Cosmology of a generalised version of holographic dark energy in presence of bulk viscosity and its inflationary dynamics through slow roll parameters
aa r X i v : . [ phy s i c s . g e n - ph ] J un Cosmology of a generalized version of holographic dark energy inpresence of bulk viscosity and its inflationary dynamics throughslow roll parameters
Gargee Chakraborty and Surajit Chattopadhyay ∗ Department of Mathematics, Amity University, Major Arterial Road,Action Area II, Rajarhat, New Town, Kolkata 700135, India. Department of Mathematics, Amity University, Major Arterial Road,Action Area II, Rajarhat, New Town, Kolkata 700135, India. (Dated: June 15, 2020)The present study reports a reconstruction scheme of a Dark Energy (DE) model withhigher order derivative of Hubble parameter, which is a particular case of Nojiri-Odintsovholographic DE [50] that unifies phantom inflation with the acceleration of te universe onlate-time. The reconstruction has been carried out in presence of bulk-viscosity, where thebulk-viscous pressure has been taken as a function of Hubble parameter. Ranges of cosmictime t have been derived for quintessence, cosmological constant and phantom behaviour ofthe equation of state (EoS) parameter. In the viscous scenario, the reconstruction has beencarried out in an interacting and non-interacting situations and in both the cases stabilityagainst small perturbations has been observed. Finally, the slow roll parameters have beenstudied and a scope of exit from inflation has been observed. Also, availability of quasiexponential expansion has been demonstrated for interacting viscous scenario and a studythrough tensor to scalar ratio has ensured consistency of the model with the observationalbound by Planck. Alongwith primordial fluctuations the interacting scenario has been foundto generate strong dissipative regime. Keywords : Holographic Dark Energy; Bulk Viscosity; Interaction; Equation of state pa-rameter; Slow Roll parameter.
I. INTRODUCTION
Riess et al.[1] and Perlmutter et al.[2] independently reported in the late 90’s that the currentuniverse is passing through a phase of accelerated expansion. Their discovery is a breakthrough inthe field of Modern Cosmology. They [1, 2] discovered this accelerated expansion by accumulating ∗ Electronic address: [email protected]; [email protected] the observational data of distant Supernovae Ia (SNeIa) and their discovery has further beensupported by other observational studies [3–7]. Some exotic matter characterised by negativepressure is thought to be responsible for driving this acceleration. This exotic matter is dubbedas ”Dark Energy” (DE) [8, 9]. The DE differs from the ordinary matter in the sense that it ischaracterised by negative pressure. A DE is described by the equation of state (EoS) parameterdefined as w = pρ , where p is the pressure and ρ is the density due to DE. From Friedmann’sequations one can easily verify that w < − is a necessary condition for the accelerated expansionof the universe. Cosmological constant (Λ), characterised by constant EoS parameter w = −
1, isthe simplest candidate for DE [44]. Although Λ is consistent with observations, other candidatesof DE have also been proposed in the literature and those candidates have time varying EoSparameter. Such candidates have been proposed in the literature to get rid of some limitations ofcosmological constant [10]. Candidates with dynamic EoS parameter can be broadly differentiatedinto (i) scalar field models; (ii) holographic models of DE; (iii) Chaplygin gas models. Various DEmodels have been reviewed in the literatures including [8, 9]. It may be noted that around 68.3%of the total energy density of the present observable universe is contributed by DE. Remainingdensities are due to dark matter (DM), ordinary baryonic matter and radiation. However, thecontribution due to baryonic matter and radiation are negligible with respect to the total densityof the universe.Nojiri and Odintsov [13, 14] developed cosmological models, where the DE and DM were treatedas imperfect fluids. Viscous fluids represent one particular case of what was presented in [13, 14]. Inrecent years, a handful of literatures have explored the possibility that the late-time accelerationis driven by a kind of viscous fluid [15–17]. In those references [15–17], a universe filled withbulk-viscous matter has been analysed through the theory for evolution of the viscous pressureunder the perview of late-time acceleration of the universe. At this juncture, it should be statedthat the late-time accelerated phase is not the only accelerated phase of the universe. There wasanother phase of acceleration in the early stage of the universe and that is called an inflationaryscenario [17]. In this very early phase of evolution, the dissipative effects including both bulk andshear viscosity are thought to play a significant role [18]. It was reported in the work of Chimentoet al. [19] that it is possible to have the accelerated expansion of the universe in presence of acombination of a cosmic fluid characterised by bulk-dissipative pressure and a quintessence matter.It was also reported in [19] that the above process involves a sequence of dissipative processes. Theintroductory attempts in the direction of creating the theory of relativistic dissipative fluids wherereported in the works of Eckart [20] and Landau and Lipshitz [21]. A time varying viscosity in DEframework was reported by Nojiri and Odintsov [22], where EoS was considered to be associatedwith an inhomogeneous and Hubble parameter dependent term. Brevik et al. [16] demonstratedthe entropy for a coupled fluid and established a relationship between the entropy of closed FRWuniverse to the energy contained in it. Brevik et al. [17] considered a DE - DM interacting scenarioin a flat FRW universe and demonstrated Little Rip, Pseudo Rip and Bounce Cosmology in BulkViscosity framework by considering the bulk-viscous pressure as a function of Hubble parameter.In a very recent work, [23] proposed an approach where an extended cosmological model wasdemonstrated in the context of viscous DE. Recent work of viscous cosmology also studied in[52, 53].It has already been mentioned in the previous