Constraining Bianchi type V universe with recent H(z) and BAO observations in Brans-Dicke theory of gravitation
EEur. Phys. J. Plus manuscript No. (will be inserted by the editor)
Constraining Bianchi type V universe with recentH(z) and BAO observations in Brans - Dicke theoryof gravitation
R. Prasad , Avinash Kr. Yadav , AnilKumar Yadav Received: 28 January 2020 / Accepted: 2 March 2020
Abstract
In this paper, we investigate a transitioning model of Bianchi type Vuniverse in Brans-Dicke theory of gravitation. The derived model not only vali-dates Mach’s principle but also describes the present acceleration of the universe.In this paper, our aim is to constrain an exact Bianchi type V universe in Brans -Dicke gravity. For this sake, firstly we obtain an exact solution of field equationsin modified gravity and secondly constrain the model parameters by boundingthe model with recent H ( z ) and Baryon acoustic oscillations (BAO) observationaldata. The current phase of accelerated expansion of the universe is also describedby the contribution coming from cosmological constant screened scalar field withdeceleration parameter showing a transition redshift of about z t = 0 .
79. Somephysical properties of the universe are also discussed.
Kewwords:
Bianchi V spaceptime; Brans-Dike gravity; Scalar field; Acceler-ating universe.
Pacs:
The supernovae Ia observations [1,2] have exhibited a strong evidence that ouruniverse is dominated by two types of dark components at present epoch. Thesetwo components of present universe are named as dark matter and dark energy.Today, it is one of the major issues in modern cosmology to describe the natureof dark matter and dark energy. The dark matter has not been directly observedbut there are many evidences such as galaxy rotation curves, gravitational effects, Department of Physics, Galgotias College of Engineering and Technology, Greater Noida -201310, India, E-mail: [email protected] Department of Mathematics, United College of Engineering and Research, Greater Noida -201310, India E-mail: [email protected] Department of Physics, United College of Engineering and Research, Greater Noida - 201310,India E-mail: [email protected] a r X i v : . [ phy s i c s . g e n - ph ] M a r R. Prasad, Avinash Kr. Yadav and Anil Kumar Yadav gravitational lensing etc which support the existence of dark matter. The dark en-ergy is an unknown form of energy that pervades the whole universe. It is believedto have negative pressure, the dark energy is causing acceleration in the presentuniverse. According to WMAP observations [3,4,5], the universe energy densityappears to consist of approximately 4 % of that of visible matter, 21 % of that ofdark matter and 75 % of that of dark energy. In the literature, the acceleration inpresent universe is described by two ways i) inclusion of dark energy in right sideof Einstein’s equation i. e. by modifying energy-momentum tensor ii) modificationin left side of Einstein’s equation i. e. geometric modification. The authors of Refs.