Anisotropic dark energy Bianchi type III cosmological models in the Brans–Dicke theory of gravity Muhammad Farasat Shamir and Akhlaq Ahmad Bhatti Abstract: The main purpose of this paper is to explore the solutions of Bianchi type III cosmological model in Brans–Dicke theory of gravity in the background of anisotropic dark energy. We use the assumption of constant deceleration parameter and power law relation between scalar field f and scale factor a to find the solutions. The physical behavior of the solutions has been discussed using some physical quantities. PACS Nos: 04.50.Kd, 98.80.–k, 98.80.Es Résumé : Le but premier de ce papier est d’explorer les solutions du modèle cosmologique de Bianchi de type III en théo- rie de la gravité de Brans Dickie dans le fond d’énergie sombre anisotrope. Afin de trouver les solutions, nous supposons que le paramètre d’accélération est constant et qu’il existe une relation en loi de puissance entre le champ scalaire f et le facteur d’échelle a. Nous analysons le comportement physique des solutions à l’aide de quelques quantités physiques. [Traduit par la Rédaction] 1. Introduction Recent experimental data [1] about late time accelerated expansion of the universe have attracted much attention in re- cent years. Cosmic acceleration can be well explained from high red shift supernova experiments. The recent results from cosmic microwave background fluctuations [2] and large scale structure [3] suggest the expansion of the uni- verse. Dark energy seems to be the best candidate to explain cosmic acceleration. It is now believed that 96% of the en- ergy of the universe consists of dark energy and dark matter (76% dark energy and 20% dark matter) [2, 4]. Dark energy is the most popular way to explain recent observations that the universe is expanding at an accelerating rate. The exact nature of the dark energy is a matter of speculation. It is known to be very homogeneous, not very dense and is not known to interact through any of the fundamental forces other than gravity. Because it is not very dense, roughly 10– 29 g/cm3, it is difficult to detect it in the laboratory. It is thought that dark energy has a strong negative pressure to ex- plain the observed acceleration in the expansion rate of the universe. Dark energy models have significant importance now as far as theoretical study of the universe is concerned. It would be more interesting to study the variable equation of state (EoS), that is, P = ru(t), where P is the pressure and r is the energy density of the universe. Usually the EoS parame- ter is assumed to be a constant with the values –1, 0, –(1/3), and +1 for vacuum, dust, radiation, and stiff matter domi- nated universes, respectively. However, it is a function of time or red shift [5] in general. Latest observations [6] from supernova Ia data indicate that u is not constant. In recent years, many authors [7–11] have shown keen interest in studying the universe with variable EoS. Sharif and Zubair [7] discussed the dynamics of Bianchi type VI0 universe with anisotropic dark energy in the presence of an electro- magnetic field. The same authors [8] explored Bianchi type I universe in the presence of magnetized anisotropic dark en- ergy with variable EoS parameter. Akarsu and Kilinc [9] in- vestigated the general form of the anisotropy parameter of the expansion for Bianchi type III model. The isotropic models are considered to be most suitable to study the large scale structure of the universe. However, it is believed that the early universe may not have been exactly uniform. This prediction motivates us to describe the early stages of the universe with the models having anisotropic background. Thus, it would be worthwhile to explore aniso- tropic dark energy models in the context of modified theories of gravity. Among the various modifications of general rela- tivity (GR), the Brans–Dicke (BD) theory of gravity [12] is a well-known example of a scalar–tensor theory in which the gravitational interaction involves a scalar field and the metric tensor. One extra parameter, 6, is used in this theory, which satisfies &f ¼ 8pT 3þ 26 where f is known as BD scalar field while T is the trace of the matter energy–momentum tensor. It is mentioned here that general relativity is recovered in the limiting case 6 → Received 9 November 2011. Accepted 6 January 2012. Published at www.nrcresearchpress.com/cjp on 7 February 2012. M.F. Shamir and A.A. Bhatti. Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore-54770, Pakistan. Corresponding author: M. Farasat Shamir (e-mail:
[email protected]). 193 Can. J. Phys. 90: 193–198 (2012) doi:10.1139/P2012-007 Published by NRC Research Press C an . J . P hy s. D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y T ex as A & M U ni ve rs ity o n 11 /1 4/ 14 Fo r pe rs on al u se o nl y. ∞. Thus we can compare our results with experimental tests for significantly large values of 6. Bianchi type models are among the simplest models with anisotropic background. Many authors [13–24] have explored Bianchi type space–times in different contexts. Moussiaux et al. [25] investigated the exact solution for vacuum Bianchi type III model with a cosmological constant. Xing-Xiang [26] discussed Bianchi type III string cosmology with bulk viscosity. He assumed that the expansion scalar is propor- tional to the shear scalar to find the solutions. Wang [27] in- vestigated string cosmological models with bulk viscosity in Kantowski–Sachs space–time. Upadhaya [28] explored some magnetized Bianchi type III massive string cosmological models in GR. Hellaby [29] gave an overview of some recent developments in inhomogeneous models and it was con- cluded that the universe is inhomogeneous on many scales. Sharif and Shamir [17, 18] have studied the solutions of Bianchi types I and V space–times in the framework of f(R) gravity. Recently, we [19, 20] explored the exact vacuum sol- utions of Bianchi types I, III, and Kantowski–Sachs space– times in the metric version of f(R) gravity. The study of Bianchi type models in the context of BD theory has attracted many authors in recent years [30]. A de- tailed discussion of BD cosmology is given by Singh and Rai [31]. Lorenz-Petzold [32] studied exact Bianchi type III solu- tions in the presence of electromagnetic field. Kumar and Singh [33] investigated perfect fluid solutions using Bianchi type I space–time in scalar–tensor theory. Adhav et al. [34] obtained an exact solution of the vacuum BD field equations for the metric tensor of a spatially homogeneous and aniso- tropic model. Pradhan and Amirhashchi [35] investigated anisotropic dark energy Bianchi type III model with variable EoS parameter in GR. Adhav et al. [36] explored Bianchi type III cosmological model with negative constant decelera- tion parameter in BD theory of gravity in the presence of perfect fluid. In this paper, we focus our attention on exploring the sol- utions of anisotropic dark energy Bianchi type III cosmologi- cal model in the context of BD theory of gravity. We find the solutions using the assumption of constant deceleration pa- rameter and power law relation between f and a. The paper is organized as follows. A brief introduction of the field equa- tions in BD theory of gravity is given in Sect. 2. In Sect. 3, the solutions of the field equations for Bianchi types III space– time are found. Some physical and kinematic parameters are also evaluated for the solutions. Singularity analysis is given in Sect. 4. A brief summary is given in the last section. 