Dynamical system analysis and thermal evolution of the causal dissipative model
Jerin Mohan N D, Krishna P B, Athira Sasidharan, Titus K Mathew
DDynamical system analysis and thermal evolutionof the causal dissipative model
Jerin Mohan N D, Krishna P B, Athira Sasidharan and Titus K MathewDepartment of Physics, Cochin University of Science and Technology,Kochi-22, [email protected];[email protected],[email protected],[email protected]
Abstract
The dynamical system behaviour and thermal evolution of a homogeneous andisotropic dissipative universe are analyzed. The dissipation is driven by the bulk vis-cosity ξ = αρ s and the evolution of bulk viscous pressure is described using the fullcausal Israel-Stewart theory. We find that for s = 1 / S (cid:48)(cid:48) < s (cid:54) = 1 / , the case s < / s > / s > / Astronomical observations ([1, 2, 3, 4, 5, 6, 7]) have shown that the current universe isexpanding at an accelerating rate. The most successful model which explains this recent1 a r X i v : . [ g r- q c ] M a r cceleration is the ΛCDM, which assumes the cosmological constant Λ , with equation ofstate ω Λ = − z T ∼ . , which shows thefeasibility of describing a late accelerating universe. The current status of the viscousmodels are described in the review [48].In the present work, our aim is two fold. Firstly to perform a dynamical system analysisof dissipative model of the late universe and secondly to study the thermodynamic evo-lution of the model based on the IS theory. In both analyses we choose the viscosity as ξ ∝ ρ s , where the parameter take values s = 1 / s (cid:54) = 1 / . The first method is aimedat finding the critical points of the autonomous differential equations which are obtainedfrom the Friedmann equations consistent with the conservation conditions. The sign andproperties of the eigenvalues corresponding to these critical points will then determine theasymptotic stability of the model. Our analysis show that, there exists an unstable criticalpoint corresponds to prior decelerated universe and an asymptotically stable critical pointcorresponding to a future accelerating epoch for s = 1 / . We also explore the status of theenergy conditions, both the strong and dominant energy conditions, to check the physicalfeasibility of the solutions corresponding to the respective critical points. Further, we anal-yses the thermal evolution of the model where we check the status of the generalized secondlaw (GSL) and the convexity condition, S (cid:48)(cid:48) < , where S is the entropy and the prime de-3otes a derivative with respect to a suitable cosmic variable. In this context we found thatthe end stage in this model is thermodynamically stable with an upper bound for entropywhen s = 1 / , which indicates that our universe behaves like an ordinary macroscopic sys-tem [49]. Authors in reference [50] have analyses the viscous model following Israel-Stewartapproach, by considering an ansatz for the Hubble parameter and with varying barotropicequation of state and have shown, in contrary, that the end stage violates the convexitycondition. However for s (cid:54) = 1 / s = 1 / s (cid:54) = 1 / s = 1 / We consider a flat FLRW universe with viscous matter as the cosmic component. The basicequations governing the evolution of the universe are,3 H = ρ m , (1)˙ H = − H −
16 ( ρ m + 3 P eff ) , (2)where H = ˙ aa is the Hubble parameter with a is the scale factor, ρ m is the matter densityand P eff = p + Π , (3)is the effective pressure, p = ( γ − ρ is the normal kinetic pressure with γ as the barotropicindex and Π is the bulk viscous pressure. The evolution of the density of the viscous fluidsatisfies the conservation equation,˙ ρ m + 3 H ( ρ m + P eff ) = 0 . (4)In the full causal IS theory, the evolution of the viscous pressure is given by, τ ˙Π + Π = − ξH − τ Π (cid:32) H + ˙ ττ − ˙ ξξ − ˙ TT (cid:33) , (5)4here τ , ξ and T are the relaxation time, bulk viscosity and temperature respectively andare generally functions of the density of the fluid, defined by the following equations [51], τ = αρ s − , ξ = αρ s , T = βρ r , (6)Here α , β and s