Viscous extended holographic Ricci dark energy in the framework of standard Eckart theory
aa r X i v : . [ g r- q c ] M a y Viscous extended holographic Ricci dark energy in the frameworkof standard Eckart theory
Surajit Chattopadhyay ∗ Pailan College of Management and Technology (MCA Division),Bengal Pailan Park, Kolkata-700 104, India. (Dated: July 2, 2018)In the present work we report a study on the viscous extended holographic Ricci darkenergy (EHRDE) model under the assumption of existence of bulk viscosity in the linearbarotropic fluid and the EHRDE in the framework of standard Eckart theory of relativisticirreversible thermodynamics and it has been observed that the non-equilibrium bulk viscouspressure is significantly smaller than the local equilibrium pressure. We have studied theequation of state (EoS) parameter and observed that the EoS behaves like “quintom” andis consistent with the constraints set by observational data sets from SNLS3, BAO andPlanck + WMAP9 + WiggleZ measurements in the reference S. Kumar and L. Xu,
Phys.Lett. B , , 244 (2014). Analysis of statefinder parameters has shown the possibility ofattainment of ΛCDM phase under current model and at the same time it has been pointedout that the for z = 0 i.e. current universe, the statefinder pair is different from thatof ΛCDM and the ΛCDM can be attained in a later stage of the universe. An analysisof stability has shown that although the viscous EHRDE along with viscous barotropic isclassically unstable in the present epoch, it can lead to a stable universe in very late stage.Considering an universe enveloped by event horizon we have observed validity of generalizedsecond law of thermodynamics. PACS numbers: 98.80.-k; 04.50.Kd
I. INTRODUCTION
Independent studies by Riess et al. [1] and Perlmutter et al. [2] of high-redshift supernavoesearch team and supernovae cosmology project team respectively reported that the current universeis expanding with acceleration. Subsequent observational studies including large-scale structure(LSS) and the cosmic microwave background (CMB) have further confirmed the accelerated ex-pansion (see [3]). In order to have this accelerated expansion there must be something to overcome ∗ Electronic address: [email protected], [email protected] the effect of gravity. “Dark energy” (DE), an exotic matter characterized by negative pressure andhaving equation of state (EoS) parameter w = p/ρ < − /
3, is believed to be responsible for thisaccelerated expansion. However, its exact nature is yet to be known. Different candidates for DEhave been proposed till date with varying behaviour of EoS parameter. The simplest candidateis the cosmological constant Λ with w Λ = − ρ Λ = 3 c M p L − , where L is the infrared cut-off. Works on HDE include [28–30, 44–47].Different variants of HDE have been proposed in the literature. In the present work we consider aspecial form of HDE [48] dubbed as “extended holographic Ricci dark energy” (EHRDE) [48]. Itsdensity has the form ρ DE = 3 M p (cid:16) αH + β ˙ H (cid:17) (1)where the upper dot represents derivative with respect to cosmic time t , M p is the reduced Planckmass, α and β are constants to be determined. Wang and Xu [61] found the best-fit values inorder to make this cutoff to be consistent with observational data as α = 0 . +0 . . − . − . and β = 0 . +0 . . − . − . . In the current work we shall take α = 0 .
