Exact solutions of a Flat Full Causal Bulk viscous FRW cosmological model through factorization O. Cornejo-P´erez Facultad de Ingenier´ıa, Universidad Auto´noma de Quer´etaro, Centro Universitario Cerro de las Campanas, 76010 Santiago de Quer´etaro, Mexico J. A. Belincho´n Departamento de F´ısica At´omica, Molecular y Nuclear. Universidad Complutense de Madrid, E-28040 Madrid, Espan˜a (Date text: January 25, 2013) We study the classical flat full causal bulk viscous FRW cosmological model through the factor- izationmethod. Themethodshowsthatthereexistsarelationshipbetweentheviscosityparameter sand theparameter γ enteringtheequationsof stateof themodel. Also, thefactorization method 3 allows to find some new exact parametric solutions for different values of the viscous parameter s. 1 Special attention is given tothewell known case s=1/2, for which thecosmological model admits 0 scalingsymmetries. Furthermore,someexactparametricsolutionsfors=1/2areobtainedthrough 2 theLie group method. Keywords: Exact solutions, Full Causal Bulk viscosity, factorization method, Lie groups. n a J I. INTRODUCTION. 4 2 Factorizationoflinearsecondorderdifferentialequationsisawellestablishedmethodtofindexactsolutionsthrough ] c algebraic procedures. It was widely used in quantum mechanics and developed since Schrodinger’s works on the q factorizationofthe Sturm-Liouvilleequation. Atthe presenttime, verygoodinformativereviewsonthe factorization - method can be found in open literature (see for instance [1, 2]). However, in recent times the factorization method r g has been applied to find exact solutions of nonlinear ordinary differential equations (ODE) [3–7]. In [4], based on [ previous Berkovich’sworks [3], it has been provideda systematic way to apply the factorizationmethod to nonlinear second order ODE. In [5], Wang and Li extended the application to more complex nonlinear second and third order 2 ODE. The factorization of some ODE may be restricted due to constraints which appear in a natural way within v 8 the factorization procedure. However, here it is shown that by performing transformation of coordinates, one can 3 be able to get exact parametric solutions of an ODE which does not allow its factorization or presents cumbersome 9 constraints. 5 Thepurposeofthepresentworkistoapplythefactorizationmethodtostudythefullcausalbulkviscouscosmolog- . 4 ical model with flat FRW symmetries. Since the Misner [8] suggestionstressingthe fact that the observedlarge scale 0 isotropy of the Universe may be due to the action of the neutrino viscosity when the Universe was about one second 2 old, there have been numerous works pointing out the importance of the physical processes involving viscous effects 1 in the evolution of the Universe (see for instance [9]). Due to such assumption, dissipative processes are supposed to : v play a fundamental role in the evolution of the early Universe. i Thetheoryofrelativisticdissipativefluids,createdbyEckart[10]andLandauandLifshitz[11]hasmanydrawbacks, X and it is known that it is incorrect in several respects mainly those concerning causality and stability. Israel [12] r a formulates a new theory in order to solve these drawbacks. This theory was latter developed by Israel and Stewart [13] into what is called transient or extended irreversible thermodynamics. The best currently available theory for analyzing dissipative processes in the Universe is the full causal thermodynamics developed by Israel and Stewart [13], Hiscock and Lindblom [14] and Hiscock and Salmonson [15]. The full causal bulk viscous thermodynamics has been extensively used to