An exact solution of the observable universe in Bianchi V space-time
Rajendra Prasad, Manvinder Singh, Anil Kumar Yadav, A. Beesham
AAn exact solution of the observable universe in Bianchi V space-time
Rajendra Prasad, ∗ Manvinder Singh, † Anil Kumar Yadav, ‡ and A. Beesham § Department of Physics, Galgotias College of Engineering and Technology, Greater Noida 201310, India Department of Applied Science, G. L. Bajaj Institute of Technology and Management, Greater Noida 201306, India Department of Physics, United College of Engineering and Research,Greater Noida - 201310, India Department of Mathematical Sciences, University of Zululand, Kwa-Dlangezwa 3886, South AfricaFaculty of Natural Sciences, Mangosuthu University of Technology, Umlazi 4026, South Africa
In this paper we investigate an observable universe in Bianchi type V space-time by taking intoaccount the cosmological constant as the source of energy. We have performed a χ test to obtainthe best fit values of the model parameters of the universe in the derived model. We have usedtwo types of data sets, viz: i) 31 values of the Hubble parameter and ii) the 1048 Phanteon dataset of various supernovae distance moduli and apparent magnitudes. From both the data sets, wehave estimated the current values of the Hubble constant, density parameters (Ω m ) and (Ω Λ ) . Thepresent value of deceleration parameter of the universe in derived model is obtained as q = 0 . +0 . − . and 0 . +0 . − . in accordance with H ( z ) and Pantheon data respectively. Also we observe that thereis a signature flipping in the sign of deceleration parameter from positive to negative and transitionred-shift exists. Thus, the universe in derived model represents a transitioning universe which is inaccelerated phase of expansion at present epoch. We have estimated the current age of the universe( t ) and present value of jerk parameter ( j ). Our obtained values of t and j are in good agreementwith its values estimated by Plank collaborations and WMAP observations. PACS numbers: 98.80.-k, 04.20.Jb, 04.50.kdKeywords: Cosmological constant; Deceleration parameter; Bianchi type V space-time; Observational con-straints.
I. INTRODUCTION
In the first decade of twenty one century, there hasbeen developments in cosmology that have reinterpretedthe cosmological constant (Λ). Firstly, the idea of in-flation gave cosmology a whole new view upon the firstsplit second of our universe. A key ingradient in the in-flationary model is the behaviour of model that have acosmological constant like behaviour. Secondly, recentastrophyical observations indicate that we live in an ac-celerated universe. The inclusion of Λ in Einstein’s fieldequation can give rise to such behaviour as we will show inin this paper. Also, the cosmological observations leadedby SN Ia groups had confirmed that the universe is in ac-celerated expansion phase at present epoch [1, 2, 3]. Thisacceleration in the universe may driven by an exotic typeof unknown fluid that have positive energy density andhuge negative pressure. This fluid is usually known asDark Enegry (DE) but its nature is still unknown. Themost suitable candidate of this DE is the Λ. However,there is a huge dissimilarity in the value of Λ predicted byobservations and particle physics ground that leads tun-ing problem. In Refs. [4, 5], the authors have addressedfine tuning problem of Λ and given a clue to solve tun-ing problem associated with Λ by assuming its dynamicalcharacter with respect to time or red-shift. The DE is less ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] effective in early universe but it dominates the presentuniverse and since it does not interact with the baryonicmatter, hence it makes difficult to detect. In Ma et al. [6],the authors have studied non interaction of DE and bary-onic matter with the cosmic expansion history and thegrowth rate of cosmic large scale structure. Several theo-retical models based on the phenomenon of late time ac-celeration in the universe have been proposed in last twodecades mostly; post supernovae observations [7]-[17]. Inrecent past, the hybrid scale factor which can mimic thecosmic transitive behaviour of the universe from early de-celerated phase to late time accelerated phase has beeninvestigated [18, 19, 20]. The cosmological models in anon-interacting two fluid scenario such as the usual DEand electromagnetic field have been studied [21, 22]. Theoutcome of this research is that the DE dominates theuniverse at present epoch and