Dark energy model with variable q and ω in LRS Bianchi-II space-time
aa r X i v : . [ phy s i c s . g e n - ph ] A p r Dark energy model with variable q and ω in LRSBianchi-II space-time Bijan Saha and Anil Kumar Yadav † Abstract
The present study deals with spatial homogeneous and anisotropic lo-cally rotationally symmetric (LRS) Bianchi-II dark energy model in gen-eral relativity. The Einstein’s field equations have been solved exactly bytaking into account the proportionality relation between one of the com-ponents of shear scalar ( σ ) and expansion scalar ( ϑ ), which, for somesuitable choices of problem parameters, yields time dependent equationof state (EoS) and deceleration parameter (DP), representing a modelwhich generates a transition of universe from early decelerating phase topresent accelerating phase. The physical and geometrical behavior of uni-verse have been discussed in detail. Key words: Dark energy, variable DP and EoS parameter.PACS Nos: 98.80Es, 98.80-k, 95.36.+x
The discovery of acceleration of the universe stands as a major breakthrough ofthe observational cosmology. The power of observations in cosmology is clearfrom the observations of supernovae of Ia (SN Ia) which dramatically changed,about a decade ago, the then standard picture of cosmology - of an expandinguniverse evolving under the rules of general relativity such that the expansionrate should slow down as cosmic time unfolds. Surveys of cosmologically distantSN Ia (Riess et al. 1998; Permutter et al. 1999) indicated the presence of new,unaccounted - for dark energy that opposes the self-attraction of matter andcauses the expansion of the universe to accelerate. When combined with indirectmeasurements using cosmic microwave background (CMB) anisotropies, cosmicshear and studies of galaxy clusters, a cosmological world model has emergedthat describes the universe at flat, with about 70% of it’s energy contained inthe form of this cosmic dark energy (Seljak et al. 2005). This acceleration Laboratory of Information Technologies, Joint Institute for Nuclear Research Dubna - 141980, Russia. E-mail : [email protected] Department of Physics, Anand Engineering College, Keetham, Agra-282 007, India.E-mail: [email protected] † corresponding author
1s realized with negative pressure and positive energy density that violate thestrong energy condition. This violation gives a reverse gravitational effect. Dueto this effect, the universe gets a jerk and the transition from the earlier decel-eration phase to the recent acceleration phase take place (Caldwell et al 2002).The cause of this sudden transition and the source of accelerated expansion isstill unknown. In physical cosmology and astronomy, the simplest candidatefor the DE is the cosmological constant (Λ), but it needs to be extremely fine-tuned to satisfy the current value of the DE density, which is a serious problem.Alternatively, to explain the decay of the density, the different forms of dynam-ically changing DE with an effective equation of state (EoS), ω = p/ρ < − / ω > −
1) (Steinhardt et al. 1999), phantom( ω < −
1) (Caldwell 2002) etc. While the the possibility ω << − ω = − − . < ω < − .
62 and − . < ω < − .
79, respectively. The latest resultsin 2009, obtained after a combination of cosmological data-sets coming fromCMB anisotropies, luminosity distances of high red-shift type Ia supernovaeand galaxy clustering, constrain the dark energy EoS to − . < ω < − .