paragraph that one of the broad type of DEcandidates is the holographic DE (HDE) model that has been extensively discussed in the references[24–27]. The HDE is based on the holographic principle [24]. The density ρ Λ of HDE is given by ρ Λ = 3 c M P L − [26] where c represents a dimensionless constant, M P is the reduced Plankmass and L stands for IR cutoff. Different modifications to the IR cutoff has been proposed inthe literature and various types of HDE have been discussed till date. Examples include modifiedHDE [28], Holographic Ricci DE [29] and generalised HDE [30]. Note that all these HDEs arejust particular cases of Nojiri-Odintsov cut-off which may even serve to get the covariant theoryfor specific Nojiri-Odintsov cut-off [49]. Since DE is responsible for about 68.3% of the totalenergy density of the late-time universe and it was negligible after the big-bang, Chen and Jing[31] argued that the DE density should be a function of the Hubble parameter H and its higherorder derivatives with respect to cosmic time t . The physical explanation behind this argument isthat the H gives us information about the expansion rate of the universe. Based on this physicalexplanation, reference [31] proposed the following form of HDE which is basically a specific case ofthe Nojiri - Odintsov HDE [11]: ρ Λ = 3( α ¨ H H − + β ˙ H + ǫH ) (1)where α , β and ǫ are three arbitrary dimensionless parameters. It may be noted that we haveassumed M P = (8 πG ) − = 1. In the limiting case with α = 0, we get the HDE with Granda-Oliveros (GO) cutoff. A detailed account in this regard has been presented in a recent work [32].Recently holographic bounce was proposed in [51].In this paper, firstly will reconstruct Hubble parameter H without any choice of scale factor, alsoreconstruct state equation w Λ , w eff and deceleration parameter. Secondly, with power law formof scale factor we will reconstruct Hubble parameter H , bulk viscous pressure Π, state equation w Λ , w eff and deceleration parameter q and the state finder parameter r and s . Thirdly, in caseof non interacting scenario in presence of bulk viscosity we will discuss equation of state w Λ , w eff and squared speed of sound. In the fourth in case of interacting scenario in presence of bulkviscosity we will discuss equation of state w Λ , w eff and squared speed of sound. In the fifth wewill discuss background evolution. Here we discus viscous interacting dark energy as scalar field.We reconstructed here the Hubble slow roll parameters ǫ H , η H and potential slow roll parameters ǫ V , η V . Then reconstructed EoS parameter w φ and reconstructed dissipative coefficient Γ andcalculated 2 V − ˙ φ . Rest of the paper is organized as follows: In section I, we have reportedreconstruction schemes for ρ Λ through reconstruction of H in presence of bulk viscosity with aswell as without any specific choice of scale-factor. Interacting as well as non-interacting scenariostaken into account. In section II, we have demonstrated the findings of background evolution inviscous interacting dark energy as scalar field. We have calculated Hubble slow roll parameters,tensor to scalar ratio and the effective EoS parameter. We have concluded in section III. II. RECONSTRUCTION SCHEMESA. Viscous scenario without dark matter
1. Without any choice of scale factor.
In the present subsection we are going to demonstrate a reconstruction scheme for the DEpresented in Eq.(1) in presence of bulk-viscosity. That is in addition to the thermodynamic pressurea bulk-viscous pressure is to be considered as Π = − Hξ where ξ = ξ + ξ H + ξ ( ˙ H + H ) (2)where ξ , ξ , ξ are all positive constraints. In presence of bulk-viscosity the Friedmann’s equationsare: H = 13 ρ Λ (3)6 ¨ aa = − ( ρ Λ + 3( p Λ + Π)) (4)We shall now demonstrate reconstruction scheme for ρ Λ in presence of bulk-viscosity as statedabove. Solving Eq.(1) and Eq.(3), we have the following solution for the Hubble parameter H = 2 αβt + C (5) FIG. 1: Evolution of reconstructed Bulk Viscous pressure without any choice of scale factor. We consider α = 0 . β = 0 . ξ = 0 . ξ = 0 . ξ = 0 . where it should be stated that we have chosen ǫ = 1 within the permissible range. A naturalconsequence of the reconstructed H in the reconstructed scale factor, whose form comes out to be a = ( βt + C ) αβ ( βt + C ) αβ (6)Using,the reconstructed Hubble parameter, we can get the reconstructed DE density as ρ Λ = 12 α ( βt + C ) (7)Also, the bulk-viscosity coefficient being dependent upon H and ˙ H , we can have the followingreconstructed bulk-viscous pressure:Π = − α (( C + tβ ) ξ + 2 α ( C + tβ ) ξ + 4 α ξ − αβξ )( C + tβ ) (8)As we are considering a D.E. dominated scenario under the assumption of negligible contributiondue to DM, we are not supposed to have any interacting scenario. In absence of an interaction theconservation equation takes the following form in bulk-viscous framework:˙ ρ Λ + 3 H ( ρ Λ + p Λ + Π) = 0 (9)From Eq.(9), we can easily write p Λ = − ( ˙ ρ Λ H + ρ Λ + Π),which can be reconstructed using Equations(5), (7) and (8).At this juncture,we have reconstructed p Λ and ρ Λ in bulk viscous scenario andhence we can have the EoS parameter w Λ = p Λ ρ Λ in presence of bulk viscosity with backgroundevolution as the holographic form of DE presented in Eq.(1). The form of w Λ is derived below: w Λ = − β + 3 ( C + tβ ) ξ α + ξ + (2 α − β ) ξ C + tβ (10)Clearly, the reconstructed behaves like w Λ is quintessence, cosmological constant or phantom ac-cordingly as t T ( β − α ) ξ C − ξ − β +3 C ξ αβξ α − (2 α − β ) ξ βC and C ξ ξ = 4 α − αβ . Now, we consider the effective EoSparameter w eff = p Λ +Π ρ Λ , which comes out to be w eff = − β α and the deceleration parameterbecomes q = − β α .Hence, it is understandable that if β α > w eff > − β α < w eff < − w Λ , its impact is neutralised in w eff . Hence, behaviour of w eff to be quintessenceor phantom would be determined by the nature of α and β as applied in Eq.(1). Furthermore,accelerated expansion would be available if q < − β α < α > β . The behaviour ofthe bulk viscous pressure reconstructed in Eq.(8) is plotted in Fig.1, where it is observed that inabsence of DM the effect of bulk viscous pressure is decaying with cosmic time t .