study the evolution of the early Universe and some astrophysicalprocess [16, 17]. The paper is organized as follows. In Section II, we start by reviewing the main components of a flat bulk viscous FRW cosmological model, and introduce the factorization technique as applied to the cosmological model. Field equations (FE) of the classical bulk viscous FRW cosmological model [17] reduce to a single nonlinear second order ODE,thefundamentaldynamicalequationfortheHubblerate. Byperformingatransformationofboththedependent andindependentvariablesandusingthefactorizationmethod,thisequationistransformedintoanonlinearfirstorder ODE.TheorderreductionoftheequationfortheHubblerateallowstofindavarietyofnewexactparametricsolutions of the FE for the viscous FRW cosmological model. Furthermore, the factorization technique provides relationships for parameters entering the factorized equation. Then, a noteworthy result is that the viscosity parameter s is not longerassumedtobeindependentofthevaluesofparameterγ. Suchparameterrelationshipshavenotbeenpreviously reported. In Section III, severalparticularmodels for s=1/2 arestudied. We obtainnew exactparametricsolutions 6 through factorization and compare with the ones obtained by several authors [18–25] who use different approaches. SectionIVisdevotedtothespecialcases=1/2,forwhichthemodeladmitsscalingsymmetries. Thescalingsolution, 2 previously studied by many authors is obtained. In order to obtain more new solutions and compare the solutions obtainedthroughfactorizationfors=1/2,weconsiderthe Lie groupmethod forthis specialcaseinSectionV.Some conclusions end up the paper in Section VI. II. THE MODEL. We consider a flat FRW Universe with line element ds2 = dt2+f2(t) dx2+dy2+dz2 , (1) − where the energy-momentum tensor of a bulk viscous cosm(cid:0)ological fluid is g(cid:1)iven by [17]: Tk =(ρ+p+Π)u uk+(p+Π)δk, (2) i i i where ρ is the energy density, p the thermodynamic pressure, Π the bulk viscous pressure and u the four-velocity i satisfying the condition u ui = 1. We use the units 8πG = c = 1. The gravitational field equations together with i − the continuity equation, Tk =0, are given as follows i;k 2H˙ +3H2 = p Π, (3) − − 3H2 =ρ, (4) 1 τ˙ ξ˙ T˙ Π+τΠ˙ = 3ξH τΠ 3H + , (5) − − 2 τ − ξ − T! ρ˙ = 3(γρ+Π)H, (6) − where H =f˙/f. In order to close the system of equations we are assuming the following equations of state [17] p=(γ 1)ρ, ξ =αρs, T =βρr, τ =ξρ 1 =αρs 1, (7) − − − whereT isthetemperature,ξthebulkviscositycoefficientandτ therelaxationtime. Theparameterssatisfyγ [1,2], ∈ s 0, and r = 1 1 . The growth of entropy has the following behavior ≥ − γ (cid:16) (cid:17) t Σ(t)≈−3kB−1 ΠHf3T−1dt. (8) Zt0 The Israel-Stewart-Hiscock theory is derived under the assumption that the thermodynamical state of the fluid is close to equilibrium, i.e., the non-equilibrium bulk viscous pressure should be small when compared to the local equilibrium pressure Π <<p =(γ 1)ρ. Then, we may define the l(t) parameter as: l = Π/p. If this condition is | | − | | violatedthenoneiseffectivelyassumingthatthelineartheoryalsoholdsinthenonlinearregimefarfromequilibrium. For a fluid description of the matter, the condition ought to be satisfied. Toseeifacosmologicalmodelinflatesornotitisconvenienttointroducethedecelerationparameterq =dH 1/dt − − 1. The positivesignofthe decelerationparametercorrespondsto standarddeceleratingmodels,whereasthe negative sign indicates inflation. The fundamental dynamical equation for the Hubble rate is given by [17] H˙2 H¨ A + 3H+CH2 2s H˙ +DH3+EH4 2s =0, (9) − − − H (cid:0) (cid:1) where 1 9 1 A=(1+r)=2 , B =3, C =31−s, D = (γ 