Bianchi Type-IX Bulk Viscous String Cosmological Model in f(r,t) Gravity with Special Form of Deceleration Parameter

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1 International Journal of heoretical and Mathematical Physics 0, (6): 0-7 DOI: 0593/jijtmp00060 Bianchi ype-ix Bul Viscous String Cosmological Model in f(r,) Gravity with Special Form of Deceleration Parameter H R Ghate *, Atish S Sontae Department of Mathematics, Jijamata Mahavidyalaya, Buldana, India Abstract A Bianchi type-ix bul viscous string cosmological model has been investigated in the frame wor of f ( R, gravity proposed by Haro et al (Phys Rev D 8:000, 0) he source for energy momentum tensor is bul viscous fluid containing one dimensional cosmic strings he field equations have been solved by applying a special form of deceleration parameter proposed by Singha and Debnath (IJP 8:35, 009) A barotropic equation of state for the pressure and energy density is assumed to get a determinate solution of the field equations Also the bul viscous pressure is assumed to be proportional to energy density he physical and geometrical properties of the models are also discussed Keywords f ( R, Gravity, Cosmic Strings, Bul Viscosity, Bianchi type-ix, Special form of Deceleration Parameter Introduction It is well nown that Einstein s general theory of relativity has been successful in developing various cosmological models to study a structure formation and early stages of evolution of the universe Einstein pointed out that general relativity does not account satisfactorily for inertial properties of matter ie Mach s principle is not substantiated by general relativity Several attempts have been made to generalize the general theory of gravitation by incorporating Mach s principle and other desired features which were lacing in the original theory Alternative theories of gravitation have been proposed to Einstein s theory to incorporate certain desirable features in the general theory In the last decades, as an alternative to general relativity, scalar tensor theories and modified theories of gravitation have been proposed Brans and Dice [], Saez and Ballester [] have much importance amongst the scalar-tensor theories of gravitation Recently f (R) gravity and f ( R, gravity [3, ] theories have much importance amongst the modified theories of gravity because these theories are supposed to provide natural gravitation alternatives to dar energy From the cosmological observations, it is nown that the energy composition of the universe has 76% dar energy, 0% dar matter and % Baryon matter his is confirmed by the high red shift supernovae experiments [5, 6] and cosmic microwave bacground radiation observations [7, 8] * Corresponding author: hrghate@gmailcom (H R Ghate) Published online at Copyright 0 Scientific & Academic Publishing All Rights Reserved Coperland et al [9] have given a comprehensive review of f (R) gravity and Haro et al [] proposed f ( R, theory of gravity as an alternative amongst the modified theory of gravity of Einstein s theory of gravitation Several authors have studied f ( R, theory of gravity in different contexts [0-5] Recently Chirde and Kadam [6] have studied A new class of Bianchi type bul viscous and string cosmological model in f ( R, theory of gravitation In f ( R, theory of gravity, the field equations are obtained from the Hilbert-Einstein type variation principle he action for this modified theory of gravity is given by S = g f ( R, + Lm d x 6 G, () π where f ( R, is an arbitrary function of the Ricci scalar R and the trace of the stress-energy tensor of the matter µν L m is the matter Lagrangian density he stress-energy tensor of matter is defined as [7] δ ( glm ) =, () g δ g µν µν ν and its trace by = g µ µ ν respectively By assuming that the Lagrangian density L m of matter depends only on the metric tensor components g µ ν and not on its derivatives, we obtain