2. With power law form of scale factor.
In the previous section, we have demonstrated the behaviour of bulk-viscous pressure, where thebackground evolution is according to the HDE type presented in Eq.(1). In the previous sectionwe didnot make any assumption regarding the choice of scale factor. Rather we have obtained thesolution for the Hubble parameter to get the scale factor. In the present section we are going todevelop a reconstruction scheme for the DE candidate presented in Eq.(1) with power-law formof scale factor and in presence of bulk-viscosity. The scale factor is chosen as a = a t n where a , n >
0. Therefore, H = nt . (11)Hence, the D.E. density becomes ρ Λ = 6 α − nβ + 3 n ǫt (12)and the bulk viscosity coefficient is ξ = t ξ + ntξ + ( − n ) nξ t (13)Finally, the bulk-viscous pressure Π = − Hξ comes out to beΠ = − n (cid:0) t ξ + ntξ + ( − n ) nξ (cid:1) t (14) FIG. 2: Evolution of reconstructed Bulk Viscous pressure with power law form of scale factor. We consider ξ = 0 . ξ = 0 . ξ = 0 . Using Eq.(12) in the conservation equation Eq.(9), we can find reconstructed p Λ in presence ofbulk-viscosity and subsequently obtained the reconstructed EoS parameter as follows: w Λ = 3 n t ξ + t ((4 − n ) α + n (( − n ) β + n (2 ǫ − nǫ + 3 nξ ))) + 3( − n ) n ξ nt (6 α − nβ + 3 n ǫ ) (15). Like the previous section,we obtained the EoS parameter as w eff = − n (16)The deceleration parameter q = − n (17)As n is always positive, w eff > − n > { r, s } where r = ... aaH (18) s = r − q − ) (19)where q = − a ¨ a ˙ a . In the present case, we calculate the state finder parameters and observed that r = ( − n )( − n ) n (20) s = 23 n (21)In order to have r = 1, we need n = <
1, which is not compatible with the requirement of thepresent acceleration. Furthermore, s = 0. Hence, for a fixed n the Λ CDM fixed { r = 1 , s = 0 } isnot attainable. However as n → ∞ we observe that r → , s →
0. Therefore, we can understandthat the ΛCDM fixed point is attainable in the limiting case. Moreover, it is also understand thatlike the previous case, the presence of bulk viscosity is not influencing the deceleration parameterand state-finder diagnostics.
B. Viscous scenario in presence of dark matter
1. non-interacting scenario
The conservation equation in a viscous scenario can be broken into two parts when we considerinteraction between DE and DM. An interaction term Q can be chosen in various forms and isadded to the right hand side in the following manner:˙ ρ Λ + 3 H ( ρ Λ + p Λ + Π) = − Q (22)˙ ρ m + 3 Hρ m = Q (23)In this subsection we consider non interacting scenario i.e., Q = 0 and hence from Eq.(23), we willhave the solution for DM as ρ m = ρ m a − . As we consider the coexistence of DE and DM, the firstFriedmann equation takes the form 3 H = ρ m + ρ Λ and hence using the Eq.(22), we have [ ǫ = 1],3 H = 3 (cid:16) α ¨ H H − + β ˙ H + H (cid:17) + ρ m a − (24)where ˙ H = aH dHda and ¨ H = aH a (cid:18) dHdt (cid:19) + H (cid:18) dHdt + a d Hdt (cid:19)! . Solving Eq.(24) we have thereconstructed H as a function of scale factor a as follows: H ( a ) = ± √ q − ρ m βa + 9 a − βα α ( − α + β ) C + 9(3 α − β ) βC p (3 α − β ) β (25) - - H FIG. 3: Evolution of reconstructed Hubble parameter in non interacting scenario. We have chosen ρ m =0 .
32 . The red ,green and blue correspond to α = 0 . β = 0 . C = 0 . C = 1 . η = 1 . α = 0 . β = 0 . C = 0 . C = 0 . η = 0 . α = 0 . β = 0 .
21 , C = 0 . C = 0 .
99 , η = 0 . We consider the the positive solution in Eq.(25). Now we will try to constrain the model parameterspresent in the solution for H . Clearly if β > H . α > β C > β (cid:18) βρ m α − β + 9 C α (cid:19) a − βα (27)We can infer from (27) that if C > C >
0. It is also understandable that it is feasible tochoose C > C . Hence we assume C = C + η, η >
0. This assumption leads us to have thefollowing constraint: C > ( β )( βρ m α − β ) a − βα − η (1 − αβ a − βα ) (28)Using the positive solution of Eq.(25) and computing its derivative with respect to cosmic time t as explained above we get the reconstructed DE density as ρ Λ = a − − βα (cid:16) a βα C (3 α − β ) β + 18 a C α ( − α + β ) + a βα ρ m β ( − − α + 3 β ) (cid:17) α − β ) β (29)0 TABLE I: Reconstructed EoS parameter for different combinations of ξ , ξ , ξ in the current universe (z=0)for non interacting viscous scenario z = 0 ξ = 0 . ξ = 0 . ξ = 0 . ξ = 0 . ξ = 0 . ξ = 0 . ξ = 0 . ξ = 0 . ξ = 0 . w Λ -1.08172 -0.904172 -0.995758 Also, we can also reconstructed the viscosity as ξ = ξ + √ r − ρm βa +9 C (3 α − β ) β +9 a − βα C α ( − α + β ) ξ √ (3 α − β ) β + a − − βα (cid:18) a βα ρ m β +18 a βα C (3 α − β ) β − a C ( α − αβ + β ) (cid:19) ξ α − β ) β (30)which helps us get the bulk-viscous pressure asΠ = − √ r − ρm βa +9 C (3 α − β ) β +9 a − βα C α ( − α + β ) √ (3 α − β ) β × ξ + √ r − ρm βa +9 C (3 α − β ) β +9 a − βα C α ( − α + β ) ξ √ (3 α − β ) β + a − − βα (cid:18) a βα ρ m β +18 a βα C (3 α − β ) β − a C ( α − αβ + β ) (cid:19) ξ α − β ) β (31)The above equations gives us the bulk-viscous pressure when the background evolution is governedby the DE given in Eq.