[6,7] have described late time acceleration of the universe by considering dark en-ergy and modified gravity respectively. Later on, numerous cosmological modelshave been investigated in General Relativity (GR) with inclusion of dark energy[8,9,10,11,12,13,48,15] and in modified theories of gravity without inclusion ofdark energy [16,17,18,19,20,21,22,23]. Even after all these attempts, the reliablenature of dark energy has not been convincingly explained yet.The Brans-dicke (BD) theory [24], which is a natural generalization of GR,provides a worthy framework for dynamical dark energy models. In this this the-ory, the scalar field φ is being time-dependent and it is equivalent to (8 πG ) − .Therefore, in BD scalar-tensor theory, the scalar field φ couples to the gravitywith a dimensionless coupling parameter ω . It is worth to note that BD theory ofgravitation commits expanding solutions for scalar field and average scale factorwhich are compatible with the solar system observations. In Refs. [25,26,27], theauthors have investigated that BD theory explains the late time accelerated ex-pansion of the universe and also conciliates the observation data. It is also to benoticed that BD theory of gravitation reduces to GR if scalar field is constant and ω → ∞ [28,29]. Some new agegraphic dark energy models in Brans-Dicke gravityhave been investigated [30,31,32,33]. These models explain the late time acceler-ated expansion of the universe with evolution of scalar field as power law of scalefactor. In the literature, BD theory is invoked to fulfill the requirement of Mach’sprinciple[24,34,35,36,37]. In Sen and Sen [38], authors have investigated that aperfect fluid cannot support acceleration but a fluid with dissipative pressure candrive late time acceleration of current universe. The present cosmic accelerationwithout resorting to a cosmological constant or quintessence matter has been in-vestigated in BD theory but then Brans-Dicke coupling constant asymptoticallyacquires a small negative value for an accelerating universe at late time[39] while inRef. [25], authors have obtained solution for accelerating universe with φ poten-tial for large BD coupling constant without considering positive energy conditionfor matter and scalar field both. Recently Akarsu et al. [40,82] have investigatedsome particular negative range of ω and positive large value of ω that lead ac-celeration in massive Brans-Dicke gravity. Some large angle anomalies viewed incosmic microwave background (CMB) radiations [5] are favoring the presence ofanisotropies in the early stage of the universe which violate the isotropical na-ture of the observable universe and hence to clearly describe the early universe - a spatially homogeneous but anisotropic Bianchi models play a significant role.In the literature, several Bianchi type models have been investigated with differ-ent matter distribution in Brans-Dicke theory of gravitation. In particular, Kiranet al [41] have investigated an interacting Bianchi V cosmological model withinthe framework of Brans-Dicke cosmology. In the recent past, some Brans-Dicke onstraining Bianchi type V universe.. 3 anisotropic models have been studied to discuss the late time accelerated expan-sion of the universe [42,43,44,45]. Some useful applications of Bianchi type modelscompatible with astrophysical observations are given in Refs. [15,46,47,48,49,83].In this paper, we have investigated a Bianchi type V model of the universefilled with pressure-less matter and cosmological constant at present in Brans-Dicke gravity. Firstly we have obtained an exact Brans-Dicke universe and thenfind constraints on model parameters by using recent H(z) and BAO observationaldata. The rest of the paper is organized as follows: in