Hypersurface Homogeneous Space Time with Anisotropic Dark Energy in Brans Dicke Theory of Gravitation

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1 Commun. Theor. Phys. 62 ( Vol. 62, No. 5, November, 204 Hypersurface Homogeneous Space Time with Anisotropic Dark Energy in Brans Dicke Theory of Gravitation S.D. Katore,, M.M. Sancheti, S.P. Hatkar, 2, and N.K. Sarkate Department of Mathematics, S.G.B. Amravati University, Amravati , India 2 Department of Mathematics, A.E.S., Arts, Commerce and Science College, Hingoli 4353, India (Received January 27, 204; revised manuscript received June 9, 204 Abstract The main purpose of the present paper is to explore hypersurface homogenous space time in Brans Dicke theory of gravitation in terms of the dark energy source. To obtain solution of the field equation, we have taken into account the relation between and the average scale factor a. The volumetric expansions are considered to get deterministic solutions. Physical properties of the models are also discussed in detail. PACS numbers: x, k, Ay Key words: hypersurface homogeneous, Brans Dicke theory, dark energy, average scale factor Introduction The theoretical predicts and recent experimental data reveal the existence of an anisotropic universe that approaches to isotropy. It has been also discovered that the expansion of the universe is accelerating. [] It is customary to research for the type of matter that can behave like a cosmological constant. [23] Observational data suggests that dark energy with negative pressure provides a mechanism for acceleration. [47] There are large number of candidates for dark energy in the literature such as a cosmological constant, [8] the X-matter, [90] quintessence. [2] In modern cosmology, there have been several attempts to identify the dark energy. The dark energy which amounts to about 70 percent and permeates homogeneously in the universe. The study of anisotropy of universe is object of researcher from long decades. By considering spatially homogeneous and anisotropic space time one can observe the amount of homogeneity and isotropy. In such models, the EoS parameter and metric are allowed to exhibit isotropy. It can be classified according to the isotropization of the universe, which can be examined from metric and the EoS of fluid. The isotropization occurs at an early time or at late time of the universe. An early time related to the inflation field and late time related to dark energy which is responsible for acceleration of the universe. This dark energy mentioned by equation of state p = ωρ where p is the pressure, ρ is density, and ω is equation of state parameter. We allow the dark energy to be time dependent. The simplest model for dark energy is a cosmological constant Λ. When ω < /3 we call the model as quiescence (Q. When w is dependent on time the dark energy model other than Q and Λ is called Kinessence (K. In recent years, many authors have shown keen interest in studying the universe with variable equation of state. [36] Shamir and Bhatti [7] investigated anisotropic dark energy Bianchi type III cosmological models in Brans Dicke theory of gravity. Katore [8] studied Einstein Rosen universe with magnetized anisotropic dark energy model. Five-dimensional dark energy models in a scalar tensor theory of gravitation are explored by Reddy et al. [9] Katore et al. [20] investigated Bianchi type III dark energy cosmological model in scalar tensor theory of gravitation. Singh and Beesham [2] in the investigation of hypersurface homogeneous space time with anisotropic dark energy obtained exact solution of Einstein s field equations under an assumption of the fluid anisotropy. Koivisto and Mota [22] studied a cosmological model where the accelerated expansion of the universe with the EoS by introducing two skewness parameter. Mahanta