectively. The four-velocity of the fluid is denoted by u µ . Also, variation of the action (2) withrespect to the scalar field φ gives the equation of motion for φ field as¨ φ " ∂ L ∂X ! + (2 X ) ∂ L ∂X ! + " H ∂ L ∂X ! + ˙ φ ∂ L ∂X∂φ ! ˙ φ − ∂ L ∂φ ! = 0 (6)Let us consider a homogeneous, isotropic and spatially flat FRW universe which is characterizedby the line element ds = dt − a ( t )[ dr + r dθ + r sin θdφ ] (7)where a (t) is the scale factor of the universe with cosmic time t . Here, we only consider thespatially flat FRW universe as indicated by the anisotropy of the CMBR measurement [50]. TheEinstein field equations for the space-time given by equation (7) with matter in the form ofpressureless perfect fluid (i.e., p m = 0) takes the form,3 H = ρ m + ρ φ , (8)2 ˙ H + 3 H = − p φ (9)˙ ρ φ + 3 H ( ρ φ + p φ ) = 0 (10)˙ ρ m + 3 Hρ m = 0 (11)Here H = ˙ aa is the Hubble parameter and an overdot indicates differentiation with respect to thetime-coordinate t . The expressions for the energy density ρ φ and the pressure p φ of the scalarfield φ are given by ρ φ = ∂ L ∂X ! X − L , p φ = L (12)where X = ˙ φ .A number of functional forms of L ( φ, X ) have been considered in the literature, see for instanceRefs. [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. In this present work, we choose aLagrangian density of the following form : L ( φ, X ) = X − V ( φ ) (13)where V ( φ ) is the potential for the scalar field φ .Equation (13) can be obtained from the general form of Lagrangian density L ( φ, X ) = X XM p ! α − − V ( φ ) (14)considered by several authors [31, 49, 51] for α = 2 and M p = 1 / √ πG = 1. It is interesting tonote that for α = 1, equation (14) reduces to the usual Lagrangian density for a canonical scalarfield model h L ( φ, X ) = ˙ φ − V ( φ ) i .From equation (12), one can obtain the energy density and the pressure for the φ -field corre-sponding to the above Lagrangian density as ρ φ = 3 X + V ( φ ) , p φ = X − V ( φ ) (15)with X = 12 ˙ φ (16)Considering equations (13), (15) and (16), equations (8) and (9) take the form3 H = ρ m + 34 ˙ φ + V ( φ ) , (17)2 ˙ H + 3 H = −
14 ˙ φ + V ( φ ) (18)Also the conservation equations (10) and (11) take the form˙ ρ φ + 3 H ˙ φ = 0 (19)˙ ρ m + 3 Hρ m = 0 . (20)With the above set of equations (17) - (20), we want to build up an accelerating model for theuniverse in which the non-canonical scalar field plays the role of dark energy such that the universesmoothly transits from a decelerating to an accelerating phase. This is a must for the structureformation of the universe. This type of transition has been achieved in a number of canonicalscalar field models [44, 52], modified gravity models or non-minimally coupled scalar field models[45, 46, 53, 54]. In this work, we try to achieve the same in non-canonical scalar field model ofdark energy.Usually the dark matter and scalar field components are considered to be non-interacting. Butas nothing is known about the nature of the dark energy, an interaction between the two mattercomponents will provide a more general scenario. At this stage we consider that the scalar fieldand the dark matter components do not conserve separately but interact with each other througha source term (say Q ). The sign of Q will determine the direction of energy flow between thesecomponents. The equations (10) and (11) under such a scenario generalize to˙ ρ φ + 3 H (1 + ω φ ) ρ φ = Q (21)˙ ρ m + 3 Hρ m = − Q (22)We choose a simple functional form of Q as Q = αH ˙ φ (23)where α is an arbitrary constant and characterizes the strength of the coupling. There is nofundamental theory concerning the form of coupling term in the dark sector till now. In addition,we have lack of information regarding the nature of the dark sectors. So, if the dark sectorsare allowed to interact among themselves, the coupling term Q can be of any arbitrary form. Itdeserves mention that the ansatz in equation (23) is purely phenomenological and is motivatedby mathematical simplicity.Out of the four equations (equations (8), (9), (21) and (22)), only three are independent andthe fourth one can be derived from the others using the Bianchi identity. So, we have threeindependent equations to solve for four unknown parameters a , ρ m , φ and V ( φ ). Hence, oneassumption can be made to match the number of unknowns with the number of independentequations.It is well known observationally that at present ω φ ≃ − a >