98 and β = 0 . t ≈ − s and a temperature of about T ≈ K [51]. First attempts towards creating atheory of relativistic dissipative fluids were made by Eckart [53] and Landau and Lifshitz [54].Israel and Stewart [55] developed a relativistic second-order theory. Nojiri and Odintsov [56]made a time dependent viscosity consideration to DE by considering EoS with inhomogeneous,Hubble parameter dependent term. Brevik et al. [57] discussed entropy of DE in the frameworkof holographic Cardy-Verlinde formula and in a relatively recent work, Brevik et al. [58] deriveda formula for the entropy for a multicomponent coupled fluid that could relate the entropy of aclosed FRW universe to the energy contained in it together with its Casimir energy. Brevik et al.[59] investigated interacting dark energy and dark matter in flat FRW universe with examples ofLittle Rip, Pseudo Rip, and bounce cosmology and expressed bulk viscosity as function of Hubbleparameter and time.Plan of the present paper is as follows: In Section II we have presented the cosmological con-sequences of existence of bulk viscosity in the linear barotropic fluid and the EHRDE in theframework of standard Eckart theory of relativistic irreversible thermodynamics through the studyof reconstructed Hubble parameter, EoS parameter and statefinder diagnostics. In Section III wehave presented the stability analysis and in Section IV we have examined the validity of general-ized second law of thermodynamics under the assumption that the universe is enveloped by eventhorizon. We have concluded in Section V. II. ECKART APPROACH
For an homogeneous and isotropic flat universe the FRW metric is ds = − dt + a ( t ) (cid:0) dr + r ( dθ + sin θdφ ) (cid:1) (2)where, a ( t ) is the scale factor, t is the cosmic time. According to the first order thermodynamictheory of Eckart [53] the field equations in the presence of bulk viscous stresses are (cid:18) ˙ aa (cid:19) = H = ρ aa = ˙ H + H = −
16 ( ρ + 3 P eff ) (4)where, P eff = p + Π in which Π is the bulk viscous pressure andΠ = − Hξ (5)where, ξ is the bulk viscosity coefficient. The condition ξ > H = ρ DE + ρ ν (6)and ρ ′ DE = − ρ DE + p DE + Π DE ) (7) ρ ′ ν = − νρ ν + Π ν ) (8)where, the barotropic fluid has the equation of state p ν = ( ν − ρ ν ; (0 ≤ ν ≤
2) and Π DE and Π ν are to be defined later based on the physically natural fact that the bulk viscosity is proportionalto the fluid’s velocity vector. The prime denotes the derivative with respect to x = ln a . FromEq.(6) we can write 3 H = 3 (cid:18) αH + β dH dx (cid:19) + ρ ν (9)that implies ρ ν = 3 (cid:18) (1 − α ) H − β dH dx (cid:19) (10)and from barotropic equation of state we get p ν = 3( ν − (cid:18) (1 − α ) H − β dH dx (cid:19) (11)Subsequently, we can haveΠ ν = − √ τ ν y = 3(1 − α ) νy + (1 − α − βν ) y ′ − β y ′′ (12)where y = H and the primes indicate derivative with respect to x = ln a . We can rewrite Eq.(12)as βy ′′ − (2(1 − α ) − βν ) y ′ − − α ) ν − √ τ ν ) y = 0 (13)solving which we get H = C e x ∆ − + ( H − C ) e x ∆ + (14)where ∆ ∓ = 12 β − α − βν ∓ r ( − α + 3 βν ) − β (cid:16) − ν + 6 αν + 6 √ τ ν (cid:17)! (15)and H = C + ( H − C ). From Eq.(15) we understand that for real ∆ ∓ we require √ τ ν ≤ − α ) ν + (2( α −
1) + 3 βν ) β (16)Thereafter, energy density of the viscous extended holographic RDE is ρ DE = 3 (cid:20) α (cid:16) C e x ∆ − + e x ∆ + (cid:0) − C + H (cid:1)(cid:17) + 12 β (cid:16) C e x ∆ − ∆ − + e x ∆ + (cid:0) − C + H (cid:1) ∆ + (cid:17)(cid:21) (17)and that of the barotropic fluid is ρ ν = − h e x ∆ + H (cid:0) − α + β ∆ + (cid:1) + C (cid:16) (cid:16) e x ∆ − − e x ∆ + (cid:17) ( − α ) + e x ∆ − β ∆ − − e x ∆ + β ∆ + (cid:17)i (18)Taking Π DE = − √ τ DE H based on Eckart approach [53] we have from Eq.(7) p DE = − C (cid:16) e x ∆ − α − e x ∆ + α − √ (cid:16) e x ∆ − − e x ∆ + (cid:17) τ DE + e x ∆ − (2 α + 3 β )∆ − + e x ∆ − β (∆ − ) − e x ∆ + α ∆ + − e x ∆ + β ∆ + − e x ∆ + β (∆ + ) (cid:17) − e x ∆ + H (cid:0) − √ τ DE + (3 + ∆ + ) (2 α + β ∆ + ) (cid:1) (19)and hence the EoS parameter w DE is w DE = p DE ρ DE = − C e x ∆ − (cid:0) √ τ DE − ∆ − (2 α + β ∆ − ) (cid:1) − e x ∆ + (cid:0) C − H (cid:1) (cid:0) √ τ DE − ∆ + (2 α + β ∆ + ) (cid:1) C e x ∆ − (2 α + β ∆ − ) − e x ∆ + (cid:0) C − H (cid:1) (2 α + β ∆ + ) (20)and w total = p DE + p ν ρ DE + ρ ν = − (cid:0) ν (1 − α ) + 6 √ τ DE − ∆ + (2 α + 3 βν + β ∆ + ) − C e x ∆ − ( ∆ − − ∆ + )( α +3 βν + β ( ∆ − +∆ + )) C e x ∆ − + e x ∆+ ( − C + H ) (cid:19) (21)Defining effective pressure ˜ p = p DE + p ν + Π DE + Π ν we have˜ p = − e x ∆ + H (3 + ∆ + ) + C (cid:16) − e x ∆ − + 3 e x ∆ + − e x ∆ − ∆ − + e x ∆ + ∆ + (cid:17) (22)In Fig.1 we have plotted the reconstructed H as a function of redshift z based on Eq. (14). Inthis and subsequent plots black, red, green and blue lines will correspond to ν = 0 . , . , .
