We discuss the field equations in Saez-Ballester scalar-tensor theory of gravitation for a Bianchi type-V model filled with viscous fluid together with heat flow. We obtain two classes of cosmological solutions by applying a special law of variation of Hubble\u27s parameter which yields a constant value of deceleration parameter. One class of solutions corresponds to a model of universe which evolves from a big-bang type singularity at t=0 and expands with power-law expansion. The other class of solutions represents an universe expanding exponentially having singularity in the infinite past. The physical and kinematical features of the models are discussed. We observe that the models of the universe in two types of cosmologies are compatible with the results of recent observations.Razmatramo jednadžbe polja u Saez-Ballesterovoj skalarno-tenzorskoj teoriji gravitacije za Bianchijev model tipa V s viskoznom tekućinom i tokom topline. Postigli smo dvije vrste kozmoloških rješenja primjenom posebne relacije za promjene Hubbleovog parametra koje vode na stalnu vrijednost parametra usporavanja. Jedna vrsta rješenja odgovara modelu svemira koji se razvija iz singularnosti tipa velikog praska u t = 0 i širi se prema potencijalnom zakonu. Druga vrsta rješenja predstavlja svemir koji se širi eksponencijalno a singularnost mu je u beskonačnoj prošlosti. Raspravljaju se fizičke i kinematičke osebine oba modela. Nalazimo da su ova dva modela svemira u skladu s nedavnim opažanjima

Ram, Shri

Verma, M. K.

Zeyauddin, Mohd.

English

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Printed ISSN 1330–0016Online ISSN 1333–9133CD ISSN 1333–8390CODEN FIZBE7ANISOTROPIC VISCOUS FLUID COSMOLOGICAL MODELS WITH HEATFLOW IN SAEZ-BALLESTER THEORY OF GRAVITATIONSHRI RAMa,1, M. K. VERMAb and MOHD. ZEYAUDDINDepartment of Applied Mathematics, Institute of Technology, Banaras Hindu University,Varanasi 2210 05, IndiaE-mail addresses: asrmathitbhu@rediffmail.com, bmanojvermaitbhu@gmail.comReceived 11 June 2009; Accepted 16 November 2009Online 7 December 2009We discuss the field equations in Saez-Ballester scalar-tensor theory of gravitationfor a Bianchi type-V model filled with viscous fluid together with heat flow. Weobtain two classes of cosmological solutions by applying a special law of variation ofHubble’s parameter which yields a constant value of deceleration parameter. Oneclass of solutions corresponds to a model of universe which evolves from a big-bangtype singularity at t = 0 and expands with power-law expansion. The other class ofsolutions represents an universe expanding exponentially having singularity in theinfinite past. The physical and kinematical features of the models are discussed. Weobserve that the models of the universe in two types of cosmologies are compatiblewith the results of recent observations.PACS numbers: 98.80.Es; 98.80.-k UDC 524.83, 52.423Keywords: Bianchi V, cosmology, Hubble’s parameter, viscous fluid1. IntroductionThe Friedmann-Robertson-Walker (FRW) models are only globally acceptableperfect fluid space-time which are spatially homogeneous and isotropic. The ad-equacy of isotropic cosmological models for describing the present state of theuniverse is no basis for expecting that they are equally suitable for describing theearly stages of evolution of the universe. At the early stages of the evolution of theuniverse, when radiation in the form of photons as well as neutrino decoupled, thematter behaved