SOME EXACT BIANCHI TYPE-I COSMOLOGICAL MODELS IN SCALAR-TENSOR THEORY OF GRAVITATION WITH TIME DEPENDENT DECELERATION PARAMETER

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1 SOME EXACT BIANCHI TYPE-I COSMOLOGICAL MODELS IN SCALAR-TENSOR THEORY OF GRAVITATION WITH TIME DEPENDENT DECELERATION PARAMETER ANIRUDH PRADHAN 1, ANAND SHANKAR DUBEY 2, RAJEEV KUMAR KHARE 3 1 Department of Mathematics,Hindu Post-graduate College, Zamania , Ghazipur, India pradhan@iucaa.ernet.in; pradhan.anirudh@gmail.com 2,3 Department of Mathematics, Sam Higginbottom Institute of Agriculture, Technology & Sciences Allahabad , India asdubey77@yahoo.com 3 Department of Mathematics, Sam Higginbottom Institute of Agriculture, Technology & Sciences Allahabad , India drrajeevkhare@gmail.com Received October 5, 2011 A new class of a spatially homogeneous and anisotropic Bianchi type-i cosmological models of the universe for perfect fluid distribution within the framework of scalar-tensor theory of gravitation proposed by Saez and Ballester (Phys. Lett. 113:467, 1986 is investigated by considering time dependent deceleration parameter. Two different physically viable models of the universe are obtained. The modified Einstein s field equations are solved exactly and the models are in good agreement with recent observations. Some physical and geometric properties of the models are also discussed. Key words: Bianchi type-i universe, exact solution, alternative gravitation theory, variable decceleration parameter. PACS: k. 1. INTRODUCTION General relativity (GR passes all present tests with flying colours, however, there are several reasons why it remains very important to consider alternative theories of gravity. The first one is that theoretical attempts at quantizing gravity or unifying it with other interactions generically predict deviation from Einstein s theory, because gravitation is no longer mediated by a pure spin-2 field but also by partners to the graviton. The second reason is that it is any way extremely instructive to contrast GR s predictions with those of alternative models, even if there were no serious theoretical motivation for them. The third reason is the existence of several puzzling experimental issues, which do not contradict GR in a direct way, but may nevertheless suggest that gravity does not behave at large distances exactly as Newton and Einstein predicted. Cosmological observations notably tell us that about 96% of the total energy density of the universe is composed of unknown, non-baryonic, RJP Rom. 57(Nos. Journ. Phys., 7-8, Vol , Nos. 7-8, (2012 P , (c Bucharest, 2012

2 2 Exact Bianchi type-i cosmological models with time dependent deceleration parameter 1223 fluids (72% of dark energy and 24% of dark matter, and the acceleration of the two Pioneer spacecrafts towards the Sun happens to be larger than what is expected from the 1/r 2 law. Therefore, alternative theories are any way important to study. Since the observed universe is almost homogeneous and isotropic, space-time is usually described by a Friedman-Lemaitre-Robertson-Walker (FLRW cosmology. But it is also believed that in the early universe the FLRW model does not give a correct matter description. The anomalies found in the cosmic microwave background (CMB and the large structure observations stimulated a growing interest in anisotropic cosmological model of the universe. Observations by the Differential Radiometers on NASA s Cosmic Background Explorer registered anisotropy in various angle scales. It is conjectured, that these anisotropies hide in their hearts the entire history of the cosmic evolution down to recombination, and they are considered to be indicative of the universe geometry and the matter composing the universe. It is expected, that much more will be known about anisotropy of cosmic microwave s background after the investigations of the microwave s anisotropy probe. There is a general agreement among cosmologists that cosmic microwave s background anisotropy in the small angle scale holds the key to the formation of the discrete structure. The theoretical argument [1] and the modern experimental data support the existence of an anisotropic phase, which turns into an isotropic one. Scalar-Tensor theories of gravitation provide the natural generalizations of general relativity and also provide a convenient set of representations for the observational limits on possible deviations from general relativity. There are two categories of gravitational theories involving a classical scalar field φ. In first category the scalar field φ has the dimension of the inverse of the gravitational constant G among which the Brans-Decke theory [2] is of considerable importance and the role of the scalar field is confined to its effect on gravitational field equations. In the second category of theories involve a dimensionless scalar field. Saez and Ballester [3] developed a scalar-tensor theory in which the metric is coupled with a dimensionless scalar field in a simple manner. This coupling gives a satisfactory description of the weak fields. In spite of the dimensionless character of the scalar field, an anti-gravity regime appears. This theory suggests a possible way to solve the missing-matter problem in non-flat FRW cosmologies. The Scalar-Tensor theories of gravitation play an important role to remove the graceful exit problem in the inflation era [4]. In earlier literature, cosmological models within the framework of Saez-Ballester scalar-tensor theory of gravitation, have been studied by Singh and Agrawal [5, 6], Ram and Tiwari [7], Singh and Ram [8]. Mohanty and Sahu [9, 10] have studied Bianchi type- VI 0 and Bianchi type-i models in Saez-Ballester theory. Recently, Tripathi et al. [11], Reddy et al. [12,13], Reddy and Naidu [14], Rao et al. [15] [18], Adhav et al. [18], Katore et al. [19], Mohanty and Sahu [20] and Pradhan and Singh [21] have obtained the solutions in Sauz-Ballester Scalar-Tensor theory of gravitation in different