and Biswal [23] investigated a dark energy model with variable the EoS parameter in self creation theory of gravitation. Amirhashchi et al. [24] studied a Bianchi type VI dark energy models with a variable equation of state. In this paper, we explore hypersurface homogeneous space time in the Brans Dicke theory of gravitation, [25] which is well known alternative scalar tensor theory of gravitation to general relativity theory. Brans Dicke theory is most natural choice as generalization of general relativity because of its simplicity and a possible reduction to general relativity in some limit. It had been shown that BD theory can potentially generate sufficient acceleration in the matter dominated era even without any help from an exotic Q-field. Sing and Debnath [26] show whether the BD parameter w is constant or not, the chaplygin gas provide early deceleration and late time acceleration of the katoresd@rediffmail.com schnhatkar@gmail.com c 204 Chinese Physical Society and IOP Publishing Ltd

2 No. 5 Communications in Theoretical Physics 769 universe. We focus our attention to obtain solutions of anisotropic dark energy. For this, we have assumed relation between and average scale factor a. Some physical properties of the model have also been discussed. 2 Field Equations We consider metrics admitting a group of motions G 4 on V 3, which are locally rotationally symmetric (LRS in the form ds 2 = dt 2 +A 2 (tdx 2 +B 2 (t[dy 2 + S 2 (y, kdz 2 ], ( where A(t, B(t are the cosmic scale functions, S(y, k = sin y, y, sinhy for k =, 0, respectively. Ram and Verma [27] studied model ( with a bulk viscous term and found some exact solutions. Singh and Beesham [2] investigated metric ( in the context of anisotropic dark energy in general relativity. Chandel et al. [28] have investigated hypersurface homogeneous bulk viscous fluid cosmological model with time dependent cosmological term. Hajj Boutros [29] have obtained exact solution of the field equation by constructing method for metric ( in the presence of perfect fluid. Stewart and Ellis [30] have discussed solutions of Einstein field equations in the presence of perfect fluid. Time like hypersurface homogeneous space time with diagonal metric is analyzed by Harness. [3] Homogeneous hypersurface with bulk viscosity in f(r gravity is studied by Sancheti et al. [32] Uggla et al. [33] have introduced a framework, which explains the existence and similarities of most exact solutions of the Einstein equations with a wide range of sources for the class of hypersurface homogeneous space times which admits a Hamiltonian formulations. The class of hypersurface homogeneous includes the astrophysically interesting static spherically symmetric models and the stationary cylindrically symmetric models as well as the spatially homogeneous cosmological models. The Einstein equations reduce to more manageable ordinary differential equations for this class. [34] The hypersurface homogeneous models admit simply transitive or multiply transitive homogeneity groups. There are two types of models. The models which admit a simply transitive three-dimensional homogeneity group are called Bianchi type models. The multiply transitive models which do not admit a simply transitive subgroup are Kantowski Sachs models and the static spherically symmetric models. Among the Bianchi type models there is a special family of Bianchi type I, II, III, V, VII 0,VII h, VIII and IX models which admit multiply transitive symmetric group. Bianchi type cosmological models are important because they are useful for describing the early stage of evolution of the universe. The isotropization of the universe is studied by using these models. They are homogeneous and anisotropic. Bianchi type metric describes spatially ellipsoidal geometry of the universe. Ellis [35] shows that Bianchi universe