Int. J. Contemp. Math. Sciences, Vol. 7, 01, no. 1, 1005-101 A Mathematical Aspect of Higher Dimensional Cosmological Models with Varying G and ΛTerm. K. Dubey Department of Mathematics, Govt. Science P.G.College ewa, Madhya Pradesh, India rkdubey004@yahoo.co.in Abhijeet Mitra Department of Mathematics, Govt. P G College Satna, Madhya Pradesh, India abhijeetmitra@rediffmail.com Bijendra Kumar Singh Department of Mathematics, Govt. P G College Satna, Madhya Pradesh, India singh.bijendrakumar@gmail.com Abstract. In this paper we have discussed the mathematical aspect of a non empty higher dimensional cosmological model with variable cosmological constant ( ) and variable gravitational constant (G ) under some suitable assumptions. The α α + ( m+ 1)( 1) gravitational constants found to decrease with time as G t γ + (where α is constant and γ is equation of state parameter), whereas cosmological constant Λ t. We obtain the exact solutions for the field equations and discuss some physical properties of the cosmological model. An expanding universe is found by using a relation between scalar potential and an equation of state. Keywords: Cosmology, higher dimension, variable gravitational coupling (G ) and Cosmological Constant term ( Λ ).
1006. K. Dubey, A. Mitra and B. K. Singh 1. INTODUCTION "Einstein Universe" is one of Friedman's solutions of Einstein's field equations for the value of cosmological constant (Λ). This is only stationary solution of all Friedman's solutions, and because it is stationary, it is thought to be non-physical by majority of astronomers. Those astronomers think that universe is expanding because there is observed a phenomenon of Hubble red shift and it is interpreted by those astronomers as a Doppler s shift caused by galaxies moving away from our own Galaxy. Therefore, it is thought that the real solution of Einstein's field equations cannot be stationary. As discussed earlier by many researchers [4] that a constant (Λ) cannot explain the huge difference between the cosmological constant inferred from observations and energy density resulting from quantum field theories. In the year 1930 and onwards eminent cosmologists such as A.S. Eddington and Abbe Lemaitre [10, 5] felt that the Λ -term introduced certain attractive features into cosmology and that model based on it should also be discussed. To solve the above discussed problem, variable Λ was introduced such that Λ was large in the early universe and then decayed with evolution. A number of models with different decay laws for the variation of cosmological term were investigated during last two decades by Chen and Wu (1991) Pavon-1991 [14] Carvalho, Lima and Waga 199- [19], Lima and Maia 1994 [0] Lima and Trodden 1996 [1] Arbab and Abdel- ahaman 1994 [5,6] Vishwakarma 001 [33] Cunha and Santos 004 [4] Carneiro and Lima (005) [34]. It was Bertolami [8, 9] who obtained Cosmological Models with time dependent G and Λ terms and suggested ~ Λ ~ t.in 001, Singh and Kotambkar [17] considered a cosmological model representing a flat viscous universe with variable G and Λ in the context of higher-dimensional space-time. A possible time variable G was suggested by Dirac [30]. Many authors [1-4, 8-9, 16, 18, 3, 6-7, 3] have proposed linking of the variation of G with that of Λ within the rules of general relativity theory. This new thought leaves Einstein's equations nearly unchanged as a variation in Λ is accompanied by the variation ing. Mathematically well-posed gravitation theories were developed in which Einstein s theory of general relativity was generalized to include a varying G by deriving it from a scalar field satisfying a conservation equation. There has been considerable interest in solutions of Einstein s equations in higher dimensions in the context of physics of early universe both from cosmological as well as mathematical point of view [11-13, 16, 18, 31]. In the present paper we have taken G = a α and tried to investigate the mathematical properties which have been discussed up to now and many other challenges which have been left and require attention to solve by peer researchers.