(1). As we already have reconstructed H , ρ Λ and Π, we can get p Λ fromthe conservation equation (see Eq.(22))taking Q = 0 with that p Λ we can have the following EoSparameter: w Λ = a βα (3 α − β ) β − a − βα C + ρ m (cid:16) − − α + β (cid:17) a + a − βα C α (3 α − β )+ ρm α − β ) βa +18 C β ( − α + β )(3 α − β ) β + √ r − ρm βa +9 C (3 α − β ) β +9 a − βα C α ( − α + β ) ζ √ (3 α − β ) β × (cid:16) a C α ( − α + β ) + a βα β (cid:0) a C (3 α − β ) + ρ m ( − − α + 3 β ) (cid:1)(cid:17) − (32)where, ζ = ξ + √ r a − βα C α ( − α + β )+ β (cid:16) − ρm a +27 C α − C β (cid:17) ξ √ (3 α − β ) β + (cid:18) C + ρ m a (3 α − β ) + a − βα C ( − α + β ) β (cid:19) ξ The Hubble parameter reconstructed in Eq.(25) is plotted in Fig.3, where against redshift z andwe observed that H >
FIG. 4: Evolution reconstructed EoS parameter in non intracting scenario. We have chosen ρ m = 0 . α = − . β = − . C = 1 . ξ = 0 . ξ = 0 . ξ = 0 . by v s = ˙ p eff ˙ ρ Λ (33)This approach for checking stability of the DE model has earlier been used in Myung [35]. Thepresent model is considering the presence of bulk-viscous pressure alongwith the thermodynamicpressure. Hence, instead of considering p Λ only. We take to compute v s . Hence, we are going toconsider v s = ˙ p eff ˙ ρ Λ and consequently the squared speed of sound comes out to be v s = − a C ( − α + β ) α (cid:16) a βα ρ m (2 + 9 α − β ) + 6 a C (3 α − β ) (cid:17) (34)Based on the constraints already obtained for the constants, the squared speed of sound isplotted in Fig.5, for a range of values of C within its permissible boundaries. It has been observedthat the squared speed of sound is positive throughout and for lower values of C it is closed tozero, for the current universe i.e., z = 0 and significantly greater than zero for higher values of C .Furthermore, it appears from Fig.5 is also apparent that for higher value of C the squared speedof sound will remain positive for a considerable period of time beyond z = 0. Hence, a very stablemodel against small perturbations can be obtained from this non-interacting scenario where thebackground evolution is holographic and the universe is under bulk-viscous pressure apart fromthe thermodynamic pressure. If we consider the physical bounds 0 ≤ v s ≤
1, it is apparent fromFig.5 that this model is not violating the bounds.2
FIG. 5: Evolution of reconstructed squared speed of sound in non-interacting scenario. We consider ρ m = 0 . α = − . β = − . C = 1 .
2. Interacting scenario in presence of bulk viscosity
In the present section we are going to demonstrate the cosmological consequences of an inter-action between the HDE with higher order derivatives as presented in Eq.(1)and the pressurelessdark matter. The interaction term Q is chosen in the form Q = 3 Hδρ m where δ is the interac-tion parameter and ρ m is the density of pressureless dark matter. The conservation equations ininteracting scenario and in presence of bulk viscosity are˙ ρ Λ + 3 H ( ρ Λ + p Λ + Π) = − Hδρ m (35)where Π, the bulk viscous pressure, has the form as descrbed in Eq.(2)˙ ρ m + 3 Hρ m = 3 Hδρ m (36)As already mentioned previously, the First Friedmann equation in presence of DM takes the form3 H = ρ m + ρ Λ , where ρ Λ comes from the solution of Eq.(36)as ρ m = ρ m a − − δ ) . With this formof DM, we obtained the solution for Hubble parameter from the Friedmann equations mentionedabove: H = r − a − δ ρ m β + 18 a − βα ( β + 3 α ( − δ ))( − δ ) (cid:16) − αC + a βα βC (cid:17) p β ( β + 3 α ( − δ ))( − δ ) (37)It may be noted that like non-interacting case, here also we take ǫ = 1. As we know, ˙ H = aH dHda ,3 - - H FIG. 6: Evolution of reconstructed Hubble parameter in interacting scenario. We have considered ρ m =0 . δ = 0 . α = 0 . β = 0 . C = − . C = 2; α = 0 . β = 0 . C = 0 . C = 8 . α = 0 . β = 0 . C = 0 . C = 22 . we obtained the two derivatives of H below:˙ H = a − − βα (cid:16) − a βα +3 δ ρ m + 3 a C ( β + 3 α ( − δ )) (cid:17) β + 3 α ( − δ )) (38)¨ H = − a − − βα β (cid:18) a C β ( β +3 α ( − δ ))+ a βα +3 δ ρ m α ( − δ ) (cid:19) α ( β ( β +3 α ( − δ ))( − δ )) / × r − a − δ ρ m β + 18 a − βα (cid:16) − C α + a βα C β (cid:17) ( β + 3 α ( − δ ))( − δ )( − δ ) (39)As we are now having reconstructed Hubble parameter in interacting scenario, we apply this formEq.