aa r X i v : . [ g r- q c ] F e b A Viable Dark Fluid Model
Esraa Elkhateeb
Physics Department, Faculty of Science, Ain Shams University., Abbassia 11566, Cairo, Egypt
November 8, 2018
Abstract
We Consider a cosmological model based on a generalization of theequation of state proposed by Nojiri and Odintsov [46] and ˇ S tefanˇ c i´ c [47],[48]. We argue that this model works as a dark fluid model which caninterpolate between dust equation of state and the dark energy equationof state. We show how the asymptotic behavior of the equation of stateconstrained the parameters of the model. The causality condition for themodel is also studied to constrain the parameters and the fixed points aretested to determine different solution classes. Observations of Hubble di-agram of SNe Ia supernovae are used to further constrain the model. Wepresent an exact solution of the model and calculate the luminosity dis-tance and the energy density evolution. We also calculate the decelerationparameter to test the state of the universe expansion. Recent cosmological observations favoring the scenario of spatially flat and accel-erated universe. Observations come from Hubble diagram of SNe Ia supernovaeare best fitted by spatially flat accelerated cosmological models [1]-[4]. Besides,other observations from Cosmic Microwave Background Radiation (CMBR) andBaryon Acoustic Oscillation (BAO) are also strong evidences for this scenario[5]-[11]. This currently observed speeding up expansion of the universe leads tothe fact that there is a repulsive force acting in the universe space-time.According to our current understanding of the universe history, the radiationera was followed by a matter dominated era during which most of the universestructures like stars, galaxies and galaxy clusters we now observe was formedby gravitational instabilities. Through this phase of the universe evolution, aparticular form of gravitating but non-baryonic form of matter is formed. Thissort of matter which is non-relativistic does not interact with radiation, and so iscalled Dark Matter (DM). Late in this matter dominated phase, a new but verypeculiar form of invisible and non-gravitating ”matter” started to dominate.This sort of ”matter” does not interact with baryons, radiations, or any othersort of visible matter. It is now known as Dark Energy (DE).1bservational evidences show that, see for ex. [1] and [12]-[14], our universeat present contains approximately 4% only of radiation and baryonic matter thatis well known by the standard model of particle physics and can be detected inlaboratory and so is considered visible. While about 26% is the non-baryonicdark matter. The rest of our universe content, which is about 70% is the exoticcomponent known as dark energy. As a result, this dominated dark energycomponent antigravitates leading to the present observed universe expansion.Although dark energy is an acceptable and effective description for suchlarge scale homogeneous and isotropic accelerated universe, identifying its ori-gin and nature is one of the most debated questions in physics and cosmology.Large variety of models are proposed to describe the nature of such dark en-ergy [15],[16]. As the energy density due to DE remains unchanged throughoutthe universe space-time, the simplest choice was to revive the idea of Einsteinabout the cosmological constant Λ [17],[18]. This together with Cold Dark Mat-ter (CDM) provides the standard model for cosmology ΛCDM [18]-[21], whichappears to be in a very good agreement with observational data. In Einstein’sfield equation, this term has the concept of intrinsic energy density of vacuum[22]. However, this arises an important question, why the observed value of vac-uum energy density is very far below that is predicted from particle physics?.This is the famous fine tuning problem. Another important question is whythe energy densities of the vacuum and the matter are of the same order ofmagnitude today?, the famous coincidence problem.Dynamical models are other alternative to the cosmological constant. Someof these are the minimal coupled scalar fields or quintessence [23]-[26], k-essence[27]-[30], Chaplygin gas [31]-[33], all have an equation of state parameter ω ≥−