section 2, we present themodel and its basic equations. In section 3, we describe the method and likelihoods.In section 4, we discuss the physical and kinematic properties of the model underconsideration. The summary of our findings is presented in section 5. The Einstein’s field equations in Brans-Dicke theory is given by R ij − Rg ij + Λg ij = 8 πφc T ij − ω φ (cid:18) φ i φ j − g ij φ k φ k (cid:19) − φ ( φ ij − g ij (cid:3) φ ) (1)and (2 ω + 3) (cid:3) φ = 8 πTc + 2 Λφ (2)where ω is the Brans-Dicke coupling constant; φ is Brans-Dicke scalar field and Λ is the cosmological constant.The Bianchi type V space-time is read as ds = dt − A ( t ) dx − e αx (cid:104) B ( t ) dy + C ( t ) dz (cid:105) (3)where A ( t ) , B ( t ) & C ( t ) are scale factors along x , y and z direction respectivelyand average scale factor is defined as a = ( ABC ) . The exponent α (cid:54) = 0 in (3) isan arbitrary constant.The energy momentum tensor of perfect fluid is given by T ij = ( p + ρ ) u i u j − pg ij (4)Here, p and ρ are the isotropic pressure and energy density of the matter underconsideration. also u i u j = − u i is the four velocity vector.The field equations (1) for space-time (3) are read as ¨ BB + ¨ CC + ˙ B ˙ CBC − α A + ω φ φ + ˙ φφ (cid:18) ˙ BB + ˙ CC (cid:19) + ¨ φφ = − πpφ + Λ (5)¨ AA + ¨ CC + ˙ A ˙ CAC − α A + ω φ φ + ˙ φφ (cid:18) ˙ AA + ˙ CC (cid:19) + ¨ φφ = − πpφ + Λ (6) R. Prasad, Avinash Kr. Yadav and Anil Kumar Yadav ¨ AA + ¨ BB + ˙ A ˙ BAB − α A + ω φ φ + ˙ φφ (cid:18) ˙ AA + ˙ BB (cid:19) + ¨ φφ = − πpφ + Λ (7)˙ A ˙ BAB + ˙ B ˙ CBC + ˙ C ˙ ACA − α A − ω φ φ + ˙ φφ (cid:18) ˙ AA + ˙ BB + ˙ CC (cid:19) = 8 πρφ + Λ (8)2 ˙ AA − ˙ BB − ˙ CC = 0 ⇒ A = BC (9)¨ φφ + (cid:18) ˙ AA + ˙ BB + ˙ CC (cid:19) ˙ φφ = 8 π ( ρ − p )(2 ω + 3) φ + 2 Λ ω + 3 (10)where over dot denotes derivatives with respect to time t.The equation of continuity is read as˙ ρ + (1 + γ ) (cid:18) ˙ AA + ˙ BB + ˙ CC (cid:19) ρ = 0 (11)where γ is the equation of state parameter of perfect baro-tropic fluid and it isdefined as γ = pρ = constant . The pressure of dark matter is zero which can berecover from baro-tropic equation of state by choosing γ = 0.2.1 Solution of Einstein’s field equationsEquations (5)-(7) lead the following system of equations¨ AA − ¨ BB + ˙ A ˙ CAC − ˙ B ˙ CBC + (cid:18) ˙ AA − ˙ BB (cid:19) ˙ φφ = 0 (12)¨ BB − ¨ CC + ˙ A ˙ BAB − ˙ A ˙ CAC + (cid:18) ˙ BB − ˙ CC (cid:19) ˙ φφ = 0 (13)¨ CC − ¨ AA + ˙ B ˙ CBC − ˙ A ˙ BAB + (cid:18) ˙ CC − ˙ AA (cid:19) ˙ φφ = 0 (14)The equations (12)-(14) are the system of three equations with four unknownvariables A , B , C and φ . So, one can not solve these equations in general. Inconnection with equation (9), one may propose the following relation among themetric functions B = AD & C = AD (15) where D = D ( t ) measures the anisotropy in universe. For D = 1 and α = 0,Bianchi V universe recovers the case of FRW universe.Equations (13) and (15) lead to¨ DD − ˙ D D + ˙ DD (cid:18) AA + ˙ φφ (cid:19) = 0 (16) onstraining Bianchi type V universe.. 