2 International Journal of heoretical and Mathematical Physics 0, (6): 0-7 µ ν Lm = gµ ν Lm µν g In the present paper, we use the natural system of units with G = c = so that the Einstein gravitational constant is defined as = 8π he corresponding field equations of the f ( R, gravity are found by varying the action with respect to the metric g µν []: { f R ( R, Rµν f ( R, gµν + ( gµν µ ν ) f R ( R, } = 8π µν f ( R, µν f ( R, Θµν, (3) δ f ( R, δ f ( R, where f R, f, µ µ, δ R δ µ is the covariant derivative and µν is the standard matter energy-momentum tensor derived from the Lagrangian L m Contracting (3) gives the relation between the Ricci scalar R and the trace of the stress-energy tensor, f R ( R, R + 3 f R ( R, f ( R, = 8π f ( R, f ( R, Θ, () µ where Θ = Θ µ Generally the field equations also depend [through the tensor Θ µν ] on the physical nature of the matter field Hence several theoretical models corresponding to different matter sources in f ( R, gravity can be obtained By considering some particular class of f ( R, modified gravity model obtained by explicitly specifying the functional form of f Assuming that the function f ( R, given by f ( R, = R +, (5) where f ( is an arbitrary function of the trace of the stress-energy tensor of matter From equation (3), the gravitational field equations are given by Gij Rij Rgij, (6) = 8πij f ( ) ij f ( ) Θij f( ) gij where the prime denotes a derivative with respect to the argument If the matter source is a perfect fluid, Θ=µν pgµν then the above equation (6) reduces to Gij Rij Rgij = 8π + f ( ) + [ pf ( ) + f( )] g ij ij ij, (7) where the prime denotes derivative with respect to the argument he Deceleration parameters (q) are useful in studying expansion of the universe We now that the universe has (i) decelerating expansion if q > 0, (ii) an expansion with constant rate if q = 0, (iii) accelerating power law expansion if < q < 0, (iv) exponential expansion (de-sitter expansion) if q = (v) super-exponential expansion if q < Many relativists have studied DE cosmological models with different form of deceleration parameters Singha and Debnath [8], Adhav et al [9, 0] have obtained DE models with special form of deceleration parameter he study of Bianchi type cosmological models are important in achieving better understanding of anisotropy in the universe Moreover the anisotropic universes have greater generality than FRW isotropic models he simplicity of the field equations made Bianchi type space-times useful Bianchi type I-IX cosmological models are homogeneous and anisotropic Bianchi type-ix universe are studied by the number of cosmologists because of familiar solutions lie Robertson Waler Universe, the de-sitter universe, the aub-nut solutions etc Charaborty [], Bali and Dave [], Bali and Yadav [3], Pradhan et al [], yagi et al [5], Ghate and Sontae [6, 7] have obtained Bianchi type-ix cosmological models in different contexts Bul viscosity plays a very important role in getting accelerated expansion of the universe he matter behaves lie viscous fluid in the early stages of the universe when neutrinos decoupling occurs he effect of viscosity on the evolution of the universe had been studied by Misner [8, 9] Several authors have studied bul viscous cosmological models in general relativity [30-37] Johri and Sudharsan [38], Pimental [39], Banerjee and Beesham [0], Singh et al [], Rao et al [], Naidu et al [3], Reddy et al [] have studied bul viscous string cosmological models in scalar-tensor theories of gravitation Recently Naidu et al [5], Reddy et al [6] have studied Bianchi type-v and Kaluza-Klein cosmological models with bul viscosity and cosmic strings in f ( R, theory of gravitation Motivated by the above discussion and investigations in f ( R, modified theory of gravity, in this paper we have investigated Bianchi type-ix bul viscous string cosmological model in f ( R, theory of gravity with special form of deceleration parameter Bianchi type-ix space-time has considered when universe

3 H R Ghate et al: Bianchi ype-ix Bul Viscous String Cosmological Model in f(r,) Gravity with Special Form of Deceleration Parameter is filled with cosmic strings and bul viscosity in f ( R, theory of gravity with special form of deceleration parameter his wor is organized as follows: In Section, the model and field equations have been presented he field equations have been solved in Section 3 by applying special form of deceleration parameter he physical and geometrical behavior of the model have been discussed in Section In the last Section 5, concluding remars have been expressed Metric and Field Equations ds Bianchi type-ix metric is considered in the form, ( sin cos ) dt a dx b dy b y a y dz =,(8) a cos ydxdz where a, b are scale factors and are functions of cosmic time t We consider the energy momentum tensor for a bul viscous fluid containing one dimensional string as ( ρ + p) u u p λ x x, (9) and i j = i j g i j p = p 3ξ H, (0) where ρ is the rest energy density of the system, ξ (t) is the coefficient of bul viscosity, 3ξ H is bul viscous i pressure, H is Hubble s parameter, x is the direction of the string and λ s is the string tension density Also i i u = δ is a four-velocity vector which satisfies i i i giju uj =, x xj =, u xi = 0 Here we consider ρ, p and λs as function of time t only We assume the string to be lying along the x-axis One-dimensional strings are assumed to be loaded with particles and the particle energy density is ρ p = ρ λs In co-moving coordinate system, we get 0 0, = ρ, = p λs = 3 3 = p, s s = ρ 3 p λ () Now we choose the function f ( of the trace of the stress-energy tensor of the matter so that f ( = µ, () where µ is constant In the co-moving coordinate system, the field equations (7) for the metric (8) and with the help of (9)-() can be written as a b b a + + = ρ(8π + 3µ ) µ p µ λs, (3) a b b b b i j b b 3a + + = µ ρ (8π + 3µ )( p + λ ) s, () b b b b a b ab a = µρ (8π + 3 µ ) p µλ s, (5) a b ab b where the overdot ( ) denotes the differentiation with respect to t 3 Solutions of Field Equations he field equations (3) (5) are a system three independent equations in six unnowns a, b, p, ρ, ξ, λs hree additional conditions relating these unnowns may be used to obtain explicit solutions of the systems (i) Firstly, we assume that the expansion θ in the model is proportional to the shear σ his condition leads to m b a =, ( m ), (6) where m is proportionality constant he motive behind assuming condition is explained with reference to horne [7], the observations of the velocity red-shift relation for extragalactic sources suggest that Hubble expansion of the universe is isotropic today within 30 percent [8, 9] Collin et al [50] have pointed out that for spatially homogeneous metric, the normal congruence to the homogeneous expansion satisfies that the σ condition is constant θ (ii) Secondly for a barotropic fluid, the combined effect of the proper pressure and the bul viscous pressure can be expressed as p = p 3 ξ H = ερ, (7) where ε = ε 0 γ ( 0 ε 0 ), p = ε 0 ρ and ρ = λs Here ε 0, γ, are constants (iii) Lastly following Singha and Debnath [7], we use a special form of deceleration parameter as: RR q = = +, (8) R + R where R is the average scale factor, > 0 is constant Solving equation (8), the average scale factor R is given by ( ) t α e α R =, (9) where α and α are constants of integration For the metric (8), the scale factor R is given by