like a viscous fluid. Since viscosity counteracts the gravitational1Corresponding authorFIZIKA B 18 (2009) 4, 207–218 207ram al.: anisotropic viscous fluid cosmological models with heat flow in . . .collapse, a different picture of the initial stage of the universe may appear due todissipative processes caused by viscosity.Misner [1,2] studied the effect of viscosity on the evolution in the cosmologicalmodels and has suggested that the strong dissipation due to the neutrino viscositymay considerably reduce the anisotropy of the black-body radiation. Murphy [3]obtained an exact cosmological model of zero-curvature of FRW type in the pres-ence of bulk viscosity alone which exhibits an interesting feature that the big-bangsingularity appears in the infinite past. Roy and Tiwari [4] presented some planesymmetric solutions to Einsteins’s field equations representing inhomogeneous cos-mological models with viscous fluid and constant bulk viscosity. Szydlowski andHeller [5] constructed models of the universe filled with interacting matter and ra-diation including bulk viscosity dissipation. Mohanty and Pradhan [6] obtained aclass of exact non-static solution in a closed elliptic Robertson-walker space-timefilled with viscous fluid in the presence of attractive scalar fields.Belinski and Khalatnikov [7], while investigating a Bianchi type I cosmologi-cal model with the influence of viscosity, have found the important property thatnear the initial singularity the gravitational field creates matter. Banerjee et al.[8] obtained some Bianchi type I solutions in the case of stiff matter by using theassumption that shear viscosity coefficient are power-law functions of the energydensity. Huang [9] presented exact solution for a Bianchi type I cosmological modelwith viscosity without using shear viscosity. Goener and Kowaleski [10] developed amethod for obtaining irrotational anisotropic viscous fluid solutions of Bianchi typeI with barotropic equation of state. Banerjee and Sanyal [11] presented an irrota-tional Bianchi type V model under the influence of both shear and bulk viscositytogether with heat flow. Coley [12], Coley and Hoogan [13], while generalizing thework of Coley and Tupper [14], studied diagonal Bianchi type V imperfect fluidmodels with both viscosity and heat conduction with and without the cosmologicalterm. Recently, Singh and Chaubey [15] investigated the evolution of Bianchi typeV model with viscous fluid and cosmological constant.In this paper, we obtain two classes of exact solutions of the field equations inthe scalar-tensor theory of Saez and Ballester [16] for a Bianchi type V space-timefilled with a viscous fluid with both bulk and shear viscosities together with heatflow. In order to find solutions of the field equations, we apply a special law ofvariation of Hubble’s parameter as proposed by Berman [17]. One class of solutionsrepresents model of the universe with power-law expansion which evolves from abig-bang singularity at t = 0. The other class of solutions corresponds to an expo-nentially expanding universe having singularity in the infinite past. The physicaland kinematical behavior of models are discussed in two types of cosmologies.2. Field equations and general expressionsWe consider the spatially homogeneous Bianchi type V model in the formds2 = dt2 −A2(t)dx2 − e2mx[(B2(t)dy2 + C2(t)dz2], (1)208 FIZIKA B 18 (2009) 4, 207–218ram al.: anisotropic viscous fluid cosmological models with heat flow