NEW EXACT SOLUTION OF BIANCHI TYPE V COSMOLOGICAL STIFF FLUID MODEL IN LYRA S GEOMETRY

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1 ASTROPHYSICS NEW EXACT SOLUTION OF BIANCHI TYPE V COSMOLOGICAL STIFF FLUID MODEL IN LYRA S GEOMETRY VINEET K. YADAV 1,, LALLAN YADAV 2, ANIL KUMAR YADAV 3 1,2 Department of Physics, D. D. U. Gorahpur University, Gorahpur , India. 1 vineety@yahoo.co.in, 2 nisaly06@rediffmai1.com 3 Department of Physics, Anand Engineering College, Keetham, Agra , India. abanilyadav@yahoo.co.in Received April 27, 2009 The new Exact solution for stiff matter Bianchi type V cosmological model is investigated in Lyra s geometry for empty universe and matter filled universe. It is observed that the displacement vector β(t) is a decreasing function of time. The model start with singular state and evolve continuously as t. The physical and geometrical aspects of the models in empty and matter filled universe are also discussed. Key words: cosmology, Bianchi Type V universe, Lyra s geometry. PACS: ; Ex 1. INTRODUCTION In recent years, our nowledge of cosmology has improved remarably by various experimental and theoretical result. Einstein introduced his general theory of relativity in which gravitation is described in terms of geometry of space time. Motivated by it, Einstein geometrized other physical fields in general relativity. One of the first attempts in this direction was made by Weyl [1] who proposed a more general theory in which gravitation and electromagnetism is also described geometrically. However, this theory was never considered seriously as it was based on the nonintegrability of length transfer. Later Lyra [2] suggested a modification of Riemannian geometry by introducing a gauge function which removes the non-integrability condition of the length of a vector under parallel transport. This modified Riemannian geometry is nown as Lyra s geometry. Subsequently, Sen et al. [3,4] proposed a new scalar tensor theory of gravitation. They constructed an analog of the Einstein s corresponding author Rom. Journ. Phys., Vol. 55, Nos. 7 8, P , Bucharest, 2010

2 2 Bianchi type V cosmological stiff fluid model in Lyra s geometry 863 field equations based on Lyra s geometry which in normal gauge may be written as R ij 1 2 g ijr φ iφ j 3 4 g ijφ φ = 8πGT ij where φ i is the displacement vector and R ij is the curvature tensor, g ij is the metric tensor and T ij is the energy momentum tensor. Halford [5] pointed out that the constant displacement vector field φ i in Lyra s geometry plays the role of a cosmological constant in the normal general relativistic treatment. Halford [6] showed that the scalar-tensor treatment based on Lyra s geometry predicts some effects within observational limits, as in Einstein s theory. Several authors, Refs[7-12], have studied cosmological models based on Lyra s geometry with a constant displacement field vector. However, this restriction of the displacement field to be a constant is a coincidence and there is no priori reason for it. Singh et al. [13-15] have studied Bianchi type I, III, Kantowsi-Sachs and a new class of models with a time dependent displacement field in Lyra s geometry. The cosmological models with variable Hubble parameter and constant decelerating parameter have studied by Berman [16] and Rao et al. [17]. Pradhan et al. [18,19] and Rahman [20,21] have investigated bul viscous cosmological models in Lyra geometry with displacement vector β as a decreasing function of time. The study of Bianchi type V cosmological models play an important role in the study of universe and it create more interest as these models contain isotropic special cases and permit arbitrary small anisotropy levels at some instant of time. This property maes them suitable as a model of our universe. A number of authors have wored out Bianchi type V cosmological models in Lyra s geometry and other context are also studied in Refs.[22-40]. Singh et al. [41] have obtained Bianchi type V and V I 0 spaces in Lyra geometry. In both the models, the gauge function β is time dependent and β = constant, are considered. Ram et al. [42] have investigated cosmological models of Bianchi Type III and V in Lyra s Geometry. They have found contracting universes from infinite volume to zero volume. Motivated by the situation discussed above, in this paper we have investigated new exact solutions of Bianchi Type V stiff fluid cosmological models in Lyra s Geometry for empty universe and matter filled universe. The physical and geometrical aspects of the models in both the empty and matter filled universe are also discussed. 2. FIELD EQUATIONS We consider the Bianchi type V metric of the form ds 2 = dt 2 A 2 (t)dx 2 e 2αx [B 2 (t)dy 2 + C 2 (t)dz 2 ] (1)