derives the late time ac-celeration of the universe. Some important applicationsof DE in anisotropic space time are given in Refs. [23, 24].In recent times, Bianchi type V cosmological modelshave commended the attention of several cosmologistsdue to the facts that its dynamical behaviour can be formore general in comparison to FRW and Bianchi type Icosmological models. The Bianchi V space-time is stillless complicated than Bianchi type II, VI and IX. Fur-ther it is observed that Bianchi type V space time per-mits small anisotropic at fixed stages in the evolutionof the universe [25]. So, Bianchi type V cosmologicalmodels create more interest and due to its some specificproperties, these models turn into a suitable model forstudy of the universe. One of the important outcomeof CMB observations is that it favors the existence ofanisotropic phase which approaches isotropic one at later a r X i v : . [ phy s i c s . g e n - ph ] O c t times. Therefore, it make sence to study the universewith anisotropic background in presence of cosmologi-cal terms Λ. Note that in recent years, Bianchi typecosmological models are playing invaluable role in ob-servational cosmology as the WMAP observational data[26, 27, 28] has confirmed an addition to standard ΛCDMmodel that bears a likeness to the Bianchi morphology[29, 30, 31, 32]. Therefore, in spite of inflation, the uni-verse has slightly anisotropic special geometry that leadsto a non-trivial isotropization history of universe due tothe presence of an anisotropic energy source. Some sen-sible research in Bianchi type V space-time are given inRefs. [33, 34, 35]. In particular, Collins [33] has in-vestigated simplest perfect fluid cosmological models inBianchi V space-time which possess a singularity. One ofthe interesting result of this study is that this model con-sist of two disjoint regions in which the matter densityis non-zero, separated by a Cauchy horizon on which thematter density is null. In Coles and Ellis [34], authorshave discussed the case of an open universe and investi-gated that observations support a model of the universewhich has low density. Maartens and Nel [35] had inves-tigated some new and exact solution in spatially homoge-neous Bianchi V space-time that admit a non-vanishingmagnetic field. In our previous work [36], we have studiedthe kinematics and fate of Bianchi V universe by takinginto account the dynamical nature of DE while in this pa-per, we have considered cosmological constant as sourceof energy. THe mechanism of obtaining Hubble’s func-tion H ( z ) are altogether different from H ( z ) given in Ref.[36].In this work, we have investigated a Bianchi type-Vmodel of the universe in which baryons also have pres-sure. It has been stated that “All of observational cosmol-ogy is the search for two numbers: i) Hubble parameter(HP) and ii) deceleration parameter (DP) H and q ”[37]. In the present scenario, higher derivatives of thescale factor, such as the jerk parameter ( j ), snap pa-rameter ( s ) and lerk parameter ( l ) do play a role. Asuccessful cosmological model will be one in which theseparameters fit best with the observational inputs. Keep-ing this as our motto, we have performed a χ test toobtain the best fit values of the model parameters of theuniverse in our derived model that leads to good con-sistency of the theoretical model with observations. Wehave used two types of data sets: i) A data set of 31Hubble parameter values and ii) The 1048 Phanteon dataset of various supernovae distance moduli and apparentmagnitudes. From both data sets, we have estimatedcurrent values of the Hubble constant, density parame-ters (Ω m ) and (Ω Λ ) . The present value of decelerationparameter of the universe in derived model is obtainedas q = 0 . +0 . − . and 0 . +0 . − . in accordance with H ( z )and Pantheon data respectively. This values of q is veryclose to its empirical value obtained by Cunha et al [41].Also we observe that there is a signature flipping in thesign of deceleration parameter from positive to negativeand transition red-shift exists. Thus, the universe in de- rived model represents a transitioning universe which isin accelerated phase of expansion at present epoch. Wehave estimated the current age of the universe ( t ) andpresent value of jerk parameter ( j ). Our obtained valuesof t and j are in good agreement with its values esti-mated by Plank collaborations and WMAP observations.The paper is organized as follows: Section I deals withthe straightforward description of the various investiga-tions and their results in the thrust area of the paper.In section II, the model and its basic equations are pre-sented. Section III is devoted to data and likelihoods.Some physical parameters and properties of the universein our derived model are described in section IV. Finally,we have given conclusion of this research in Section V. II. THE MODEL AND BASIC EQUATIONS