anisotropies give rise to the cosmological microwave background anisotropies. Camponelli et al. [36] investigated that allowing large scale spatial geometry of the universe to be plane symmetric with eccentricity at decoupling of order 0 2 can bring the quadruple amplitude in accordance with observations without power spectrum of the temperature anisotropy. Also, the Sloan Digital Sky Survey of spiral galaxies indicates that there is a symmetry axis in the large scale geometry of the universe. [37] This motivates us to study metric ( in the context of Brans Dicke theory of gravitation. Many cosmological issues such as universe inflation and its late time behavior, cosmic acceleration, coincidence problem etc are solved by using Brans Dicke theory of gravitation. [3839] Some major advantages and features of the Brans Dicke theory are dynamical gravitational constant, Mach s hypothesis, weak equivalence principal non-minimal interaction of scalar field with geometry, compatibility with the Dirac large number. [2,404] In Literature various version of Brans Dicke theory like generalized and Chameleonic Brans Dicke theory etc are available for different cosmological implications. The modified version of Brans Dicke theory can be obtained by introducing variable Brans Dicke parameter i.e. w( and a self interacting potential form. In a simple Brans Dicke theory the field equations are R ij 2 g ijr = 8π T ij w 2(,i,j 2 g ij,k,k ( ;ij g ij,k ;k, (2 =,k ;k = 8πT (3 + 2w, (3 where w is a dimensionless constant of coupling. The simplest generalization of the EoS parameter of perfect fluid may be to determine the EoS parameter separately on each spatial axis by preserving the diagonal form of the energy momentum tensor in a consistence way with the considered metric. [23] Here we take energy momentum tensor for anisotropic dark energy fluid in the following form T i j = diag[t, T 2 2, T 3 3, T 4 4 ] = diag[p x, p y, p z, ρ], (4 where ρ is the energy density of the fluid p x, p y, and p z are pressures on x, y, and z axes respectively. The anisotropic fluid is characterized by the EoS p = ωρ, where ω is not necessarily constant. Now parametrizing the deviation from isotropy by introducing skewness parameter γ i.e. the deviation from ω respectively on both the y and z axes, which is also not necessarily constant and can be function of the cosmic time t. Then the energy momentum tensor can be written as T i j = diag[ω, ω + γ, ω + γ, ]ρ. (5 The general form of the anisotropy parameter of the expansion for Bianchi type III metric in the presence of a single diagonal imperfect fluid with a dynamically

3 770 Communications in Theoretical Physics Vol. 62 anisotropic equation of state parameter and a dynamical energy density in general relativity is obtained by Akarsu and Kilinc. [5] Singh and Sharma [42] studied spatially homogeneous and anisotropic Bianchi type II space time with variable equation of stage parameter and deceleration parameter in the context of scale covariant theory of gravitation. Yadav et al. [43] found that the isotropic distribution of magnetized dark energy leads to the present acceleration expansion of the universe. Rao et al. [44] investigated Bianchi type II, VIII and IX perfect fluid dark energy cosmological models in Saez Ballester and general relativity theory of gravitation. They found that the EoS parameter may be attributed to the current acceleration expansion of the universe. The energy momentum tensor consists of anisotropic fluid with anisotropic the EoS and a uniform magnetic field of energy density in the context of general relativity studied by Sharif and Zubair. [45] Bianchi type III anisotropic dark energy models with constant deceleration parameter are investigated by Yadav and Yadav. [46] The Brans Dicke field equations for the metric ( with Eqs. (2, (3, and (5 