Higher dimensional cosmological models 1007. The Field equations and their solutions We consider 5-D obertson-walker metric dr ds = dt () t + r ( dθ + sin θdφ ) (1) 1 kr Where, (t), k =0, + 1, D = m + Stand for scale factor, curvature parameter and dimension respectively. The energy-momentum tensor of a perfect fluid is Tij = ( p+ ρ) uu i j pgij Where, ρ is the energy density of the cosmic matter and p its pressure and u i is j the unit flow vector such thatuu i = 1. The Einstein s field equations with time varying cosmological and gravitational constants are given by 1 ik gik = 8 πg[( ρ + p) uiuk ρgik ] +Λ gik () Where the cosmological term Λ is time- dependent and c, the velocity of light in vacuum is assumed to be unity. For the metric (1) yields two independent Friedman equations mm ( + 1) & k + 8πGρ = +Λ (3) m&& m( m 1) & k + + 8π GP = +Λ (4) Here equation (3) & (4) are time and space component of the field equation ().The dot ( ) denotes derivative with respective to t. & k (8 πgρ + Λ) From (3) + =, differentiating w.r.t. t m( m+ 1) 3 3 {8 π G ρ + 8 π G& ρ +Λ& } & k& && & = + + (5) & m ( m + 1) & & Substituting the value of && obtained in (5) to (4) m {8πG & ρ + 8 πg& ρ +Λ& } & k m( m 1) & k + + + + 8π GP = +Λ m( m+ 1) & Using equation (3) in the above equation
1008. K. Dubey, A. Mitra and B. K. Singh {8πG & ρ+ 8 πg& ρ+λ & } + 8 πgρ + 8 πgp = 0 ( m+ 1) & Above equation reduces to G Λ & ρ + ρ + + ( P+ ρ)( m+ 1) = 0 G 8π G (6) From equation (6) it can be seen that in this case the energy density of the matter fields is not conserved because of the varying nature of scalars G & Λ. The principle of equivalence requires only g ik (not Λ andg ) which should be involved in the equations of motion of particles and photons. Thus in the present case also the conservation law of energy-momentum (u i T ik ; k=0) holds and its suggests & & ρ+ ( m+ 1)( ρ+ P) = 0 (7) From equation (6) and (7), we have 8πG& ρ+ Λ= & 0 Λ G& = & 8πρ (8) Assuming the equation of state as P = γρ where 0 γ 1, Equation (7) reduces to & ρ ( m 1)( γ 1) ρ = + + & (7.1) Integrating the above relation ( m+ ρ = a 1 (9) ( m+ Where a1= ρ00 and suffix 0 represents the present value of the parameters. Eliminating & 1 & ρ from equation (3) & (7) and using = ( m+ ρ, substituting value of & in (3), we obtain & ρ 16πG Λ k = ( m + 1) ( γ + 1) 3 ρ + mm ( + 1) mm ( + 1) ρ ρ Again differentiating (10) and using equation (8) & (7.1) [( ( m+ ) ] & ρ k 3 = ( m + && ρ Λ ρ ρ ( γ + 1) m & 1 & ρ From (7.1) = ( m+ ρ (10) (11)
Higher dimensional cosmological models 1009 1 H& ρρ&& & ρρ& = So, H& && ρ & ρ = + (1) ( m + ρ ( m+ ρ ( m+ ρ From (1) and (11) we obtain ( ( m 1)( 1)) k ( 1) H& + γ + Λ γ + + ( m+ H + = 0 (13) m Where H= & is the Hubble parameter In most of the investigations a power law relation between the scale factor and scalar field is assumed. Cosmological models with the gravitational and cosmological constants generalized as coupling scalars whereg = λ α, Where λ, α are constants have been discussed by S. Weinberg & E.B. Norman [36-40]. Let us consider: G = a α (14), Where a is the proportionality constant andα o, Using the values of ρ and G given by equations (9) & (14) equation (8) reduces to (8 πρ) G= Λ (8 πρ) G& = Λ& (Using 8) ( 1) But G & = aα α + (From 14) ( α+ 1) ( m+ 1)( γ+ 1) So Λ= & 8παa a1 8π α aa 1 (( α+ 1) + ( m+ 1)( γ+ 1)) + 1 Integrating, Λ= (15) (( α + 1) + ( m + ) + 1 Where the integration constant is taken to be zero as in the beginning of the universe a = 0 & Λ = 0 From equations (3), (9), (10) m( m + 1) & k + = 8πGρ + Λ m & (!)( γ + 1) + ( + = 16πa1 a k m m m ( + 1)( α + 1) + ( + + 1) (16) For a general k, it is difficult to integrate equ. (16). Under the assumption k =0 for flat model equ. (16) yields the solution. 16π a1a ( + ( m + ) & = m( m + 1)(( α + 1) + ( m + + 1) 1/ 16πa1a ( + ( m + Let A= m( m + 1)(( α + 1) + ( m + ) + 1) ( ) 1/