5 After integration of equation (18), we obtain D = exp (cid:20)(cid:90) kA φ dt (cid:21) (17)Now, the average scale factor is computed as a = ABC = A ⇒ a = A (18)Therefore, the Friedmann equations (5) and (8) respectively recast as following2 ¨ aa + ˙ a a − α a + k a φ + ω φ φ + 2 ˙ φφ ˙ aa + ¨ φφ = − πpφ + Λ (19)3 ˙ a a − α a − k a φ − ω φ φ + 3 ˙ φφ ˙ aa = 8 πρφ + Λ (20)2.2 The model: Brans-Dicke anisotropic universeThe density parameters are read as Ω m = 8 πρ m H φ , Ω Λ = Λ H , Ω σ = k H a φ , Ω α = α a H (21)where ρ m = ( ρ m ) a − is the energy density of pressure-less matter and Ω m , Ω Λ and Ω σ represent the dimensionless density parameters for dark matter, Λ - energy,shear anisotropy and α parameter respectively. H is Hubble’s parameter and it isdefined as H = ˙ aa .The deceleration parameter q and scalar field deceleration parameter q φ areread as q = − ¨ aaH , q φ = − ¨ φφH (22)Dividing equations (20) by 3 H and then using equation (21), we have Ω m + Ω Λ + Ω σ + Ω α = 1 + ψ − ω ψ (23)where ψ = ˙ φφH .After some algebra in equations (10), (19) and (20), finally we obtained φ = φ (cid:18) a a (cid:19) ψ & ψ = 1 ω + 1 (24) where a is the present value of scale factor.Thus, equation (23) reduces to Ω m + Ω Λ + Ω σ + Ω α = 1 + 5 ω + 66( ω + 1) (25) R. Prasad, Avinash Kr. Yadav and Anil Kumar Yadav
If we define the density of scalar field φ as Ω φ = − ω + 66( ω + 1) , (26)then, equation (25) is recast as Ω m + Ω Λ + Ω σ + Ω α + Ω φ = 1 (27)The scale factor a and φ in connection with z are read as a = a z , φ = 1(1 + z ) ω , (28)Equations (21), (27) and (28) leads to H σBD = H (1 − Ω φ ) (cid:104) Ω m (1 + z ) . ( √ − Ωφ − ) +3 + Ω σ (1 + z ) ( √ − Ωφ − ) +6 + Ω α (1 + z ) + Ω Λ (cid:105) (29)where H , Ω m , Ω σ and Ω Λ denote present values of Hubble constant and densi-ties parameters due to dark matter, anisotropy and cosmological constant respec-tively. In this section, we briefly describe the observational data and the statisticalmethodology to constrain the Bianchi V universe as discussed in the previoussection. – Observational Hubble Data (OHD) : We adopt 46 H ( z ) datapoints overthe redshift range of 0 ≤ z ≤ .
36 obtained from cosmic chronometric (CC)technique. We have compiled all 46 H ( z ) datapoints in table 1. – Baryon acoustic oscillations (BAO) : We use 10 baryon acoustic oscilla-tions data extracted from the 6dFGS [50], SDSS-MGS [51], BOSS [52], BOSSCMASS [53], and WiggleZ [54] surveys.Note that in the above References, H ( z ) and error σ i are in the unit of km s − M pc − .In this paper, we have converted these quantities in the unit of Gyr − .For all analysis, we have defined a χ for parameters with the likelihood givenby ζ ∝ e − χ . Therefore, the χ function for H ( z ) data is written as χ H = (cid:88) i =1 (cid:20) H ( z i , s ) − H obs ( z i ) σ i (cid:21) (30) where s and σ i denote the parameter vector and standard error in in experimentalvalues of Hubble’s function H respectively.Similarly, the joint χ is read as χ joint = χ H + χ BAO (31) onstraining Bianchi type V universe.. 7
Table 1
Hubble parameter versus redshift data.
S.N. z H(z)[
Gyr − ] σ i [ Gyr − ] References1 0 0.069 0.0013 [55]2 0.07 0.069 0.020 [56]3 0.09 0.071 0.012 [57]4 0.01 0.071 0.012 [58]5 0.12 0.071 0.027 [56]6 0.17 0.07 0.0081 [58]7 0.179 0.085 0.0041 [59]8 0.1993 0.077 0.0051 [59]9 0.2 0.077 0.030 [56]10 0.24 0.075 0.0026 [60]11 0.27 0.081 0.014 [58]12 0.28 0.079 0.035 [56]13 0.35 0.091 0.0085 [61]14 0.352 0.085 0.0143 [59]15 0.38 0.085 0.0019 [62]16 0.3802 0.083 0.0137 [63]17 0.4 0.097 0.0173 [57]18 0.4004 0.079 0.0104 [63]19 0.4247 0.089 0.0114 [63]20 0.43 0.088 0.0038 [60]21 0.44 0.084 0.008 [64]22 0.4449 0.095 0.013 [63]23 0.47 0.091 0.051 [65]24 0.4783 0.083 0.009 [63]25 0.48 0.099 0.061 [58]26 0.51 0.092 