yield 2B 44 B + B2 4 B 2 + k B 2 = 8π ωρ w 2 A 44 A + B 44 B + A 4B 4 AB = 8π (ω + γρ w 2 ( 4 2A 4 B 4 AB + B2 4 B 2 + k B 2 = 8π ρ + w 2 ( B 4 ( 4 B 44, (6 2 ( 4 A4 A + B 4 44 B, (7 2 4 ( A4 A + 2B 4 B, (8 where subscript 4 indicates differentiation with respect to t. Subtracting (6 from (7, we get A 44 A B 44 B + A 4B 4 AB B2 4 = 8π γρ 4 B 2 k B 2 ( A4 A B 4 B. (9 Now the average scale factor and the volume of the universe are defined as Equation (9 further reduces to A 4 A B 4 B = V + V a 3 = V = AB 2. (0 [ k B 2 8πγρ ] V dt, ( where is the constant of integration. The integral term in Eq. ( vanishes for γ = k 8πB 2 ρ. (2 The directional Hubble parameter in the direction of x, y, and z axes respectively are as follows H x = A 4 A, H y = H z = B 4 B. (3 The mean Hubble parameter is given by H = V 4 3 V = ( A4 3 A + 2B 4. (4 B The anisotropy of the expansion is defined as = 3 ( Hi H 2, (5 3 H i= where H i (i =, 2, 3 are the directional Hubble parameters in the direction of x, y, and z axes respectively. Using Eqs. (, (2, (3, and (5, we get = H 2 V 2. (6 For = the value of the mean anisotropy parameter is similar to the value obtained by some other authors [2,42] in general relativity theory. Using Eqs. ( and (2, we obtain A ( = exp B V dt, (7 where is the constant of integration. Now we have the system of three independent equations in six unknown A, B, ρ,, ω, γ therefore we use the power law relation between and average scale factor a. It has been used by Johri and Desikan [47] in the context of Robertson Walker Brans Dicke model. Shamir and Bhatti [7] used it to obtain solution of anisotropic dark energy Bianchi type III cosmological model in Brans Dicke theory of gravity. The power law relation implies that = ba n, (8 where b, n are constant. To get deterministic solution, we consider volumetric expansion V = c e 3mt, (9 V = c t 3m, (20 where c and m are arbitrary positive constant. The model with exponential expansion exhibit accelerating volumetric expansion, where as the model of power law gives constant deceleration parameter q for 0 < m < and accelerated expansion for m >. For m =, the average scale factor has a linear growth with constant velocity and q = 0 that is universe in inflationary phase. 3 Exponential Expansion Model Using Eqs. (7, (8 and (9, we obtain A=(c 2 /3 exp mt e m(n+3t}, (2 3bm(n+3c (n+3/3 ( c /3 B= exp mt+ 3bm(n+3c (n+3/3 e m(n+3t}.(22

4 No. 5 Communications in Theoretical Physics 77 For this model the value of becomes = bc n/3 e mnt. (23 The directional Hubble parameter is given by H x = m + 3bc (n+3/3 H y = H z = m e m(n+3t, (24 3bc (n+3/3 rho = 3bcn/3 8π e m(n+3t. (25 emnt m 2( + n n2 w 6 m(n 2t + kb 8π c2/3 2 c (n2/3 exp It is clear that directional Hubble parameters are constant at t = 0 and when time increases their values decrease. At large time they tend to constant. The mean Hubble parameter H of this model is found to be constant showing that the universe expands with constant value. The energy density of the fluid using Eq. (8 is obtained as 9b 2 (n+3/3 e 2m(n+3t} 3bm(n + 3c (n+3/3 e m(n+3t}. (26 The contribution of to the energy density is negative. For the consistency of the model with real world equation (26 leads to useful restriction on the values of constant. This natural and physical fact restricts the constant to n < 0. The energy density remains positive for the allowed range of constant as shown in Fig.. Initially it is constant and decreases gradually with time. At large time, it tends to zero thus representing an empty universe in the future. In general relativity [2] initially energy density increases for k = whereas in this model of Brans Dicke theory is always decreasing. Thus, the scalar field decreases the energy density. Fig. Energy density versus time for w = 0, m = 2, n = 0.5, b = c = = =. The skewness parameter of the model