Bianchi Type VI0 Inflationary Universe with Constant Deceleration Parameter and Flat Potential in General Relativity

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1 Advances in Astrophysics, Vol., No., May Bianchi ype VI Inflationary Universe with Constant Deceleration Parameter and Flat Potential in General Relativity Raj Bali and Parmit Kumari Department of Mathematics, University of Rajasthan, Jaipur-, India balir5@yahoo.co.in Abstract. Inflationary universe scenario with constant deceleration parameter in the presence of massless scalar field and flat potential taking Bianchi ype VI space time as a source is discussed. We find that the rate of expansion slows down with increase of time. It is also observed that the ratio of shear and expansion is non-zero for all values of where at + b =, t being cosmic time, a and b being constants. hus the universe remains anisotropic throughout the evolution. he Higgs field is constant for large values of when α < and the Higgs field evolves slowly but the universe expands for α > where α is a constant. It may be positive and negative both. he model represents decelerating and accelerating phases of universe and has Point ype singularity at = (MacCallum [5]). In special case i.e. if N= and α > then the model isotropizes at late time, N being constant of integration. Keywords: Bianchi VI, Inflationary, deceleration parameter, flat potential Introduction he primordial acceleration in which the universe undergoes rapid exponential expansion is known as inflation. he notion of inflation is so far one of the best mechanism at the early stages of evolution of the universe to explain the flat, homogeneous and isotropic nature of the present day universe. In these models, the universe undergoes a phase transition characterized by the evolution of Higgs field φ. he inflation will take place if potential V(φ) has flat region and in this region, the Higgs field φ evolves slowly but the universe expands in an exponential way due to vacuum field energy. he flat part of the potential is naturally associated with a vacuum energy that is identified as an effective cosmological constant (Ʌ) which makes the universe enter an inflationary period. Historically, a model closely related to the inflationary universe was suggested by Starobinsky [] but the inflationary cosmological models became popular after an important paper of Guth [] who proposed inflationary model in the context of grand unified field theory (GU). here are two basic types of inflationary models: one is due to appearance of a flat potential e.g. in GU s phase transitions and the other is due to a scalar curvaturesquarred term. Barrow and urner [] pointed out that a universe with a large amount of anisotropy will not undergo the inflationary phase. A universe with only moderate anisotropy will undergo inflation and will be rapidly isotropized. Sato [] in a study has shown that the most prevailing inflationary models are investigated through the scalar field which acts as a source of inflation and generates cosmic acceleration. Inflationary scenario for homogeneous and isotropic models (FRW models) has been studied by many authors viz. Linde [5], Wald [6], Barrow [7], Abrecht and Steinhardt [8], Abbott and Wise [9], La and Steinhardt [], Mataresse and Lucchin []. Rothman and Ellis [] have pointed out that we can have solution for isotropic problem if we work with anisotropic metric but isotropize in special cases. In view of these observations Singh [] investigated Bianchi ype II inflationary models with constant deceleration parameter in general relativity. Bali [] investigated inflationary scenario in Bianchi ype I space time with flat potential. he universe in smaller scale is neither homogeneous nor isotropic nor do we expect the universe to have these properties in its early stages. Also Astronomical observations in late eighties revealed that the predictions of FRW models do not always meet our requirements as were believed earlier (Smoot et. al. [5]). herefore, spatially homogeneous and anisotropic Bianchi space times (I-IX) are considered to study the universe in its early stages of evolution. Among these, Bianchi ype VI space time is of Copyright 7 Isaac Scientific Publishing

2 68 Advances in Astrophysics, Vol., No., May 7 particular interest because this is simple generalization of Bianchi ype I space time. Barrow [6] pointed out that Bianchi ype VI cosmological models give a better explanation of some of the cosmological problems like helium abundance and these can be istropized in special cases. Seeing the importance of these models, various authors viz. Ellis and MacCallum [7], Collins [8], Roy and Singh [9], ikekar and Patel [], Bali et. al. [,], Ram and Singh [] have investigated cosmological models considering Bianchi ype VI space time in different contexts. Metric and Field Equations We consider Bianchi ype VI metric in the form x x ds = dt + A dx + B e dy + C e dz () where A, B, C are metric potentials and functions of t-alone. We assume the co-ordinates to be comoving so that v = = v = v,v = he action of the gravitational field minimally coupled to a scalar field with potential V(φ) is given by Stein-Schabes [] as ij S= g R g φ φ V( φ i j ) dx () he Einstein s Field equations (in gravitational units G = c=) in the case of massless scalar field φ with potential V(ϕ) are given by R g = 8π ij ij ij R () with ρ = φ φ φ φ + V ρ ( φ) ij i j g () ij j he conservation relation = leads to i;j ( g ) φ dv = µ µ g dφ he Einstein s Field equation () for the metric () leads to BC B C = 8π φ V ( φ BC B C A A AC C + + = 8π φ V ( φ A AC C A A AB B + + = 8π φ V ( φ A AB B A AB AC BC + + = 8π φ + V ( φ AB AC BC A B C = B C Equation () leads to where m is constant of integration. Solution of Field Equation he equation (5) for the scalar field (φ) leads to (5) (6) (7) (8) (9) () B = mc () Copyright 7 Isaac Scientific Publishing