92 at68% confidence level (Komatsu et al. 2009; Hinshaw et al. 2009)Moreover, in recent years Bianchi universes have been gaining an increas-ing interest of observational cosmology, since the WMAP data (Hinshaw et al.2003, 2007; Jaffe et al. 2005) seem to require an addition to the standard cos-mological model with positive cosmological constant that resembles the Bianchimorphology (Jaffe et al. 2006a, 2006b; Campanelli et al. 2006, 2007; Hoftuft etal. 2009). According to this, the universe should achieve a slightly anisotropicspecial geometry in spite of the inflation, contrary to generic inflationary modelsand that might be indicating a nontrivial isotropization history of universe dueto the presence of an anisotropic energy source. The Bianchi models isotropizeat late times even for ordinary matter, and the possible anisotropy of the Bianchimetrics necessarily dies away during the inflationary era (Ellis 2006). In factthis isotropization of the Bianchi metrics is due to the implicit assumption thatthe DE is isotropic in nature. Therefore, the CMB anisotropy can also be finetuned, since the Bianchi universe anisotropies determine the CMB anisotropies.The price of this property of DE is a voilation of null energy condition (NEC)since the DE crosses the phantom divide line (PDL), in particular dependingon the direction.The anomalies found in the cosmic microwave background (CMB) and large2cale structure observations stimulated a growing interest in anisotropic cosmo-logical model of universe. Here we confine ourselves to models of Bianchi-typeII. Bianchi type-II space-time has a fundamental role in constructing cosmo-logical models suitable for describing the early stages of evolution of universe.Asseo and Sol (1987) emphasized the importance of Bianchi type-II Universe.Recently Pradhan et al (2011) and Kumar and Akarsu (2011) have dealt withBianchi-II DE models by considering the spatial law of variation of Hubble’s pa-rameter which yields the constant value of deceleration parameter (DP). Someauthors (Akarsu and Kilinc 2010a, 2010b; Yadav et al. 2011; Yadav and Yadav2011; Kumar and Yadav 2011; Yadav 2011; Adhav et al. 2011 and recentlyYadav and Saha 2011) have studied DE models with variable EoS parameter.In this paper, we presented general relativistic cosmological model with time de-pendent DP in LRS Bianchi-II space-time which can be described by isotropicand variable EoS parameter. The paper is organized as follows: The metric andfield equation are presented in section 2. Section 3 deals with the solution offield equations and physical behavior of the model. Finally the findings of paperare discussed in section 4.
The gravitational field in our case is given by a Bianchi type-II (BII) metric ds = − dt + a ( dx − x dx ) + a dx + a dx , (1)with a , a , a being the functions of time only. In what follows, we considerthe LRS BII model setting a = a .Given the fact that the dark energy is isotropically distributed, it is enough toconsider only three Einstein equations (Saha 2011) corresponding to the metric(1), namely 2 ¨ a a + (cid:16) ˙ a a (cid:17) − a a = − ωρ, (2a)¨ a a + ¨ a a + ˙ a a ˙ a a + 14 a a = − ωρ, (2b)2 ˙ a a ˙ a a + (cid:16) ˙ a a (cid:17) − a a = ρ. (2c)Here over dots denote differentiation with respect to time ( t ).Let us introduce a new function V = a a = √− g. (3)The expressions for expansion and shear for BII metric given by (1) read: ϑ = u µ ; µ = Γ µµ = ˙ a a + 2 ˙ a a = ˙ VV , (4)3nd σ = ˙ a a − ϑ, σ = σ = ˙ a a − ϑ, σ = z h ˙ a a − ˙ a a + z a ˙ a a i . (5)Let us now define the generalized and directional Hubble parameters. As inknow, the Hubble parameter was defined by E. Hubble for the FRW model ds = − dt + a ( dx + dx + dx ) , (6)as H = a dadt . Taking into account that √− g = a it can be defined as H =
13 1 √− g d √− gdt , and the directional Hubble parameters as H i = √ g ii d √ g ii dt or H i =
12 1 g ii dg ii dt .Taking into account that for BII metric (1) √− g = a a a = a a = V and g = a , g = x a + a and g = a , analogically we define H = ˙ a a , H = x a ˙ a + a ˙ a x a + a , H = ˙ a a (7)and H = 13 ˙ VV = 13 (cid:0) ˙ a a + 2 ˙ a a (cid:1) . (8)It should be noted that though for a = a we have H = H = H as in isotropiccase, the present definition does not lead to H = ( H + H + H ) /