4 International Journal of heoretical and Mathematical Physics 0, (6): R = ( a b ) 3 (0) Solving field equations (3)-(5), with the help of equations (9) and (0), we obtain 3m α ( m+ ) ( ) t e α a =, () b 3 = α m ( + ) () ( ) t e α Using equations () and (), the metric (8) taes the form ds = 6m αt dt + ( α ( ) e m+ ) dx 6 αt ( ) + ( αe m+ ) dy 6 αt ( ) ( αe m+ ) sin y + dz 6m αt ( ) + ( αe m+ ) cos y 6m α t ( ) ( αe m+ ) cos ydxdz (3) Some Physical Properties of the Model For the cosmological model (3), the physical quantities Spatial volume V, Hubble parameter H, Expansion scalar θ, Mean Anisotropy parameter A m, Shear scalar σ, Energy density ρ, String tension density λ s, Bul viscosity pressure p, Coefficient of bul viscosity ξ and Particle energy density Spatial volume, Hubble parameter, H Expansion scalar, ρ p are obtained as follows : ( ) t α e α 3 V = () t ( α e α ) = αα (5) t ( e α ) θ= 3H = 3αα α (6) Mean Anisotropy Parameter, ( ) ( + ) m A m = = constant ( 0 for m ) (7) m Shear scalar, σ αt ( α e ) 3 αα ( m ) =, (8) ( m + ) Equation (3) represents Bianchi type-ix bul viscous σ ( m ) strings cosmological model in f ( R, gravity with special where = 0 (9) θ 3( m + ) form of deceleration parameter Energy density, 6 6( ) m 9(m + ) αα αt αt m ( ) αt ρ = ( ) ( α e ) + ( α e + ) ( αe m+ ) (30) ( m + ) 8π + µ 3ε String tension density, 6 6( ) m 9(m + ) αα αt αt ( m ) αt λ ( ) ( ) ( ) ( m s = α e + α e + α e + ) ( m + ) (3) 8π + µ 3ε Particle energy density, 6 6( m) ( ) 9(m + ) αα αt αt m ( ) αt ρ ( ) ( ) ( ) ( m p = α e + α e + α e + ) ( m + ) (3) 8π + µ 3ε Bul viscous pressure,

5 H R Ghate et al: Bianchi ype-ix Bul Viscous String Cosmological Model in f(r,) Gravity with Special Form of Deceleration Parameter 6 6( m ) ε 9(m + ) αα αt αt ( m ) αt p = ( ) ( α e ) + ( αe + ) ( αe m+ ) ( m + ) 8π + µ 3ε Coefficient of bul viscosity, ( ε ε0) α ( + ) ξ = 6 6( m) 3αα 8π + µ 3ε αt m ( ) αt ( ) ( α e + ) ( αe m+ + ) 9(m + ) αα αt αt ( ) ( α e e ) m (33) (3) Physical Behavior of the Model For the cosmological model (3), we observed that, the spatial scale factors has finite value at the initial epoch t = 0 and increases for large values of t In particular for α = and t = 0, the model has a big bang type singularity It can also be observed that, at t = 0, the universe starts evolving with finite volume and expands infinitely with the increase in cosmic time t (Figure ) he values of expansion θ and shear σ tend to infinity for large values of t ( t ) showing that the universe is expanding with increase of time (Figure 3, ) Figure he Plot of Volume verses time Figure he Plot of Hubble parameter verses time

6 International Journal of heoretical and Mathematical Physics 0, (6): Figure 3 he Plot of Expansion scalar verses time Figure he Plot of Shear scalar verses time Figure 5 he Plot of deceleration parameter verses time