1010. K. Dubey, A. Mitra and B. K. Singh So, ( α+ ( m+ 1)( γ+ 1) & = A, Integrating = B. t (17) A( α + ( m + Where B= From equation (17) we can say that for expanding model of the universe ( m + + α <.With the help of the result (17) energy density ρ, q and Λ may be obtained from equations (9), (14) and (15) respectively. ρ = a G = a B ( m+!)( γ + 1) 1 Β. α t t α α ( m+ i.e. G t 8π aa 1 ( α+ ( m+ 1)( γ + )) Λ= B t ( α + ( m + (0) i.e. Λ t From above equations it can be easily seen that energy density ( ρ ) and cosmological term ( Λ ) are decreasing while gravitational constant (G ) is also decreasing during the expansion of the universe. In most variable (G ) cosmologies, (G ) is a decreasing function of time. In this case, the deceleration parameter q is given by: && q = & Using (17) we find, ( m + α 1 q = = 1+ [( m + + α ], For ( m + + α < the deceleration parameter q < 0,this is satisfied with observations (Knop et al., iess et al). (18) (19) 4. CONCLUSION In this paper we have tried to present the cosmological models with varying gravitational coupling G and cosmological term Λ in higher dimensions. It is clear from the values obtained of ρ(), t G(), t Λ (), t and q () t that these quantities depend on the dimensionality of space-time. The results obtained
Higher dimensional cosmological models 1011 in this paper are in favor of the views of astronomical observations.. If we select α = 0, γ = 1, then from (17),(18),(19) we have, G = constt., Λ t, ρ t. The cosmological term Λ decides the behavior of the universe in the model. In this paper we have obtained a negative value of Λ which will correspond to positive effective mass density. Hence, we can get a universe that expands and then recontract. The observations on magnitude and red shift of type Ia supernova suggest that our universe may be an accelerating one or otherwise with induced cosmological density, through the cosmological Λ -term [15, 35,].Thus our models are consistent with the results of the observations made in recent times. eferences 1. Abbussattar,.G. Vishwakarma, Class. Quantum Grav., 14, 945 (1997).. A. Beesham, Phys. ev., D 48, 3539 (1993). 3. A. Beesham, Gen. el. Grav., 6, 159, (1994). 4. A.D Dolgov: In the very early Universe. G.W. Gibbons, S.W. Hawking, and S.T.C. Siklos,, Cambridge University press Cambridge, p. 449 (1983). 5. A.I Arbab,. And A.M.M Abdel ahaman,.: Phys. ev. D. 50, 775 (1994). 6. A.M.M. Abdel- ahaman, Gen. el. Grav., 665 (1990) 7. A.Pradhan., A.K. Singh, S.Otarod, om. J. Phys., 5, 415 (007). 8. A.Pradhan., P. Pandey, G.P. Singh..V. Deshpandey, Spacetime & Sulstance, 6 (116) (003) 9. A.Pradhan., V.K. Yadav, Int. J. Mod. Phys., D 11, 893 (00). 10. A.S. Eddington, Mon. Not. oy. Astron. Soc. 90, 668. (1930) 11. B. atra, and P.J.E. Peebles, Phys. ev. D 37, 3406 (1988). 1. C.P.Singh, S. Kumar, A.Pradhan, Class. Quantum Grav., 4, 455 (007). 13. D.Kalligas, P. Wesson, C.W.f. Everitt, Gen., el. Grav., 4, 315 (199). 14. D. Pavon: Physics. ev. D43, 375 (1991) 15. E.P. Hubble, and.c. Tolman Ap. J. 8, 30 (1935). 16. G.P Singh, S.Kotambkar, and A.Pradhan, Int. J. Mod. Phys., D 1, 941 (003). 17. G.P Singh and S.Kotambkar, Gen. el. Grav. 33, 61 (001). 18. I.Chakraborty, A.Pradhan, Grav. & Cosmo, 7, 55 (001). 19. J.A.S. Lima, and J.C.Carvalho: Gen. el. Grav. 6, 909 (1994). 0. J.A.S. Lima. And J.M.F Maia,.: Phys. ev. D 49, 5579 (1994) 1. J.A.S. Lima. And M Trodden: Phys. ev. D 53, 480 (1996). J.Narlikar: An Introduction to Cosmology, Cambridge University Press (00). 3. J.P.Singh, A.Pradhan., A.K. Singh, Gr. Qe./0705-0459 (007) 4. J.V. Cunha, and.c Santos: Int. J. Mod. Phys. D 13, 131 (004).
101. K. Dubey, A. Mitra and B. K. Singh 5. Lemaitre, G.Abbe, Mon. Not. oy. Astron. Soc. 91, 483(1931). 6. M.S.Berman, Gen. el. Grav. 3, 465 (1991). 7. M.S.Berman, Phys. ev., D 43, 1075 (1991). 8. O.Bertolami: Fortschr. Phys. 34, 89 (1986) 9. O.Bertolami: Nuovo Cimento 1393 (36 (1986). 30. P.A.M. Dirac, Nature, 61, 33 (1937). 31. P. Astier, et al., Astrono. Astrophys. 447, 31 (006). 3..F.Sistero, Gen. el. Grav., 3, 165 (1991). 33..G. Vishwakarma : Gen. elativ. Gravit. 33, 1973 (001). 34. S.Carneiro, and J.A.S. Lima: Int. J. Mod. Phys. A 0, 465 (005). 35. S.Perlmutter, et al., Astrophys. J. 483, 565 (1997). 36. S.Weinberg, Phys. ev. Lett. 19, 164 (1967). 37. S.Weinberg. Phys. ev. Lett. 43, 1566 (1979). 38. S.Weinberg, Physics in higher dimensions, world scientific, Singapore (1986). 39. S.Weinberg, ev. Mod. Phys, 61, 1 (1989). 40. S.Weinberg, Gravitation and Cosmology, Wiley, New York (1971). eceived: November, 011