3. For thisequality to held, one must set H = ˙ a a . Unfortunately, there is no uniquedefinition for directional Hubble parameters. Finally we define the decelerationparameter (DP) as q = − V ¨ V ˙ V . (9)Imposing the proportionality condition, i.e., assuming that the expansion ϑ is proportional to say σ : ϑ ∝ σ , (10)one finds the following relations between the metric functions a = a n , (11)with n being some constant. Inserting (11) into (3) we obtain a = V / (2 n +1) , a = V n/ (2 n +1) , (12)Subtraction of (2b) from (2a) gives¨ a a − ¨ a a + (cid:0) ˙ a a (cid:1) − ˙ a a ˙ a a − a a = 0 . (13)Inserting a and a from (12) into (13) we find equation for defining V :¨ V = 2 n + 1 n − V (3 − n ) / (1+2 n ) , (14)4ith the solution in quadrature Z dV √ V / (2 n +1) + C = 2 n + 1 p n − t. (15)Eq. (15) imposes some restriction on the choice of n , namely, n >
1. Thus we seethat the proportionality condition (10) in our case does not allow isotropizationof the initially anisotropic space-time.Once V is defined, we can define DP from (9) and EoS parameter from ω = − n + 1)(2 n + 1) V ¨ V − n + 2 n ) ˙ V + (2 n + 1) V / (2 n +1) n + 2 n ) ˙ V − (2 n + 1) V / (2 n +1) = 1 − n + 1)(2 n + 1) V ¨ V n + 2 n ) ˙ V − (2 n + 1) V / (2 n +1) (16)Thus we see that V plays central role here in defining all physical quantities. Inwhat follows we find V from (14) or (15) for some concrete values of n or C . One can not solve equation (15) in general. So, in order to solve the problemcompletely, we have to choose either C or n in such a manner that equation(15) be integrable. The easiest way is to set C = 0 in (15). In that case onedully obtains V = C t (2 n +1) / (2 n − , C = h (2 n + 1) (2 n − p n − i (2 n +1) / (2 n − . (17)As one sees, in this case V is an increasing function of time, but this solutionleads to the constant DP.Since, we are looking for a model explaining an expanding universe withacceleration, we consider the case for nontrivial C , which for a suitable choiceof n gives the time dependent DP. The motivation for time dependent DP isbehind the fact that the universe is accelerated expansion at present as observedin recent observations of Type Ia supernova (Riess et al. 1998, 2004; Perlmutteret al. 1999; Tonry et al. 2003; Clocchiatti et al. 2006) and CMB anisotropies(Bennett et al. 2003; de Bernardis et al. 2000; Hanany et al. 2000) and de-celerated expansion in the past. Also, the transition redshift from decelerationexpansion to accelerated expansion is about 0.5. Now for a Universe whichwas decelerating in past and accelerating at the present time, the DP mustshow signature flipping (see Padmanabhan and Roychowdhury 2003; Amendola2003; Riess et al. 2001). So, there is no scope for a constant DP at presentepoch. So, in general, the DP is not a constant but time variable.Thus we consider the Eq. (15) with a nontrivial C . For C = 0 Eq. (15)allows exact solution only when 4 / (2 n + 1) = N , where N is an integer number.5n this case N can be integer only for n = 1 / n = 3 /
2. Since n > n = 3 /
2. In this case Eq. (15) reducesto Z dV √ V + C = 4 t (18)which after integration leads V = 4 t + 2 βt + γ (19)where β is the integrating constant and γ = β − C Inserting equation (20) into (12), we obtain a = (4 t + 2 βt + γ ) (20) a = (4 t + 2 βt + γ ) (21)The physical parameters such as directional Hubble’s parameters ( H , H , H ),average Hubble parameter ( H ), expansion scalar ( θ ) and scale factor ( a ) are,respectively given by H = 4 t + β t + 2 βt + γ ) (22) H = (4 t + β )( x + 12 t + 6 βt + 3 γ )2(4 t + 2 βt + γ )( x + 4 t + 2 βt + γ ) (23) H = 3(4 t + β )4(4 t + 2 βt + γ ) (24) H = 2(4 t + β )3(4 t + 2 β + γ ) (25) θ = 2(4 t + β )(4 t + 2 β + γ ) (26) a = (4 t + 2 βt + γ ) (27)The components of shear scalar are given by σ = − t + β t + 2 βt + γ ) , (28) σ = σ = 4 t + β t + 2 βt + γ ) , (29) σ = x (4 t + β )4(4 t + 2 βt + γ ) − x (4 t + β )4(4 t + 2 βt + γ ) (30)The value of DP ( q ) is found to be q = − h C t +2 βt + γ i = − (cid:2) Ca (cid:3) (31)6 tq Figure 1: Plot of deceleration parameter ( q ) versus time ( t ). tω Figure 2: Plot of EoS parameter ( ω ) versus time ( t ).7he sign of q indicates whether the model inflates or not. A positive sign of q corresponds to the standard decelerating model whereas the negative sign of q indicates indicates inflation. The recent observations of SN Ia (Riess et al.1998, Perlmutter et al. 1999) reveal that the present universe is acceleratingand the value of DP lies somewhere in the range − < q <
0. Figure 1 depictsthe variation of DP versus cosmic time as representative case with appropriatechoice of constants of integration and other physical parameters.The energy density of the cosmic fluid ( ρ ), EoS parameter ( ω ) and densityparameter (Ω) are found to be ρ = 21(4 t + β ) t + 2 βt + γ ) − t + 2 βt + γ ) (32) ω = 1 − t + 2 βt + γ )21(8 t + 2 β ) − t + 2 βt + γ ) (33)Ω = 6364 − t + 2 βt + γ )16(4 t + β ) (34)From Eq. (33) follows that at large t when only the quadratic terms stayalive, from EoS parameter we find ω → − . t . (8 t ) − . t = 1 − . − .