are all positive constant parameters and r = γ − γ . For τ = 0, the differentialequation (5) reduces to the simple Eckart equation, Π = − ξH. Friedmann equation (1)can be combined with (4) and (3) to express the bulk viscous pressure Π as,Π = − (cid:104) H + 3 H + ( γ − ρ (cid:105) . (7)Following this, the bulk viscosity evolution in (5) can be expressed as,¨ H + 32 [1 + (1 − γ )] H ˙ H + 3 − s α − H − s ˙ H − (1 + r ) H − ˙ H + 94 ( γ − H + 12 3 − s α − γH − s = 0 . (8)For γ = 1 corresponding to non-relativistic matter and taking s = [52], the above equationadmits solution [47] of the form, H = H (cid:0) C a − m + C a − m (cid:1) , (9)where H is the present Hubble parameter and the other constants are [47], C = ± √ α ∓ √ α ˜Π √ α , (10) m √ α (cid:16) √ α + 1 ∓ (cid:112) α (cid:17) . (11)Here ˜Π = Π H is the dimensionless bulk viscous pressure parameter, with Π as the presentvalue of Π . The model parameters up to 1 σ level were estimated by contrasting the modelwith the supernovae data [47] and are given in table 1. We find m = 0 . , m = 5 . . Since m < m > , the expansion rate will be dominated by a − m in the early epoch, whilethe term a − m dominates in the late epoch. Hence in the limit a → , the decelerationparameter q, becomes q = − − ˙ H/H → − m > , which implies a prior decelerated5 α ˜Π χ min χ d.o.f. .
29 0 . +0 . − . − . +0 . − . .
29 1 . χ minimum value inthe bulk viscous matter dominated universe using the full IS theory as per the earlier work[47]. We have used the Supernovae data.expansion phase. But in the limit a → ∞ , it turn out that q → − m < , whichimplies a late accelerating phase of expansion and therefore the model predicts a transitioninto the late accelerating epoch. However, since m is a positive quantity, the decelerationparameter will general be greater than − , but owing to the smallness of m it can approacha value near to − total ∼ Ω darkmatter . From (9) the matter density parameter Ω m is obtained asΩ m = ρ m ρ critical = H H = ( C a − m + C a − m ) . (12)The matter density parameter in the present time Ω m , corresponding to a = 1 and is,Ω m = ( C + C ) = 1 . (13) We will now consider the dynamical system analysis [53] of the model. For this we definethe following dimensionless variables,Ω = ρ m H , ˜Π = Π3 H , and H ( t ) dt = d ˜ τ , (14)6here the last relation is equivalent to a new time variable. The (2), (4) and the IS equation(5) can then be re-written as, H (cid:48) = − H (cid:20) (cid:21) , (15)Ω (cid:48) = (Ω − , (16)and ˜Π (cid:48) = − − ˜Π (cid:34) (cid:32) (cid:33) + H − s α (3Ω) s − − Ω − − (cid:35) , (17)where the (cid:48) prime (cid:48) denotes a derivative with respect to the new variable ˜ τ . Since H is alwayspositive for an expanding flat universe, the above equations are well defined. The abovethree dynamical equations constitute the evolution of the system in a phase space describedby the variables ( H, Ω , Π) . We are considering a universe with single component, the viscousmatter, implying that Ω = 1 . Then the phase space becomes two dimensional with variables( H, ˜Π) . The critical parameter in studying the evolution is s. We have found exact solutionsfor s = 1 / s (cid:54) = 1 / s = 1 / For this choice (15) and (17) decouple from each other and as a result the phase spacewill effectively reduces to one dimension and (17) represents the evolution of this singledimensional phase space. More over in the present case, since Ω = 1 , (17) can be expressedin a much simpler form in terms of the equation of state, ω = ˜Π / Ω as, ω (cid:48) = 32 ( ω − ω + )( ω − ω − ) , (18)where ω ± = 1 √ α [1 ± (cid:112) α ] (19)are the fixed points. The equation (19) implies that, ω + > ω − < α > . Theearly phase corresponding to ω + is decelerating. If α is sufficiently large then ω − < − / α the equation of state can assume values accordingly. For a range √ ≤ α ≤ √ the equation of state vary between to − / > ω ≥ − . So an asymptoticde Sitter epoch ( ω → −
1) is possible only if α assumes the upper limit value around √ . For the best estimated value of the model parameter, we have obtained that ω + = 2 .