2. Some basics of BD theory of gravity The line element for the spatially homogeneous and aniso- tropic Bianchi type III space–time is given by ds2 ¼ dt2 � A2ðtÞdx2 � e�2mxB2ðtÞdy2 � C2ðtÞdz2 ð1Þ where A, B, and C are cosmic scale factors, and m is a posi- tive constant. The energy–momentum tensor for anisotropic dark energy is given by Tij ¼ diag½r;�px;�py;�pz� ¼ diag½1;�ux;�uy;�uz�r ð2Þ where r is the energy density of the fluid; px, py, and pz are the pressures on the x, y, and z axes, respectively; u is the EoS parameter of the fluid with no deviation; and ux, uy, and uz are the EoS parameters in the directions of x, y, and z axes, respectively. The energy–momentum tensor can be parameterized as Tij ¼ diag½1;�u;�ðuþ gÞ;�ðuþ dÞ�r ð3Þ For the sake of simplicity, we choose ux = u and the skewness parameters g and d are the deviations from u on y and z axes, respectively. The BD field equations are Rij � 1 2 Rgij � 6 f2 f;if;j � 1 2 gijf;kf ;k � � � 1 f ðf;ij � gij&fÞ ¼ 8pTij f ð4Þ and &f ¼ f;k;k ¼ 8pT 3þ 26 ð5Þ where 6 is a dimensionless coupling constant. For Bianchi type III space–time, the field equations take the form _A _B AB þ _B _C BC þ _C _A CA � m 2 A2 �6 2 _f f � �2 þ _f f _A A þ _B B þ _C C � � ¼ 8pr f ð6Þ €B B þ €C C þ _B _C BC þ €f f þ6 2 _f f � �2 þ _f f _B B þ _C C � � ¼ � 8pur f ð7Þ €C C þ €A A þ _C _A CA þ €f f þ6 2 _f f � �2 þ _f f _C C þ _A A � � ¼ � 8pðuþ gÞr f ð8Þ €A A þ €B B � m 2 A2 þ _A _B AB þ €f f þ6 2 _f f � �2 þ _f f _A A þ _B B � � ¼ � 8pðuþ dÞr f ð9Þ Also, the 01-component can be written in the following form: _A A � _B B ¼ 0 ð10Þ Integrating this equation, we obtain B ¼ c1A ð11Þ where c1 is an integration constant. Without loss of any gen- erality, we take c1 = 1. Using (5), we get 194 Can. J. Phys. Vol. 90, 2012 Published by NRC Research Press C an . J . P hy s. D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y T ex as A & M U ni ve rs ity o n 11 /1 4/ 14 Fo r pe rs on al u se o nl y. €f þ _f _A A þ _B B þ _C C � � ¼ 8pð1� 3u� d� gÞr fð3þ 26Þ ð12Þ Now we define some physical parameters before solving the field equations. The average scale factor, a, and the volume scale factor, V, are defined as a ¼ ffiffiffiffiffiffiffiffiffi A2C 3 p V ¼ a3 ¼ A2C ð13Þ The generalized mean Hubble parameter H is given in the form H ¼ 1 3 ðH1 þ H2 þ H3Þ ð14Þ where H1 ¼ _A A ¼ H2 H3 ¼ _C C are the directional Hubble parameters in the directions of x, y, and z axes, respectively. Using (13) and (14), we obtain H ¼ 1 3 _V V ¼ 1 3 ðH1 þ H2 þ H3Þ ¼ _a a ð15Þ The expansion scalar q and shear scalar s are defined as follows: q ¼ um;m ¼ 2 _A A þ _C C ð16Þ s2 ¼ 1 2 smns mn ¼ 1 3 _A A � _C C � �2 ð17Þ where smn ¼ 1 2 ðum;ahan þ un;ahamÞ � 1 3 qhmn ð18Þ hmv = gmn – umun is the projection tensor while um = (g00)1/2 (1, 0, 0, 0) is the four-velocity in comoving coordinates. The mean anisotropy parameter Am is defined as Am ¼ 1 3 X DHi H � �2 ð19Þ where DHi = Hi – H, (i = 1, 2, 3). 3. Solution of the field equations Subtracting (7) from (8), we get €B B � €A A þ _C C _B B � _A A � � þ _f f _B B � _A A � � ¼ 8pgr f ð20Þ Using (11), this equation gives g = 0. Thus (6)–(9) and (12) reduce to _A A � �2 þ 2 _C _A CA � m 2 A2 �6 2 _f f � �2 þ _f f 2 _A A þ _C C � � ¼ 8pr f ð21Þ €C C þ €A A þ _C _A CA þ €f f þ6 2 _f f � �2 þ _f f _C C þ _A A � � ¼ � 8pur f ð22Þ 2 €A A þ _A A � �2 � m 2 A2 þ €f f þ6 2 _f f � �2 þ 2 _f _A fA ¼ � 8pðuþ dÞr f ð23Þ €f þ _f 2 _A A þ _C C � � ¼ 8pð1� 3u� dÞr fð3þ 26Þ ð24Þ Integration after subtracting (22) from (23) yields _A A � _C C ¼ 1 A2Cf Z m2 A2 � 8pdr f � � fA2Cdt þ l A2Cf ð25Þ where l is an integration constant. The integral term in this equation vanishes for d ¼ m 2f 8prA2 ð26Þ Using (26) in (25), it follows that A C ¼ c2 exp l Z dt a3f � � ð27Þ where a3 = A2C and c2 is an integration constant. Here we use the power law assumption to solve the integral part in the preceding equations. The power law relation between scale factor a and scalar field f has already been used by Johri and Desikan [37] in the context of Robertson–Walker– BD models. Thus the power law relation between f and a, that is, f ∝ am, where m is any integer, implies that f ¼ bam ð28Þ where b is the constant of proportionality. The deceleration parameter, q, in cosmology is the measure of the cosmic ac- celeration of the universe expansion and is