2), E = 32−sγ. (10) − γ 4 − 2 Let us perform the following transformation of the dependent and independent variables H =y1/2, dη =y1/2dt, (11) then Eq. (9) turns into d2y A dy 2+ 3+Cy21−s dy +2y(D+Ey12−s)=0. (12) dη2 − 2y dη dη (cid:18) (cid:19) (cid:16) (cid:17) 3 Let us consider now the following factorization scheme [4, 5]. The nonlinear second order equation y +f(y)y2+g(y)y +h(y)=0, (13) ′′ ′ ′ where y = dy =D y, can be factorized in the form ′ dη η [Dη φ1(y)y′ φ2(y)][Dη φ3(y)]y =0, (14) − − − under the conditions f(y)= φ , (15) 1 − dφ 3 g(y)=φ φ y φ φ y, (16) 1 3 2 3 − − − dy h(y)=φ φ y. (17) 2 3 If we assume [D φ (y)]y =Ω(y), then the factorized Eq. (14) can be rewritten as η 3 − y′ φ3y =Ω, (18) − Ω (φ y +φ )Ω=0. (19) ′ 1 ′ 2 − We can introduce the functions φi by comparing Eqs. (12) and (13). Then, φ1 = 2Ay, φ2 = a−11 and φ3 = 2a1(D+Ey12−q), where a1(=0) is an arbitrary constant, are proposed. 6 Eq. (19) can be easily solved for the chosen factorizing functions obtaining as result Ω=κ eη/a1yA/2, where κ is 1 1 an integration constant. Then, Eq. (18) turns into the equation y′ 2a1 D+Ey12−s y κ1eη/a1yA/2 =0, (20) − − (cid:16) (cid:17) whose solution is also solution of Eq. (12). Furthermore, the following relationship is obtained from Eq. (16), Aa1D−a−11−2a1D+a1E(A−3+2s)y21−s =3+Cy12−s. (21) Eq. (21) is a noteworthy result which provides the explicit form of a and the relationship among the parameters 1 entering Eq. (12). Then, the viscous parameter s as a function of parameter γ is obtained. By comparing both sides of Eq. (21) and assuming r =1 1, leads to obtain: − γ √2+γ3/2 s(γ) = ± . (22) 2γ3/2 ± Then, s [0,.25] γ [1.2599,2],and s (.75,1.2071068] γ [1,2). Also, the explicit form of a is + 1 − ∈ ∀ ∈ ∈ ∀ ∈ 2γ1/2 a(γ) = . (23) 1± ±3(√2 γ1/2) ∓ Then, a [ 1/3, .29499] γ [1.2599,2],and a [1.60947, ) γ [1,2). 1 1+ We fin−d t∈he−follow−ing signifi∀cat∈ive values ∈ ∞ ∀ ∈ γ s a s a 1 + 1+ − − 1 1.2071 1.6095 4 4.0721 10 2 0.29966 0.95928 2.9663 3 × − − 2 1 1 4 −3 The main difference of these results from other approaches is expressed through Eq. (22), which represents an advantage of the factorization method as opposed to different approaches studied by other authors. This equation provides the relationship between the parameters s and γ in such a way that by fixing s we get a particular value of γ. 4 The main dynamical variables of the FE are given in parametric form as follows f(η)=f exp(η η ), (24) 0 0 − H(η)=y1/2(η), (25) d 1 q(η)=y1/2(η) 1, (26) dη H(η) − (cid:18) (cid:19) ρ(η)=3y(η), (27) p(η)=3(γ 1)y(η), (28) − dy Π(η)= 3γy(η)+ , (29) − dη (cid:18) (cid:19) Π l(η)= | |, (30) p Σ(η)= 3k Π(η)f3(η)H(η)T(η) 1y(η) 1/2dη. (31) B − − − Z TheauthorshavenotbeenabletofindthemostgeneralsolutionofEq. (20). However,thisequationcanbestudied for some specific cases providing particular solutions of physical interest. In Sections III and IV, the cosmological solutions as obtained for the viscosity parameter s=1/2 and s=1/2 are studied. 6 III. SOLUTION WITH s6=1/2. In this section, some particular cases of Eq. (20) for s=1/2 are studied to obtain exact particular solutions of FE 6 (3)-(6). By setting κ =0, Eq. (20) simplifies as 1 y′ 2a1 D+Ey21−s y =0, (32) − (cid:16) (cid:17) whose solution is given by 2/(2s 1) E − y(η)= κ ea1D(2s 1)η , (33) 2 − − D (cid:18) (cid:19) where κ is an integration constant. Therefore, the parametric form of the time function is obtained from Eq. (11) 2 as follows t(η)= y 1/2(η)dη = κ ea1D(2s 1)η E 1/(1−2s)dη. (34) − 2 − − D Z Z (cid:18) (cid:19) A. Case s=0. The first special case considered corresponds to s = 0, which means that the bulk viscosity coefficient ξ = const. The following particular solution is obtained 2 E − 1 y(η)= κ e a1Dη , and t(η)= κ e ηDa1 +ηEa . (35) 2 − 2 − 1 − D −Da (cid:18) (cid:19) 1 (cid:0) (cid:1) Eqs. (22) and (23) provide the corresponding constant parameters a = 0.295 and γ = √32, respectively. A 1 particular equation of state is obtained once again through Eq. (22). In F−igs. −1 and 2, the behavior of the FE main quantities for different values of constant κ is plotted. 