0) gives the effective EoS parameter: − < ω φ < − . Motivated by these restrictions whichshows that the range of allowed values of ω φ is very small, we consider that the EoS parameter ω φ is constant (say, ω ) and is given by the following simple relation ω φ = p φ ρ φ = X − V X + V = ω (a constant) (24)This makes the system of equations closed now.The above equation leads to X = 1 + ω − ω V ⇒ ˙ φ = 4 (cid:18) ω − ω (cid:19) V ( φ ) (25)From equations (15) and (25), we obtain the expressions for the energy density and pressure forthis interacting model as ρ φ = 4(1 − ω ) V ( φ ) , p φ = 4 ω (1 − ω ) V ( φ ) (26)Using equation (26) and changing the argument from t to a , equation (21) can be re-written as dV ( a ) da ˙ a + ǫ ˙ aa V ( a ) = 0 ⇒ V ( a ) = V a − ǫ (27)where V is a positive constant and ǫ = (3 − α )(1 + ω ).From equation (9), (26) and (27), by simple algebra one can obtain the form of the Hubbleparameter for this model as H = γa − ǫ + Ba − (28)where γ = ωV (3 − ǫ )(3 ω − and B is a positive integration constant.Finally using equations (25) and (27), one can obtain the expression for evolution equation of thenon-canonical scalar field φ as φ ( a ) = A Z f ( a ) da + φ (29)where f ( a ) = 1 a vuut a − ǫ γB a (3 − ǫ ) ! , (30) A = q B h V ω − ω i and φ is the constant of integration. It is worth mentioning that for somespecific choices of α , ω and B , which will determine the value of ǫ and eventually will provide aspecific functional form of f ( a ), equation (29) can be integrated to obtain an analytical expressionfor φ ( a ). So it is found that in principle, a potential V ( a ) leads to a number of possible solutionsfor the scalar field for different values of the model parameters.The evolutionary trajectory for the potential V ( φ ) corresponding to the scalar field φ is shownin Figure 1. It has been found that the nature of the potential does not crucially depend on thevalues of ǫ , ω , V , B etc.From equation (8), one can also obtain the expressions for the energy densities ρ φ and ρ m as ------> Φ ------ > V Figure 1:
Plot of V vs. φ for ǫ = 1 . (thick curve), ǫ = 1 . (dashed curve), ǫ = 1 . (dotted curve), ω = − . , V = 2 , B = 0 . and φ = 3 . ρ φ ( z ) = 4(1 − ω ) V = 4 V − ω (1 + z ) ǫ (31) ρ m ( z ) = 3 H − ρ φ = (cid:18) γ − V − ω (cid:19) (1 + z ) ǫ + 3 B (1 + z ) (32)where 1 + z = a a and a = 1 is the present value of the scale factor.Now, from equation (32) it is found that in order to ensure the positivity of the energy densityfor the matter field, the value of ǫ is constrained as, ǫ < γ ).For plotting the figures, we have chosen the value of ǫ ∼ . . ǫ .Also for the sake of completeness one can find out the expressions for the density parameters forthe scalar field (Ω φ ) and the matter field (Ω m ) asΩ φ ( z ) = 4 V B (1 − ω ) " (1 + z ) ǫ − γB (1 + z ) ǫ − , Ω m ( z ) = 1 − Ω φ ( z ) (33) ------> z ------ > W Φ , W m ------> z ------ > W Φ , W m Figure 2:
Plot of Ω m (dashed curve) and Ω φ (solid curve) as a function of z for ǫ = 1 . (upperpanel) and ǫ = 1 . (lower panel). We have chosen ω = − . , V = 2 and B = 0 . . Here, γ = ωV (3 − ǫ )(3 ω − . Figure (2) shows the variation of density parameters for the two fields with redshift z . It showsthat Ω φ remains sub-dominant in the early epoch and starts dominating in the recent past. Ithas been also found that the nature of the plot is insensitive to slight variation of the values of ǫ , B etc.As mentioned earlier we are mainly interested in a model for the universe which smoothly transitsfrom a decelerating to an accelerating phase. This can be best understood from the evolution ofthe deceleration parameter of the universe. The deceleration parameter q for this model comesout as q ( z ) = − ¨ aaH = − − ˙ HH = 12 + 2 ωV B (1 − ω ) " (1 + z ) ǫ − γB (1 + z ) ǫ − (34)It is clearly evident from figure (3) that q undergoes a smooth transition from a deceleratedto an accelerated phase of the universe. The signature flip in q takes place at around z ≈ . q against z is also hardly affected by small change in the value of ǫ and other model parameters. Determination of sign of Q : For an accelerated universe, the EoS parameter ω must satisfy the inequality, − < ω < − - - ------> z ------ > q Ε= Ε= Ε= Figure 3:
Plot of q vs. z for different values of ǫ , ω = − . , V = 2 and B = 0 . . Here, α = 3 − ǫ ω and γ = ωV (3 − ǫ )(3 ω − . (except for tachyon models in which ω < − ǫ = (3 − α )(1 + ω ) as ǫ <
3. As α and ǫ are related as α = 3 − ǫ ω , it is evidentthat α will be negative. As a consequence the parameter Q = αH ˙ φ also turns out to be negativefor the present model. Equations (21) and (22) thus indicates that the energy gets transferredfrom the dark energy (DE) to the dark matter (DM) sector during the cosmic evolution. This isindeed counter intuitive as the dark energy component should dominate at later stages of cosmicevolution. A large number of DE models [38, 39, 40, 41, 42, 43] have been proposed where energyflows from the DM sector to the DE sector such that the DE dominates over the DM at later timesand drives the late-time acceleration of the universe. This provides a solution to the cosmologicalcoincidence problem. In our model, however the direction of flow of energy is reverse. But recentlyPavon and Wang [55] have shown that as long as dark energy is amenable to a fluid descriptionwith a temperature not far from equilibrium, the overall energy transfer should be from DE to DMsector if the second law of thermodynamics and Le Chatelier-Braun principle are to be fulfilled.Also in the present model, we have the expressions for energy densities of the scalar field, ρ φ (DE) and the normal matter field ρ m (DM) given by equations (31) and (32) respectively. A plotof ρ φ and ρ m vs. z (Fig 4) shows that inspite of the fact that energy flows from the DE to theDM sector, the evolution dynamics of the universe for the present non-canonical model is suchthat the energy density of the DE sector dominates over the DM sector at late times. The reasonbehind this is that the rate of flow of energy is very less and thus energy density of the DM sectorfalls off more quickly than the DE sector. However it may happen that because of this reverseflow of energy, the DM sector may dominate over the DE sector in future and this acceleratingphase of the universe may come to an end.Also the plot of (Ω m , Ω φ ) vs. z (Figure 2) indicates that Ω φ increases as z decreases. This isintriguing as we have obtained the direction of flow of energy from the DE to the DM sector.But figure 4 shows that ρ φ falls off with evolution. This unusual behaviour of Ω φ may arise ifthe square of the Hubble parameter H falls off more rapidly than ρ φ (since Ω φ = ρ φ H ) whichmakes Ω φ increasing with the evolution. Figure 5 exhibits a similar behaviour where ρ φ and H are plotted as a function of z for the same fixed values of the model parameters used in plotting ------> z ------ > Ρ Φ , Ρ m Figure 4:
Plot of ρ φ (solid line) and ρ m (dashed line) vs. z for ǫ = 1 . , ω = − . , V = 2 and B = 0 . . figure 2. ------> z ------ > H , Ρ Φ Figure 5:
Plot of ρ φ (solid line) and H (dashed line) vs. z for ǫ = 1 . , ω = − . , V = 2 and B = 0 . . In this section we rewrite the Einstein field equations for the interacting non-canonical scalar fieldmodel as a plane-autonomous system and study the stability of the critical points for the system.For this purpose, we define three new variables : x = ˙ φ H , y = √ V √ H and λ = − φV dVdφ .For the present non-canonical scalar field model having a constant equation of state parameter ω , the choice dVdφ = − λ ˙ φV leads to a simple power law potential [ V ( φ ) = V ( φ − φ ) − ]. So thepresent analysis is valid for power-law potentials in particular. However other choices of potentialsare also possible depending upon the dynamics of the φ -field.0In terms of the new variables, the evolution equations for