The anisotropic and homogeneous Bianchi type Vspace-time is read as ds = dt − X ( t ) dx − exp (2 αx ) (cid:2) Y ( t ) dy − Z ( t ) dz (cid:3) (1)where X ( t ), Y ( t ) and Z ( t ) are scale factors along the x , y and z axes. The exponent α is an arbitrary constant.Einstein’s field equation is read as R ij − Rg ij − Λ g ij = 8 πGT ij (2)where R is the Ricci scalar and T ij is energy-momentumtensor of perfect fluid.The energy-momentum tensor ( T ij ) of the perfect fluid isgiven as T ij = ( ρ + p ) v i v j − pg ij (3)where v i is the four velocity vector which satisfies v i v i =1. ρ and p are the energy density and pressure of theperfect fluid.Solving (2) with the space-time metric (1), we get thefollowing system of equations¨ YY + ¨ ZZ + ˙ Y ˙ ZY Z − α X = − πGp + Λ (4)¨ XX + ¨ ZZ + ˙ X ˙ ZXZ − α X = − πGp + Λ (5)¨ XX + ¨ YY + ˙ X ˙ YXY − α X = − πGp + Λ (6)˙ X ˙ YXY + ˙ Y ˙ ZY Z + ˙ Z ˙ XZX − α X = 8 πGρ + Λ (7)2 ˙ XX − ˙ YY − ˙ ZZ = 0 ⇒ X = κ Y Z (8)where κ is constant of integration and we have taken κ = 1 without loss of generality.Equations (4)-(6) lead to the following system of equa-tions ¨ XX − ¨ YY + ˙ X ˙ ZXZ − ˙ Y ˙ ZY Z = 0 (9)¨ YY − ¨ ZZ + ˙ X ˙ YXY − ˙ X ˙ ZXZ = 0 (10)¨ ZZ − ¨ XX + ˙ Y ˙ ZY Z − ˙ X ˙ YXY = 0 (11)If ξ is an arbitrary function of t , then equation (8)satisfiesthe following B = Aξ & C = Aξ (12)where ξ = ξ ( t ) relates to the anisotropy in the universe.Using equation (12) in any one of the equations (9) -(11), we obtain ¨ ξξ − ˙ ξ ξ + ˙3 ξξ ˙ XX = 0 (13)After integration of equation (13), we obtain˙ ξξ = KX (14)Now, the average scale factor a ( t ) is defined as a = ( XY Z ) (15) Finally, the field equations of Bianchi type V space-timein term of average scale factor are read as2 ¨ aa + ˙ a a − α a = − πGp + Λ − K a (16)3 ˙ a a − α a = 8 πGρ + Λ + K a (17)From equations (16) and (17), we observe that it is asystem of two equations with three variables. Hence,one con not solve it in general. Therefor, to get an ex-plicit solution, we have to adopt an additional but physi-cally reasonable condition that is why we take well knownbarotropic equation of state which quantifies the relationbetween energy density ( ρ ) and pressure ( p ). i.e.p = ωρ (18)where 0 ≤ ω ≤ aa + (3 ω + 1) ˙ a a − K ( ω − a − (3 ω + 1) α a = Λ( ω + 1)(19)Using the standard definition, H = ˙ aa and a = z , equa-tion (19) leads to2( z + 1) HH (cid:48) − ω + 1) H + Λ( ω + 1) + K ( ω − z + 1) + (3 ω + 1) α (1 + z ) = 0 (20)where H (cid:48) is the first order derivative of H with respect to z .Solving equation (20), we obtain H = H (cid:113) (Ω m ) (1 + z ) ω +1) + (Ω α ) (1 + z ) + (Ω σ ) (1 + z ) + (Ω Λ ) (21)where (Ω m ) = 1 − Λ3 H − K H − α H , (Ω Λ ) = Λ3 H , (Ω α ) = α H & (Ω σ ) = K H Thus, we have(Ω m ) + (Ω Λ ) + (Ω σ ) + (Ω α ) = 1where H is the present value of Hubble’s parameter. Now, the expressions for luminosity distance ( D L ) anddistance modulus ( µ ) of any distant luminous object aredetermined as D L ( z ) = (1 + z ) H (cid:90) z dzh ( z ) ; h ( z ) = HH (22)and µ ( z ) = m b − M = 5 log (cid:18) D L ( z ) M pc (cid:19) + µ (23)where m b and M are apparent magnitude and absolutemagnitude of any distant luminous object respectively. µ is the zero point offset. III. DATA AND LIKELIHOOD
In this section, we use the observational H ( z ) dataand recent SN Ia Pantheon data. Also, we describe herethe statistical methodology for constraining differentmodel parameters of the derived universe. • Observational Hubble Data (OHD) : For HubbleH(z) data, we adopt 31 H ( z ) observational dat-apoints in the range of 0 ≤ z ≤ .
96 taken table1 of Ref. [38]. The cosmic chronometric (CC)technique is adopted to determined these uncor-related data. There reason behind to take thisdata is the fact that OHD data obtained from CCtechnique is model-independent. In fact, the mostevolving galaxies based on the “galaxy differentialage” method is used to determine this CC data [39]. • Pantheon Data : For the SN Ia data, we considerthe recent Pantheon sample compiled in Scolnic etal. [40]. This Pantheon SN Ia catalogue is publiclyavailable in Scolnic et al. 2018.