7 6 H R Ghate et al: Bianchi ype-ix Bul Viscous String Cosmological Model in f(r,) Gravity with Special Form of Deceleration Parameter From equations (7) and (9), the mean anisotropy parameter A σ m = constant ( 0 for m ) and = θ constant ( 0 for m ) indicates that the model is anisotropic throughout the evolution of the universe except at m = (ie the model does not approach isotropy) In figure 5, the plot of deceleration parameter verses time is given from which we conclude that the model is decelerating at an initial phase and changes the expansion from decelerating to accelerating Hence the model is consistent with the recent cosmological observations [5-6, 5-55] Bul viscosity in the model decreases with increase in time which is in accordance with the well-nown fact that bul viscosity decreases with time and leads to inflationary model [56] For illustrative purposes, evolutionary behaviors of some cosmological parameters are shown graphically (Figure -5) 5 Conclusions Bianchi type-ix bul viscous strings cosmological model has been discussed in the frame wor of f ( R, gravity proposed by Haro et al (Phys Rev D 8:000, 0) when the source for energy momentum tensor is bul viscous fluid containing one dimensional cosmic strings he solution of the field equations has obtained by choosing the special form of deceleration parameter q = + It + R is observed that in early phase of universe, the value of deceleration parameter is positive while as t, the value of q = Hence the universe had a decelerated expansion in the past and has accelerated expansion at present which is in good agreement with the recent observations of SN Ia It is worth to mention that, the model obtained is point type singular, expanding, shearing, non-rotating and do not approach isotropy for large t Further the models are anisotropic throughout the evolution his shows that the present model not only represent accelerating universe but also confirms the well-nown fact that the bul viscosity will play a vital role in getting the accelerated expansion We hope that our model will be useful in the discussion of structure formation in the early universe and an accelerating expansion of the universe at present REFERENCES [] Brans, CH, Dice, RH: Phys Rev, 95, 96 [] Saez, D, Ballester, VJ: Phys Lett A 3, 67, 986 [3] Abar, M, Cai, RG: Phys Lett B 635, 7, 006 [] Haro,, Lobo, FSN, Nojiri, S, Odintsov, SD: Phys Rev D 8, 000, 0 [5] Riess, A, et al: Astron J 6, 009, 998 [6] Perlmutter, S, et al: Astrophys J 57, 565, 999 [7] Spergel, S, et al: Astrophys J 8, 75, 003 [8] Spergel, S, et al: Astrophys J 70, 377, 007 [9] Coperland, et al: Int J Mod Phys D 5, 753, 006 [0] Adhav, KS: Astrophys Space Sci 339, 365, 0 [] Katore, SD, Shaih, A Y, Prespacetime 3(), 087, 0 [] Reddy, DRK, Santiumar, R, Naidu, RL: Astrophys Space Sci 3, 9, 0 [3] Shri Ram, Priyana: Astrophys Space Sci 37, 389, 03 [] Rao, VUM, Neelima, D, Sireesha, KVS: Prespacetime (3), 98, 03 [5] Sharif, M, Zubair, M: Astrophys Space Sci 39, 59, 0 [6] Chirde, VR, Kadam, VP: Int J Scientific Research 3(5), 85, 0 [7] Landau, LD, Lifshitz, EM: he Classical heory of Fields (Butterworth-Heinemann, Oxford, 998) [8] Singha, A K, Debnath, U: Int J heor Phys, 8, 35, 009 [9] Adhav, KS, Wanhade, RP, Bansod, AS: IJIRSE, (5), 656, 03a [0] Adhav, KS, Wanhade, RP, Bansod, AS: Journal of Modern Physics,, 037, 03b [] Charaborty, S: Astrophysics and Space Science, 80(), 93, 99 [] Bali R, Dave, S: Pramana Journal of Physics, 56(), 53, 00 [3] Bali, R, Yadav, MK: Pramana Journal of Physics, 6(), 87, 005 [] Pradhan, A, Srivastav, SK, Yadav, MK: Astrophys Space Sci, 98, 9, 005 [5] yagi, A, Chhajed, D: American Journal of Mathematics and Statistics, (3), 9, 0 [6] Ghate, HR, Sontae, AS: IJSER, (6), 769, 03a [7] Ghate, HR, Sontae, A S: IJAA, (), 8, 0 [8] Misner, CW: Nature, 0, 967 [9] Misner, CW: Astrophys J 5, 3, 968 [30] Barrow, JD: Phys Lett B 80, 335, 986 [3] Pavon, D, Bafluy, J, Jou, D: Class Quantum Gravity 8, 37, 99 [3] Singh, JK, Shriram: Astrophys Space Sci 36, 77, 996 [33] Bali, R, Dave, S: Astrophys Space Sci 8, 6, 00 [3] Singh, JK: ILNuoam 08, 5, 005

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