52 and ω − = − .
79 and are corresponding to a prior decelerated phase in which the viscous matterassumes a stiff fluid nature and a late accelerated epoch, in which the matter assumes aquintessence nature respectively. So for the case with γ = 1 (the barotropic index) and (cid:15) = 1( γ and (cid:15) appears in the general equation of relaxation time (51) given in the Appendix) thelate universe with bulk viscous matter can be accelerating but it will not approach a purede Sitter epoch like the standard ΛCDM.The deceleration parameter corresponding to the equilibrium points can be obtainedusing the relation 1 + q = (1 + ω ) , through which we arrive at q + ∼ .
28 and q − ∼ − . . Taking account of these facts, it is possible to re-write the general solution given in (9) as, H = H (cid:16) C a − (1+ q − ) + C a − (1+ q + ) (cid:17) . (20)Using this the transition from the decelerated to the current accelerated phase of expansioncan easily be explained. The transition redshift z T . can be obtained using (20) as, z T = (cid:18) − C q + C q − (cid:19) − q + − q − ∼ . , (21)where the numerical value is corresponding to the best estimated values of the model pa-rameters and is found to be in the WMAP range z T = (0 . − .
73) [54].Without knowing the analytical solution, it is possible to analyze the cosmic evolutionfrom (18) in a transparent way by drawing the phase diagram of ω, namely plotting ω (cid:48) versus ω. Since the phase space is one dimensional, we interpret (18) as a vectorfield ona single line [55]. The evolution of ω is represented by the direction of the change of ω along the axis and is determined by the sign of ω (cid:48) . A small variation in ω is expressed as δω = ω (cid:48) δ ˜ τ , so that for ω flows towards the increasing direction of ˜ τ (right) if ω (cid:48) > ω (cid:48) < . Perturbations in the ω space aroundthe critical point, ω c ( ω + or ω − ) propagates with a rate dd ˜ τ ( δω ) = ω (cid:48) = f ( ω ) = f ( ω c + δω ) . (22)8he Taylor series expansion around ω c can be written as, f ( ω c + δω ) = f ( ω c ) + δωf (cid:48) ( ω c ) + O ( δω ) , where f (cid:48) ( ω c ) = ddω f ( ω ) | ω c , from which we get ddω ( δω ) = δωf (cid:48) ( ω c ) . By linearising δω about the critical point ω c we get, δω (˜ τ ) ∝ e f (cid:48) ( ω )˜ τ . (23)The above equation tells us that the stability of critical points is determined by the slope, f (cid:48) ( ω ) . If f (cid:48) ( ω c ) > , then any small disturbance around the critical point grow exponentiallyand hence it becomes unstable (repeller). On the other hand, if f (cid:48) ( ω c ) < , all smalldisturbances around critical point decay exponentially and it will be a stable one(attractor).The critical point will be semi stable, if the slope f (cid:48) ( ω c ) changes its sign at the critical point.The slope corresponding to (18) can be obtained as, f (cid:48) ( ω ) = 32 (cid:2) ω − ω + − ω − (cid:3) . (24)For the best estimated values of the model parameters, it is clear that the condition, ω − <ω < ω + is always be satisfied. Then, at the critical points ω ± the slope will satisfy theconditions, f (cid:48) ( ω + ) = 32 (cid:2) ω + − ω − (cid:3) > f or α > , (25) f (cid:48) ( ω − ) = 32 (cid:2) ω − − ω + (cid:3) < f or α > , (26)which indicates that ω + is an unstable fixed point while ω − is a stable fixed point. Hencethe universe will evolves from an unstable decelerated epoch to the stable accelerated epoch.So in effect we get a qualitative description of the behaviour of the cosmological evolutionwithout relying on the exact solution. The phase portrait is shown in figure 1.The exact solutions corresponding to the fixed points ω ± follows from (15) are, H ω ± = 1(1 + q ± ) t , a ω ± = a t q ± ) . (27)For ω + we have q + ) < , indicating a decelerating solution, while for ω − we have q − ) > ρ ω ± = 3(1 + q ± ) t , ˜Π ω ± = 3 ω ± (1 + q ± ) t . (28)9 ' ΩΩ (cid:43) Ω (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Figure 1: The one dimensional phase portrait of evolution of ω (cid:48) versus ω in the bulk viscousmatter dominated universe using the full causal IS theory for best estimated value of themodel parameter, when s = 1 / . For ω + the pressure ˜Π > ω − it becomes negative, ˜Π < , implying the generationof negative pressure in the late acceleration epoch.It is essential to know the status of the energy conditions [56] which characterize thefeasibility of different solutions. The strong energy condition (SEC) implies that ρ +3 P eff ≥