1. These models adopt the idea of the possibility of unification of the two darksectors, dark energy and dark matter. Models with ω ≺ − ω with the red shift and with time [38],[39]. Modification of the governing equation, or the so called modified gravitytheory [40], [41], is also one way of thinking.The idea of unification of DE and DM is a promising one and Chaplygingas model is one of the most acclaimed models representing it. This model wasproposed by Kamenshchik et. al. [42] as early as 2001 where they assumedthat the universe, within the framework of standard cosmology, is filled with anexotic fluid obeys the Chaplygin gas equation of state (EoS) p = − A/ρ α (1.1)where A is a positive constant and α = 1. This model has the advantage ofsmooth transition between different phases of the universe. Many authors thenextended this model using the idea of generalizing or modifying the EoS ofthe background fluid in order to improve the behavior of the fluid throughoutthe universe evolution. In 2002, Bento et. al. [43] extended the model to a2ore general form known as the generalized Chaplygin gas (GCG) model wherethe α constant is considered in the range 0 ≺ α ≤
1. Other various modelsbased on Chaplygin gas were also proposed such as the modified Chaplygin gasmodel introduced in 2002 by Benaoum [44] where he could interpolate betweenstandard fluids at high energy densities and Chaplygin gas fluids at low energydensities using the EoS p = Aρ − Bρ α (1.2)where α ≥ A and B are positive constants, and the hybrid Chaplygin gasmodel introduced by Bili´ c et. al. in 2005 [45] and leads to transient acceleration.In this work, we have adopted a class of dynamical models in which the EoShas the advantage of interpolation between two different Chaplygin Gas models.Our fluid is a barotropic one with an EoS can be considered as a correction to thevacuum EoS by a power law. This power law has the advantage of interpolationbetween the two equations of state of the DE and DM which might describesome sort of smooth phase transition. Another advantage of this model is thatit is not only interpolating between dust and DE in early and late times, butalso it has a more general EoS for DE that enables the cosmological constantas a special case. This has the more advantage of a general asymptotic solutionthan Chaplygin gas which goes to the cosmological constant. We present anexact solution of the equations which enables more accurate results for thecalculations of the luminosity distance, while observations of Hubble diagramof SNe Ia supernovae are used to further constrain the model parameters. Westudy the energy density and the equation of state parameter evolutions. Theanalysis of the EoS shows that the dynamical properties of both dark sectors canbe described through this model which lets our choice seems natural because ofthe smooth transition. We also calculate the deceleration parameter to examinethe expansion of the universe due to the model.The rest of the article is organized as follows; in the following section wehave formulated the model. In section 3 the mathematical treatment of themodel is made. Section 4 treats the first case of study of our model where thecausality constraints are examined and the dynamics of the universe is studiedwhere the equation of the luminosity distance as a function of red shift is solvedand parameters are constrained through Hubble diagram observations, the en-ergy density and the deceleration parameter evolutions due to the model areexamined. Section 5 treats the second case of study of our model. Section 6concludes our work. We are working through the standard FRW cosmology where the metric in thespatially flat geometry is ds = dt − a ( t )( dr + r d Ω ) (2.1)3here we consider units with c = 1. Using the energy momentum tensor givenby T µν = ( ρ + P ) U µ U ν − P g µν (2.2)where in comoving coordinates U µ = δ µ , the Einstein’s equation will lead tothe Friedman equations ˙ a a = 8 πG ρ (2.3)¨ aa = − πG ρ + 3 P ) (2.4)while the conservation equation T νµ ; µ = 0 gives˙ ρ = − aa ( ρ + P ) (2.5)Now we have to consider an EoS to close the system. It is a fact that darkenergy dominates the universe today and an adequate EoS for this componentis p = ωρ . Observations indicate that the EoS parameter ω is close to − ω is constant or varies as theuniverse expands. Owing to this fact, Nojiri and Odintsov [46] considered ageneral EoS of the form p = − ρ − f ( ρ ) (2.6)where f ( ρ ) is an arbitrary function may be considered as a correction to thestandard DE equation of state. ˇ S tefanˇ c i´ c [47], [48] studied in details the caseof f ( ρ ) ∝ ρ α with a constant α = 1, and Nojiri et. al. [49] studied the futuresingularity associated with this model.In this work we adopted a barotropic fluid with an EoS which is also acorrection to the standard dark energy EoS. The fluid pressure has a generalpower law form for the density dependence P = − ρ + γρ n δρ m (2.7)Where γ , δ , n , and m are constants which are considered as free parameters.This enables interpolation between different powers for the density, so that thephase transitions during the universe evolution are described smoothly. TheEoS parameter is given by ω = − γρ n − δρ m (2.8) Let’s now study the evolution of the physical quantities of the universe due toour model. Considering units with 8 πG = 1, eqns (2.3)-(2.5) reduce to4 = 13 ρ (3.1)¨ aa = H + ˙ H = −