3 1224 Anirudh Pradhan, Anand Shankar Dubey, Rajeev Kumar Khare 3 context. Recently, Kumar and Singh [22] obtained exact Bianchi type-i cosmological models in Saez and Ballester Scalar-Tensor theory of gravitation by assuming the constant deceleration parameter. In literature it is common to use a constant deceleration parameter, as it duly gives a power law for metric function or corresponding quantity. In 1998, published observations of Type Ia supernovae by the High-z Supernova Search Team (Riess et al. [23] followed in 1999 by Supernova Cosmology Project (Perlmutter et al. [24] suggested that the expansion of the universe is accelerating. Recent observations of SNe Ia of high confidence level (Tonry et al. [25]; Riess et al. [26]; Clocchiatti et al. [27] have further confirmed this. Also, the transition redshift from deceleration expansion to accelerated expansion is about 0.5. Now for a Universe which was decelerating in past and accelerating at the present time, the DP must show signature flipping (see the Refs. Padmanabhan and Roychowdhury [28], Amendola [29], Riess et al. [30]. So, there is no scope for a constant DP at present epoch. So, in general, the DP is not a constant but time variable. Motivated by the discussions, in this paper, we have obtained a new class of exact solutions of field equations given by Saez and Ballester [3] in Bianchi type-i space-time by considering a time dependent deceleration parameter. The out line of the paper is as follows: In Section 2, the metric and the field equations are described. Section 3 deals with the solutions of the field equations and its physical significance. Section 4 deals with an other solutions of the field equations and physical and geometric behaviour of the model are also discussed. Finally, conclusions are summarized in the last Section THE METRIC AND FIELD EQUATIONS We consider spatially homogeneous and anisotropic Bianchi type-i space-time given by ds 2 = dt 2 + A 2 dx 2 + B 2 dy 2 + C 2 dz 2, (1 where the metric potentials A, B and C are functions of cosmic time t alone. This ensures that the model is spatially homogeneous. The field equations given by Saez and Ballester [3] for combined scalar and tensor fields are G ij ωφ m ( φ,i φ,j 1 2 g ijφ,k φ,k = 8πT ij, (2 where the scalar field φ satisfies the equation 2φ m φ,i ;i + mφ,kφ,k = 0, (3

4 4 Exact Bianchi type-i cosmological models with time dependent deceleration parameter 1225 where G ij = R ij 1 2 Rg ij is the Einstein tensor; ω and m are constants; T ij is the stress-energy tensor of the matter; comma and semicolon denotes partial and covariant differentiation, respectively. The energy-momentum tensor T ij for a perfect fluid has the form T ij = (p + ρu i u j pg ij, (4 where p is the thermodynamical pressure, ρ the energy density, u i the four-velocity of the fluid satisfying g ij u i u j = 1. (5 In co-moving system of coordinates, we have u i = (0,0,0,1. For the energy momentum tensor (4 and Bianchi type-i space-time (1, Einstein s modified field equations (2 and (3 yield the following five independent equations Ä A + B B + AḂ AB = 8πp ωφm φ2, (6 Ä A + C C + AĊ AC = 8πp ωφm φ2, (7 B B + C C + ḂĊ BC = 8πp ωφm φ2, (8 ȦḂ AB + AĊ AC + ḂĊ BC = 8πρ 1 2 ωφm φ2, (9 ( φ + φ A A + Ḃ B + Ċ + m φ 2 = 0, (10 C 2φ where an over dot denotes derivative with respect to cosmic time t. The law of energy-conservation equation (T ij ;j = 0 gives ( A ρ + (ρ + p A + Ḃ B + Ċ = 0. (11 C It is worth noting here that our approach suffers from a lack of Lagrangian approach. There is no known way to present a consistent Lagrangian model satisfying the necessary conditions discussed in this paper. The spatial volume for the model (1 is given by V 3 = ABC. (12 We define a = (ABC 1 3 as the average scale factor so that the Hubble s parameter is anisotropic and may be defined as ( H = ȧ a = 1 A 3 A + Ḃ B + Ċ. (13 C