3 864 Vineet K. Yadav, Lallan Yadav, Anil Kumar Yadav 3 where α is the constant. The field equations in the normal gauge for Lyra s manifold, as obtained by Sen are R ij 1 2 g ijr φ iφ j 3 4 g ijφ φ = 8πGT ij (2) where φ i is the displacement field vector defined as φ i = (0,0,0,β(t)) (3) and other symbols have their usual meaning as in Riemannian geometry. We tae the perfect fluid form for the energy momentum tensor T ij = (p + ρ)u i u j pg ij (4) together with the comoving co-ordinate u i u i = 1. The equation of state for the fluid is taen as p = γρ (5) where γ(0 γ 1) is a constant. For metric (1), the field equation (2) with the equation (3) and equation (4) tae the form A + B B + AḂ AB α2 A 2 = χp 3 4 β2 (6) A + C C + AĊ AC α2 A 2 = χp 3 4 β2 (7) B B + C C + ḂĊ BC α2 A 2 = χp 3 4 β2 (8) A + B B + AḂ AB α2 A 2 = χρ β2 (9) 2 A A Ḃ B Ċ C = 0 (10) The energy conservation equation gives χ ρ + 3 [ 2 β β + χ(ρ + p) + 3 ] ( ) A 2 β2 A + Ḃ B + Ċ = 0 (11) C where χ = 8πG. The quantities with dots overhead refer to their partial derivatives with respect to time coordinate.

4 4 Bianchi type V cosmological stiff fluid model in Lyra s geometry SOLUTION OF THE FIELD EQUATIONS By combination of field equation (6)-(9) we obtain A + B B + 2 AḂ AB + AĊ AC + ḂĊ BC 4α2 = χ(ρ p) A2 (12) A + C C + AḂ AB + 2 AĊ AC + ḂĊ BC 4α2 = χ(ρ p) A2 (13) B B + C C + AḂ AB + AĊ AC + 2ḂĊ BC 4α2 = χ(ρ p) A2 (14) A + B B + C C + 2 AḂ AB + 2 AĊ AC + 2ḂĊ BC 6α2 A 2 = 3 χ(ρ p) 2 (15) using equation of state (5) and defining Integrating equation (10) we get using equation (16) in equation (15), we have V 3 = ABCe 2αx (16) A 2 = BC (17) V V + 2 V 2 2α 2 e 4αx 3 = 1 χ(ρ p) (18) 2 Now we consider some cases of physical interest CASE I. (EMPTY UNIVERSE) In this case p = ρ = 0; then equation (18) will be The solution of equation (19) is given by V V + 2 V 2 2α 2 e 4αx 3 = 0 (19) V = αe 2αx 3 t + t0 (20) where t 0 is the constant of integration. Taing αe 2αx 3 =, then from equation (16), (17) and (21) we get V = t + t 0 (21) A = αt + b (22)

5 866 Vineet K. Yadav, Lallan Yadav, Anil Kumar Yadav 5 where b = αt 0. From equation (12) and (13), we have which on integration yields 2 B B + ) 2 (Ḃ = 2 C B C + Ḃ B Ċ C = (BC) 3 2 ) 2 (Ċ (23) C by using equation (17) Ḃ B Ċ C = A 3 (24) Solving equation (10) and (24), we have B = (αt + b)e C = (αt + b)e The metric (1) can now be written in the form 2α(αt+b) 2 (25) 2α(αt+b) 2 (26) ds 2 = dt 2 (αt + b) 2 [dx 2 + e 2αx {e α(αt+b) 2 dy 2 + e α(αt+b) 2 dz 2 }] (27) Physical Behaviour of the model For empty universe the equation (11) is given by On integrating equation (28), we have β + 3β V V = 0 (28) β = a V 3 = a (t + t 0 ) 3 (29) where a is the constant of integration. When t t 0,V 0,β. Thus the model has singularity at t = t 0. Halford has described that the constant displacement field vector φ i plays the role in Lyra s geometry as the cosmological constant Λ in general relativity [5,6]. From eqn. (29) we observe that the displacement vector β is a decreasing function of time and it approaches a small value. It is large in the beginning and decreases fast with the evolution of the universe. For time dependent creation field, the similar result has been seen in Hoyle s creation field theory. Figure (1) clearly shows the behavior of β as decreasing function of time.