16 ( ρ + 3 P ) (3.2)˙ H = −
12 ( ρ + P ) (3.3)where H = ˙ aa is the Hubble parameter. Combining eq(2.7) to eq(3.3) we get˙ H = − αH r βH s ) (3.4)where again α , β , r , and s are constants related to those of (2.7) through therelations r = 2 n ; s = 2 m ; α = 3 n γ ; and β = 3 m δ (3.5)The solution of equation(3.4) indicates the evolution of the Hubble parameterwith time. This equation is integrated to give α t − t ) = 11 − r (cid:0) H − r − H − r (cid:1) + βs − r + 1 (cid:0) H s − r +1 − H s − r +1 (cid:1) (3.6)However, our model is not describing a pure dark energy, in which case thevalues of the parameters are totally free and are not constrained. Instead, ourmodel able to follow the smooth phase transition which takes place throughthe universe evolution. This restricts the parameter values to obey the limitssatisfied by observations so that for early time we must have the equation ofstate of the perfect fluid.Accordingly, we first have to study the asymptotic behavior of ˙ H . Fromone side, this indicates to what extent our model agrees the real evolution ofthe universe, and from the other side may also fixes the limits of some of ourmodel parameters. Table 1 shows this behavior for the different ranges of of theparameters r and s .Table 1: Asymptotic behavior of the equation of state
Case ranges of r and s large H small HI r ≻ s ≻ H → − α β H r − s ˙ H → − α H r II r ≻ s ≺ H → − α H r ˙ H → − α β H r − s III r ≺ s ≻ H → − α β H r − s ˙ H → − α H r IV r ≺ s ≺ H → − α H r ˙ H → − α β H r − s One can see from the table that the values of negative r must be precluded inour calculations as a result of the asymptotic behavior of the ˙ H function. This isbecause for asymptotically perfect fluid behavior of our universe, the ˙ H function5 h dh / d -ve s h dh / d +ve s Figure 1:
Fixed points phase diagram. at large H forces r to be positive if s is negative, and forces ( r − s ) to be positiveif s is positive, which does not satisfy constraints for the pre-assumptions.On the other hand, the asymptotic behavior of ˙ H for the two cases of positive r can attain the perfect fluid behavior at large H if r − s = 2 and α/β = 3 forthe case of + ve s and if r = 2 and α = 3 for the case of − ve s . These two caseswill be studied in details.Finally let’s study the different classes of solutions due the fixed points inthe model. This general class of models can be studied using the phase spacemethod developed in [50]. The phase diagram in Fig.1 shows that there are nofixed points in the solution so that there are no different classes, here h = HH and τ = tt . s is positive This is the first case that match the dark fluid dynamics. The asymptoticbehavior of the EoS at large H , early times, has the form ˙ H → − α β H r − s , so thatfor r − s = 2 and α/β = 3 we have the perfect fluid EoS at early times. Let’snow examine the constraints that can be obtained by applying the causalitycondition to this model. Spherical sound waves are lunched as a result of the overpressure due to theinitial overdensity in the DM and gas. These Oscillations have a speed whichis an indication to the velocity by which such perturbations are transmitted.Accordingly, it must not exceed the speed of light, which is a condition ofcausality. In barotropic cosmic fluid, the speed of these sound waves is defined6y the relation c s = dpdρ (4.1)This must satisfy the condition c s ≤ ρ . Accordingly, applying causality constraints, one may get an indication forthe available ranges for the values of the free parameters in the EoS. Now using(2.7), we get c s = − n − m ) δρ m + n ] γρ n − (1 + δρ m ) ≤ n and m are + ve , relation (4.2) for large ρ gives thecondition ( n − m ) γ δ ρ n − m − ≤ n ≤ m + 1 (4.4)While for small ρ , the condition will be( n − m )2 γδρ n + m − + n γρ n − ≤ n + m ≥ and n ≥ − m ≤ n ≤ m (4.7)Which means 2(1 − s ) ≤ r − s ≤ ρ we also have alarge dependence on γ and δ . Accordingly, we have to make a check using theobtained optimized values for the parameters. In this section we study the ability of our model to produce the physical quanti-ties that can be really measured. This, of course, depends on the compatibilityof the results of our model with the astrophysical data that are indeed observed.7 .2.1 The Luminosity Distance