25 and - z H FIG. 1: Reconstructed Hubble parameter H based on Eq.(14). .
27 respectively. The reconstructed Hubble parameter is found to be decreasing with evolutionof the universe and this behaviour is consistent with the accelerated expansion of the currentuniverse. However, in a later stage − . . z . − .
25, the reconstructed H is found to startincreasing. Thus, in a later stage ˙ a ( t ) may dominate a ( t ). Equation of state (EoS) parameterfor the viscous EHRDE based on Eq.(20) is presented in Fig. 2 and we observe that a transitionfrom EoS > − EoS < − z ≈ .
01 i.e. in an earlierstage of the universe and as seen in Table I the current value of the EoS parameter for viscousEHRDE is favouring the ΛCDM model and the w DE for the current model is consistent with resultsobtained by [9] through observational data sets from SNLS3, BAO and Planck+WMAP9+WiggleZmeasurements. If we give a minute look at Fig.2 we can observe that for ν = 0 .
15 the crossing ofphantom boundary is occurring at earlier stage i.e. z & ν = 0 .
25 and 0 .
27 the crossing of phantom divide (
EoS = −
1) is occurring atlater stage i.e. z < ν the transition from quintessence to phantom is getting delayed. However, irrespective of the valuesof ν the viscous EHRDE is behaving like “quintom” i.e. transiting from quintessece to phantom.If we look at Fig.3 we can understand that the behaviour of w total is largely similar to that of w DE as far as the “quintom” behaviour is concerned. However, the transition to phantom is occurringin a later stage z ≈ − . ν . Hence, for w total the time point atwhich the universe is transiting to phantom from quintessence is not influenced by the value of ν .In Table I we have computed different values of w total and w DE for the current universe( z = 0) and for different choices of ν . Studying observational data sets from SNLS3, BAO andPlanck+WMAP9+WiggleZ measurements of matter power spectrum [9] have fixed observational - - - - - - - z w D E FIG. 2: Plot of EoS parameter w DE based on Eq.(20). - - - - - - z w t o t a l FIG. 3: Plot of EoS parameter w total based on Eq.(21). - z È p Ž È FIG. 4: Plot of absolute value of effective pressure | ˜ p | = | p DE + p ν + Π DE + Π ν | basedon Eq.(22). - z È Π È FIG. 5: Plot of absolute value of the bulk viscous pres-sure | Π | = | Π DE + Π ν | . - z È p D E + p Ν È - È Π È FIG. 6: Plot of the difference between | Π | = | Π DE +Π ν | and | p DE + p ν | . TABLE I: w total and w DE at z = 0 for different values of ν .EoS parameter ν = 0 . ν = 0 . ν = 0 . ν = 0 . w total -0.988403 -0.961182 -0.934464 -0.923836 w DE -1.0018 -0.979765 -0.95919 -0.9513 constraint on the EoS parameter for the current universe at − . +0 . − . and found the best fit valueof w DE to be − .
01. Comparing our results with that of [9] we observed that the EoS parameterdue to viscous EHRDE is well within the range specified by [9] for all values of ν . Moreover, w DE (= − . ν = 0 .