5 1226 Anirudh Pradhan, Anand Shankar Dubey, Rajeev Kumar Khare 5 We also have H = 1 3 (H x + H y + H z, (14 where H x = Ȧ A, H y = Ḃ B and H z = Ċ C. The deceleration parameter q, the scalar expansion θ, shear scalar σ 2 and the average anisotropy parameter A m are defined by q = aä ȧ 2, (15 A θ = A + Ḃ B + Ċ C, (16 ( 3 σ 2 = 1 Hi 2, θ2 (17 i=1 3 ( 2 Hi, (18 A m = 1 3 where H i = H i H(i = x,y,z. i=1 H 3. SOLUTIONS OF THE FIELD EQUATIONS The field equations (6-(10 are five equations necessitating six unknowns A, B, C, p, ρ and φ. One additional constraint relating these parameters are required to obtain explicit solutions of the system. We consider the deceleration parameter as time dependent to get the deterministic solution. We define the deceleration parameter q as (Ḣ q = aä + H 2 ȧ 2 = H 2 = b(t say. (19 As discussed in previous section of introduction, the deceleration parameter is considered as time dependent. The equation (19 may be rewritten as ä bȧ2 a + = 0. (20 a2 In order to solve the Eq. (20, we assume b = b(a. It is important to note here that one can assume b = b(t = b(a(t, as a is also a time dependent function. It can be done only if there is a one to one correspondences between t and a. But this is only possible when one avoid singularity like big bang or big rip because both t and a are increasing function.

6 6 Exact Bianchi type-i cosmological models with time dependent deceleration parameter 1227 The general solution of Eq. (20 with assumption b = b(a, is given by e b a da = t + m, (21 where m is an integrating constant. One can not solve Eq. (21 in general as b is variable. So, in order to solve the problem completely, we have to choose b ada in such a manner that Eq. (21 be integrable without any loss of generality. Hence we consider b da = lnl(a. (22 a which does not affect the nature of generality of solution. Hence from Eqs. (21 and (22, we obtain L(ada = t + m. (23 Of course the choice of L(a, in Eq. (23, is quite arbitrary but, since we are looking for physically viable models of the universe consistent with observations, we consider 1 L(a = α 1 + a, (24 2 where α is an arbitrary constant. In this case, on integrating, Eq. (23 gives the exact solution a(t = sinh(αt. (25 where T = t + m. Recentlly, relation (25 is also used by Amirhashchi et al. [31] to study the evolution of dark energy models in a spatially homogeneous and isotropic FRW space-time filled with barotropic fluid and dark energy by considering a time dependent deceleration parameter. Substituting (25 into (15, we obtain q = tanh 2 (αt, (26 which is a variable deceleration parameter. Now subtracting (6 from (7, (6 from (8, (7 from (8 and taking second integral of each expression, we obtain the following three relations, respectively A B = d 1 exp (x 1 a 3 dt, (27 A C = d 2 exp (x 2 a 3 dt, (28 B C = d 3 exp (x 3 a 3 dt, (29 where d 1, d 2, d 3, x 1, x 2 and x 3 are integrating constants.

7 1228 Anirudh Pradhan, Anand Shankar Dubey, Rajeev Kumar Khare 7 From above Eqs. (27-(29, the metric functions A(t, B(t and C(t are explicitly obtained as A(t = a 1 aexp (b 1 a 3 dt, (30 B(t = a 2 aexp (b 2 C(t = a 3 aexp (b 3 a 3 dt, (31 a 3 dt, (32 where a 1 = (d 1 d 2 1 3, a2 = ( d3 d 1 1 3, a3 = (d 2 d 3 1 3,b1 = 1 3 (x 1 + x 2, b 2 = 1 3 (x 3 x 1, b 3 = 1 3 (x 2 + x 3. It is worth mentioned here that these constants also satisfy the following two conditions The second integral of (10 leads to [ k(m + 2 φ(t = 2 a 1 a 2 a 3 = 1, b 1 + b 2 + b 3 = 0. (33 ] 2 a 3 (m+2 dt, (34 where k is a constant due to first integral while the constant of second integral is taken as zero for simplicity without any loss of generality. Using Eq. (25 in (30-(32, we get the following expressions for metric coefficients A = a 1 sinh(αt exp (b 1 B = a 2 sinh(αt exp (b 2 C = a 3 sinh(αt exp (b 3 cosech 3 (αtdt, (35 cosech 3 (αtdt, (36 cosech 3 (αtdt. (37 Again substituting (25 in (34, the scalar field is obtained as [ ] 2 k(m + 2 φ = cosech 3 (m+2 (αtdt, (38 2 Eq. (38 implies that φ m φ2 = k 2 cosech 6 (αt (39