in . . .where A(t), B(t) and C(t) are the cosmic scale factors and m is a constant.The energy-momentum tensor Tij in a viscous fluid with heat conduction isgiven byTij = (ρ+ p̄)uiuj−p̄gij+ηδkj(ui;k + uk;i − uiukuj;k − ujukui;k)+hiuj+hjui , (2)where ρ is the energy density and p̄ the effective pressure given byp̄ = p−(ξ −23η)uk;k. (3)Here ui is the 4-velocity of the fluid, ξ and η are coefficients of bulk and shearviscosity, and hi is the heat flow vector orthogonal to ui. If we assume that theheat flow is in the x-direction only, then hi = (h1, 0, 0, 0), h1 being a function oftime.The field equations in Saez-Ballester scalar-tensor theory of gravitation areRij −12Rgij − ωφr(φ;iφ;j −12gijφ;kφ;k) = −Tij . (4)The scalar field φ satisfies the equation2φrφ;k;k + rφr−1φ;kφ;k = 0, (5)where r is an arbitrary constant and ω is a dimensionless coupling constant. Asemicolon denotes covariant derivative. We have taken 8πG = c = 1 in Eq. (4).The energy-momentum tensor Tij satisfies the conservation equationT ij;j ui = 0. (6)For the metric (1), the field equations (2) – (4), in comoving coordinates, yieldB̈B+C̈C+ḂBĊC−m2A2= −(p̄− 2ηȦA)+12ωφrφ̇2 , (7)ÄA+C̈C+ȦAĊC−m2A2= −(p̄− 2ηḂB)+12ωφrφ̇2 , (8)ÄA+B̈B+ȦḂAB−m2A2= −(p̄− 2ηĊC)+12ωφrφ̇2 , (9)ȦAḂB+ȦAĊC+ḂBĊC−3m2A2= ρ−12ωφrφ̇2 , (10)FIZIKA B 18 (2009) 4, 207–218 209ram al.: anisotropic viscous fluid cosmological models with heat flow in . . .m(2ȦA−ḂB−ĊC)= h1 , (11)where an over dot denotes differentiation with respect to time t. The conservationequation (6) givesρ̇+ (ρ+ p̄)(ȦA+ḂB+ĊC)− 2η(ȦA)2+(ḂB)2+(ĊC)2 =2mA2h1. (12)The scalars of expansion and shear are calculated asθ = ui;i =(ȦA+ḂB+ĊC), (13)2σ2 =(ȦA)2+(ḂB)2+(ĊC)2−θ23. (14)For the metric (1), the average scale factor R is defined byR = (ABC)1/3. (15)The spatial volume V is given byV = R3 = ABC. (16)The generalized mean Hubble’s parameter H is defined byH =ṘR=13(ȦA+ḂB+ĊC), (17)where H1 = Ȧ/A, H2 = Ḃ/B and H3 = Ċ/C are the directional Hubble’s param-eters in the directions of x, y and z, respectively. The deceleration parameter q isgiven byq = −RR̈Ṙ2. (18)Subtracting Eqs. (7) and (8), Eqs. (8) and (9), and Eqs. (7) and (9), and integratingthe resulting equations, we obtain the quadrature solutions for the average scalefactors A, B and C as follows:A(t) = l1R exp[X1∫exp (−2∫ηdt)R3dt], (19)210 FIZIKA B 18 (2009) 4, 207–218ram al.: anisotropic viscous fluid cosmological models with heat flow in . . .B(t) = l2R exp[X2∫exp (−2∫ηdt)R3dt], (20)C(t) = l3R exp[X3∫exp (−2∫ηdt)R3dt], (21)where constants X1, X2, X3 and l1, l2, l3 satisfy the relationsX1 +X2 +X3 = 0 and l1l2l3 = 1. (22)For the metric (1), Eq. (5) providesφ̈+ φ̇(ȦA+ḂB+ĊC)+r2φφ̇2 = 0 , (23)which, on integration yieldsφ =[h(r + 2)2∫dtR3]2/(r+2). (24)From Eqs. (7) – (10), we obtain the expressions for energy density and effectivepressure as given byρ = 3H2 − σ2 −3m2A2+12ωφrφ̇2 , (25)p̄ = H2 (2q − 1)− σ2 +m2A2+23ηθ +12ωφrφ̇2. (26)Clearly, we can find the solutions of the Eqs. (19) – (21) for the scale factors A, B,C if the shear viscosity coefficient η and R are known. Regarding η, we make thephysically valid assumption that the shear is proportional to the expansion, η ∝ θ,i.e.