15. Thus, ΛCDMscenario is expected to be favoured by the viscous EHRDE considered here.In Fig.4 we have plotted | ˜ p | = | p DE + p ν + Π DE + Π ν | based on Eq.(22). This figure shows thatthe magnitude of the effective pressure is decreasing with the expansion of the universe and it maybe noted that this decrease is occurring till z ≈ − .
4. In Fig.5 we have plotted the magnitude ofbulk viscous pressure Π = Π DE + Π ν . Comparing Figs. 4 and 5 we can interpret that both | ˜ p | and | Π | have approximately similar decaying pattern. Although the rate of decrease in magnitudeof | Π | is higher than | ˜ p | , it may be noted that this pattern wise similarity indicates a significantcontribution of the bulk viscosity to the effective pressure. Secondly, we further observe from Fig.6 that | p DE + p ν | − | Π | ≫ | Π | ≪ | p DE + p ν | . This implies that the non-equilibrium bulkviscous pressure is significantly smaller than the local equilibrium pressure [51].To further consolidate observations made through EoS parameter a pair of cosmological pa-rameters { r, s } , the so-called “statefinder parameters”, introduced by Sahni et al. [66] and andAlam et al. [67] used to discriminate between the various candidates of dark energy. If the { r − s } trajectory meets the point { r = 1 , s = 0 } then the model is said to attain ΛCDM phase of theuniverse. Statefinder parameters for different dark energy candidates have been studied in [68–71].The { r, s } parameters are given by r = q + 2 q + ˙ qH (23) s = r − (cid:0) q − (cid:1) . (24)where, q is the deceleration parameter. Hence, in the current framework Eq.(23) and (24) take the Today's point L CDM r < > <8 r < > < 8 r > < < - r s FIG. 7: The time evolution of the { r, s } trajectory. Today's point L CDM - - - - - r q FIG. 8: The time evolution of the { r, q } trajectory. form r = − (cid:18) − − α ) ν + 6 √ τ DE − ∆ + (2 α + 3 βν + β ∆ + ) − C e x ∆ − ( ∆ − − ∆ + )( α +3 βν + β ( ∆ − +∆ + )) C e x ∆ − + e x ∆+ ( − C + H ) − C e x ∆ − ∆ − ( ∆ − − ∆ + )( α +3 βν + β ( ∆ − +∆ + )) C e x ∆ − + e x ∆+ ( − C + H ) + C e x ∆ − ( ∆ − − ∆ + ) (cid:16) C e x ∆ − ∆ − + e x ∆+ ( − C + H ) ∆ + (cid:17) ( α +3 βν + β ( ∆ − +∆ + ))( C e x ∆ − + e x ∆+ ( − C + H )) + (cid:18) − α ) ν − √ τ DE + ∆ + (2 α + 3 βν + β ∆ + ) + C e x ∆ − ( ∆ − − ∆ + )( α +3 βν + β ( ∆ − +∆ + )) C e x ∆ − + e x ∆+ ( − C + H ) (cid:19) ! (25) s = 8 − (cid:0) − α ) ν + 6 √ τ DE − ∆ + (2 α + 3 βν + β ∆ + ) − C e x ∆ − ( ∆ − − ∆ + )( α +3 βν + β ( ∆ − +∆ + )) C e x ∆ − + e x ∆+ ( − C + H ) − C e x ∆ − ∆ − ( ∆ − − ∆ + )( α +3 βν + β ( ∆ − +∆ + )) C e x ∆ − + e x ∆+ ( − C + H ) + C e x ∆ − ( ∆ − − ∆ + ) (cid:16) C e x ∆ − ∆ − + e x ∆+ ( − C + H ) ∆ + (cid:17) ( α +3 βν + β ( ∆ − +∆ + ))( C e x ∆ − + e x ∆+ ( − C + H )) + (cid:18) − α ) ν − √ τ DE + ∆ + (2 α + 3 βν + β ∆ + ) + C e x ∆ − ( ∆ − − ∆ + )( α +3 βν + β ( ∆ − +∆ + )) C e x ∆ − + e x ∆+ ( − C + H ) (cid:19) (cid:19)(cid:30)(cid:18) (cid:18) − α ) ν − √ τ DE + ∆ + (2 α + 3 βν + β ∆ + ) + C e x ∆ − ( ∆ − − ∆ + )( α +3 βν + β ( ∆ − +∆ + )) C e x ∆ − + e x ∆+ ( − C + H ) (cid:19)(cid:19) (26)In Fig.7 we plot the time evolution of { r − s } trajectory and observe that the ΛCDM fixed pointi.e. { r = 1 , s = 0 } is attainable by the statefinder trajectory. Moreover, it is worth noting thatcrossing the ΛCDM fixed point the trajectory is reaching the fourth quadrant i.e. { r > , s < } .Table II shows that location today’s point is { r = 0 . , s = 0 . } for ν = 0 .