0. The violation of SEC indicates an accelerating expansion of the universe. The dominantenergy condition (DEC) implies that ρ + P eff ≥ . A violation of DEC causes the breakdownof the generalised second law of thermodynamics in normal case. However, if there occurdissipative effects in the cosmic fluid, the GSL can still be satisfied even when DEC isviolated [57]. In term of equation of state SEC and DEC can translated as, 1 + 3 ω ≥ ω ≥ ω + . In the case of the fixed point ω − , SEC is violated, since it represents an accelerating solution, but DEC is satisfied as it is aphysically feasible epoch. All these facts are summarized in table 2.A similar case of the non-validity of the strong energy condition for a future accelerating10ritical points → ω + ω − ω . − . q . − . ω + and ω − in the dynamic system ofthe bulk viscous model using the full causal Israel-Stewart theory for the best estimatedvalues of the model parameters, when s = 1 / s (cid:54) = 1 / Unlike in the case of s = 1 / s (cid:54) = 1 / . However, we can get a qualitative description of the evolution by extracting theinformation from the equilibrium points. In this case we have a two dimensional space( H, Π) . For simplicity we define a variable. h = H − s . (29)The dynamical equations (15) and (17) then become, h (cid:48) = −
32 (1 − s )(1 + ω ) h, (30) ω (cid:48) = − (cid:34) ω (cid:32) ( − s ) α h − ω (cid:33)(cid:35) . (31)having two critical points, P : h = 0 , ω = √ , (32)11 : h = 3 s α , ω = − . (33)For s < / P with h = 0 implies a static universe with the Hubbleparameter H = 0 . The critical point P corresponds to a de Sitter epoch at which theHubble parameter is a non-zero constant. Since the first phase P is a static one it will notimply any further evolution. Hence the case s < / s > / , the fixed point P is representing a prior decelerated epoch with infinitely large Hubble parameter and P corresponds to a late de Sitter epoch. However the equation of state corresponding to theprior decelerated epoch is greater than one, implying that the matter is of stiff nature atthis epoch. We will restrict to the case s > / P , and hence itcorresponds to physically feasible decelerating epoch. The fixed point P , satisfies DEC butviolates SEC as it is corresponding to an accelerating epoch.To determine the stability property of the critical points, we first linearize (30) and (31)about the critical points and obtain the Jacobian matrix as, J ( h, ω ) = (cid:34) − (1 − s ) h − (1 − s )(1 + ω ) − (cid:16) − s hα − ω (cid:17) − − s ωα (cid:35) (34)Diagonalising the Jacobian matrix, we obtain the eigenvalues λ ± = 3 − s √ α (cid:20) − ± (cid:113) s ( √ s − α (cid:21) , (35) λ +2 = 3 − s α , λ − = 3 s s − α, (36)for P and P respectively. Here we restrict α to the range 0 < α < . The fixed point P is a saddle one, since the eigenvalues are, λ +1 > , λ − < , while P is found to be unstable,since its eigenvalues are both positive, λ +2 > , λ − > . The saddle nature of the earlydecelerated phase implies that the system will continue the evolution further. For the sakeof completeness, it may be noted that, for s < / , the fixed point P , which correspondsto a static universe, is found to be stable since λ +1 < λ − < P is a saddle pointas the eigenvalues satisfies, λ +2 > λ − < . All these facts are summarised in table 3.12ritical points → P P q . − s > / s < / P and P in the dynamic system ofthe bulk viscous model using the full causal Israel-Stewart theory, when s (cid:54) = 1 / . The fixed point P corresponds to a solution given by, a = a e ¯ H t , (37)where ¯ H = (cid:0) s α (cid:1) − s with s > / . This is a de Sitter type solution, ensuring acceleratedexpansion. It is not possible to get any corresponding exact solution for P as the Hubbleparameter in this case is infinity. Even though an exact solution for P is impossible, anapproximate solution can be obtained. For this, first express the (30) in terms of H andthen through a simple integration we arrive at, H ∼ e − (1+ q ) τ , (38)from which it is evident that