0.0019 [62]27 0.57 0.106 0.0035 [53]28 0.593 0.106 0.0132 [59]29 0.6 0.089 0.0062 [64]30 0.61 0.099 0.0021 [62]31 0.68 0.094 0.0082 [59]32 0.73 0.099 0.0072 [64]33 0.781 0.107 0.012 [59]34 0.875 0.128 0.0173 [59]35 0.88 0.092 0.041 [58]36 0.9 0.120 0.0234 [58]37 1.037 0.157 0.020 [59]38 1.3 0.172 0.0173 [58]39 1.363 0.164 0.0343 [66]40 1.43 0.181 0.0183 [58]41 1.53 0.143 0.0143 [58]42 1.75 0.207 0.041 [58]43 1.965 0.191 0.0514 [66]44 2.3 0.229 0.0082 [67]45 2.34 0.227 0.0072 [68]46 2.36 0.231 0.0082 [69] Figures 1 and 2 exhibit the one-dimensional marginalized distribution and two-dimensional contours with 68% CL and 95% CL for parameter space Θ σBD using H(z) and combined
H(z)+ BAO data respectively. The numerical result of sta-tistical analysis is listed in table 2.We have summarized the numerical result of statistical analysis in table 2. Fromtable 2, it has been observed that the estimated constraints on H as 0 . Gyr − ∼ . km s − M pc − and 0 . Gyr − ∼ . km s − M pc − are closer to R. Prasad, Avinash Kr. Yadav and Anil Kumar Yadav
Fig. 1
One-dimensional marginalized distribution and two-dimensional contours with 68%CL and 95% CL for parameter space Θ σBD using H(z) data.
Table 2
Summary of statistical analysis
Model parameters H ( z ) H ( z ) + BAOH . Gyr − ) 0 . Gyr − ) Ω m Ω Λ Ω φ χ min χ ν other investigations [70,71,72,73]. The best fit curve of Hubble rate versus redshiftof derived model is shown in Fig. 3. In this paper, our aim is also to constrain the density of scalar field Ω φ and the estimated constraints on Ω φ with H(z) and
H(z)+ BAO data are as Ω φ = 0 . Ω φ = 0 .
014 respectively. InAmirhashchi and Yadav [74], we also find constraint on scalar field density as Ω φ = 0 .
010 by using different observational data sets. In table 2, χ ν is read as χ ν = χ min /dof where dof is abbreviation of degree of freedom and it is defined onstraining Bianchi type V universe.. 9 Fig. 2
One-dimensional marginalized distribution and two-dimensional contours with 68%CL and 95% CL for parameter space Θ σBD using H(z)+ BAO data. as the difference between all observational data points and the number of freeparameters. It should be noted that for χ ν ≤
1, the fitting of model with observeddata is considered as the best fitting model. q ( z ) = − z ) H (cid:48) σBD H σBD (32)Here, H (cid:48) σBD denotes first derivative of H σBD with respect to z . Fig. 3
The plot of Hubble rate versus the red-shift z. The points with error bars indicate theexperimental data summarized in Table 1. H(z) is in unit of
Gyr − . Fig. 4
The plot of deceleration parameter versus the red-shift z. The transition redshift is z t = 0 . Using equation (29), equation (32) is recast as q ( z ) = −
1+ ( z + 1) (cid:16) . Ω m0 (cid:0) . (cid:112) − Ω φ + 0 . (cid:1) ( z + 1) . √ − Ω φ − . + 2( z + 1) Ω α + q (cid:17) (cid:16) Ω Λ + Ω m0 ( z + 1) . √ − Ω φ +0 . + ( z + 1) Ω α + Ω σ ( z + 1) √ − Ω φ +1 (cid:17) (33)where q = Ω σ (cid:0) (cid:112) − Ω φ + 1 (cid:1) ( z + 1) √ − Ω φ .The present value of deceleration parameter is obtained as q = − Ω α + Ω m0 (cid:0) . (cid:0)(cid:112) − Ω φ − (cid:1) + 3 (cid:1) + Ω σ (cid:0) (cid:0)(cid:112) − Ω φ − (cid:1) + 6 (cid:1) Ω α + Ω Λ + Ω m0 + Ω σ ) (34)The dynamics of deceleration parameter with the age of universe id depictedin Fig. 4. The derived model represents a transitioning universe with a transitionredshift of about z t = 0 .