the scalar field can be written as aplane-autonomous system: x ′ = − W x + 32 x (1 + x − y ) + λy (35) y ′ = 32 y (1 + x − y ) − λxy (36)where W = (3 − α ) and a prime indicates differentiation with respect to N ( N = lna ).Also the constraint equations come out asΩ φ = ρ φ H = x + y (37)and Ω m = ρ m H = 1 − x − y (38)The total EoS parameter can be written as ω tot = p φ ρ m + ρ φ = x − y (39)and the condition for acceleration is ω tot < − .At critical points, say ( x ∗ , y ∗ ), both x ′ and y ′ becomes zero. In order to study the stability of thecritical points, the system is perturbed about the fixed points by small amounts u and v as x = x ∗ + u, y = y ∗ + v. (40)Putting these in x ′ and y ′ , one obtains first order differential equations of the form u ′ v ′ = M uv (41)where M = ∂x ′ ∂x ∂x ′ ∂y∂y ′ ∂x ∂y ′ ∂y . M is a called Jacobian matrix at the fixed points. The physical stability of an autonomous systemis determined completely by the eigenvalues of the matrix M .(i) If the eigenvalues are real and have opposite signs, then the fixed point is a saddle point.(ii) If both the eigenvalues are real and negative, then the fixed point is stable.(iii) If the eigen values are real and positive, the fixed point is unstable.(iv) If the eigen values are complex but real parts of the eigen values are negative, then the fixedpoint is a stable spiral.(v) For complex eigen values if the real parts of the eigen values are positive, then the fixed pointis a unstable spiral. A detailed analysis of the stability criteria is given in the Refs. [56, 57, 58].In order to obtain a stable solution we require that all the eigenvalues of M must have negativereal part. For the present system, the Jacobian matrix is1 M = (cid:16) − W + ( x ∗ − y ∗ ) (cid:17) (2 λy ∗ − x ∗ y ∗ )( x ∗ y ∗ − λy ∗ ) (cid:16) − λx ∗ + ( x ∗ − y ∗ ) (cid:17) where W = (3 − α ).The critical points and the corresponding eigen values for the present autonomous system arelisted in Table 1. x ∗ y ∗ Eigenvaluesi 0 0 − W, ii √ W − W − W − λ √ W − −√ W − W − W + λ √ W − b + p √ λ [( k + k ) ± q ( k + k ) − k ]v b + − p √ λ [( k + k ) ± q ( k + k ) − k ]vi b − q √ λ [( k + k ) ± q ( k + k ) − k ]vii b − − q √ λ [( k + k ) ± q ( k + k ) − k ]Table 1: Critical points and the corresponding eigen values for the present systemHere for simplicity we have introduced the following parameters : b ± = β ± √ − λ +(3 W +2 λ ) λ , β = (3 W +2 λ )8 λ , c = W λ + 24 λ − W λ − λ , p = q c + ( b + − β )(8 W − λ ), q = q c + ( b − − β )(8 W − λ ), k = − W + b − p λ , k = − λb + + b − p λ , k = k k − p λ ( b + − λ )(2 λ − b + ), k = − W + b − − q λ , k = − λb − + b − − q λ , k = k k − q λ ( b − − λ )(2 λ − b − ),2It is not very straight forward to analyse the nature of the eigen values (positive or negative)from the above table as too many parameters are involved. Infact the stability criteria willcrucially depend on the values of λ and α - the strength of the coupling factor between the DEand the DM components which is again dependent on the values of the model parameters ǫ and ω φ . So for simplicity, we choose the values of the above mentioned parameters of the model as λ = 1, ǫ = 1 . ω φ = − .
9, which are the values for which we have plotted the graphs in theprevious sections. With these values chosen, we try to analyse the nature of stability of the criticalpoints. The critical points, the nature of stability and acceleration ( ω tot < − ) are summarizedin Table 2 for the chosen values of the model parameters. x ∗ y ∗ Nature of eigenvalues Stability? ω ∗ tot Acceleration?i 0 0 real, unequal and opposite signs Saddle point 0 No( µ = − . , µ = +1 . √ W − µ = +11 . , µ = +3 . −√ W − µ = +11 . , µ = +10 . b + p √ λ real, unequal and opposite signs Saddle point 2.87 No( µ = +17 . , µ = − . b + − p √ λ real, unequal and opposite signs Saddle point 2.87 No( µ = +17 . , µ = − . b − q √ λ real, unequal and negative Stable node -0.898 Yes( µ = − . , µ = − . b − − q √ λ real, unequal and negative Stable node -0.898 Yes( µ = − . , µ = − . Table 2: The properties of the critical points. This is for λ = 1, ǫ = 1 . ω φ = − .