6 6 Bianchi type V cosmological stiff fluid model in Lyra s geometry 867 Fig. 1 The plot of displacement vector β vs. time for model (27) with parameters a=.15, =1, t 0 =.20 The physical quantities expansion scalar θ and shear scalar σ 2 have the following expression θ = u i A ;i = A + Ḃ B + Ċ (30) C [ σ 2 = 1 2 σ ijσ ij = 1 θ 2 AḂ 3 AB AĊ ] AC ḂĊ (31) BC hence σ 2 = 1 3 [ θ = 3α (αt + b) 6α 2 (αt + b) 2 2 (αt + b) 6 The cosmological parameter H and q are given by ] (32) (33) In this model particle horizon exists because t t H = V V = (34) t + t 0 q = 2 = constant (35) 3 dt V = [ 1 log(t + t 0) ] t t (36)

7 868 Vineet K. Yadav, Lallan Yadav, Anil Kumar Yadav 7 is a convergent integral. This model has singularity at t = t 0. It has a particle horizon. As t, the shear dies out and the expansion stops. Thus the gauge function β(t) is large in the beginning of the model but decays continuously during its evolution CASE II. (MATTER-FILLED UNIVERSE) In the case of dust filled and radiation dominated universe, the solution of equation (18) is very tedious. However, in the case of Zeldovich fluid or stiff matter, the energy density and pressure of perfect fluid are connected by a linear equation of state given by ρ = p. We can see that in this case equation (18) is integrable and we can find out a physically meaningful solution of the Einstein s field equations. Using equation (5) and (16) equation (11) becomes χ ρ + 3 [ 2 β β + 3 χ(1 + γ)ρ + 3 ] V 2 β2 V = 0 (37) In this case (for stiff-fluid) γ = 1, hence equation (37) becomes χ ρ + 3 [ 2 β β + 3 2χρ + 3 ] V 2 β2 V = 0 (38) We have found that in this case the solution is same as empty space solution, discussed in case (I) but equation (38) yields p = ρ = 1 χ(t + t 0 ) 6 3 4χ β2 (39) From equation (39) it is obvious that the pressure and density depend on time as well as gauge function β. Thus, Equation (27) together with (39) is an exact Bianchi Type V stiff fluid cosmological model in frame wor of Lyra s manifold Physical Behaviour of the model From equations (21) and (39), we have ρ + 3 4χ β2 1 V 6 (40) When t t 0, V 0 and β. Also, when t, V and β 0. The scalar expansion and shear scalar are given by θ = 3α (αt + b) (41)

8 8 Bianchi type V cosmological stiff fluid model in Lyra s geometry 869 σ 2 = 1 [ 6α 2 3 (αt + b) 2 2 ] (αt + b) 6 The cosmological parameters H and q are given by (42) H = t + t 0 (43) q = 2 = constant (44) 3 The model has singularity at t = t 0. It has a particle horizon. At t, shear dies out and the expansion ceases. The gauge function β(t) is large in the beginning but decreases fast with the evolution of the universe. Similar result can be obtained for Hoyle s creation field theory, if the creation field is time dependent. 4. CONCLUDING REMARK In the present study we have investigated new exact solution of Einstein s field equations in the vacuum and in the presence of stiff matter for spatially homogeneous cosmological models of Bianchi type V in normal gauge for Lyra s geometry. Since σ lim 2 t = constant in both the cases, the models do not approach isotropy for θ 2 large value of t. The Bianchi type V cosmological model in Lyra s geometry starts with a singular state and evolve continuously, as t. We have obtained superdense and smooth universe with shear. The particle horizon has been found in empty and matter dominated universe. The gauge function β(t) is large in the beginning and reduces fast with the evolution of the universe. It should be noted that the model in both the cases have no initial singularities while in case II the energy density and pressure of the model has singularity at t = t 0. The deceleration parameter q is found to be negative. Its value is constant and less than unity which is supported by the current observations of SNe Ia and CMBR, favoring an accelerating model (q < 0). Acnowledgements: One of the authors (V. K. Yadav) would lie to than to Inter-University Centre for Astronomy and Astrophysics, Pune, India for its ind hospitality and providing facility where a part of this wor was carried out during a visit. REFERENCES 1. H. Weyl, Sitz. ber. Preuss Aad. Wiss., Berlin, 1918, G. Lyra, Math. Z., 54, 52, (1951). 3. D. K. Sen, Z. Phys., 149, 311, (1957).

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