3 Advances in Astrophysics, Vol., No., May 7 69 A C dv φ + + φ = () A C dφ where suffix indicates ordinary partial derivatives with respect to t. We are interested in inflationary solution so flat region is considered. hus V(ϕ) = K is constant. Now equation () leads to A C φ + + φ = () A C From equation (), we have l φ = () AC where l is constant of integration. he scale factor R for line-element () is given by R = ABC = mac (5) o find the deterministic model of the universe, we assume that deceleration parameter is constant. he deceleration parameter in Cosmology is a dimensionless measure of the cosmic acceleration of the expansion of space in FRW universe. It is defined as R R q = = α ( Constant ) (6) R R where α > or α < and R is scale factor of the universe. If constant deceleration is taken then accelerating and decelerating phases are included in the following manner. he expansion of the universe is said to be accelerating if R > and in this case, q < as per recent astronomical observations. Observations of the cosmic microwave background demonstrate that the universe is very nearly flat, so q >. his implies that the universe is decelerating. However, observations of distant I a Supernovae indicate that q < and the expansion of universe is accelerating. hus, it is interesting to assume that decelerating parameter is constant with the help of which we can explain the decelerating and accelerating phases of universe. From equation (6), we have R R + α = R R which leads to ( ) α + R = at + b (7) where a =β(αα+), β is constant of integration. From equation (5) and (7), we have AC = ( at + b) α + m (8) Adding equation (6) and (9), we have C C AC + + C C AC = 8π K (9) Multiplying equation (9) by AC, we have ACC + AC + A C C = 8π KAC () After using (8), we have 8π K ( ACC ) = ( at + b) α + m () hus, we have Copyright 7 Isaac Scientific Publishing

4 7 Advances in Astrophysics, Vol., No., May 7 ( α + ) ( ) ( α + )α + ACC = at + b α + + N ma where N is constant of integration. Dividing equation () by A C and using equation (8), we have Equation () leads to which leads to where and L is constant of integration. Now From equation (8), we have C C ( α + ) ( at b) mn ( at b) ( ) α + α + = a ( + ) ( ) ( ) ( ) ( ) α mn + α + α + a( α ) α α logc = at + b + at + b + log L a () () α α + C = Lexp γ + λ () ( ) ( ) α + mn α + γ =, λ = a + a α α + B = mc = ml exp γ + λ (5) α γ λ α + α + + e A = ml where at+b =. After suitable transformation of coordinates, the metric () leads to the form where 6 α γ λ α + α + + = α a γ + λα + ml X ml X { } ds d dx e e dy e dz e x = X, mly = Y, Lz = Z ml Physical and Geometrical Features he rate of Higgs fields (ϕ) is given by equation () as l lm φ = = AC α + which leads to lm α ( α + ) α + φ = + M a α ( ) where M is constant of integration. he spatial volume ( R ), the Hubble parameter (H), the expansion (θθ) and shear (σσ) for the model (7) are given by (6) (7) (8) (9) Copyright 7 Isaac Scientific Publishing