as τ → −∞ , H → ∞ . Integrating the above equation bychanging the variable from τ to t using (14), we get the scale factor as a ∼ t q ) and thecorresponding pressures is ˜Π ∼ ω (1+ q ) t . This section is devoted to the analysis of the evolution of entropy. Viscosity can causeentropy generation and the local entropy thus generated can be obtained as [20], T ∇ ν S ν = ξ ( ∇ ν u ν ) = 9 H ξ, (39)13here T is the temperature and ∇ ν S ν is the rate of generation of entropy in unit vol-ume. According to second law of thermodynamics, the entropy must always increase, i.e. T ∇ ν S ν ≥ , which implies that ξ ≥ . For s = 1 / , from (6) and (9), it follows ξ = √ αH. Since both, α and H are always positive definite in the present case the local second lawwill be satisfied. Then it is easy to conclude that the local second law will be satisfied atthe critical points ω + and ω − since they are the critical points corresponding to the case s = 1 / . As there are no analytical solutions for s (cid:54) = 1 / , it is impossible to make a similaranalysis.Now we turn to the more general aspects of the entropy evolution, namely the statusof the generalised second law (GSL) and the behaviour of the second order derivative ofentropy. An ordinary macroscopic system evolving towards a state of stable thermodynamicequilibrium must satisfy the conditions, S (cid:48) ≥ , and S (cid:48)(cid:48) < , at least in the long run (40)where (cid:48) prime (cid:48) denotes a derivative with respect to suitable cosmological variable like cosmictime or scale factor. The first condition refers to the GSL and the second one is the convexitycondition implying an upper bound to the growth of entropy. In reference [49], the authorshave shown that our universe seems to behave like an ordinary macroscopic system whichobeys the above conditions. The consideration of the entropy evolution in the standardΛCDM model also supports this [59].According to GSL, the total entropy must always increase, i.e., S (cid:48) = S (cid:48) m + S (cid:48) h ≥ , (41)where S m and S h are the matter entropy and horizon entropy respectively and the (cid:48) prime (cid:48) denotes the derivative with respect to scale factor. The entropy of the Hubble horizon isdefined as [60], S h = A l p k B = πc l p H k B , (42)where A = 4 πc /H is the area of the Hubble horizon of a spatially flat FLRW universe, k B is the Boltzmann constant, l p is the Planck length and c is the velocity of light. We havethe derivative of the horizon entropy with respect to the scale factor as, S (cid:48) h = − πc H (cid:48) l p H k B . (43)14 ' a (cid:180) (cid:180) (cid:180) (cid:180) (cid:180) (cid:180) Figure 2: The evolution of S (cid:48) in units of k B with scale factor a in the bulk viscous matterdominated universe using the full causal IS theory for best estimated values of the modelparameters, when s = 1 / S (cid:48) m can be obtained from the Gibb’s relation, T m S (cid:48) m = E (cid:48) + P eff V (cid:48) , (44)where T m is the temperature of the viscous matter, E = ρ m V is its total energy and V = πc H is the volume enclosed by the Hubble horizon. Using the Friedmann equation and assumingthermal equilibrium so that T m = T h , where T h = H ¯ h π k B the Hawking temperature of thehorizon, we get S (cid:48) m = − c H (cid:48) GH qT h . (45)Adding (43) and (45), we get the rate of change of total entropy as, S (cid:48) = − πc H (cid:48) l p H ( q + 1) . (46)For s = 1 / , it is evident from general solution (9) that H (cid:48) < q ) > S (cid:48) is shown in figure 2 and is such that the15rst increase occurs during the decelerated epoch and then it decreases during the accel-erated epoch. The figure 2 shows that the slope of the curve changes drastically aroundthe transition redshift. The maximum of S (cid:48) corresponds to the transition from decelerationto acceleration epoch. It is then quite natural to expect that GSL will be satisfied at thecorresponding critical points, ω + and ω − . The Hubble parameter corresponding to thesefixed points is H ω ± = q ± ) a q ± , implying that H (cid:48) ω ± < s > / , (we restrict to this case, since as noted earlier H = 0 for the critical point corresponding to s < /
2) we change the variable from scalefactor to newly defined time, ˜ τ .