9. Here, α = 3 − ǫ ω φ = − . W = (3 − α ) = 7 . x ∗ = b − = 0 .
12 and y ∗ = ± q √ λ ∼ ± . x ∗ and y ∗ , equation (37) readily givesΩ φ ∼ . m ∼ . q = − − ˙ HH = − h x − y i = − . . These values are little bit higher than the ones suggested by recent observations, but it hasbeen found that as we increase the value of λ , the calculated values for various parameters ofthe model approach the observationally suggested values. It deserves mention that these valuesprovide us information about various cosmological parameters at the stable fixed points (vi) and(vii) for the chosen values of λ , ǫ and ω φ and does not provide us with the complete evolutionof the universe. However the evolution of the scalar field (DE) can be obtained by numerically3solving equations (35) and (36). The behaviour of x and y against N for different values of λ are shown in figure 6. It is evident from figure 6 that the evolution of the system is not verysensitive to the value of λ and the at the early phase of evolution ( N <
0) the kinetic term ofthe scalar field (demonstrated by parameter x ) was dominating and at present the potential termis dominating over the kinetic term, as expected for a scalar field model of dark energy with ω φ > −
1. This reassures the results obtained in the previous section. An almost similar resulthas been obtained for slight variation in the values of ǫ and ω φ also.The evolution of the deceleration parameter q and the equation of state parameter ω φ against xy - - - - Λ =1-------- Λ =1.5 Λ =2 - - N x , y < Figure 6:
Plot of x and y against N for λ = 1 . , . , . , ǫ = 1 . and ω φ = − . . N for λ = 1 is shown in figure 7. It is evident from figure 7 that the universe was undergoinga decelerated expansion phase and enters into an accelerated phase in the recent past and theevolution of ω φ indicates that the value of ω φ was positive initially, at present it is close to − . − ω φ over the total span is verysmall and thus it is justified to consider ω φ constant over the entire span of evolution. Furthermoreas ω φ settles to a value close to − −
1, this indicates that the present modelwill behave like a ΛCDM model in future and the universe will evolve to the asymptotic de Sitterspace-time. This is in agreement with the various observational results [59, 60, 61]. As ω φ or ω tot never crosses −
1, this indicates that the non-canonical scalar field considered in the presentmodel is not phantom and thus there is no future singularity in this model [63, 64].One can also draw the phase portrait for the system for the chosen values of the model parameters q Ω Φ - - - N Figure 7:
Evolution of parameters q and ω φ for λ = 1 . (i)(vi)(vii) (iv)(v)(ii)(iii) - - - - x y Figure 8:
Phase diagram of the autonomous system in the x-y plane for λ = 1 , ǫ = 1 . and ω φ = − . . (vi)(i) (vii) - - - - - x y Figure 9:
Phase diagram of the autonomous system in the x-y plane near the stable points.
In this paper, we have described a cosmological model with a non-canonical scalar field in whichthe scalar field is allowed to interact with the matter component of the universe. We haveconsidered a specific form for the coupling function as Q = αH ˙ φ . As mentioned earlier, the formof Q chosen is quite arbitrary, but as nothing specific is known about the nature of dark energy ,any coupling term can be considered phenomenologically. Next we have obtained analyticalsolutions for various cosmological parameters of the model with a constant equation of stateparameter for the φ -field. This toy model is somewhat restricted in this sense but observationaldata suggests that the allowed values of the equation of state parameter is − . ≤ ω φ ≤ − . ω φ as aconstant parameter. However, it would be interesting to study the properties of the model with5a varying equation of state parameter for the scalar field.Furthermore, it has been found that for this interacting model, the deceleration parameter q undergoes a smooth transition from a decelerated to an accelerated phase of expansion driven bythe non-canonical scalar field φ (see Figure 3). This is essential for the structure formation of theuniverse.In the coupling term Q , the parameter α was initially kept arbitrary and its value has been foundout from other parameters of the model. It has been found that the α < One of the authors (AAM) acknowledges UGC, Govt. of India for financial support throughMaulana Azad National Fellowship. SD wishes to thank IUCAA, Pune for the associateshipprogramme where part of this work has been carried out.
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