FIG. 1: Two dimensional (2D) contours at σ , σ and σ confidence regions by bounding the derived model with latest 31observational Hubble data compiled from CC technique. Theestimated values of H = 67 . ± . km s − Mpc − , (Ω Λ ) = 0 . ± . and (Ω m ) = 0 . ± . . FIG. 2: Two dimensional (2D) contours at σ , σ and σ confidence regions by bounding the derived model with Pantheondata. The estimated values of H = 70 . ± . km s − Mpc − , (Ω Λ ) = 0 . ± . and (Ω m ) = 0 . ± . . For probing the model parameters, we have defined χ forthe parameters with the likelihood given by ϕ ∝ e − χ . χ OHD = (cid:88) i =1 (cid:20) H ( z i , Ψ) − H obs ( z i ) σ i (cid:21) (24)where Ψ and σ i represent the parameter vector andstandard error in experimental values of the Hubblefunction H respectively.Now, we assume following uniform priors enforced onthe model parameters for statistical analysis. H ∼ (50 , , (Ω m ) ∼ (0 , . , & (Ω Λ ) ∼ (0 , H = 67 . ± . km s − M pc − , (Ω Λ ) = 0 . ± . m ) = 0 . ± . χ P antheon = (cid:88) i =1 (cid:20) µ ( z i , Ψ) − µ obs ( z i ) σ i (cid:21) (25)where Ψ and σ i are parameter vector and standard errorin experimental values of µ ( z ) respectively.Figure 2 depicts 2D contours at 68%, 95% and 99%confidence regions by bounding the model underconsideration with recent Pantheon data. The esti-mated values of H = 70 . ± . km s − M pc − ,(Ω Λ ) = 0 . ± .
014 and (Ω m ) = 0 . ± . IV. PHYSICAL PROPERTIES OF THE MODELA. Deceleration parameter
The deceleration parameter (DP) is read as q = − a ¨ a ˙ a = − z ) H (cid:48) H (26)where ˙ z = − (1 + z ) H .The the expression of q of our derived model of the uni-verse is obtained as q = − ω )(Ω m ) (1 + z ) ω ) + 6(Ω σ ) (1 + z ) + 2(Ω α ) (1 + z ) m ) (1 + z ) ω ) + (Ω Λ ) + (Ω σ ) (1 + z ) + (Ω α ) (1 + z ) ] (27)Therefore, the present value of DP ( q ) is computed byputting z = 0 in equation (27) q = − ω )(Ω m ) + 6(Ω σ ) + 2(Ω α ) m ) + (Ω Λ ) + (Ω σ ) + (Ω α ) ] (28)Thus, the present values of DP of the universe in de-rived model are estimated as q = − . +0 . − . and q = − . +0 . − . fit with OHD and Pantheon data respectively.In Fig. 3, the best fit curve of q at 68% confidence levelis shown. Note that the estimated value of q in thispaper is nicely tally with its value obtained by obser-vational researches [41, 42, 43]. Therefore, the universein derived model is in good agreement with recent ob-servations. Further, we observe that, the early universewas in decelerated phase of expansion while the currentuniverse repels its ingredient matter/energy with acceler-ation. Hence, the universe in derived model represents amodel of transitioning universe that have signature flip-ping at z t = 0 . ± .
06 and z t = 0 . ± .
04 with respectto OHD and pantheon data respectively. Some sensibleresearches for transition red-shift ( z t ) are as follows: Fa-rooq et al [44] have estimated z t = 0 . ± .
05 by usingthe 38 H(z) data points (Also see [45] for the results ob-tained from the 28 H(z) data). Recently, in [46], theauthors have given a comparison between transition red-shift of the ΛCDM and ΛBI models. So, the obtainedvalues of transition red-shift ( z t ), in this paper, is veryclose to the previous results. FIG. 3: Plots of q ( z ) versus z for the model parameters obtainedfrom bounding the derived model with OHD (left panel) andPantheon data (right panel). The transition red-shift is obtainedas z t = 0 . ± . and z t = 0 . ± . for OHD and Pantheondata respectively. B. Age of the universe
The age of the universe is obtained as dt = − dz (1 + z ) H ( z ) ⇒ (cid:90) t t dt = (cid:90) z z ) H ( z ) dz (29)where t denotes present age of the universe in our de-rived model.Thus, the present age of the universe is read as t = lim z →∞ (cid:90) z dzH (1 + z ) (cid:112) (Ω m ) ( z + 1) ω +1) + (Ω Λ ) + (Ω σ ) ( z + 1) + (Ω α ) (1 + z ) (30)It has been observed that the present agre of theuniverse is 14 . +0 . − . Gyrs in accordance with OHD.Similarly when we bound our model with Pantheondata then t = 14 . +0 . − . Gyrs. It is important tonote that the empirical values of age of the universe inWMAP observation [47] is read as t = 13 . ± . t in thispaper. In some other cosmological investigations, age ofthe universe is estimated as 13 . ± .