η = η0θ , (27)where η0 is a positive constant.In the next section, we find two explicit forms of the average scale factor R byapplying a special law of variation for Hubble’s parameter. Using the explicit formsof R, we find the solutions of Eqs. (19) – (21) in two type of cosmologies.FIZIKA B 18 (2009) 4, 207–218 211ram al.: anisotropic viscous fluid cosmological models with heat flow in . . .3. Variation law for Hubble’s parameterIn order to obtain explicit solutions of the Eqs. (19) – (21), we assume that themean Hubble’s parameter H is related to the average scale factor R by the relationH = lR−n , (28)where l and n are non-negative constants. Such type of relation has already beenused by Berman [17], Berman and Gomide [18] for solving field equations in FRWcosmologies, Singh and Kumar [19], Kumar and Singh [20], for Bianchi type Ispace-times and Singh et al. [21], Shri Ram et al. [22] for Bianchi type V models indifferent physical contexts. This relation yields a constant value of the decelerationparameter. The positive value of q corresponds to standard decelerating universe,whereas the negative value indicates inflation.Substituting Eq. (28) in Eq. (17), we obtainṘ = lR−n+1 , (29)which, on differentiation, givesR̈ = −l2(n− 1)R−2n+1. (30)From Eqs. (18), (29) and (30), we find thatq = n− 1. (31)Thus, the variation law Eq. (28) of Hubble’s parameter gives a constant value ofdeceleration parameter. For n > 1, we have a decelerating model and for 0 ≤ n < 1,we have an accelerating model of the universe.Integration of Eq. (29) providesR = (nlt+ k)1/n, n /=0 , (32)R = c exp (lt) , n = 0 , (33)where k and c are integration constants. Without loss of any generality, we cantake k = 0. Then Eq. (32) becomesR = (nlt)1/n. (34)We now use Eqs. (33) and (34) to solve Eqs. (19) – (21) for scale factors A, Band C in two types of cosmology.212 FIZIKA B 18 (2009) 4, 207–218ram al.: anisotropic viscous fluid cosmological models with heat flow in . . .3.1. Cosmological model with power-law expansion (n =/ 0)Using the power-law solution to the average scale factor R, given in Eq. (34),we obtainH = (nt)−1, (35)θ = 3 (nt)−1, (36)η = 3η0(nt)−1. (37)Substituting the value of R and η in Eqs. (19) – (21) and integrating, we obtainA(t) = l1 (nlt)1/nexp[X1l(n− 3− 6η0)(nlt)(n−3−6η0)/n], (38)B(t) = l2 (nlt)1/nexp[X2l(n− 3− 6η0)(nlt)(n−3−6η0)/n], (39)C(t) = l3 (nlt)1/nexp[X3l(n− 3− 6η0)(nlt)(n−3−6η0)/n], (40)provided n /=(3 + 6η0). The directional Hubble’s parameters H1,H2,H3 are ob-tained asH1 = (nt)−1+X1 (nlt)−(3+6η0)/n , (41)H2 = (nt)−1+X2 (nlt)−(3+6η0)/n , (42)H3 = (nt)−1+X3 (nlt)−(3+6η0)/n . (43)The shear scalar σ is given byσ2 =12(X21 +X22 +X23)(nlt)−(6+12η0)/n . (44)Eq. (24), on integration, yieldsφ =[h(r + 2)2l(n− 3)]2/(r+2)(nlt)2(n−3)/{(n(r+2)}, n /=3 . (45)Using Eqs. (38) – (40) in Eq. (11), we find thath1 = 3mX1 (nlt)−(3+6η0)/n . (46)FIZIKA B 18 (2009) 4, 207–218 213ram al.: anisotropic viscous fluid cosmological models with heat flow in . . .With these solutions, the conservation equation (12) is identically satisfied.From Eqs. (25) and (26), the energy density and the effective pressure areobtained asρ = 3(nt)−2 −12(X21 +X22 +X23)(nlt)−(6+12η0)/n−3m2l12 (nlt)−2/n exp[2X1l(3 + 6η0 − n)(nlt)(n−3−6η0)/n]+12ωh2(nlt)−6/n (47)p̄ = (2n+ 6η0 − 3)(nt)−2 −12(X21 +X22 +X23)(nlt)−(6+12η0)/n+m2l12 (nlt)−2/n exp[2X1l(3 + 6η0 − n)(nlt)(n−3−6η0)/n]+12h2ω(nlt)−6/n (48)provided n /=(3 + 6η0).If we assume that the energy density ρ and pressure p satisfy the barotropicequation of state p = γρ, 0 ≤ γ ≤ 1, then, from Eq. (3), we find thatξ =(3γ − 2n+ 3)l3(nlt)−1 +(X12 +X22 +X32)(1− γ)6l(nlt)(n−6−12η0)/n−m2(3γ + 1)3l12l(nlt)(n−2)/n exp[2X1l(3 + 6η0 − n)(nlt)(n−3−6η0)/n]+ω(γ − 1)h26l(nlt)(n−6)/n. (49)The coefficients of bulk and shear viscosities are time-dependent.For