One of the most important observed data is that of the Hubble diagram oftype SNe Ia supernovae, which is a plot of the luminosity distance D L , thatdetermines the flux of the source, as a function of the red shift. We will usethese data to constrain the parameters in our model. Due to the asymptoticand causality constraints, we have now only two out of four parameters that arefree and needed to be optimized due to observations.To satisfy the causality and asymptotic behavior constraints, we choose ( r − s ) as 2 while ( α/β ) has to be 3, so that we can attain the perfect fluid behaviorof ˙ H at large H . Let’s now start with our model by using the relation (3.4) for( r − s ) = 2 in (3.3), so that we have˙ ρ = − H αH s +2 βH s (4.9)which in turn gives dHda = − a αH s +1 βH s (4.10)This has the solution H ( a ) = a − α/ β exp (cid:20) s W (cid:18) β a αs/ β e Cαs/ β (cid:19) e − α β C (cid:21) (4.11)The function W ( x ) is the Lambert W-function. To fix the constant C we con-sider that at some time where t = t we have a = a and H = H , this gives C = − ln (cid:16) a H β/α (cid:17) + 2 αs H − s (4.12)So that H ( a ) = (cid:16) a a (cid:17) α/ β H e − H − s /βs exp (cid:20) s W (cid:18)(cid:16) a a (cid:17) − αs/ β H − s β e H − s /β (cid:19)(cid:21) (4.13)Accordingly H ( z ) = ( z + 1) α/ β H e − H s /βs exp (cid:20) s W (cid:18) ( z + 1) − α/ β H − s β e H − s /β (cid:19)(cid:21) (4.14)the luminosity distance D L is then given by D L ( z ) = (1 + z ) D p ( z ) (4.15)where D p ( z ) is the proper distance given by D p ( z ) = Z z H ( ζ ) dζ (4.16)8ccordingly D p ( z ) = 1 H e H − s /βs Z z ( ζ + 1) − α/ β exp (cid:20) − s W (cid:18) ( ζ + 1) − α/ β H − s β e H − s /β (cid:19)(cid:21) dζ (4.17)The integration of the above equation can not be solved analytically. Accord-ingly, we will use numerical methods for solving the integral. This enables usto get an exact solution for the relation controlling the dependence of the lu-minosity distance on the red shift. In Fig.2 we plot the apparent magnitude, m = M + 5 log (cid:0) D L MP c (cid:1) + 25, of the data set from Conley et al. (2011)[51] ver-sus their red shifts z together with the fitted results of our model. Data set areavailable at Supernova Cosmology Project[52]. Using H = 67 . Km/s/M P c due to Plank+ WMAP+ BAO measurements[53], our χ minimization fit re-sults in the parameters s = 1 . α = 0 . χ value of χ red = 1 . D L ( z ) redshift Observations model Figure 2: model versus observations for SNe Ia supernovae in case of + ve s
Having the full set of parameters, let’s test the causality condition usingthese parameters. A plot of causality relation (4.2) using our parameters forthis case is shown in Fig(3). We can see from the figure that the parameterssatisfy the causality condition for the model throughout the whole stages of theevolution of the universe. 9 -1 dp / d Figure 3:
Causality test using our optimized free parameters for + ve s
Let us see how the energy will evolve with time due to the model in a homoge-nous and isotropic universe. Relations (2.5) and (2.7) give dρ = 1 a − γρ n δρ m da (4.18)To simplify our way of solution to this equation let’s use our parameters. Inthis case we have n = m + 1, so that the solution of relation (4.18) will take theform ρ ( a ) = ρ (cid:18) aa (cid:19) − γ/δ exp (cid:18) W ( x ) m − mδρ m (cid:19) (4.19)Where x = 1 δρ m e δρ m (cid:18) aa (cid:19) γm/δ (4.20)To study the properties of relation (4.19) at early times we make use of theproperty exp ( nW ( x )) = (cid:18) xW ( x ) (cid:19) n (4.21)then exp (cid:18) W ( x ) m (cid:19) = (cid:18) xW ( x ) (cid:19) /m (4.22)Now, in the past, where a ≺≺ a , so that x ≺≺ m is + ve ,Taylor expansion of W -function can be approximated to W ( x ) ≈ x , so that the10xponential term of the W -function converges to 1, then the energy density forthat time will take the asymptotic form ρ ( a ) → A (cid:18) aa (cid:19) − (4.23)as we have γ/δ = 1 for this case, where A = ρ e − δmρ m is a constant. Accordingly,the fluid of the model behaves as a matter in the earlier times.On the other hand, for a ≻≻ a , x ≻≻
1, the energy density function dueto relation (4.22) will tend to ρ ( a ) = ( δx ) − /m (cid:16) xW (cid:17) /m (4.24)Accordingly ρ ( a ) = (cid:18) δW ( x ) (cid:19) /m (4.25)Using the value of m we can see that this function is a very slowly decreasingfunction of x for x ≻
1, and tends to be nearly constant at x ≻≻
1, i.e. at a ≻≻ a , a signature for a cosmological constant towards the future.The equation of state parameter ω for this case, where m = n −
1, will begiven, from (2.8), by ω = − γρ − m + δ = − ρ ) − m /γ + δ/γ (4.26)Using our parameters for this case where δ/γ = 1, we finally get ω = − ρ ) − m /γ + 1 (4.27)Indicating that for earlier times, where we have large values of ρ , ω →
0, asignature for matter era, while changes to − The acceleration of the universe is related to a dimensionless parameter knownas the deceleration parameter q defined as q = − ¨ aa H (4.28)Since you have to have ¨ aa ≻ q ≺
0, the expansion is accelerating, while if q ≻ aa = H + ˙ H (4.29)11hen q = − − ˙ HH (4.30)Now using (3.4), one gets the deceleration parameter for our model to be q = − αH r − βH s ) (4.31)Using r = s + 2, relation (4.31) gives q = − α H − s + β ) (4.32)Using the values of our parameters with H = 67 .