(37) in Eq.(1) to have the reconstructed HDE density and it comes out to be ρ Λ = a − − βα (cid:18) − a βα +3 δ ρ m β ( β ( − δ )+9 α ( − δ ) ) − a C α ( β +3 α ( − δ ))( − δ )+18 a βα C β ( β +3 α ( − δ ))( − δ ) (cid:19) β ( β +3 α ( − δ ))( − δ ) (40)Furthermore, through this reconstructed H , we have the following forms of bulk viscosity coefficientand bulk viscous pressure respectively as ξ = ξ + s − a − δ ρ m β +18 a − βα (cid:18) − C α + a βα C β (cid:19) ( β +3 α ( − δ ))( − δ ) ξ √ β ( β +3 α ( − δ ))( − δ ) + a − − βα (cid:18) − a C (2 α − β )( β +3 α ( − δ ))( − δ )+18 a βα C β ( β +3 α ( − δ ))( − δ ) − a βα +3 δ ρ m β ( − δ ) (cid:19) ξ β ( β +3 α ( − δ ))( − δ ) (41)4Π = − √ β ( β +3 α ( − δ ))( − δ ) r − a − δ ρ m β + 18 a − βα (cid:16) C α + a βα C β (cid:17) ( β + 3 α ( − δ ))( − δ ) ξ + s − a − δ ρ m β +18 a − βα (cid:18) − C α + a βα C β (cid:19) ( β +3 α ( − δ ))( − δ ) ξ √ β ( β +3 α ( − δ ))( − δ ) + a − − βα (cid:18) − a C (2 α − β )( β +3 α ( − δ ))( − δ )+18 a βα C β ( β +3 α ( − δ ))( − δ ) − a βα +3 δ ρ m β ( − δ ) (cid:19) ξ β ( β +3 α ( − δ ))( − δ ) (42)Since, ρ Λ , ρ m , H and Π are all having their reconstructed forms, we can reconstruct the thermo-dynamic pressure p Λ by putting the corresponding forms in Eq.(35) and hence the reconstructedEoS parameter w = p Λ ρ Λ comes out to be w Λ = (cid:16) a βα β ( β + 3 α ( − δ ))( − δ ) a − − βα (cid:18) − a C1( β +3 α ( − δ ))+ a βα +3 δ ρ m ( β ( − δ )+9 α ( − δ ) ) (cid:19) β +3 α ( − δ )) − a − − βα (cid:18) − a βα +3 δ ρ m β ( β ( − δ )+9 α ( − δ ) ) − a C α ( β +3 α ( − δ ))( − δ )+18 a βα C β ( β +3 α ( − δ ))( − δ ) (cid:19) β ( β +3 α ( − δ ))( − δ ) − a − δ ) ρ m s − a − δ ρ m β +18 a − βα (cid:18) − C α + a βα C β (cid:19) ( β +3 α ( − δ ))( − δ ) δ √ β ( β +3 α ( − δ ))( − δ ) + √ β ( β +3 α ( − δ ))( − δ ) r − a − δ ρ m β + 18 a − βα (cid:16) − C α + a βα C β (cid:17) ( β + 3 α ( − δ ))( − δ ) ξ + s − a − δ ρ m β +18 a − βα (cid:18) − C α + a βα C β (cid:19) ( β +3 α ( − δ ))( − δ ) ξ √ β ( β +3 α ( − δ ))( − δ ) + a − − βα (cid:18) − a C1(2 α − β )( β +3 α ( − δ ))( − δ )+18 a βα C β ( β +3 α ( − δ ))( − δ ) − a βα +3 δ ρ m β ( − δ ) (cid:19) ξ β ( β +3 α ( − δ ))( − δ ) h(cid:16) − a βα +3 δ ρ m β (cid:0) β ( − δ ) + 9 α ( − δ ) (cid:1) − a C α ( β + 3 α ( − δ ))( − δ )+18 a βα C β ( β + 3 α ( − δ ))( − δ ) (cid:17)i − (43)The reconstructed Hubble parameter and w Λ are now plotted in Fig.6 and Fig.7 and Fig.7respectively with choice of parameters in their acceptable ranges. In Fig.6, we observed that theHubble parameter is increasing with the evolution of universe and in Fig.7, we observe that theEoS parameter is tending to -1 and is behaving like phantom. Nevertheless it is not crossing thephantom boundary. As already discussed in the previous section, the squared speed of sound for5 FIG. 7: Evolution of reconstructed EoS parameter in interacting scenario. We consider α = 0 .
09 , β = 0 . ρ m = 0 .
32 , C = − . C = 2, ξ = 0 . ξ = 0 . ξ = 0 . ξ , ξ , ξ in the current universe(z=0) for interacting scenario z = 0 ξ = 0 . ξ = 0 . ξ = 0 . ξ = 1 . ξ = 0 . ξ = 0 . ξ = 1 . ξ = 0 . ξ = 0 . w Λ -1.00392 -0.955097 -0.81857 the present case comes out to be v s = (cid:0) ( β ( β + 3 α ( − δ ))( − δ )) / (cid:16) − √ a C (3 α − β )( β + 3 α ( − δ )) √ ∆ p β ( β + 3 α ( − δ ))( − δ )+ a βα +3 δ ρ m α (cid:16) − C β ( β + 3 α ( − δ )) ( − δ ) + √ √ ∆ (2 + 3 β ( − δ )+9 α ( − δ ) (cid:1) p β ( β + 3 α ( − δ ))( − δ ) (cid:17) δ + 9 a βα +6 δ ρ m αβ ( β + 3 α ( − δ ))( − δ ) δ +9 a δ C ρ m α ( β + 3 α ( − δ )) ( − β + 6 α ( − δ ))( − δ ) δ (cid:1)(cid:1)h(cid:16) a αβ (cid:16) a C ( β + 3 α ( − δ )) − a βα +3 δ ρ m (cid:0) β ( − δ ) + 9 α ( − δ ) (cid:1)(cid:17) ( β + 3 α ( − δ )) r − a − δ ρ m β + 18 a − βα (cid:16) − C α + a βα C β (cid:17) ( β + 3 α ( − δ ))( − δ )( − δ ) (cid:19) ] − (44)where∆ = r a − − βα (cid:16) − a βα +3 δ ρ m β − a C α ( β + 3 α ( − δ ))( − δ ) + 9 a βα C β ( β + 3 α ( − δ ))( − δ ) (cid:17) Eq.(44) is plotted against redshift for a range of values of δ . It is observed in this figure thatthe physical bounds 0 ≤ v s ≤ FIG. 8: Evolution of reconstructed squared speed of sound for interacting viscous scenario. We considered α = 0 . β = 0 . ρ m = 0 . C = 2. III. VISCOUS INTERACTING DARK ENERGY AS SCALAR FIELD.