8 8 Exact Bianchi type-i cosmological models with time dependent deceleration parameter 1229 Substituting (35-(39 in (8 and (9, the expressions for isotropic pressure (p and energy density (ρ for the model are obtained as ( 1 p = 2 ωk2 β 3 cosech 6 (αt α 2 { 2 + coth 2 (αt }, (40 ( 1 ρ = 3α 2 coth 2 (αt + 2 ωk2 + β 2 cosech 6 (αt, (41 where β 1 = b b b 2 3, β 2 = b 1 b 2 + b 2 b 3 + b 3 b 1, β 3 = b b b 1 b 2. (42 In view of (33, it is observed that the solutions given by (35-(42 satisfy the energy conservation equation (11 identically and hence represent exact solutions of the Einstein s modified field equations (6-(10. It is evident that the energy conditions ρ 0 and p 0 are satisfied under the appropriate choice of constants. Due to exponential nature of scale factor we are not in position to derive analytical conditions exactly but apparent that such feature are there in present study. From Eq. (41, it is observed that ρ is a positive decreasing function of time and it approaches to zero as T. This behaviour is clearly depicted in Figure 1 as a representative case with appropriate choice of constants of integrations and other physical parameters using reasonably well known situations. The rate of expansion H i in the direction of x, y, and z read as H x = αcoth(αt + b 1 cosech 3 (αt, (43 H y = αcoth(αt + b 2 cosech 3 (αt, (44 H z = αcoth(αt + b 3 cosech 3 (αt. (45 The Hubble parameter, expansion scalar and shear of the model are, respectively given by H = αcoth(αt, (46 θ = 3αcoth(αT, (47 σ 2 = 1 2 β 1cosech 6 (αt. (48 The spatial volume (V and anisotropy parameter (A m are found to be V = sinh 3 (αt, (49 A m = 1 3 β cosech 6 (αt 1 α 2 coth 2 (αt. (50 From the above results, it can be seen that the spatial volume is zero at T = 0, and it increases with the cosmic time. The parameter H i, H, θ, and σ diverge at the

9 1230 Anirudh Pradhan, Anand Shankar Dubey, Rajeev Kumar Khare 9 Fig. 1 The plot of energy density ρ versus T. Here α = 1, k = β 2 = 1. Fig. 2 The plot of anisotropic parameter A m versus T. Here α = 1. initial singularity. There is a Point Type singularity (MacCallum [31] at T = 0 in

10 10 Exact Bianchi type-i cosmological models with time dependent deceleration parameter 1231 the model. The dynamics of the mean anisotropic parameter depends on the constant β 1 = b 2 1 +b2 2 +b2 3. From Eq. (50, we observe that at late time when T, A m 0. Thus, our model has transition from initial anisotropy to isotropy at present epoch which is in good harmony with current observations. Figure 2 depicts the variation of anisotropic parameter (A m versus cosmic time T. From the figure, we observe that A m decreases with time and tends to zero as T. Thus, the observed isotropy of the universe can be achieved in our model at present epoch. From Eq. (26, it can be seen that for T, q = 1 and for T 0, q = 0. Hence our universe is in accelerating phase during the evolution of the universe which is consistent with recent observations of Type Ia supernovae (Riess et al. [23]; Perlmutter et al. [24]; Tonry et al. [25]; Riess et al. [26]; Clocchiatti et al. [27]. 4. OTHER EXACT SOLUTION OF THE FIELD EQUATIONS AND THEIR PHYSICAL ASPECTS For a suitable choice of L(a, Eq. (23 gives after integration the exact solution for the scale factor, where the increase in terms of time evolution is We define the deceleration parameter q as usual, Using (51 into (52, we find a(t = te t. (51 q = äa ȧ 2 = ä ah 2. (52 q = (1 + t 2. (53 Using Eq. (25 in (30-(32, we get the following expressions for metric coefficients A = a 1 (te t exp (b 1 (te t 3 dt, (54 B = a 2 (te t exp (b 2 C = a 3 (te t exp (b 3 (te t 3 dt (te t 3 dt, (55. (56 Again substituting (25 in (34, the scalar field is obtained as [ ] 2 k(m + 2 φ = (te t 3 (m+2 dt, (57 2 Eq. (57 implies that φ m φ2 = k 2 (te t 6 (58