79. We observe that the current universe is in acceleratingphase while it was in decelerating phase of expansion in past. The present value ofdeceleration parameter q is about − .
61. This value of q is in excellent agreementwith recent observations.4.2 The age of universeThe age of universe is obtained as dt = − dz (1 + z ) H σBD ⇒ (cid:90) t t dt = − (cid:90) z z ) H σBD dz (35)Equations (29)and equation (35) lead to t − t = (cid:90) z (1 − Ω φ ) dzH (1 + z ) (cid:104) Ω m (1 + z ) . ( √ − Ωφ − ) +3 + Ω σ (1 + z ) ( √ − Ωφ − ) +6 + Ω α (1 + z ) + Ω Λ (cid:105) (36)Here, t is the present age of the universe. Hence t = lim x →∞ (cid:90) z (1 − Ω φ ) dzH (1 + z ) (cid:104) Ω m (1 + z ) . ( √ − Ωφ − ) +3 + Ω σ (1 + z ) ( √ − Ωφ − ) +6 + Ω α (1 + z ) + Ω Λ (cid:105) (37)Integrating equation (37), we get H t = 0 . t = 0 . H − ∼ H ( t − t ) versus redshift z is graphed in Fig. 5. FromFig. 5, we observe that at present time i.e. for z = 0, H ( t − t ) is null which turninto imply t = t . Fig. 5
The plot of H ( t − t ) versus the red-shift z for Ω m = 0 . Ω Λ = 0 .
733 and Ω φ = 0 . t = 0 to t = t .Here, we assume light signal emits from a source along x-axis. The properdistance of the source will be a x and we are receiving that signal at present time t . Thus, the proper distance of the source from us is calculated as a (cid:82) t t p dta ( t ) where t p is the time in past at which the light signal was transmitted from source.Therefore, the particle horizon is computed as R p = lim t p → a (cid:90) t t p dta ( t ) = lim z →∞ (cid:90) z dzH σBD (39)Using equation (29), equation (39) becomes R p = lim z →∞ (cid:90) z (1 − Ω φ ) dz H (cid:104) Ω m (1 + z ) . ( √ − Ωφ − ) +3 + Ω σ (1 + z ) ( √ − Ωφ − ) +6 + Ω α (1 + z ) + Ω Λ (cid:105) (40)Integrating equation (40) for Ω m = 0 . Ω Λ = 0 .
733 and Ω φ = 0 . R p = 2 . H (41) onstraining Bianchi type V universe.. 13 Fig. 6
The plot of proper distance a H x versus the redshift z for Ω m = 0 . Ω Λ = 0 . Ω φ = 0 . Fig. 6 shows variation of proper distance versus redshift. From Fig. 6, we observethat at present i.e. for z = 0, a H x is null which turn into imply that x → ∞ .Thus we are at infinite distance from the first event occurred in past.4.4 The jerk parameterThe jerk parameter (j) [76], in terms of red-shift is given by j = 1 − (1 + z ) H (cid:48) σBD H σBD + 12 (1 + z ) [ H (cid:48)(cid:48) σBD ] [ H σBD ] (42)Equations (29) and (42) lead to j = 1 − (1 + z ) ξ + (1 + z ) ξ (43)where ξ = . Ω m ( √ − Ω φ +0 . ) ( z +1) . √ − Ωφ − . +2( z +1) Ω α + Ω σ ( √ − Ω φ +1 ) ( z +1) √ − Ωφ √ − Ω φ (cid:113) Ω Λ + Ω m ( z +1) . √ − Ωφ +0 . +( z +1) Ω α + Ω σ ( z +1) √ − Ωφ +1 ξ = ( w w − w ) w w = . (cid:16) Ω m ( . √ − Ω φ +0 . ) ( z +1) . √ − Ωφ +( z +1) . (cid:16) (0 . z +0 . Ω α + Ω σ ( . √ − Ω φ +0 . ) ( z +1) √ − Ωφ (cid:17)(cid:17) ( z +1) Fig. 7