In the literature (e.g.[37, 38]), a number of inflationary self-interaction potentials have beenproposed to date to explain the inflation. The self-interacting potentials result in different infla-tionary scenarios. In particular, if we talk about density fluctuations, then we will find that theyhave different observational consequences for the CMB radiation. In GR scalar field cosmology,different inflation potentials have been proposed in the literature [39]. In a very recent work, Nojiriet al. [11] applied the holographic principle at the early universe, obtained an inflation realizationof holographic origin to calculate the Hubble slow-roll parameters and obtained the scalar spectralindex, the tensor-to-scalar ratio, and the tensor spectral index. Bamba et al. showed the equiva-lence between scalar field theories and the fluid description [8]. In the process firstly we take thedescribed fluid and then derived a scalar field theory with the same EoS as that in a fluid descrip-tion. By following this process, we got constraints on a coefficient function in the φ and V ( φ ) ofthe scalar field. Therefore ,derived the expression of φ and V ( φ ) in the scalar field theory for afluid model. Again, we have a scalar field theory described by φ and V ( φ ) and by the solution of φ , V ( φ ), H , we get w φ . By the expression of ρ Λ with w φ , we acquire f ( ρ ) in the fluid description.Therefore, it implies that the scalar field theory and fluid model description is equivalent. In aa very significant work, [40] demonstrated a phantom cosmology having a dynamics that allowsthe the universe to trace back the evolution to the inflationary epoch and developed the unifiedphantom cosmology where the same scalar field is capable of explaining the early time (phan-7tom) inflation and late-time accelerated DE phase of the universe. Considering the inflationarydynamics in modified gravity framework of f ( R, G ), the authors [41] could demonstrate a doubleinflationary scenario. Inflationary dynamics through scalar field have also been demonstrated in[42], where inflationary solutions could be obtained that followed neither from any effective scalarfield potential nor from a cosmological constant.In the present section we consider flat FRW universe to get the viscous interacting dark energyin scalar field framework. Denoting φ as a scalar field and V ( φ ) as a potential we have the followingequations: H = 13 (cid:20)
12 ˙ φ + V ( φ ) (cid:21) (45)˙ H = −
12 ˙ φ (46)The equation of motion for the scalar field is¨ φ + 3 H ˙ φ + ∂ φ V ( φ ) = 0 (47)The basic purpose for this section is to demonstrate whether it is possible to have inflationaryexpansion from the interacting viscous holographic dark energy in scalar field formalism. To dothe same we consider slow roll parameters ǫ H and η H given by the following equations:In Eq. (47), if we consider standard approximation technique for analysing inflation in slow rollapproximation, then we have [45] 3 H ˙ φ ≈ − ∂ φ V ( φ ) and from Eq. (45), we have H ≈ V ( φ ). Forthis approximation to be valid we need to have ǫ H ≪ , p η H ( φ ) p ≪ ǫ H and η H are given by ¨ aa = H (1 − ǫ H ) (49)and η H = − ¨ φH ˙ φ (50)These are Hubble slow roll parameters and in terms of potential, they can be written as [] ǫ V = 12 " dVdφ V (51)8 η V = ( dVdφ ) V (52)The slow roll parameters in term of potential become equal to the Hubble slow roll parameter inthe following situation [46] ǫ V ≈ ǫ H (53) η V ≈ ǫ H + η H (54)It may be noted that ǫ V is positive by definition and slow-roll approximation is valid and inflationis guaranteed if the slow roll roll approximation i.e., ǫ V ≪ ǫ H as perEq. (49), we need the scale factor and Hubble Parameter, which is already obtained in equationEq. (37) for interacting DE in presence of bulk viscosity. The expression of ǫ H for the scenariounder consideration is presented below: ǫ H = 3 β (cid:16) a βα +3 δ ρ m − a C ( β + 3 α ( − δ )) (cid:17) ( − δ ) − a βα +3 δ ρ m β − a (cid:16) C α − a βα C β (cid:17) ( β + 3 α ( − δ ))( − δ ) (55)Eq. (55) makes it apparent that the following constraints on C as in Eq.(37) are to be satisfied:For ǫ H > β > C > a ( βα + 3 δ − ρ m β + 3 α ( − δ )) (56)For ǫ H > β < C < a ( βα + 3 δ − ρ m β + 3 α ( − δ )) (57)For ǫ H < C < a ( βα +3) C β Ω( − δ ) − a ( βα +3 δ ) ρ m β − βa ( βα +3 δ ) ρ mo ( − δ )9 a Ω( − δ )[2 α − β ] (58)where Ω = β + 3 α ( − δ )In the above constraints,(56) and (57) encompass the cases of positive as well as negative β .However both of them satisfy the positivity of ǫ H . The third inequation (58) constraints C , whichis not effected by the sign of β .The evolution of Hubble slow roll parameter ǫ H is studied through Eq.(55) and Fig.9. Basedon the constraints presented for 0 < ǫ H <
1, the ǫ H is plotted against the red shift z for a rangeof values of δ in Fig.9. In this figure we observe that for the entire range of values of δ , ǫ H ≪ FIG. 9: Evolution reconstructed of ǫ H in case of viscous interacting DE as scalar field. We consider ρ m = 0 . α = 0 . β = 0 . C = − C = 0 . and is always positive. Hence it is clear that under these constraints ǫ H is an infinitesimally smallpositive number. Hence, we may look into the primordial inflation using this ǫ H by describingthe primordial inflation through quasi - de Sitter geometry with the EoS w = − ǫ H . Clearly,because of the infinitesimally small ǫ H we have w ≈ −
1. Therefore, it is possible to infer triviallythat the effect of bulk-viscosity is not dominant during the early universe and during this phasethe thermodynamic pressure will dominate and make the equation of state close to −