2 As we can see, the solution for κ = 1 is non-singular since ρ(0)=const. For κ = 2 and κ = 3, the energy 2 2 2 − − − densityhasasingularbehaviorwhent=0,sinceitrunstoinfinity whentimetendstozero,i.e., ρ(0) . Thebulk viscosity, Π, is negative for all values of t, i.e., Π(t) < 0 t R+, which is a thermodynamically con→sist∞ent result as ∀ ∈ expected for κ = 1. For κ = 2 and κ = 3, the solution is valid only when t>t , i.e., Π(t)<0 t>t , while 2 2 2 0 0 − − − ∀ Π(t 0)>0. Then, for this interval of time, t (0,t ), the solution has no physical meaning. The entropy behaves 0 → ∈ like a strictly growingtime function; then, there are a large amountof comovingentropy during the expansion of the 5 10 1 100 0.5 8 80 0 L 6 L-0.5 L 60 ΡHt 4 PHt-1.-51 SHt 40 2 -2 20 0 -2.5 0 0 2 4 6 8 10 2 4 6 8 10 0 2 4 6 8 10 t t t FIG.1: Solution withs=0.Plots ofenergydensityρ(t),bulkviscosity Π(t)andentropyΣ(t). Dashedlineforκ2 =−1.Solid line for κ2 =−2. Long dashed line for κ2 =−3. 1 6 5 0.5 4 Lt 0 Lt3 Hq Hl2 -0.5 1 0 -1 0 2 4 6 8 10 12 14 2 4 6 8 10 t t FIG. 2: Solution with s = 0. Plots of the deceleration parameter q(t) and parameter l(t). Plots of energy density ρ(t), bulk viscosity Π(t) and entropy Σ(t). Dashed line for κ2 =−1. Solid line for κ2 =−2. Long dashed line for κ2 =−3. universe. Thedecelerationparameterrunsfromq(0)= 0.5toq(t)= 1. Then,thesolutionisaccelerating,i.e.,itis − − inflationary. Thedecelerationparametertendsto 1ast (acceleratingsolutions)butshowsasingularbehavior − →∞ when time runs to zero. The parameter l(t) shows that all the plotted solutions are far from equilibrium since they are inflationary solutions, which is a consistent result. To the best of our knowledge this solution is new. B. Case s=1/4. The second case consideredcorresponds to s=1/4. In this case, Eqs. (22) and (23) provide a = 1/3 and γ =2. 1 − Therefore, Eq. (20) simplifies as y′+2(3)3/4y5/4 κ1e−3ηy3/4 =0. (36) − If we perform the transformation z =y1/4 in Eq. (36), then we get the Riccati equation 33/4 1 z + z2 κ e 3η =0, (37) ′ 1 − 2 − 4 whose general solution is given in terms of Bessel J and Neumman N functions, n n J (ξ(η))+κ N (ξ(η)) 1 2 1 z(η)= ξ(η) , (38) − J (ξ(η))+κ N (ξ(η)) 0 2 0 where ξ(η)= √2κ1e 3η/2 and κ is an integration constant. Therefore, the following special solution for Eq. (36) is 233/8 − 2 obtained: · J1(ξ(η))+κ2N1(ξ(η)) 4 η J1(ξ(η))+κ2N1(ξ(η)) −2 y(η)= ξ(η) , t(η)= ξ(η) dη. (39) J (ξ(η))+κ N (ξ(η)) J (ξ(η))+κ N (ξ(η)) (cid:18) 0 2 0 (cid:19) Z (cid:18) 0 2 0 (cid:19) In order to study the behavior of the FE dynamical variables in their parametric form, the calculation of Eq. (39) has been numerically addressed. The solution depends strongly on the value of the numerical constants, in such a way that our solutionis physicalonly for κ <0 andfor negativeand relatively smallvalues (<20)of κ . Numerical 2 1 analysis of the solution plotted in Fig. 3 shows that the solution is singular since the energy density tends to infinity 6 whent 0.The bulkviscosityispositive,Π>0,intheregion(0,t )sothe solutionhasphysicalmeaningonlywhen t > t , →for this era Π becomes negative as expected from the therm∗odynamical point