this model, the spatial volume V tends to zero as t → 0. The energy density,pressure, scalar expansion and shear scalar all assume infinite values at this epoch.Heat function is also infinite at t = 0. Thus, the model evolves from a big-bang typesingularity at t = 0 and eventually expands with power-law expansion. The scalarfunction φ, viscosity coefficients ζ and η are also infinite at t = 0. As the cosmic timeincreases, the volume and all scale factors will increase, but the expansion scalardecreases. The physical and kinematical quantities σ2, H1,H2, H3, H, ρ and p aredecreasing functions, as t increases. As t → ∞, the spatial volume becomes infiniteand other physical and kinematical quantities tend to zero. The heat function diesout for large time. Also σ2/θ tends to zero as t → ∞ provided n < 6(1 + 2η0).Thus, the model gives essentially empty universe for large time.214 FIZIKA B 18 (2009) 4, 207–218ram al.: anisotropic viscous fluid cosmological models with heat flow in . . .3.2. Cosmological model with exponential expansion (n = 0)Using the exponential form of R given in Eq. (33), we obtainH = l , (50)θ = 3l . (51)The shear viscosity coefficient η and volume scalar V are given byη = 3lη0 , (52)V = c3 exp (3lt). (53)Integrating Eqs. (19) – (21), we obtain the solutions for the metric functions asA(t) = l1c exp[lt−X13lc3(1 + 2η0)exp {−3l(1 + 2η0)t}], (54)B(t) = l2c exp[lt−X23lc3(1 + 2η0)exp {−3l(1 + 2η0)t}], (55)C(t) = l3c exp[lt−X33lc3(1 + 2η0)exp {−3l(1 + 2η0)t}]. (56)From Eq. (24), the scalar function φ has the solutionφ(t) =[h(r + 2)6l]2/(r+2)exp(−6ltr + 2). (57)The directional Hubble’s parameters H1, H2 and H3 have the valuesH1 = l +X1c3exp {−3l(1 + 2η0)t} , (58)H2 = l +X2c3exp {−3l(1 + 2η0)t} , (59)H3 = l +X3c3exp {−3l(1 + 2η0)t} . (60)The shear scalar σ has the value given byσ2 =(X21 +X22 +X23)2c6exp {−6l(1 + 2η0)t} . (61)FIZIKA B 18 (2009) 4, 207–218 215ram al.: anisotropic viscous fluid cosmological models with heat flow in . . .The heat function h1 is given byh1 =3mX1c3exp {−3l(1 + 2η0)t} . (62)The expressions for the energy density and effective pressure are obtained asρ = 3l2 −(X21 +X22 +X23)2c6exp {−6l(1 + 2η0)t}−3m2l21c2exp[2X13c3l(1 + 2η0)exp {−3l(1 + 2η0)t} − 2lt]+ωh22exp (−6lt) (63)p̄ = l2 (6η0 − 3)−(X21 +X22 +X23)2c6exp {−6l(1 + 2η0)t}+m2c2l21exp[2X13c3l(1 + 2η0)exp {−3l(1 + 2η0)t} − 2lt]+ωh22exp (−6lt). (64)By a straightforward calculation it can easily be seen that the conservation Eq. (12)is identically satisfied.If the energy density ρ and pressure p satisfy the gamma law of equation ofstate p = γρ, 0 ≤ γ ≤ 1, then the bulk viscosity coefficient ξ has the value given byξ = (γ + 1)l +(X21 +X22 +X23)(1− γ)6lc6exp {−6l(1 + 2η0)t}−m2(3γ + 1)3ll12c2exp[2X13c3l(1 + 2η0)exp {−3l(1 + 2η0)t} − 2lt]+(γ − 1)6lωh2 exp (−6lt). (65)Thus, in this model of the universe, bulk viscosity coefficient ξ is function of timeand shear viscosity coefficient is constant.The spatial volume V tends to zero, and energy density and pressure becomeinfinite as t tends to −∞. This means that the model has a singularity in the infinitepast. All physical and kinematical quantities are well behaved for −∞ < t < ∞.At t → ∞, the spatial volume becomes infinite, and energy density and pressuretend to constant values. Also σ2/θ → 0 as t → ∞. Therefore the model becomesisotropic for large t.216 FIZIKA B 18 (2009) 4, 207–218ram al.: anisotropic viscous fluid cosmological models with heat flow in . . .4. ConclusionsWe have