1. The scalarfield φ and potential V ( φ ) are now expressed in terms of scale factor a as follows:˙ φ = s − a − βα C + 2 a − δ ρ m β + 3 α ( − δ )) (59) V ( φ ) = 6 C + a − βα (cid:18) C − C αβ (cid:19) − a − δ ρ m (1 + δ )3( β + 3 α ( − δ ))( − δ ) (60)Using the solution of Hubble parameter as in Eq.(37) for interacting viscous scenario in Eq.(50),we obtain the slow roll parameter as a function of scale factor as follows: η H = 3 a C β ( β + 3 α ( − δ )) + 3 a βα +3 δ ρ m α ( − δ ) − a βα +3 δ ρ m α + 6 a C α ( β + 3 α ( − δ )) (61)From the Fig.9 and Fig.10, we observe that η H and ǫ H both are increasing. Therefore,the modelhas the scope of exit from inflation. Now we consider Eqns. (51) and (52) to obtain the potential0 FIG. 10: Evolution reconstructed η H in case of viscous interacting DE as scalar field. We consider ρ m = 0 . α = 0 . β = 0 . C = − slow roll parameters in an interacting viscous scenario, where H has been reconstructed in Eq.(37): ǫ V = a − − βα (cid:16) − a − δ ρ m β + 18 a − βα (cid:16) − C α + a βα C β (cid:17) ( β + 3 α ( − δ ))( − δ ) (cid:17)(cid:16) a C (6 α − β )( β + 3 α ( − δ )) − a βα +3 δ ρ m α (1 + δ ) (cid:17) × (cid:16) α β (cid:16) − a − βα C + a − δ ρ m β +3 α ( − δ )) (cid:17) ( β + 3 α ( − δ )) ( − δ ) (cid:16) C + a − βα (cid:16) C − C αβ (cid:17) − a − δ ρ m (1+ δ )3( β +3 α ( − δ ))( − δ ) (cid:17) (cid:19) − (62) η V = a − − βα (cid:16) − a − δ ρ m β + 18 a − βα (cid:16) − C α + a βα C β (cid:17) ( β + 3 α ( − δ ))( − δ ) (cid:17)(cid:16) a C (6 α − β )( β + 3 α ( − δ )) − a βα +3 δ ρ m α (1 + δ ) (cid:17) × (cid:16) α β (cid:16) − a − βα C + a − δ ρ m β +3 α ( − δ )) (cid:17) ( β + 3 α ( − δ )) ( − δ ) (cid:16) C + a − βα (cid:16) C − C αβ (cid:17) − a − δ ρ m (1+ δ )3( β +3 α ( − δ ))( − δ ) (cid:17)(cid:17) − (63)Relations (56)- (58),we have derived conditions for 0 < ǫ H < η H ≪
1. Hence, using Eq.(61) we can constrain C as follows: C ≪ a ( βα + 3 δ − ρ mo α (3 δ − α − β )( β + 3 α ( − δ )) (64)The presence of promordial tensor fluctuations is predicted by many inflationary models. As withscalar fluctuations, tensor fluctuations are expected to follow a power law and are parametrised bythe tensor index. We consider the tensor to scalar ratio [43] Tensor to scalar ratio: r = 16 ǫ [1 − ( C E + 1) ǫ ] [1 − (2 C E + 1) ǫ + C E ηH ] (65)1 × - × - × - × - n s r FIG. 11: Evolution of reconstructed tensor-to-scalar ratio r in case of viscous interacting DE as scalar field.We have considered ρ m = 0 . γ E = 0 . δ = 0 . α = − . β = 0 . C = 0 . C = − . α = − . β = 0 . C = 0 . C = − . α = − . β = 0 . C = 0 . C = − . where, C E = 4[ ln γ E ] − γ E = 0 . η depends upon second derivative of the potentialand is having the form η = V ′′ ( φ ) V ( φ ) − (cid:20) V ′ ( φ ) V ( φ ) (cid:21) . The scalar spectral index can now be expressed interms of other slow roll parameters: ǫ = − ˙ HH , ψ = ǫ − ˙ ǫ Hǫ as follows: n s = 1 − ǫ + 2 ψ (66)In Fig. 11 we have plotted the tensor-to-scalar (Eq.(65))ratio predicted by this model as a functionof scalar-spectral index n s (Eq.(66)). It may be noted that the spectral index n s is obtained byusing the reconstructed H in the slow roll parameters ǫ and ψ . It is observed that the trajectoriesin the n s − r plane exhibit a decreasing behaviour, which is consistent with the observation of Jawadet al.[12]. It is also observed that r < .
168 (95% CL, Plank TT + LowP ) the observational boundfound by Plank. Hence, the tensor-to-scalar ratio for this model is consistent with the observationalbound due to Plank. Hence this model can explain the primordial fluctuation in the early universe.Now, we attempt to examine the availability of quasi exponential expansion for the viscousinteracting scenario under consideration. In view of that we compute the effective EoS parameterfor inflation and also investigate the behaviour of 2 V − ˙ φ . Using the equation of motion, the slowroll parameters can be written as [47].2 FIG. 12: Evolution of reconstructed effective EoS parameter in case of viscous interacting DE as scalarfield. We have considered ρ m = 0 .
32 , α = 0 . β = − . C = 0 . C = 0 . ǫ = 32 ˙ φ [ 12 ˙ φ + V ( φ )] − (67) ψ = − ¨ φH ˙ φ (68)Also , the effective EoS parameter for the inflation is [48] w φ = − ǫ (69)Using the expressions of V ( φ ) and ˙ φ in Eq.(69),we obtained the expression of w φ as follows: w φ = − − β (cid:16) a βα +3 δ ρ m − a C ( β + 3 α ( − δ )) (cid:17) ( − δ ) a βα +3 δ ρ m β + 9 a (cid:16) C α − a βα C β (cid:17) ( β + 3 α ( − δ ))( − δ ) (70)and obtained 2 V − ˙ φ V − ˙ φ = − a − − βα C (cid:0) a C (3 α − β )( β + 3 α ( − δ ))( − δ ) − a βα C β ( β + 3 α ( − δ ))( − δ )+ a βα +3 δ ρ m βδ (cid:17) × ( β ( − δ )) − (71)Therefore, 2 V − ˙ φ > δ in the range 0 . ≤ δ ≤ . FIG. 13: Evolution of reconstructed 2 v − ˙ φ in case of viscous interacting DE as scalar field. We consider ρ m = 0 . α = − . β = 0 . C = 0 . C = 0 . For the entire range, 2 V − ˙ φ >
0, the difference goes higher with lowering in the value of δ . Hence, it can be interpreted that quasi exponential expansion is available for the interactingviscous holographic dark energy with higher order derivative. Furthermore, it is also observed thatincrease in the strength of interaction lowers the rate of expansion in presence of bulk viscosity.¨ φ + (3 H + Γ) ˙ φ + dVdφ = 0 (72)Γ represents the inflation decay rate or dissipative coefficient which is responsible for the decay ofthe scalar field into radiation during the inflationary expansion. We calculate Γ from the followingequation: Γ = h a ( βα ) (cid:16) − a − α + β α C (cid:16) a C β ( β + 3 α ( − δ )) + a βα +3 δ ρ m α ( − δ ) (cid:17)(cid:18)r − a − δ ρ m β + 18 a − βα (cid:16) − C α + a βα C β (cid:17) ( β + 3 α ( − δ ))( − δ ) (cid:19)(cid:18) α q a βα +3 δ ρ m − a C ( β + 3 α ( − δ )) p β ( β + 3 α ( − δ ))( − δ ) (cid:19) − + (cid:16) a (cid:16) − a − δ ρ m β + 18 a − βα (cid:16) − C α + a βα C β (cid:17) ( β + 3 α ( − δ ))( − δ ) (cid:17)(cid:16) − a C (6 α − β )( β + 3 α ( − δ )) + a βα +3 δ ρ m α (1 + δ ) (cid:17)(cid:17)(cid:16) αβ (cid:16) a βα +3 δ ρ m − a C ( β + 3 α ( − δ )) (cid:17) ( β + 3 α ( − δ ))( − δ ) (cid:17) − (cid:21) × (cid:18) C q a βα +3 δ ρ m − a C ( β + 3 α ( − δ )) (cid:19) − (73)Evolution of Γ is observed in Fig.14. We observed that | Γ | >
1, this implies that decay of scalar4
FIG. 14: Evolution of reconstructed Γ in case of viscous interacting DE as scalar field. We consider ρ m = 0 . α = − . β = 0 . C = 0 . C = − . field into radiation during inflationary phase i.e., warm inflation. | Γ3 H | measures the relativestrength of thermal damping compared to an expansion damping. In warm inflation, | Γ3 H | << | Γ3 H |≥
1, thenit is strong dissipative regime. The evolution of | Γ3 H | aginst n s is observed in Fig.15. We observethat | Γ3 H |≥
1, hence it is a strong dissipative regime.