25 and for ν = 0 .
25 this point is located very close to the { r = 1 , s = 0 } | Λ CDM fixed point. For other values0
TABLE II: Values of statefinder parameters at z = 0 for different values of ν .Statefinder parameters ν = 0 . ν = 0 . ν = 0 . ν = 0 . r s q -0.982605 -0.941773 -0.901696 -0.885755 of ν , location of today’s point is situated at larger distance from the { r = 1 , s = 0 } point. Thus,it is observed that variation in the value of ν has a significant impact on the extent of separationof the current viscous EHRDE and increase in ν takes the model away from ΛCDM, where α and β are fixed at 0 .
97 and 0 .
37 respectively.We consider the case ν = 1, which corresponds to the EoS of dark matter w m = 0. In thesituation of coexistence of viscous EHRDE and viscous dark matter the current value of statefinderpair is { r = − . , . } and it far away from the ΛCDM fixed point. However, in late timeuniverse the trajectory reaches { r = 1 , s = 0 } . Considering the smaller values ν = 0 . , . , . .
27 it is noteworthy that for different values of ν evolution of the { r, s } trajectory starts fromdifferent points in the region { r < , s > } and for ν = 0 .
15 the evolution begins from a pointcloser to ΛCDM fixed point compared to the higher values of ν . However, all the trajectories afterpassing through their current values (see Table II) converge to the ΛCDM point in late time. Itmay further be noted that for ν = 0 .
15 the today’s ( z = 0) point is very close to ΛCDM point.Therefore, although ΛCDM is favoured by the EoS parameter, the viscous EHRDE along withbarotropic fluid can be discriminated (although very close) from ΛCDM point through statefinderparameters. It may be further noted that the evolution of the trajectories are not ending at { r = 1 , s = 0 } , rather they are going to the region r > , s <
0. Before reaching { r = 1 , s = 0 } the trajectories are traversing through the region { r < , s > } only.In Fig.8 we consider the evolution of { r, q } trajectory, where q is the deceleration parameter.For all the values of ν the { r, q } trajectory is starting its evolution from { r < , q > − } . For ν = 0 .
15 the trajectory is reaching { r = 1 , q = − } i.e. ΛCDM is attainable by the model.However, the today’s ( z = 0) point is not coincident (although) very close to this fixed point. Thisfurther supports the observations through { r − s } trajectory. Moreover, for ΛCDM the trajectoryends at { r = 1 , q = − } , whereas for the current model the trajectory is going beyond that. Thecase q < − ν = 0 .