64 = 0 , (35)i.e., under the present assumption the universe is ultimately filled with dustonly at remote future.The initial time of the universe is t = − β + √ β − γ . Therefore, at t = − β + √ β − γ , the spatial volume vanishes while all other parameter diverge.Thus the derived model starts expanding with big bang singularity at t = − β + √ β − γ which can be shifted to t = 0 by choosing γ = 0. This singu-larity is point type because the directional scale factors a ( t ) and a ( t ) vanishat initial moment. The components of shear scalar vanish at t → ∞ . Thus inderived model the initial anisotropy dies out at later time.Figure 2 depicts the variation EoS parameter ( ω ) versus cosmic time as rep-resentative case with appropriate choice of constants of integration and otherphysical parameters. It is shown that the growth of ω takes place with negativesign. It should be emphasized that there is a number of models for dark energy(quintessence, Chaplygin gas, phantom and many more) and quest for the rightone is still going on. The main idea for the DE is a negative pressure, so one cantry with a negative EoS parameter. It should be noted that the quintessence isgiven by a barotropic EoS only with negative parameter. We don’t call fluid aDE, we just construct DE in analogy with fluid. Figure 3 demonstrates the be-havior of density parameter (Ω) versus cosmic time in the evolution of universe8 t Ω Figure 3: Plot of density parameter (Ω) versus time ( t ).as representative case with appropriate choice of constants of integration andother physical parameters In this paper, we have investigated LRS Bianchi II DE model under the assump-tion that ϑ ∝ σ . Under some specific choice of problem parameters the presentconsideration yields the variable DP and EoS parameter. It is to be noted thatour procedure of solving the field equations are altogether different from whatPradhan et al (2011) have adapted. Pradhan et al (2011) have solved the fieldequations by considering the variation law for generalized Hubble’s parameterwhich gives the constant value of DP and only the evolution takes place eitherin accelerating or decelerating phase whereas we have considered the propor-tionality condition ϑ ∝ σ in such a way that gives variable DP which evolvesfrom decelerating phase to current accelerating phase (see Fig. 1). Thus thepresent DE model has transition of universe from the early deceleration phase tocurrent acceleration phase which is in good agreement with recent observations(2006). The model has singular origin and the universe is ultimately filled withdust only at remote future.The theoretical arguments suggest and observational data show, the universe9as anisotropic at the early stage. Here we are dealing not only with the presentstate of the universe, but drawing a picture of the universe from the remote pastto present day. We use the Bianchi model as one of many models able to describeinitial anisotropy that dies away as the universe evolves. So though the modelis anisotropic in the past for small t but it becomes isotropic as t → ∞ . In thederived model, the EoS parameter ( ω ) is evolving with negative sign which maybe attributed to the current accelerated expansion of universe. Hence from thetheoretical perspective, the present model can be a viable model to explain thelate time acceleration of the universe. In other words, the solution presentedhere can be one of the potential candidates to describe the present universe aswell as the early universe. Acknowledgments
One of authors (AKY) is thankful to The Institute of Mathematical Science(IMSc), Chennai, India for providing facility and support where part of thiswork was carried out. Bijan Saha is thankful to joint Romanian-LIT, JINR,Dubna Research Project, theme no. 09-6-1060-2005/2013. Finally We wouldlike to thank the anonymous referee for his valuable questions which helped usto understand the depth of the problem.
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