5 Advances in Astrophysics, Vol., No., May 7 7 α + R = mac = () H H = = a ( α + ) a ( α + ) λ( α ) A B a a σ = = 6aλ A B + + α + () () () 5 Conclusion he spatial volume increases with time. he rate of expansion slows down with the increase of time and finally drops to zero when. It is also observed that the ratio of shear and expansion is non-zero for all values of. hus universe remains anisotropic throughout the evolution and the Higgs field (ϕ) is constant for large value of when αα <. When α > then the Higgs field (ϕ) evolves slowly but the universe expands. hus the model represents shearing, non-rotating and expanding universe. Also since deceleration parameter q = αα (constant). herefore, the model represents decelerating and accelerating phases of universe when α > and α < respectively and the model (7) indicates the results as per astronomical observations. At the time of evolution of universe, the anisotropy is constant and during the inflation, it is still homogeneous and anisotropic. hus the results obtained in this paper can be interpreted in the framework of perfect fluid scenario. he model (7) has Point ype singularity at = (MacCallum[5]). During an initial period, the universe is assumed to be dominated by constant potential term V(φ) with scalar field φ giving rise to power law inflation. Hence inflationary scenario is observed in Bianchi ype VI space time in presence of massless scalar field with flat potential. Acknowledgments. he authors are thankful to the Referee for his valuable comments and suggestions. References. A.A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B, vol.9, pp. 99-, 98.. A.H. Guth, Inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D, vol., pp. 7-56, 98.. D. Barrow and M.S. urner, Inflation in the universe, Nature, vol.9, pp. 5-8, 98.. K. Sato, First order phase transition of vacuum and expansion of the universe, Monthly Not. Roy. Astron. Soc. Vol.95, pp , 98a. 5. A.D. Linde, A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, Phys. Lett. B, vol.8, pp. 89-9, R.M. Wald, Asymptotic behavior of homogeneous cosmological models in presence of a positive cosmological constant, Phys. Rev. D, vol.8, pp. 8, J.D. Barrow, Cosmic no hair theorems and inflation, Phys. Lett. B, vol.87, pp. -6, A. Albrecht and P.J. Steinhardt, Cosmology for grand unified theories with radiatively induced symmetry breaking, Phys. Rev. Lett., vol.8, pp. -, L.F. Abbott and M.B. Wise, Constraints on generalized inflationary cosmologies, Nucl. Phys. B, vol., pp. 5-58, 98.. D. La and P.J. Steinhardt, Extended inflationary cosmology, Phys. Rev. Lett., vol.6, pp. 76, S. Matarrese and F. Lucchin, Power law inflation, Phys. Rev. D, vol., pp. 6-, Rothman and G.F.R. Ellis, Can inflation occur in anisotropic cosmologies, Phys. Lett. B, vol.8, pp. 9-, 986. Copyright 7 Isaac Scientific Publishing

6 7 Advances in Astrophysics, Vol., No., May 7. C.P. Singh, Bianchi ype II inflationary models with constant deceleration parameter in general relativity, Pramana-J of Phys., vol.68, pp. 77-7, 7.. R. Bali, Inflationary scenario in Bianchi ype I space-time, Int. J. heor. Phys., vol. 5, pp. -8,. 5. Smoot et. al., Structure in the COBE differential microwave radiometer first year map, Astrophys. J. Part - Letters 96, pp. L-L5, J.D. Barrow, Helium formation in cosmologies with anisotropic curvature, Roy. Astron. Society, vol., pp. 8-, G.F.R. Ellis and M.A.H. MacCallum, A class of homogeneous cosmological models, Comm. In Math. Phys., vol., pp. 8-, C.B. Collins, More qualitative cosmology, Comm. in Math. Phys., vol., pp. 7-58, S.R. Roy and J.P. Singh, Some Bianchi ype VI cosmological models with free gravitational field of magnetic type, Acta Physica Austriaca, vol.55, pp , 98.. R. ikekar and L.K. Patel, Some exact solutions in Bianchi VI string cosmology, Pramana-J. Phys., vol., pp. 8-89, 99.. R. Bali, R. Banerjee and S.K. Banerjee, Some LRS Bianchi ype VI cosmological models with special free gravitational fields, Electron. J. heor. Phys., vol.6, pp. 65-7, 9.. R. Bali, R. Banerjee and S.K. Banerjee, Bianchi ype VI bulk viscous massive string cosmological models in general relativity, Astrophys. Space-Sci., vol. 7, pp. -6, 8.. S. Ram and P. Singh, Bianchi ype VI cosmological model in certain theories of gravitation, Astrophys. Space-Sci., vol., pp. 5-6, 99.. J.A. Stein-Schabes, Inflation in spherically symmetric inhomogeneous models, Phys. Rev. D., vol.5, pp. 5-5, M.A.H. MacCallum, A class of homogeneous cosmological models III, asymptotic behavior, Comm. in Math. Phys., vol., pp. 57-7, 97. Copyright 7 Isaac Scientific Publishing