8, the present day decelerationparameter due to this model case has the value q = − .
553 (4.33)Relation (4.32) also shows that deceleration parameter evolves towards morenegative values as the Hubble parameter decreases with time. s is negative For this case ˙ H → − α H r asymptotically at large H , which means that we canattain the perfect fluid EoS at early time if r = 2 and α = 3. However, a firststep again is the examination of the causality constraints for this case of themodel. In this case m is − ve , so that relation (4.2) at large values of ρ tends to12 nγρ n − ≤ ≺ n ≤ ρ we have (cid:16) n − m (cid:17) γδ ρ n − m − ≤ m is − ve , this is always satisfied under the condition n ≥ n = 1 (5.5)which means r = 2 (5.6)in full agreement with the asymptotic behavior of the EoS at early time for thiscase. 12 .2 The dynamics of the universe We’ll now study the physical quantities representing the dynamics of our uni-verse due to this case of the model and comparing our results with observations.
We could now constrain two parameters out of four through asymptotic behaviorof the model and the causality condition, where r has to be 2 while α has tobe 3. Let’s then calculate the luminosity distance function D L ( z ) and use theobserved data of this function to constrain the remaining two parameters of themodel. We start by using relation (3.4) for r = 2 in (3.3), so that we have˙ ρ = − H αH βH s (5.7)which in turn gives dHda = − a αH βH s (5.8)This has the solution H ( a ) = a − α/ exp (cid:20) − s W (cid:16) βa − αs/ e − Cαs/ (cid:17) − Cα (cid:21) (5.9)And let’s assume that at some time t = t we have a = a and H = H . Thisgives C = − ln (cid:16) a H /α (cid:17) − αs βH s (5.10)So that H ( a ) = (cid:16) a a (cid:17) α/ H exp (cid:18) βH s s (cid:19) exp (cid:20) − s W (cid:18) β (cid:16) a a (cid:17) αs/ H s e βH s (cid:19)(cid:21) (5.11)In turn, we can write H ( z ) as H ( z ) = ( z + 1) α/ H exp (cid:18) βH s s (cid:19) exp (cid:20) − s W (cid:16) β ( z + 1) αs/ H s e βH s (cid:17)(cid:21) (5.12)The luminosity distance D L , Eqs (4.15), is then calculated using the properdistance which due to (4.16) will be given by D p ( z ) = 1 H exp (cid:18) − βH s s (cid:19) Z z ( ζ + 1) − α/ exp (cid:20) s W (cid:16) β ( ζ + 1) αs/ H s e βH s (cid:17)(cid:21) dζ (5.13)Again, the integration of relation (5.13) can not be solved analytically so thatwe use numerical methods. The apparent magnitude of Conley et al. (2011)[51]data set is plotted in Fig.4 together with our results. Our χ minimization fitresults in the parameters s = − . β = 1 . × . These parameters aredue a χ value of χ red = 1 .