defined as q ¼ � €aa _a2 ð29Þ It is mentioned here that q was supposed to be positive in- itially but recent observations from the supernova experi- ments suggest that it is negative. Thus, the behavior of the universe models depend upon the sign of q. The positive de- celeration parameter corresponds to a decelerating model while the negative value provides inflation. We also use a well-known relation [38] between the average Hubble param- eter H and average scale factor a given as H ¼ la�n ð30Þ where l > 0 and n ≥ 0. This is an important relation because it gives the constant value of the deceleration parameter. From (15) and (30), we get Shamir and Bhatti 195 Published by NRC Research Press C an . J . P hy s. D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y T ex as A & M U ni ve rs ity o n 11 /1 4/ 14 Fo r pe rs on al u se o nl y. _a ¼ la�nþ1 ð31Þ Using this value, we find that the deceleration parameter is constant (i.e., q = n – 1). Integrating (31), it follows that a ¼ ðnlt þ k1Þ1=n n 6¼ 0 ð32Þ and a ¼ k2 expðltÞ n ¼ 0 ð33Þ where k1 and k2 are constants of integration. Thus we obtain two values of the average scale factor that correspond to two different models of the universe. 3.1 Dark energy model of the universe when n ≠ 0 Now we discuss the model of the universe when n ≠ 0 (i.e., a = (nlt + k1)1/n). For this model, f becomes f ¼ bðnlt þ k1Þ�ð2=nÞ ð34Þ Using this value of f in (27), the metric coefficients A, B, and C turn out to be A ¼ B ¼ c1=32 ðnlt þ k1Þ1=n exp lðnlt þ k1Þðn�1Þ=n 3bðn� 1Þ � � n 6¼ 1 ð35Þ C ¼ c�2=32 ðnlt þ k1Þ1=n exp lðnlt þ k1Þðn�1Þ=n 3bðn� 1Þ � � n 6¼ 1 ð36Þ The directional Hubble parameters, Hi (i = 1, 2, 3), take the form H1 ¼ H2 ¼ l nlt þ k1 þ ll 3bðnlt þ k1Þ1=n ð37Þ H3 ¼ l nlt þ k1 � 2ll 3bðnlt þ k1Þ1=n ð38Þ The mean generalized Hubble parameter becomes H ¼ l nlt þ k1 ð39Þ while the volume scale factor turns out to be V ¼ ðnlt þ k1Þ3=n ð40Þ The expansion scalar, q, and shear scalar, s, take the form q ¼ 3l nlt þ k1 ð41Þ s2 ¼ l 2l2 3b2ðnlt þ k1Þ2=n ð42Þ The mean anisotropy parameter Am becomes Am ¼ 2l 2 9b2 ðnlt þ k1Þ2�ð2=nÞ ð43Þ Moreover, (21)–(23) take the form 8pr f ¼ �l2ð3þ 26Þðnlt þ k1Þ�2 � l 2l2 3b2 þ m 2 c2=32 exp �2lðnlt þ k1Þðn�1Þ=n 3bðn� 1Þ � �� � � ðnlt þ k1Þ�ð2=nÞ ð44Þ � 8pur f ¼ l2 � ð3þ 2nþ 26Þðnlt þ k1Þ�2 þ l 2 3b2 ðnlt þ k1Þ�ð2=nÞ � � ð45Þ � 8pðuþ dÞr f ¼ l2ð3þ 2nþ 26Þðnlt þ k1Þ�2 þ l 2l2 3b2 � m 2 c2=32 exp �2lðnlt þ k1Þðn�1Þ=n 3bðn� 1Þ � �� � � ðnlt þ k1Þ�ð2=nÞ ð46Þ 3.2 Dark energy model of the universe when n = 0 The average scale factor for this model of the universe is a = k2 exp(lt) and hence f takes the form f ¼ b k22 expð�2ltÞ ð47Þ Inserting this value of f into (27), the metric coefficients A, B, and C become A ¼ B ¼ c1=32 k2 expðltÞ exp � l expð�ltÞ 3blk2 � � ð48Þ C ¼ c�2=32 k2 expðltÞ exp 2l expð�ltÞ 3blk2 � � ð49Þ The directional Hubble parameters Hi become H1 ¼ H2 ¼ lþ l expð�ltÞ 3bk2 ð50Þ H3 ¼ l� 2l expð�ltÞ 3bk2 ð51Þ while the mean generalized Hubble parameter and the vo- lume scale factor turn out to be H ¼ l V ¼ k32 expð3ltÞ ð52Þ The expansion scalar, q, and shear scalar, s, take the form q ¼ 3l s2 ¼ l 2 expð�2ltÞ 3b2k22 ð53Þ The mean anisotropy parameter, Am, for this model be- comes Am ¼ 2l 2 expð�2ltÞ 9b2l2k22 ð54Þ 196 Can. J. Phys. Vol. 90, 2012 Published by NRC Research Press C an . J . P hy s. D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y T ex as A & M U ni ve rs ity o n 11 /1 4/ 14 Fo r pe rs on al u se o nl y. For this exponential model of the universe, (21)–(23) take the form 8pr f ¼ �l2ð3þ 26Þ � l 2 expð�2ltÞ 3b2k22 � m 2 c2=32 k 2 2 exp �2lt þ 2l expð�ltÞ 3blk2 � � ð55Þ � 8pur f ¼ l2ð3þ 26Þ þ l 2 expð�2ltÞ 3b2k22 ð56Þ � 8pðuþ dÞr f ¼ l2ð3þ 26Þ þ l 2 expð�2ltÞ 3b2k22 � m 2 c2=32 k 2 2 exp �2lt þ 2l expð�ltÞ 3blk2 � � ð57Þ 4. Singularity analysis The Riemann tensor is a useful tool to determine whether a singularity is essential or coordinate. If the curvature be- comes infinite at a certain point, then