of view and tending to zero in the la∗rge time limit. In the same interval of time (0,t ) the entropy production is negative, Σ(t) < 0 (unphysical situation), nevertheless when t > t , a large amount of∗comoving entropy is produced during the expansion of the universe. ∗ 3 1 20 2.5 0.75 15 2 0.5 1.5 0.25 10 L L L t 1 t 0 t H H H Ρ 0.5 P-0.25 S 5 0 -0.5 0 -0.5 -0.75 2 4 6 8 10 2 4 6 8 10 12 14 2.5 5 7.5 10 12.5 15 17.5 20 t t t FIG. 3: Solution with s =1/4 and γ = 2. Plots of energy density ρ(t), bulk viscosity Π(t) and entropy Σ(t). Dashed line for κ1 =4, κ2 =−10. Solid line for κ1 =19, κ2 =−3. Long dashed line for κ1 =3, κ2 =−0.7. Regarding the dynamical behavior of solution (39), in Fig. 4 the behavior of parameters q and l has been plotted. As we can see, the deceleration parameter shows that the universe starts in a non-inflationary phase, but quickly entering a inflationary one since q <0. The plots of l(t) are consistent with this behavior, showing that the solution starts in a thermodynamicalequilibrium but in a finite time they are far from equilibrium since they are inflationary solutions. 2 3 1.5 2.5 1 2 L L Ht 0.5 Ht1.5 q l 0 1 -0.5 0.5 10 20 30 40 50 2.5 5 7.5 10 12.5 15 17.5 20 t t FIG. 4: Solution with s=1/4 and γ =2. Plots of thedeceleration parameter q(t) and parameter l(t). Dashed line for κ1 =4, κ2 =−10. Solid line for κ1 =19, κ2 =−3. Long dashed line for κ1 =3, κ2 =−0.7. AsimilarsolutionhasbeenobtainedbyMaketal[25]but,aswehaveshown,oursolutionisqualitativelydifferent, with a very different physical meaning. 1. A particular solution for the case s=1/4. If we set κ =0 in Eq. (36), then we get the very simple ODE 1 y +2(3)3/4y5/4 =0, (40) ′ whose solution is given as 4 y(η)= (3)23/4η+κ2 − , and t(η)= 14√3η3+ 21343η2κ2+ηκ22, (41) ! where κ is an integration constant. In Figs. 5 and 6 the behavior of the FE main quantities has been plotted for 2 different values of the constant κ . 2 7 5 2 300 4 1.5 250 1 L3 L 0.5 L200 t t 0 t150 H H H Ρ2 P-0.5 S100 -1 1 50 -1.5 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 0 5 10 15 20 25 30 t t t FIG. 5: Particular solution for s = 1/4. Plots of energy density ρ(t), bulk viscosity Π(t) and entropy Σ(t). Dashed line for κ2 =0. Long dashed line for κ2 =1. Solid line for κ2 =2. 1 2 0.75 1.5 0.5 0.25 1 L L t 0 t 0.5 H H q-0.25 l 0 -0.5 -0.75 -0.5 2 4 6 8 10 0 1 2 3 4 5 t t FIG. 6: Particular solution for s=1/4. Plots of the deceleration parameter q(t) and parameter l(t). Dashed line for κ2 =0. Long dashed line for κ2 =1. Solid line for κ2 =2. The solution has been plotted for three different values of constant κ . The energy density presents a singular 2 behavior only for κ = 0, while the other two solutions show a non-singular behavior when t = 0. The solution for 2 κ = 2 runs quickly to zero. The bulk viscosity is always a negative time function for κ = 1 and κ = 2, but the 2 2 2 solution for κ = 0 is valid only for t > t since Π(t 0) > 0, which means that it lacks of physical meaning in 2 0 → the interval of time t (0,t ). The entropy always behaves like a growing time function but for the case κ =0 the 0 2 ∈ universe starts with a non-vanishing entropy, i.e., Σ(0) = const., while for the other two solutions Σ(0) 0. The → plots in Fig. 5 show that a large amount of entropy is produced during the expansion of the universe. Regarding the decelerationparameter,theplottedsolutionsruntoanaccelerationregionsinceq(t) 1/2inafinitetime. Forthis →− reason, the solution starts in an equilibrium regimen but quickly run to a non-equilibrium state as shown by plots of l(t). A particular solution of this case has been