020 Gyrs [48],14 . ± . . ± . C. Jerk parameter
The jerk parameter ( j ), a dimensionless third orderderivative of the scale factor a ( t ) with respect to time t is an important term of cosmographic series which is usedto investigate the deviations of any model of the universefrom standard concordance model or ΛCDM model. It isdefined as [51, 52]. FIG. 4: Plot of j versus z for the model parameters obtainedfrom bounding the derived model with OHD (left panel) andPantheon data (right panel) at confidence level. The solidgray line denotes the best fit curve of jerk parameter of theuniverse in our model. j = ... aaH (31)where ... a = d a ( t ) dt .The deceleration parameter q is read as q = − a ¨ a ˙ a (32)Using ˙ z = − (1 + z ) H , the reflection for j in terms of q and z is obtained as j = q (2 q + 1) + (1 + z ) dqdz = 1 − (1 + z ) [ H ( z ) ] (cid:48) H ( z ) +12 (1 + z ) [ H ( z ) ] (cid:48)(cid:48) H ( z ) (33)Therefore, the present value of the jerk parameter is readas j = q + 2 q + (cid:18) dqdz (cid:19) z =0 (34)The behaviour of jerk parameter j versus red-shift z is shown in Fig. 5. The solid gray line in the Fig. 5represents bet fit curve of j at 68% confidence region.The present values of jerk parameter in our model areestimated as j = 1 . ± .
001 and j = 1 . ± . j = 1 for the ΛCDM model.However, in this paper, j is found with little deviationfrom its ΛCDM value which implies that our modeldoes not behave like the standard ΛCDM universe. Inthe recent past, Zhai et al [54] have parameterized j ( z )phenomenologically aiming at measuring the departure of j from the ΛCDM value. Later on, some otherparametric reconstructions of the cosmological jerk havebeen investigated in different physical contexts [55, 56].Note that the Refs. [57, 58] deal with various DEmodels and role of jerk parameters to discriminate thesecosmological models. Recently Singh and Nagpal [59]have investigated some values of j which is not equal to1 by using some observational data sets. V. CONCLUSION
In this paper, firstly, we have investigated an exactobservable universe in Bianchi type V space time. Sec-ondly, we have constrained the various model parametersof the universe in derived model by executing statistical χ tests. The main result of statistical analysis is givenin Table I. TABLE I: The present values of model parameters
S. N. Model parameters H ( z ) data Pantheon data1 H . ± . . ± .
82 (Ω m ) . ± .
005 0 . ± . Λ ) . ± .
01 0 . ± . q − . +0 . − . − . +0 . − . t . +0 . − . . +0 . − . j . ± .
001 1 . ± . The characteristics of the universe in derived model areas follows:i) We have obtained an exact solution of Einstein’sfield equations in Bianchi type V space-time ratherthan an adhoc parametric reconstruction of j or q .ii) The universe in derived model is able to describethe dynamics of universe in its early phase as wellas the late time acceleration. We observe thatthere is a signature flip at transition red-shift ofthe derived universe. The result of this flipping,the decelerated universe turns into acceleratinguniverse and continue to accelerate at presentepoch. Also we have estimated the present valueof the deceleration parameter which is in goodagreement with recent astrophysical observations.iii) The age of the universe in derived model is in goodconsistency with its empirical value obtained fromWMAP observations and Plank collaborations.iv) The analysis of the jerk parameter shows that ourmodel has little deviation from ΛCDM universe.As a final comment, we have investigated an exactobservable universe in Bianchi type V space-time by tak-ing into account the cosmological constant as source ofenergy. The universe in derived model is a transitioninguniverse with transition red-shift z t = 0 . ± .
06 and z t = 0 . ± .
04 in accordance with OHD and pantheondata respectively.. It is important to note here thatin Akarsu et al. [60], the Bianchi I universe has beeninvestigated as an extension of the ΛCDM model. Inthis paper, we have used an entirely different mechanismfor solving the field equations as adopted in Ref. [60]and investigated an extension of the ΛCDM model in Bianchi type V space-time.
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