the singularity is essen- tial. We can construct different scalars from the Riemann tensor and thus it can be verified whether they become infin- ite somewhere or not. An infinite number of scalars can be constructed from the Riemann tensor, however, symmetry considerations can be used to show that there are only a fi- nite number of independent scalars. All others can be ex- pressed in terms of these. In a four-dimensional Riemann space–time, there are only 14 independent curvature invari- ants. Some of these are R1 ¼ R ¼ gabRab R2 ¼ RabRab R3 ¼ RabcdRabcd R4 ¼ RabcdRcdab Here we give the analysis for the first invariant commonly known as the Ricci scalar for both models. For the model of the universe with a power law expansion, we can write Ricci scalar R ¼ �2 l 2ð6� 3nÞ a2n þ l 2l2 3b2a2 � � m 2 c2=32 a 2 exp 2lan�1=3bðn� 1Þ� ð58Þ while for the exponential model, it is given by R ¼ �2 6l2 þ l 2 3b2a2 � m 2 c2=32 a 2 expð2la=3blÞ � � ð59Þ Both of these models show that a singularity occurs at a = 0. 5. Concluding remarks This paper is devoted to exploring the solutions of Bianchi type III cosmological models in BD theory of gravitation in the background of anisotropic dark energy. We use the power law relation between f and a to find the solution. The as- sumption of constant deceleration parameter leads to two models of the universe, that is, the power law and exponen- tial models. Some important cosmological physical parame- ters for the solutions, such as expansion scalar q, shear scalar s2, mean anisotropy parameter, and average Hubble parame- ter, are evaluated. First we discuss the power law model of the universe. This model corresponds to n ≠ 0 with average scale factor a = (nlt + k1)1/n. It has a point singularity at t ≡ ts = –(k1/nl). The physical parameters H1, H2, H3, and H are all infinite at this point but the volume scale factor vanishes here. The met- ric functions A, B, and C vanish at this point of singularity. Thus, it is concluded from these observations that the model starts its expansion with zero volume at t = ts and it contin- ues to expand for 0 < n < 1. The exponential model of the universe corresponds to n = 0 with average scale factor a = k2 exp(lt). It is nonsin- gular because an exponential function is never zero and hence there does not exist any physical singularity for this model. The physical parameters H1, H2, and H3 are all fi- nite for all finite values of t. The mean generalized Hubble parameter, H, is constant while metric functions A, B, and C do not vanish for this model. The volume scale factor increases exponentially with time, which indicates that the universe starts its expansion with zero volume from the in- finite past. The isotropy condition (i.e., s2/q → 0 as t → ∞) is also satisfied in each case. It is mentioned here that the behavior of these physical parameters is consistent with the results al- ready obtained in GR [7]. The variable EoS parameter u for both models turn out to be u ¼ l 2 ð3þ 2nþ 26Þðnlt þ k1Þ�2 þ l2=3b2 � �ðnlt þ k1Þ�ð2=nÞ� l2ð3þ 26Þðnlt þ k1Þ�2 þ l2l2=3b2 � �þ m2=c2=32� � exp �2lðnlt þ k1Þðn�1Þ=n=3bðn� 1Þ� �ðnlt þ k1Þ�ð2=nÞ u ¼ l 2ð3þ 26Þ þ l2 expð�2ltÞ=3b2k22 � l2ð3þ 26Þ þ l2 expð�2ltÞ=3b2k22 � þ m2=c2=32 k22� � expf�2lt þ ½2l expð�ltÞ=3blk2�g Shamir and Bhatti 197 Published by NRC Research Press C an . J . P hy s. D ow nl oa de d fr om w w w .n rc re se ar ch pr es s. co m b y T ex as A & M U ni ve rs ity o n 11 /1 4/ 14 Fo r pe rs on al u se o nl y. Both of these equations suggest that at t = 0, u has a pos- itive value, which indicates that the universe was matter- dominated in the early phase of its existence. At t → ∞, the value of u turns out to be zero, which indicates that the pressure of the universe vanishes at that epoch. Acknowledgment The authors would like to thank National University of Computer and Emerging Sciences, Pakistan for funding the Ph.D. program. 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