15, we find that { r = 0 . , q = − . } (see Table II) that is very close to ΛCDM.1 - - z Ν - c s FIG. 9: Plot of c s based on Eq.(27). III. PERTURBATION EQUATIONS
We consider the linear perturbation of the current viscous EHRDE model towards a dark energydominated universe. For this purpose, squared speed of sound c s = dp total dρ total is crucial. A negative c s implies classical instability of a given perturbation. Myung [72] used squared speed of soundto discriminate between holographic dark energy, Chaplygin gas, and tachyon model and found anegative squared speed for holographic dark energy under the assumption of future event horizonas the IR-cutoff. In the present case squared speed of sound takes the form c s = ˙ p total ˙ ρ total = − (cid:18) − α ) ν + 6 √ τ DE − ∆ + (2 α + 3 βν + β ∆ + ) − C e x ∆ − ∆ − ( ∆ − − ∆ + )( α +3 βν + β ( ∆ − +∆ + )) C e x ∆ − ∆ − − e x ∆+ ( C − H ) ∆ + (cid:19) (27)In Fig.9 the squared speed of sound as derived in Eq.(27) is plotted against redshift z for a range ofvalues of ν . It is observed that | c s | < z ≈ − . c s <
0. This implies that fromthe early to some later stage the universe is unstable against small perturbation. In other wordsthis negative c s indicates that even small perturbation can grow with time leading to an unstableuniverse. However, for z . − . c s >
0. Thus, at this stage the universe has gainedstability. Hence, it may be interpreted that a viscous EHRDE along with viscous barotropic fluidcan lead to a stable universe in very late stage.Matter density perturbation given by δ = δρ m ρ m , where fluctuation in matter density is given by2 δρ m . In the linear regime the perturbation δ satisfies [73]¨ δ + 2 H ˙ δ − πGρ m δ = 0 (28)Growth function is given by f = d log δd log a . Considering ρ m = ρ ν as available in Eq.(18) and H as in log a ∆ FIG. 10: Evolution of matter density perturbation δ based on Eq.(29) with scale factor. Eq.(14) the Eq. (28) takes the form2 (cid:16) C e x ∆ − + ( H − C ) e x ∆ + (cid:17) δ ′′ ( x )+ (cid:16) C e x ∆ − (4 + ∆ − ) + ( H − C ) e x ∆ + (4 + ∆ + ) (cid:17) δ ′ ( x ) − δ ( x ) = 0 (29)Eq.(29) is numerically solved and δ is plotted against x = log a in Fig.10. It is observed that | δ | <<
1, which indicates linear growth of fluctuations. In Fig.11 we have plotted the growthfunction f = d log δd log a for a range of values of ν . This figure indicates that f is a decreasing functionof z . We observe that for z = 0 .
35 the f ≈ .
88, which is consistent with [74] and z = 0 .
22 the f ≈ .
72, which is consistent with [75].
IV. SECOND LAW OF THERMODYNAMICS
Discovery of black hole thermodynamics in 1973 [76] prompted physicists to study the thermo-dynamics of cosmological models of the universe [77–83]. Bekenstein [76]associated event horizonand the thermodynamics of a black hole by showing that event horizon of the black hole is ameasure of the entropy of it. In subsequent studies this idea has been generalized to horizons ofcosmological models by connecting each horizon to an entropy [77]. This modified the second lawof thermodynamics to its generalized form, in which the time derivative of the total entropy i.e.the sum of the time derivative of the entropy on the horizon and the fluid inside the horizon must3 z Ν f FIG. 11: Evolution of growth factor f = d log δd log a . be non-negative i.e. ˙ S total ≥
0. In the present work we consider event horizon R E as the envelopinghorizon of the universe. The event horizon is given by [77] R E = a Z t s t dta = a Z t s a daHa (30)where for different spacetimes t s has different values, e.g. for de Sitter spacetime t s = ∞ . Theevent horizon satisfies ˙ R E = HR E − H (Eq.(14)) in Eq.(31) we get the solution for R E as R E = e x C + 2 r C e x ( ∆ −− ∆+ )( H − C ) F (cid:20) − + − − + , , − + − − + , − C e x ( ∆ −− ∆+ ) ( H − C ) (cid:21) (2 + ∆ + ) q C e x ∆ − + ( H − C ) e x ∆ + (32)Hawking temperature on the horizon is [84] T E = H R E π (33)and subsequently entropy on the event horizon is [84]˙ S E = 8 π R E ( p total + ρ total ) H (34)To determine the entropy variation of the fluid inside R h we start with the Gibbs relation T f dS f = dE + p total dV (35)4 z Ν ´ ´ ´ ´ d dt H S E + S f L FIG. 12: Time derivative of total entropy ˙ S total . where, volume of the fluid is V = πR E and total energy of the fluid is E = ρ total V . UsingEqs.(17), (18) and (19) in Eqs. (34) and (35) we get the time variation of total entropy as˙ S E + ˙ S f = π ( C ( e x ∆ − − e x ∆+ ) + e x ∆+ H ) / (2+∆ + ) (cid:16) C e x ∆ − ∆ − − e x ∆ + (cid:0) C − H (cid:1) ∆ + (cid:17) (2 + ∆ + ) × F (cid:20) , − + − − + , − + − − + , C e x ( ∆ −− ∆+ ) C − H (cid:21) r − C (cid:16) − e x ( ∆ −− ∆+ ) (cid:17) + H C − H +2 C e x q C (cid:0) e x ∆ − − e x ∆ + (cid:1) + e x ∆ + H + C e x q C (cid:0) e x ∆ − − e x ∆ + (cid:1) + e x ∆ + H ∆ + (cid:17) − × F (cid:20) − + − − + , , − + − − + , C e x ( ∆ −− ∆+ ) C − H (cid:21) r − C (cid:16) − e x ( ∆ −− ∆+ ) (cid:17) + H C − H +2 C e x q C (cid:0) e x ∆ − − e x ∆ + (cid:1) + e x ∆ + H + C e x q C (cid:0) e x ∆ − − e x ∆ + (cid:1) + e x ∆ + H ∆ + (cid:17) (cid:17) (36) V. CONCLUDING REMARKS
In the present work we have extended the study of [62] on viscous holohraphic Ricci darkenergy to extended holographic Ricci dark energy (EHRDE) under the influence of bulk viscosity.Considering a coexistence of viscous EHRDE and viscous barotropic fluid with equation of state p ν = ( ν − ρ ν we have reconstructed Hubble parameter H (see Eq. (14)) corresponding to ν = 0 . , . , .