23. 13 .2 0.4 0.6 0.8 1.0 1.2 1.4343638404244 D L ( z ) redshift Observations model Figure 4: model versus observations for SNe Ia supernovae for − ve s A last step for this case now is to test the causality condition using theseparameters. A plot of causality relation (4.2) using our parameters is shown inFig(5). The figure shows that the causality is satisfied for all values of ρ , i.e., itis ensured by the model in this case through the universe evolution. Let’s solve relations (4.18) for n = 1, the value of the parameter n in this case.This gives ρ ( a ) = ρ (cid:18) aa (cid:19) − γ exp (cid:18) − W ( x ) m + δρ m m (cid:19) (5.14)Where x = δρ m e δρ m (cid:18) aa (cid:19) − γm (5.15)Studying the properties of relation (5.14) we see that due to relation (4.21) wehave exp (cid:18) − W ( x ) m (cid:19) = (cid:18) xW ( x ) (cid:19) − /m (5.16)so that in the past where a ≺≺ a , so that x ≺≺ m is − ve , we can consider the W -function as approximated to W ( x ) ≈ x and itsexponential term converges to 1, the energy density for that time will then hasthe asymptotic form ρ ( a ) ⇒ A (cid:18) aa (cid:19) − (5.17)14 -1 dp / d Figure 5:
Causality test using our optimization parameters for − ve s Where we have γ = 1 for this case and A = ρ e δρ m /m . This shows that, thefluid of our model behaves as a matter in the earlier times.On the other hand, to study the fluid for a ≻≻ a , x ≻≻
1, we make useof relation (5.15) together with the property of relation (5.16) in the energydensity function, relation (5.14), so that the energy density function takes theform ρ ( a ) = (cid:16) xδ (cid:17) /m (cid:16) xW (cid:17) − /m (5.18)Accordingly ρ ( a ) ⇒ (cid:18) δW ( x ) (cid:19) − /m (5.19)Using the value of the parameter m we can see that this function is again avery slowly decreasing function of x for x ≻ x ≻≻
1, i.e., at a ≻≻ a , which is again a signature for the cosmologicalconstant towards the future.Finally, the equation of state parameter ω for this case will be given from(2.8) by ω = − δρ m (5.20)Indicating that for large values of ρ , where we are deep in the matter era, ω = 0as m is − ve , while changes to − .2.3 Deceleration Parameter and the Universe Acceleration Using r = 2 and α = 3 in relation (4.31) we get q = − βH s ) (5.21)and using the values of s and β together with H = 67 .
8, this relation gives apresent day deceleration parameter of q = − .
551 (5.22)Relation (5.21) also shows that as s is negative, the deceleration parameterevolves towards more negative values as the Hubble parameter decreases to thefuture. A barotropic fluid model is considered which initially can be described as acorrection to the dark energy regime by a power law. The power law enablesinterpolation between the DM and DE equations of state, and so guarantees asmooth phase transition from matter era to DE era. Accordingly, our model is adark fluid model provides a unification for the two dark sectors in one EoS thathas the advantage of the general asymptotic solution. This means that it candescribe dark matter at early times while describing dark energy at late times,while adopts a general EoS for DE which enables the cosmological constant asa special case.Studying the asymptotic behavior of the equation of state constrains theparameters of the equation and restricts the model. It also clarifies the casesthat can be physically acceptable as dark fluid which are found to be two cases.On the other hand, studying the phase diagram of the model for each of thesetwo cases shows that there are no fixed points for both, with the result thatthere is only one class of solution for each of them.Causality condition further constrains the parameters of the model, so thatat the end, only two out of four parameters are free. These two parameters areconstrained using the Hubble diagram of SNe Ia supernovae, where we presentan exact solution for the red shift dependence of the luminosity distance in eachmodel, or model case, avoiding the approximate formulas that are accurate onlyat small red shifts. The χ − fitting of the two models shows that they bothfit the data well. For the model with + ve − s we got a χ = 1 .
02, and for themodel with − ve − s we got a χ = 1 .
23. However, it is clear that observationaldata prefer the + ve − s model than the − ve − s one. More analysis such asdensity perturbation analysis may differentiate them and may favor one of themodels than the other.We also studied the energy density evolution which ensures the smooth tran-sition between the two dark sectors. Furthermore, we examined the evolutionof the EoS parameter ω that again ensures its two extremes of 0 and − q andexamined its evolution. Calculations resulted in a negative q for both modelcases with the present day values of − .
553 and − .
551 due to the two models,while assures evolution towards more negativity to the future as the Hubbleparameter decreases, a clear evidence for accelerated expansion in agreementwith observations.Due to the above mentioned results, our model can be considered as a darkfluid model unifying the two dark sectors of the universe and succeeded indescribing the evolution of the universe.
The author would like to thank Dr. Adel Awad for valuable discussions andguidance.
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