IV. CONCLUDING REMARKS
In the work reported above, we aimed at reconstructing ρ Λ through H in non-interacting andinteracting scenario and holographic background evolution. The bulk viscous pressure has beentaken as Π = − Hξ , where ξ = ξ + ξ H + ξ ( ˙ H + H ). In the reconstruction scheme reportedhere, firstly we choose viscous scenario neglecting the contribution of dark matter and without anychoice of scale-factor. Considering the DE density mentioned in Eq.(1), we have reconstructed H through the First Friedmann’s equation. As H = ˙ aa , we derived a solution for the scale factor (seeEq.(6))and also reconstructed DE density (see Eq.(7)). We then got reconstructed bulk viscouspressure Π in Eq. 8 and then plotted | Π | (see Fig. 1) and found that the effect of bulk viscosityis decreasing with expansion of the universe. We then got EoS and from there we can see w Λ is quintessence,cosmological constant or phantom accordingly as t ⋚ ( β − α ) ξ C − ξ − (2 β +3 C ξ αβξ α − (2 α − β ) ξ β ) C and C ξ ξ = 4 α − αβ and we got deceleration parameter as q = − β α . If β a > - - - - (cid:1)(cid:2) n s Γ (cid:4) H FIG. 15: Evolution of reconsructed | Γ3 H | in case of viscous interacting DE as scalar field. We consider ρ m = 0 . δ = 0 . α = − . β = 0 . C = 0 . C = − . α = − . β = 0 . C = 0 . C = − .
06 and α = − . β = 0 . C = 0 . C = − .
08 respectively. quintessence and if β a <
0, then it is phantom.Next, we choose viscous scenario neglecting the contribution of dark matter and with choice ofscale-factor, then we got reconstructed Hubble parameter H (see Eq.(11)), Bulk viscous pressureΠ (see Eq.(14)), density ρ Λ (see Eq. (12)), w Λ (see Eq.(15)) and w eff (see Eq. (16)). Plotted | Π | (see Fig.2) and found that the effect of bulk viscosity is decreasing with expansion of the universe.We also derived deceleration parameter, q = − n (see Eq.(17)) and the state finder parameter: r = ( − n )( − n ) n (see Eq. (20)) and s = n (see Eq. (21)). As n is always positive it implies that w eff > −
1, so it is always quintessence. If n > δ = 0 and ρ m = ρ m a − and we get reconstructed H (see Eq.(25)) andthen plotted graph of the reconstructed H against z (see Fig. (3)) and the plot shows the universeis expanding. We also get reconstructed bulk viscous pressure Π (see Eq.(31)), ξ (see Eq.(30)),density ρ Λ (see Eq.(29)), w Λ (see Eq. (32)) and squared speed of sound v s (see Eq.(33)). We plot v s against z (see Fig.5)and the plot is feasible. As v s > w Λ against z (see Fig.(4)) and we observe that w Λ ≤ − ξ , ξ and ξ have been computed6for the current universe ( z = 0) and presented in TableI and TableII for non-interacting andinteracting scenario respectively. Comparing the values with the values of EoS by the observationalscheme Plank + WP+ BAO i.e. -1.13 +0 . − . , we observe that the reconstructed EoS parameter p isconsistent with the observation in both the cases [54].Next, we choose presence of dark matter in interacting scenario in presence of bulk-viscosity.So,we take ρ m = ρ m a − − δ ) , we get reconstructed H (see Eq.(37)). We plotted H against z (seeFig. (6))and the plot shows that universe is expanding. We then derived reconstructed density ρ Λ (see Eq.(40)) bulk viscous pressure Π (see Eq.(42)), ξ (see Eq.(41)), w Λ (see Eq.(43)) and squaredspeed of sound v s (see Eq.(44)). We plotted v s against z (see Fig.8) and this plot is feasible.As v s > w Λ against z (see Fig.(7)) andwe observe that w Λ ≤ − ǫ V (see Eq.(62)), η V (see Eq.(63)), ǫ H (see Eq.(55)) and plotted ǫ H against z (see Fig.9) and wecan see from the plot that the universe is expanding and calculated η H (see Eq.(61)) and plotted η H against z (see Fig.10). From the Fig.9 and Fig.10 we see that the Hubble slow roll parameters.Therefore, this model ǫ H and η H are increasing respectively. Therefore, this model has the scopeof exit from inflation. We calculated tensor to scalar ratio ’ r ’(see Eq.(65)) and plotted r against z (see Fig.11). It is observed that r < .
168 which is consistent with the data by Plank Satellite.Hence it can explain the primordial fluctuation in the early universe. We calculated 2 V − ˙ φ (see Eq.(71)) and plotted 2 V − ˙ φ against z (see Fig.13). We have observed that 2 V − ˙ φ > w φ (see Eq.(70)) and plotted w φ (seeFig.12).We observed that w φ > −
1, hence it is quintessence. We calculated Γ (see Eq.(73)) andplotted | Γ | vs z (see Fig.14). We observed that | Γ | >
1, this implies that it is warm inflation.Plotting | Γ3 H | against n s (see Fig.15), we observed that | Γ3 H | >
1, hence it is strong dissipativeregime.
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