Following (38) satisfied by P we can rewrite the entropyderivative in (46) as, dSd ˜ τ = 2 πc l p (1 + q ) e q )˜ τ , (47)and is always greater than zero. Hence GSL is satisfied at P . The validity of GSL at P isstraight forward since it represents a de Sitter epoch at which the Hubble parameter is aconstant implying S (cid:48) = 0 . Now will check the status of the convexity condition of entropy, S (cid:48)(cid:48) < , in this model.This condition should be satisfied at least in the final stage of the evolution for the maximi-sation of entropy [49]. Taking the derivative of S (cid:48) in (46) with respect to the scale factor,we get S (cid:48)(cid:48) = − πc l p (cid:20) H (cid:48) H q (cid:48) + ( q + 1) (cid:18) H (cid:48)(cid:48) H (cid:48) − H (cid:48) H (cid:19)(cid:21) . (48)For s = 1 / S (cid:48)(cid:48) can be obtained by substituting the Hubble parameterfrom (9). The net result is plotted in figure 3. It shows that S (cid:48)(cid:48) > S (cid:48)(cid:48) < S (cid:48)(cid:48) changes its sign around the transition period. Hence the convexitycondition is fulfilled in the long run of the expansion of the universe. This indicates themaximisation of entropy of the universe and hence entropy is bounded. The boundedness ofthe entropy rules out the presence of any instabilities at the end stage [61]. The behaviourof S (cid:48)(cid:48) at the critical points ω + and ω − is evident from the above analysis. The fixed point ω + represents the earlier epoch and ω − represents the later epoch for s = 1 / . However, asa matter of simple academic interest, the evolution equation of S (cid:48)(cid:48) at the critical points can16 S'' (cid:45) (cid:180) (cid:45) (cid:180) (cid:180) Figure 3: The evolution of S (cid:48)(cid:48) in units of k B with scale factor a in the bulk viscous mat-ter dominated universe using the full causal IS theory for estimated values of the modelparameters, when s = 1 / . be expressed as, S (cid:48)(cid:48) ω ± = 2 πc l P (1 + 2 q ± )(1 + q ± ) a q ± . (49)From the figure 4, it is clear that convexity condition is violated at the critical point ω + but satisfied at ω − as expected. This indicates that the first critical point ω + is an unstablethermodynamic equilibrium and the second point ω − is thermodynamically stable.Fro s > / P can be obtainedusing (38) and (48) as, d Sd ˜ τ = 4 πc l p (1 + q ) e q ) τ . (50)For P , which represents the deceleration parameter q > , the term d S P d ˜ τ > , and theconvexity condition is hence violated. At the equilibrium point P , representing a de Sitterepoch, we observe that the S (cid:48)(cid:48) will vanish hence the convexity condition is not strictlysatisfied. These results are summarised in table 4.17 '' S Ω (cid:43) (cid:162)(cid:162) a S Ω (cid:45) (cid:162)(cid:162) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:180) (cid:180) (cid:180) Figure 4: The evolution of S (cid:48)(cid:48) in units of k B with scale factor a at the critical points ω ± for the best estimated value of the model parameters, when s = 1 / ↓ LSL GSL S (cid:48)(cid:48) < ω + Yes Yes No ω − Yes Yes Yes P Yes Yes No P Yes Yes NoTable 4: Thermal properties of the critical points ω + , ω − , P and P of the model. In the ta-ble LSL and GSL denote the local second law and generalized second law of thermodynamicsrespectively and S (cid:48)(cid:48) < s = 1 / p = ωρ, with ω varying in the range 0 < ω < , has been analyses in reference [50]. Theauthors assumed an ansatz for Hubble parameter of the form, H ( t > t s ) = | A | / ( t − t s ) , where | A | is a positive coefficient depending on ω and the viscous coefficient ξ. The IS transportequation will then give rise to a quadratic equation for | A | , with two possible solutions,say | A | + and | A | − . By considering only the solution corresponding to | A | + , the authorshave argued that the GSL is satisfied in both the prior decelerated and later acceleratedphases, but the convexity condition is satisfied by the early phase but violated in the lateraccelerated epoch. In contrast to this, the analytical solutions that we have obtained forthe IS equation with s = 1 / ω = 0 , predicts an