presented two classes of exact solutions to the field equations in theframework of Saez-Ballester scalar-tensor theory for a Bianchi type V model in thepresence of a viscous fluid together with heat flow. We have applied a special law ofvariation for Hubble’s parameter to generate models of the universe in two type ofcosmologies. One class of models represents a singular universe which evolves froma big-bang singularity at t = 0 and expands with expansion rate of power-law type.The other class of models with negative deceleration parameter corresponds to ex-ponentially expanding model having singularity in the infinite past. The evolutionof the universe in such a scenario is consistent with the present observation predict-ing an accelerated expansion. We have also discussed the physical and kinematicalbehavior of the models of the universe in two types of cosmologies. These modelscould give appropriate description of the universe at its early stages of evolution.References[1] W. Misner, Nature 214 (1967) 40.[2] W. Misner, Astrophys. J. 151 (1968) 431.[3] L. Murphy, Phys. Rev. D 8 (1973) 4231.[4] S. R. Roy and O. P. Tiwari, Ind. J. Pure Appl. Math. 14 (1983) 233.[5] M. Syzlowski and M. Heller, Acta Phys. Polonica B 14 (1983) 303.[6] G. Mohanty and B. D Pradhan, Int. J. Theor. Phys. 14 (1983) 233.[7] V. A. Belinskii and I. M. Khalatnikov, Sov. Phys. JETP 42 (1976) 205.[8] A. Banerjee, S. B. Duttachoudhury and A. K. Sanyal, J. Math. Phys. 26 (1985) 11.[9] W. H. Huang, Phys. Lett. A 129 (1988) 429.[10] F. M. Goenner and F. Kowalewsky, Gen. Rel. Grav. 21 (1989) 467.[11] A. Banerjee and A. K.Sanyal, Gen. Rel. Gravit. 20 (1988) 103.[12] A. A. Coley, Gen. Rel. Gravit. 22 (1990) 3.[13] A. A. Coley and R. J. Hoogan, J. Math. Phys. 35 (1994) 4117.[14] A. A. Coley and B. O. J. Tupper, Astrophys. J. 222 (1978) 405.[15] T. Singh and R. Chaubey, Pramana J. Phys. 68 (2007) 721.[16] D. Saez and V. J. Ballester, Phys. Lett. A 113 (1985) 467.[17] M. S. Berman, Nuovo Cimento B 74 (1983) 182.[18] M. S. Berman and F. M. Gomide, Gen. Rel. Grav. 20 (1988) 191.[19] C. P. Singh and S. Kumar, Pramana. J. Phys. 68 (2007) 707.[20] S. Kumar and C. P. Singh, Astrophys. Space Sci. 312 (2007) 57.[21] C. P. Singh, M. Zeyauddin and Shri Ram. Int. J. Mod. Phys. A 23 (2008) 1.[22] Shri Ram, M. Zeyauddin and C. P. Singh, Int. J. Mod. Phys. A 23 (2008) 4991.FIZIKA B 18 (2009) 4, 207–218 217ram al.: anisotropic viscous fluid cosmological models with heat flow in . . .NEIZOTROPNI KOZMOLOŠKI MODELI S VISKOZNOM TEKUĆINOM ITOKOM TOPLINE U SAEZ-BALLESTEROVOJ TEORIJI GRAVITACIJERazmatramo jednadžbe polja u Saez-Ballesterovoj skalarno-tenzorskoj teoriji gravi-tacije za Bianchijev model tipa V s viskoznom tekućinom i tokom topline. Postiglismo dvije vrste kozmoloških rješenja primjenom posebne relacije za promjene Hub-bleovog parametra koje vode na stalnu vrijednost parametra usporavanja. Jednavrsta rješenja odgovara modelu svemira koji se razvija iz singularnosti tipa ve-likog praska u t = 0 i širi se prema potencijalnom zakonu. Druga vrsta rješenjapredstavlja svemir koji se širi eksponencijalno a singularnost mu je u beskonačnojprošlosti. Raspravljaju se fizičke i kinematičke osebine oba modela. Nalazimo da suova dva modela svemira u skladu s nedavnim opažanjima.218 FIZIKA B 18 (2009) 4, 207–218

Neizotropni kozmološki modeli s viskoznom tekućinom i tokom topline u Saez-Ballesterovoj teoriji gravitacije

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