yields γ = [ k + ( c 2/3 exp 2mt + 3bm(n+3c (n+3/3 k Fig. 2 EoS parameter versus time for w = 0, m = 2, n = 0.5, b = c = = =. e m(n+3t ][3m 2( + n n2 w 2 9b 2 (n+3/3 e m(n+3t ]. (27 The value of skewness parameter shows that γ = 0 for k = 0 and for k 0, γ 0 as t. Therefore the fluid remains isotropic for k = 0 at all stage of evolution of the universe and at large time, it isotropizes for k 0. The deviation free part of the anisotropic equation of state parameter is m 2[ 3 + 2n + n 2( ] } + w 2 k( c /3 exp 2mt e m(n+3t 2 e 2m(n+3t 3bm(n+3c (n+3/3 3b 2 (n+3/3 ω = 3m 2( + n n2 w 6 9b 2 (n+3/3 e 2m(n+3t } + k( c2 c 2/3 exp 2mt 3bm(n+3c (n+3/3 e m(n+3t }. (28 The time dependent of the EoS parameter allows it to transit from ω > to ω <. Recently, some observations suggest that there is some dark energy models whose EoS parameter crosses the phantom divide ω = in the near past. [48] In Fig. 2, we observe that the EoS parameter value fall in the phantom region which means that the universe is phantom dominated at large time. Since the EoS parameter is time dependent and ω <, therefore, the scale factors diverge at a finite future time i.e. the Big Rip. [4950] The mean anisotropic parameter is given by 2 = e 2m(n+3t (29 9b 2 m 2 (n+3/3 The shear scalar of the model is obtained as σ 2 = e 2m(n+3t. (30 3b 2 (n+3/3 From Eq. (29, we observe that at t = 0 the mean

5 772 Communications in Theoretical Physics Vol. 62 anisotropy parameter is not zero i.e. in the early stage the universe found to be anisotropic. As time increases, it tends to zero i.e. the universe tends to isotropy. The shear scalar σ 0 at t = 0 and σ 0 as t i.e. at late time matter has no shear. 4 Power Law Model Using Eqs. (7, (8, and (2, we obtain A = (c 2 /3 t m exp B = ( c /3 t m exp 3b( m(n + 3c (n+3/3 3b( m(n + 3c (n+3/3 t m(n+3}, (3 t m(n+3}. (32 The value of the for this model becomes = bc n/3 t mn. (33 The directional Hubble parameter of the model is obtained as H x = m t + 3bc (n+3/3 H y = H z = m t t m(n+3, (34 3bc (n+3/3 t m(n+3. (35 The values of the directional parameters are infinite at t = 0 and tend to zero as t. The mean Hubble parameter becomes H = m/t which is function of time and H 0 as t i.e. the rate of expansion of the universe is decreasing. The mean anisotropic parameter is found to be 2 = t 22m(n+3. (36 9b 2 m 2 (n+3/3 The shear scalar for this model is given by σ 2 = t 2m(n+3. (37 3b 2 (n+3/3 From Eq. (36, it is observed that mean anisotropic parameters are very large at t = 0 and their value tends to zero as t. This means that initially the universe is anisotropic and it tends to isotropy as time increases. From the value of the shear scalar it is clear that σ 0 as t i.e. at late time matter has no shear. The energy density of the fluid using Eqs. (8, (3, (32, (33 becomes ρ = bcn/3 m 2 (3 + 3n wn2 2 8πt 2mn + kb 8π c2/3 2 c n2/3 t m(n2 exp t m(n+6 24πbc (n+6/3 3b( m(n + 3c (n+3/3 t m(n+3}. (38 The energy density graph shows the nature of the density with respect to the time. Initially it is constant and as time increases it tends to zero which means that the universe approaches to vacuum state at late time for an accelerating universe. It is also observed that in this case scalar field decreases the energy density when compared with general relativity model. [2] The skewness parameter yields γ = m 2 (3+3n n2 w 2 t 2mn k( c 2/3 t m(n2 exp 3b 2 (n+3/3 3b(m(n+3c (n+3/3 t m(n+6 + k( c2 c 2/3 t m(n2 exp } t m(n+3 3b(m(n+3c (n+3/3 t m(n+3 }. (39 The skewness parameter is non zero that means the pressure on the y and z axes are different than axis x. Therefore the fluid remains anisotropic. The deviation free equation of state parameter is given by ω = bc n/3 m(3m+2mn+mn 2 + mn2 w 2 n 2t mn2 2 3bc (n+6/3 t m(n+6 kbc (n2/3 /3 2 t m(n2 exp /3 t exp m(n2 bc n/3 m 2 (3+3n n2 w 2 t 2mn 2 3bc (n+6/3 t m(n+6 + k( 2 c 6n 