earlydecelerated epoch which satisfies GSL but violate convexity condition and a late acceleratedphase which satisfies both GSL and convexity conditions. In this work we have analysed the dynamical system behaviour and thermodynamic char-acteristics of the late universe with a dissipative fluid using the full Israel-Stewart theory.Assuming the bulk viscosity as ξ = αρ s , we consider two separate cases one with s = 1 / s (cid:54) = 1 / . For s = 1 / ω + and ω − corresponding to an early decelerated and late accelerating phases respectively. It emergesfrom our analysis that ω + is a past attractor hence unstable while the late acceleratingepoch corresponding to ω − is a stable one. We have seen that the effective equation ofstate indicates a stiff nature for the viscous matter in the neighbourhood of the fixed point ω + . At ω − , corresponding to the the late accelerating phase the equation of state become ω ∼ − . , implying a quintessence nature but not pure de Sitter. Regarding energyconditions, it is easy to see that both fixed points satisfy the dominant energy condition,but the strong energy condition is satisfied only by ω + as a consequence of its deceleratingnature. When s = 1 / , a general behaviour of the bulk viscous model has analyzed using ageneral relaxation time expression (51), by varying (cid:15) and barotropic index γ. For the bestestimated parameter values, the models exhibits the quintessence evolution, however thelate phase stabilizing values of equation state is close to − . s (cid:54) = 1 / . When s < / s < / s > / h = H − s and ω and having two critical points, out of which the first one, P represents a decelerated epoch and the second one P indicating the de Sitter epoch. Ouranalysis on stability shows that, P is a saddle point and P is a repeller, hence unstable.Hence a stable evolution towards an end de Sitter epoch is unlikely for s > / . In theenergy condition analysis, for the case s > / , we found that both SEC and DEC aresatisfied at P , which is corresponding to a prior decelerated phase of expansion. In thecase of P , DEC is satisfied while SEC is violated as is representing a late accelerated epoch.In the analysis of the thermodynamic characteristics, we have shown that, for s = 1 / S (cid:48) ≥ S (cid:48)(cid:48) < ω + and ω − but convexity condition is satisfied only by the later critical point ω − . This indicatesthat the expansion is tending towards a state of a maximum entropy as in the evolution ofan ordinary macroscopic system.For s (cid:54) = 1 / s > / . The GSL is validat both the critical points in this case. Among these we already noted that P representsa prior decelerated epoch and P corresponds the future de Sitter epoch. Regarding theconvexity condition, our result is that, it is violated at both the fixed points prohibiting anupper bound for the growth of entropy. Hence the case s > / s = 1 / , the present dissipative model described using theIsrael-Stewart theory predicts a stable evolution of the late universe with prior deceleratedepoch followed by an accelerated epoch. The GSL is valid throughout the evolution andthe entropy is bounded for the end phase with the convexity condition satisfied. We canalso infer that the thermal properties of the bulk viscous universe, especially the entropy,exhibits drastic change during the phase transition period. For the choice s (cid:54) = 1 / , thecase with s < / s > / − Pa sec (1 σ level) in the dark matter sectorcan cure the σ − Ω m tension ( σ is the r.m.s. fluctuations of perturbations at 8 h − Mpcscale) and the H − Ω m tension occurred when one analyse the Planck CMB parametersusing the standard ΛCDM model. Acknowledgments
We are also thankful to IUCAA, Pune for the hospitality during the visits. We are alsothankful to the referees for the comments, which helped to improve the manuscript.Authorsare grateful to Prof. M. Sabir for the careful reading of the manuscript. Author JMNDacknowledges UGC - BSR for the fellowship, author KPB acknowledges KSCSTE, Govern-ment of Kerala for financial assistance and author AS is thankful to DST for fellowshipthrough the INSPIRE fellowship.