studied by Harko et al [22] obtaining different behavior of the FE main quantities. C. Case s=1. The secondimportant case consideredcorrespondsto s=1. According to Eqs. (22) and (23), this solutionis valid only for the equation of state with γ = √32 ≈ 1.25992. Other authors have already studied similar cases for s = 1, but with different equation of state (see for instance [23] with γ =2) obtaining different results. Then, according to Eqs. (33) and (34), the following particular parametric solution is obtained: 2 E 1 E y(η)= κ ea1Dη , and t(η)= ln +eηDa1 ηDa . (42) 2 1 − D Ea −Dk − (cid:18) (cid:19) 1 (cid:20) (cid:18) 2 (cid:19) (cid:21) Then, the FE main dynamical variables can be explicitly obtained through Eqs. (24)-(31). InFigs. 7and8,the behaviorofthe mainquantities bygivingdifferent valuesto the constantκ hasbeen plotted. 2 As we can see the solution is valid only for t > t . The energy density is a decreasing function, but the function 0 behaves like a constant for a t>t . The behavior of the bulk viscous parameter shows that the solution is valid only c for t > t since the solution is positive when t 0, decreasing and going to a negative constant value during the 0 → cosmological evolution, which is consistent from the thermodynamical point of view. In the same way, the entropy behaves like a growing function only for t > t , showing that a large amount of comoving entropy is produced. 0 Nevertheless, the deceleration parameter shows that the universe starts in a non-inflationary phase, but quickly entering a inflationary one since q 1 κ . The plots of l(t) show that plotted solutions are far from equilibrium 2 → − ∀ since they are inflationary solutions. 8 44 10 1000 42 0 800 40 -10 ΡHLt3368 PHLt-20 SHLt 460000 -30 34 32 -40 200 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 t t t FIG.7: Solutionwith s=1.Plotsof energydensityρ(t),bulkviscosityΠ(t)andentropyΣ(t). Dashedlineforκ2 =0.1. Long dashed line for κ2 =10. Solid line for κ2 =100. 1 7 6 0.5 5 4 L L t 0 t 3 H H q l 2 -0.5 1 0 -1 0.1 0.2 0.3 0.4 0.5 0 0.5 1 1.5 2 t t FIG.8: Solutionwiths=1.Plotsofthedecelerationparameterq(t)andparameterl(t). Dashedlineforκ2 =0.1.Longdashed line for κ2 =10. Solid line for κ2 =100. IV. SOLUTION WITH s=1/2. Weconsidernowtheveryspecialcases=1/2. Thishasbeenthemostimportantandstudiedcase(seeforexample example[18],[19],[20],[21])within the frameworkofthe bulk viscouscosmologicalmodels since,asithas beenpointed out for several authors, this solution is stable from the dynamical systems point of view [26] as well as from the renormalizationgroup approach [27]. In this case, Eq. (9) reduces to: H˙2 H¨ A +(3+C )HH˙ +(D +E )H3 =0, (43) 1 1 1 1 − H where A = (1+r) = 2 1, C = √3, D = 9(γ 2), E = 3√3γ, and r = 1 1/γ. Since the coordinate 1 − γ 1 1 4 − 1 2 − transformation given by Eq. (11) leads to obtain several unphysical solutions for s = 1/2, we perform the more suitable change of variables given as follows (see also [18]), 1 H =y1/2, dη =3 1+ Hdt. (44) √3 (cid:18) (cid:19) Then, Eq. (43) turns into A y 1y2+y +2γby=0, (45) ′′ ′ ′ − 2y where γb = √3(γ+6) 3. Eq. (45) can be solved by factorization providing new exact parametric solutions for 8 − 2 s=1/2. Eq. (45) admits the factorization A D− 2yy′−a−11 [D−2a1γb]y =0, (46) (cid:20) (cid:21) 9 which can be rewritten in the form y 2a γby =Ω, (47) ′ 1 − A Ω′− 2yy′−a−11 Ω=0, (48) (cid:18) (cid:19) or equivalently, y 2a γby k eη/a1yA/2 =0, (49) ′ 1 1 − − where k is an integration constant, with solution given as 1 y(η)=e2a1γbη a1k1e(a−11−a1b)η +C 2γ, (50) 2γ(1 a2b) 1 − 1 ! where C is an integration constant, and the