3b(m(n+3c (n+3/3 } 3b(m(n+3c (n+3/3 t m(n+3 }. t m(n+3 It is observed that the density, skewness parameter, and the EoS parameter are dynamical quantities ρ 0, γ 0, ω as t i.e. the model represents vacuum universe and mathematically equivalent to cosmological constant. The variation of the EoS parameter with time is shown in Fig. 4 with an appropriate choice of constants. We find that at early stage of evolution the EoS parameter was zero i.e. the model behaves as dust fluid at an early stage whereas at late time ω becomes negative. This shows that the expansion of the universe may occur in the quintessence region. (40

6 No. 5 Communications in Theoretical Physics 773 Fig. 3 Energy density versus time for w =, m = 2, n =, b = c = = =. Fig. 4 The EoS parameter versus time for w =, n =, m = 2, b = c = = =. 5 Conclusion We have presented two models of the dark energy. In both model, we observe that at early stage the universe is anisotropic and as time increases it tends to isotropy. At large time, the universe is tending to empty state. The matter has no shear at large time. In case of Exponential law model, it is found that the energy density remains positive for the allowed range of constant. The EoS parameter of the fluid isotropizes and approaches a value in phantom region. In case of power law model, it is observed that the rate of expansion of the universe decreases. The EoS parameter of the fluid evolves into quintessence region. Initially the EoS parameter is zero i.e. the model behaves as dust fluid. The scalar field decreases the energy density. It should be noted that similar conclusion is drawn by Akarsu and Kilinc [5] in the context of Bianchi type III with anisotropic dark energy. It is also interesting to note that the result resembles to Singh and Beesham [2] in general relativity when the scalar field tends to constant. Acknowledgments The authors would like to acknowledge the deep sense of gratitude to the anonymous referees for valuable suggestion for improvement and up gradation of the manuscript. References [] N.A. Bahcall, J.P. Ostriker, S. Perlmutter, and P.J. Steinhardt, Science 284 ( [2] R.R. Caldwell, R. Dave, and P.J. Steinhardt, Phys. Rev. Lett. 80 ( [3] V. Sahni and A.A. Starobinsky, Int. J. Mod. Phys. D 9 ( [4] S. Perlmutter, et al., Nature (London 39 ( [5] S. Perlmutter, G. Aldering, D. Goldhaber, et al., Astrophys. J. 57 ( [6] Riess, et al., Astrono. J. 6 ( [7] Riess, et al., Astrophys. J. 560 ( [8] S.M. Carroll, W.H. Press, and E.L. Turner, Ann. Rev. Astron. and Astrophys. 30 ( [9] M.S.T urner and M. White, Phys. Rev. D 56 ( [0] Z.H. Zhu, M.K. Fujimoto, and D. Tatsumi, Astron. and Astrophys. 372 ( [] B. Ratra and P.J.E. Peeble, Phys. Rev. D 37 ( [2] V. Sahni and L.M. Wang, Phys. Rev. D 62 ( [3] M. Sharif and M. Zubair, Int. J. Mod. Phys. D 9 ( [4] M. Sharif and M. Zubair, Astrophys. Space Sci. 330 ( [5] O. Akarsu and C.B. Kilinc, Gen. Rel. Grav. 42 ( [6] F. Rahaman, B. Bhui, and B.C. Bhui, Astrophys. Space Sci. 30 ( [7] M. Shamir and A.A. Bhatti, arxiv: v[gr-qc] (202. [8] S.D. Katore, Bulg. J. Phys. 39 ( [9] D.R.K. Reddy, B. Satyanarayana, and R.L. Naidu, Astrophys. Space Sci. 339 (202 40, DOI0.007/s [20] S.D. Katore, A.Y. Shaikh, and M.M. Sancheti, Int. J. Math. Archive 3 ( [2] C.P. Singh and A. Beesham, Gravitation and Cosmology 7 ( [22] T. Koivisto and D.F. Mota, J. Cosm. Astrophys ( [23] K.L. Mahanta and A.K. Biswal, Romanian J. Phys. 58 ( [24] H. Amirhashchi, A. Pradhan, and B. Saha, Astrophys. Space Sci. 333 ( [25] C. Brans and R.H. Dicke, Phys. Rev. 24 ( [26] A.K. Singh and U. Debnath, Int. J. Theor. Phys. 50 (20 536, DOI0. 007/s [27] S. Ram and M.K. Verma, Astrophys. Space Sci. 326 ( [28] S. Chandel, M.K. Singh, and S. Ram, Adv. Studies Theor. Phys. 6 ( [29] J. Hajj-Boutros, J. Math. Phys. 26 (

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