AppendixThe possibilities of attaining a pure de Sitter epoch for s = 1 / ω = − . . Now we check possibilities of improving this value so that themodel can predict a pure de Sitter epoch with ω = − τ = αρ s − with s = 1 / . Since this doesn’t gives an asymptoticde Sitter epoch, let us relax this condition by assuming a more general relation for therelaxation time as [62], τ = α(cid:15)γ (2 − γ ) ρ s − . (51)We first fix the barotropic index as, γ = 1 and allow to vary the parameter (cid:15) in the range0 < (cid:15) ≤ . The solution for the Hubble parameter is obtained in [63], which have the same form as inthe previous case, H = H ( C a − m + C a − m ) , (52)21ut with different coefficients, C = ± (cid:15) + √ (cid:15)α + (cid:15) ∓ √ α ˜Π √ (cid:15)α + (cid:15) , (53) m = √ α (cid:16) √ α + (cid:15) ∓ (cid:112) (cid:15)α + (cid:15) (cid:17) , (54)which are satisfying the conditions C + C = 1 and m > m . Using the Supernovae type Iadata we have extracted the parameter values in the present case as α = 169 . , ˜Π = − . ,(cid:15) = 0 .
39 and H = 69 .
99 with χ d.o.f. = 0 . . We have then concentrated on the evolutionof the equation of state which can be analytically obtained as, ω = − C m a − m + C m a − m )3( C a − m + C a − m ) . (55)To get the late phase behaviour, consider the asymptotic limit of equation of state parameter(55) when the scale factor a → ∞ . In the late phase evolution, the equation of stateparameter (55) can takes the form, ω ∼ − m . (56)For the new best estimated parameter values, the constant, m = 0 . , and hence theequation of state parameter will stabilizes around ω ∼ − . . So the values has beenimproved slightly but still not represent a pure de Sitter case.As a further move we extend the analysis by varying the parameter γ also. By consid-ering this, a more general solution for the Hubble parameter can be obtained as discussedin [64] as, H = C (1 + z ) α (cid:48) cosh γ [ β ( ln (1 + z ) + C )] , (57)where C = H (cid:20) − ( q + 1 − α (cid:48) ) γ β (cid:21) γ/ ,C = 1 β arctanh (cid:20) ( q + 1) − α (cid:48) γβ (cid:21) ,α (cid:48) = √ γ ξ (cid:104) √ ξ + (cid:15)γ (2 − γ ) (cid:105) , = √ ξ (cid:113) ξ (cid:15) (2 − γ ) + (cid:15) γ (2 − γ ) . where q is the present value of deceleration parameter and ξ is the viscosity constantparameter ( ξ = α in our analysis). Following reference [64], the model parameters take thevalues as ξ = 245 . , (cid:15) = 0 .
601 and γ = 1 .
26 with χ d.o.f. = 1 .
07 for H = 70 km/M pcs and q = − . . Using the equation of parameter evaluating equation [47], we have obtainedthe the asymptotic limit of equation of state parameter for the estimated parameter values,when a → ∞ , the equation of state ω ∼ − . , and is very close to the de Sitter epochvalue. Therefore, we have conclude that, even though the model will not attain the pure deSitter epoch ( ω = −
1) as the end phase, it attains a quintessence epoch which very close tothe de Sitter phase.
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