The variation of jerk parameter versus redshift. w = (cid:16) Ω Λ + Ω m ( z +1) . √ − Ωφ +0 . +( z +1) Ω α + Ω σ ( z +1) √ − Ωφ +1 (cid:17) ( z +1) . ( z +1) w = Ω m w ( z +1) . √ − Ω φ +( z +1) . (cid:16) ( z + 1) Ω α + Ω σ (cid:0) − . Ω φ + 2 . (cid:112) − Ω φ + 12 . (cid:1) ( z + 1) √ − Ω φ (cid:17) w = 32 w (cid:16) Ω Λ + Ω m ( z + 1) . √ − Ω φ +0 . + ( z + 1) Ω α + Ω σ ( z + 1) √ − Ω φ +1 (cid:17) w = ( − . z − . Ω φ + 3 z + 3 w = (cid:16) Ω Λ + Ω m ( z + 1) . ( √ − Ω φ − ) +3 + ( z + 1) Ω α + Ω σ ( z + 1) ( √ − Ω φ − ) +6 (cid:17) In 2004, Blandford et al. [77] have described the features of the jerk parameter-ization which gives an alternative approach to describe cosmological models closeto Λ CDM model. A powerful feature of of the jerk parameter is that for the CDMmodel j = 1. In Refs. [78,79], the authors have investigated the important featuresof j for discriminating different dark energy models. The value j (cid:54) = 1 would favora non- Λ CDM model. In the considered model, the explicit behavior of j is shownin Fig. 7. We observe that the jerk parameter of considered model does not have j = 1. σ = 12 σ ij σ ij (44) onstraining Bianchi type V universe.. 15 Fig. 8
The plot of relative anisotropy A m versus z . where σ ij = u i ; j − θ ( g ij − u i u j )In derived model, the shear scalar is given by σ = ˙ D D = k (1 + z ) √ − Ω φ − (45)Thus the relative anisotropy is obtained as A m = σ ρ m (46)From equation (46), it is clear that relative anisotropy depends on red-shift z . Forhigh value of red-shift, the relative anisotropy is large and it decreases as with lowvalue of z and finally becomes null at z →
0. This behavior of relative anisotropy A m is depicted in Fig. 8. In this paper, we have investigated a transitioning model of an-isotropic universein Brains-Dicke theory of gravitation. We describe that the current phase of ac- celerated expansion of the universe is due to contribution coming from Λ screenedscalar field and the transition redshift is z t = 0 .
79. For redshift z > z t , the universewas in decelerating phase of expansion. Some important features of derived modelare as follows: i) The derived model obeys Mach’s principle.ii) We find constraints on H , Ω m and Ω Λ by bounding the model under con-sideration with recent OHD and BAO data. The best fit values of H are closerto other investigations [70,71,72,73]. Thus, we conclude the present OHD andBAO data provides well constrained values of H and our model have goodconsistency with recent observations.iii) We have estimated the present age of universe as t = 13 .
65 Gyrs. This age ofuniverse is nicely matches with those obtained by Plank collaboration.iv) The dynamics of deceleration parameter is showing a signature flipping fromearly decelerating phase to current accelerating phase at z t = 0 .
79. The presentvalue of deceleration parameter is computed as q = − . Λ CDM model of universe.vi) In the derived model, j (cid:54) = 1. Therefore, the derived solution describes the modelof universe other than Λ CDM and the deviation from j = 1 investigates thedynamics of different kinds of dark energy models other than Λ CDM. Someimportant applications of non Λ CDM model of the universe are given in Refs.[80,81].
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