25 and 0 .
27 respectively. The reconstructed Hubble parameter is found to bedecreasing with evolution of the universe and in a later stage − . . z . − .
25, the reconstructed H is found to start increasing. Thus, in a later stage ˙ a ( t ) has been found to have the possibilityof dominating a ( t ). Equation of state (EoS) parameter for the viscous EHRDE based on Eq.(20)has been observed to exhibit a transition from EoS > − EoS < − z ≈ .
01 i.e. in an earlier stage of the universe and the current value of the EoS parameterfor viscous EHRDE is found to favour the ΛCDM model and the w DE for the current modelis consistent with results obtained by [9] through observational data sets from SNLS3, BAO andPlanck+WMAP9+WiggleZ measurements. It has further been noted that irrespective of the valuesof ν the viscous EHRDE is behaving like “quintom” i.e. transiting from quintessece to phantom.It has also been observed that the behaviour of w total (see Fig.3 ) is largely similar to that of w DE as far as the “quintom” behaviour is concerned. However, the transition to phantom has beenfound to occur in a later stage z ≈ − . ν . To further consolidate the resultsfrom EoS study we have studied the statefinder diagnostics of the viscous dark energy-barotropicfluid model by deriving expressions for the { r − s } parameters (see Eqs. (25) and (26)) for thecurrent model. Plotting { r − s } for different values of ν the trajectories are found to pass throughtoday’s point ( z = 0) (see Table II) and subsequently converge to the ΛCDM point in a later stage.It has been noted that for ν = 0 .
15 the today’s point is very close to ΛCDM point. From thisobservation it may be interpreted that although ΛCDM is favoured by the EoS parameter, thecurrent model can be discriminated (although very close) from ΛCDM point through statefinderparameters. Moreover, evolution of the trajectories have not ended at { r = 1 , s = 0 } , rather theyreached the region r > , s <
0. Similar behaviour has been observed through a study of { r − q } trajectory too along with attainment of a super-accelerated phase of the universe in a later stage.In Fig. 6 we have observed | p DE + p ν | − | Π | ≫ | Π | ≪ | p DE + p ν | , which implies that thenon-equilibrium bulk viscous pressure is significantly smaller than the local equilibrium pressure.In the next phase we have studied behaviour of squared speed of sound c s to study the stabilityof the model. After obtaining the expression for c s for the current model in Eq. (27) and plottingin Fig.9 we have observed that currently c s < z . − . c s > δ = δρ m ρ m and the growth function f for the current model. Referring toFig.10 it is observed that | δ | << f = d log δd log a plotted in Fig.11 for a range of values of ν it has been observed that f is a decreasingfunction of z . It has been noted that for z = 0 .
35 the f ≈ .
88, which is consistent with [74] andfor z = 0 .
22 the f ≈ .
72, which is consistent with [75].Finally we have considered the generalized second law of thermodynamics (GSL) for the currentmodel considering event horizon as the enveloping horizon of the universe. Under the assumption6thermal equilibrium we have observed that time derivative of total entropy ˙ S total ≥ VI. ACKNOWLEDGEMENT
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