11 1232 Anirudh Pradhan, Anand Shankar Dubey, Rajeev Kumar Khare 11 Substituting (54-(58 in (8 and (9, the expressions for isotropic pressure (p and energy density (ρ for the model are obtained as 2 3(t + 12 p = t 2 + ωk2 2β 3 2(te t 6, (59 ( t + 1 ρ = 3 + ωk2 + 2β 2 t 2(te t 6. (60 where β 2 and β 3 are already defined in previous section. Fig. 3 The plot of energy density ρ versus t. Here k = β 2 = 1. In view of (33, it is observed that the solutions given by (54-(60 satisfy the energy conservation equation (11 identically and hence represent exact solutions of the Einstein s modified field equations (6-(10. It is evident that the energy conditions ρ 0 and p 0 are satisfied under the appropriate choice of constants. From Eq. (60, it is observed that ρ is a positive decreasing function of time and it approaches to zero as t 0. Figure 3 shows this behaviour of ρ. The rate of expansion H i in the direction of x, y, and z read as ( t + 1 H x = + b 1 (te t 3, (61 t ( t + 1 H y = + b 2 (te t 3, (62 t

12 12 Exact Bianchi type-i cosmological models with time dependent deceleration parameter 1233 Fig. 4 The plot of anisotropic parameter A m versus t. Here β 1 = 1. ( t + 1 H z = + b 3 (te t 3. (63 t The Hubble parameter, expansion scalar and shear of the model are, respectively given by ( t + 1 H = (64 t ( t + 1 θ = 3, (65 t σ 2 = 1 2 β 1(te t 6 (66 The spatial volume (V and anisotropy parameter (A m are found to be A m = 1 3 β 1 t where β 1 is already defined in previous section. V = (te t 3, (67 ( t (te t 6, (68 From the above results, it can be seen that the spatial volume is zero at t = 0, and it increases with the cosmic time. The parameter H i, H, θ, and σ diverge at the initial singularity. There is a Point Type singularity (MacCallum [32] at t = 0 in the model. The mean anisotropic parameter is a decreasing function of time. Figure

13 1234 Anirudh Pradhan, Anand Shankar Dubey, Rajeev Kumar Khare 13 4 plots the variation of anisotropic parameter (A m versus cosmic time t. From the figure, we observe that A m decreases with time and tends to zero as t. Thus, the observed isotropy of the universe can be achieved in our model at present epoch. 5. CONCLUDING REMARKS In this paper we have studied a spatially homogeneous and anisotropic Bianchi type-i space-time within the framework of the scalar-tensor theory of gravitation proposed by Saez and Ballester [3]. To find the deterministic solution, we have considered a time dependent deceleration parameter. In the first case, it is observed that as T, q = 1. This is the case of de Sitter universe. For T 0, q = 0. This shows that in the early stage the universe was decelerating where as the universe is accelerating at present epoch which is corroborated from the recent supernovae Ia observation (Riess et al. [23]; Perlmutter et al. [24]; Tonry et al. [25]; Riess et al. [26]; Clocchiatti et al. [27]. The parameter H i, H, θ, and σ diverge at the initial singularity. There is a Point Type singularity (MacCallum [32] at T = 0 in the model. The rate of expansion slows down and finally tends to zero as T 0. The pressure, energy density and scalar field become negligible where as the scale factors and spatial volume become infinitely large as T, which would give essentially an empty universe. We also obtain similar type of characteristic of the model described in second case (Section 4. Both the models represent expanding, shearing and non-rotating universe, which approach to isotropy for large value of t. This is consistent with the behaviour of the present universe as already discussed in introduction. The scalar field decreases to zero as t in both models. If we set k = 0, then the solutions reduce to the solution in general relativity and the Saez-Ballester scalar-tensor theory of gravitation tends to standard general theory of relativity in every respect. In literature we can get the solutions of the field equations of Saez-Ballester scalar-tensor theory of gravitation by using a constant deceleration parameter. So the solutions presented in this paper are new and different from other author s solutions. Our solutions may be useful for better understanding of the evolution of the universe in Bianchi-I space-time within the framework of Saez-Ballester scalar-tensor theory of gravitation. Acknowledgments. Author (AP would like to thank the Inter-University Centre for Astronomy and Astrophysics (IUCAA, Pune, India for providing facility and support where part of this work was carried out.

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