parameter a is restricted to values given by 1 1 4√3γ γ2 72√3 60 9γ3+γ 432√3 756 a = ± − − − , (51) 1± − q (cid:0) 3 γ 4(cid:1)√3+6 (cid:0) (cid:1) − (cid:0) (cid:1) i.e. , a [ 64.31, 8.38], and a [ 0.23, 0.01]. In the following Subsections IV.A and IV.B several possible 1+ 1 cases of in∈ter−est are s−tudied. − ∈ − − A. General solution. In this case it is possible to find a explicit parametric equation for t (from Eq. (50)) with C = 0. It is given as 1 6 follows γ t(η)= √3−3 a1(cid:18)1+ C1exp(cid:16)2aηa11γk(12γ−a1B)(cid:17)(cid:19) F −2γ2 ,γ,1 2γ2 , C1exp 2aη1γ (2γ−a1B) . (cid:0) (cid:1) 6γy1/2 2 1a B 2γ − a B 2γ − (cid:16) a k (cid:17) 1 1 1 1 − −  (52) To the best of our knowledge the solution given by Eqs. (50) and (52) has not been previously reported. The FE main dynamical variables are given in parametric form as follows f(η)=f exp(η η ), (53) 0 0 − H(η)=y1/2(η), (54) (2+B)C +2(a +γ)k exp η (2γ a B) 1 1 1 2a1γ − 1 q(η)= , (55) − 2 C +a k exp η (2(cid:16)γ a B) (cid:17) 1 1 1 2a1γ − 1 ρ(η)=3y(η), (cid:16) (cid:16) (cid:17)(cid:17) (56) p(η)=3(γ 1)y(η), (57) − C (B+3γ)+(2+3a )γk exp η (2γ a B) 1 1 1 2a1γ − 1 Π(η)= y(η), (58) C +a k exp η (2γ(cid:16) a B) (cid:17) 1 1 1 2a1γ − 1 Σ(η)=γe3η(3y)1/γ, (cid:16) (cid:17) (59) Π(η) l(η)= . (60) p(η) (cid:12) (cid:12) (cid:12) (cid:12) In Figs. 9 and 10, the behavior of (cid:12)the FE(cid:12)main quantities has been plotted. The following constantvalues have been (cid:12) (cid:12) chosen: a as given in Eq. (51) while B = √3a1 (γ+6) 3a , k = 2, C = 1, and γ = 1,4/3,2 as usual. The 1+ 4 − 1 1 1 − solutions with a are unphysical. 1 − 10 20 0 140 120 15 -5 100 HLt10 HLt-10 HLt 6800 Ρ P S 5 -15 40 20 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 2.5 5 7.51012.51517.520 t t t FIG. 9: Solution for s = 1/2. Plots of energy density ρ(t), bulk viscosity Π(t) and entropy Σ(t). Solid line for γ = 2. Long dashed line for γ =4/3. Dashed line for γ =1. 1 5 4 0.5 3 L L t 0 t 2 H H q l 1 -0.5 0 -1 5 10 15 20 25 30 0 5 10 15 20 25 30 t t FIG. 10: Solution for s=1/2. Plots of the deceleration parameter q(t) and parameter l(t). Solid line for γ =2. Long dashed line for γ =4/3. Dashed line for γ =1. Theenergydensityshowsasingularbehaviorast 0,butinafinitetimeitbehavesasadecreasingtimefunction. → Thissolutionisvalidforallvaluesoftime andγ. The bulkviscouspressure,Π,isanegativedecreasingtime function during the cosmological evolution, Π < 0 t R+, as it is expected from a thermodynamical point of view. The ∀ ∈ viscouspressurealsoevolvesfromasingularerabutitquicklytendstozero,i.e.,inthelargelimittheviscouspressure vanishes as the viscous coefficient, which also becomes negligible small. The comoving entropy behaves as a growing time function. There exists a fast growth of entropy for γ = 4/3, while for γ = 1 the entropy grows slowly. The entropy evolves from a non-singular state, i.e., Σ(0) = 0, but it quickly grows in such a way that a large amount of entropy is produced during the cosmological evolution. The picture of parameter q(t) shows that all the plotted solutions start in a non-inflationary phase, but they quickly run to an inflationary era since this quantity runs to 1 − for all the equations of state. For this reason, the parameter l(t) shows that the solutions are far from equilibrium since they are inflationary solutions. B. Particular solution In the case, it is possible to find a particular solution for t from Eq. (50) with C = 0. For this case, the solution 1 simplifies as follows 2γ a k exp η (2γ a B) y(η)=exp(Bη) 1 1 2a1γ − 1 , and t(η)= √3 3 a1y 1/2. (61)  (cid:16)2γ a B (cid:17) − 6γ − 1 − (cid:16) (cid:17)  