Existence of bulk viscous universe in f(R,T) gravity and confrontation with observational data
Anil Kumar Yadav, Lokesh Kumar Sharma, B. K. Singh, P. K. Sahoo
EExistence of bulk viscous universe in f ( R, T ) gravity and confrontation withobservational data Anil Kumar Yadav ∗ , Lokesh Kumar Sharma † , B. K. Singh ‡ , P. K. Sahoo § Department of Physics, United College of Engineering and Research, Greater Noida - 201306, India Department of Physics, GLA University, Mathura - 281406 India and Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad-500078, India
Abstract:
In this paper we have investigated a bulk viscous universe in f ( R, T ) gravity where R and T are the Ricci scalar and trace of energy momentum tensor respectively. We have obtainedexplicit solutions of field equations in modified gravity by considering the power law form of scalefactor. The Hubble parameter and deceleration parameter are derived in terms of cosmic timeand redshift both. We have estimated the present values of these parameters with observationalHubble data and SN Ia data sets. At 1 σ level, the estimated values of q and m are obtained as q = − . ± .
05 & m = 0 . ± .
02 where q is the present value of deceleration parameter and m is the model parameter. The energy conditions and Om(z) analysis for the anisotropic LRS Bianchitype I model are also discussed. Keywords:
Cosmological parameters; bulk viscosity; f ( R, T ) gravity; LRS Bianchi-I space-time.
PACS numbers: 04.50.kd.
I. INTRODUCTION
From the recent astrophysical observations: H ( z ) from Ia supernova [1–5], CMB [6, 7], baryon acoustic oscillations(BAO) [8–10] and PLANK [11], it has been confirmed that we are lived in an accelerating universe. The mainconclusion of these observations are strongly suggest that nearly two-third of the total energy density is in the formof unknown/mysterious energy. This late time acceleration of the universe is considered to be driven dark energy(DE) however the actual nature of DE is yet to be investigate. In the literature, numerous models with different DEcandidates are proposed. Among these models, ΛCDM model is more acceptable by theoretical physicists to explainthe behavior of DE and late-time acceleration of universe. The ΛCDM models have two fundamental problems -fine-tuning at plank scale and cosmic coincidence [12]. The problems associated with ΛCDM model have led tosearch for the reconstruction of gravitational field theories which could be capable of reproducing late time cosmicacceleration without inclusion of cosmological constant in Einstein’s field equation. However one can not claim thatthe idea of modification in general relativity was en-lighted just after the discovery of accelerating universe [13, 14].A number of modified theories such as f ( R ) theory [15, 16], f ( G ) gravity [17], f ( T ) theory [18, 19] and f ( R, T )theory [20] exist since a long time due to the combined need of astrophysics, high energy physics and cosmology.At beginning, the quest in modification of general relativity was focused on the change of geometrical part ofEinstein-Hilbert gravitational action. In [20] the authors have investigated a non-minimal coupling between matterand geometry in the framework of an effective gravitational Lagrangian consisting of R & T : R is an arbitraryfunction of Ricci scalar and T is the trace of energy-momentum tensor and thus introduced f ( R, T ) theory ofgravitation. In this theory, the choice of T is due to the existence of some imperfect fluids. Thus, the f ( R, T ) theoryof gravitation may give a complete description of late time acceleration of universe without resorting the existenceof dark energy. This extraordinary features of f ( R, T ) theory of gravitation has attracted researchers to study andreconstruct this theory in various contexts of astrophysics and cosmology [21, 22]. Recently, Nojiri et al. [23] havestudied inflation, bounce and late time evolution in modified theory of gravity. It is important to mentioning that f ( R, T ) theory is also applicable to investigate the effects of expansion-free condition in the formulation of structurescalars. Some useful applications and existence of relativistic stellar objects in f ( R, T ) theory of gravity are given inreferences [24–26].After discovery of Wilkinson Microwave Probe [27, 28], the homogeneous and anisotropic models have been gainingan increasing attention and tremendous momentum in observational cosmology in the search of relativistic picture ofthe universe in its early stages. A spatially homogeneous Bianchi I (BI) space-time necessarily have three dimensionalgroup, which acts as transitively on space-like three dimensional orbits. Therefore, the universe should achievefollowing two features: (i) a slightly anisotropic geometry in spite of inflation, and (ii) a non trivial isotropization ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] § Email: [email protected] a r X i v : . [ phy s i c s . g e n - ph ] M a r history of the universe due to the presence of an anisotropic energy source. The advantage of these anisotropic modelare that they have a significant role in the description of evolution of early phase of the universe and capable infinding more general cosmological models in comparison to FRW model. The LRS BI universe is homogeneous andanisotropic model of universe and it is differ from FRW model: a popular model to describes the fate of universe.But, in the recent time, CMB observation indicates very tiny variations in the intensities of microwaves coming fromdifferent direction in the sky [33]. This observation challenges the isotropic assumptions in spatial directions so theidea of anisotropic space-time comes forward with Bianchi type models [34–42]. Recently, in [43, 44] the authors haveinvestigated some non-isotropic cosmological models in different physical contexts.In the realm of cosmology, especially bulk viscous phenomena have attracted considerable interest [29] because itis only possible dissipative mechanism in Bianchi type space-time [30, 31]. The coefficient of bulk viscosity vanishesboth for actual relativistic and non-relativistic equation of state. In the era of inflationary phase, the contributionof bulk viscosity is well recognized. The basic concept of bulk viscous driven inflation is that the bulk viscositycontributes a negative pressure and this negative pressure simulating a repulsive gravity of the matter and givesan impetus for rapid expansion of the universe. Relativistic cosmological solutions and the different phases of theuniverse with non-causal viscous fluid are studied in f ( R, T ) gravity theory [32]. In this extended theory of gravity,ideal fluids play major role to contribute acceleration but on the hydrodynamics scale, it is impossible to defineturbulence phenomena without inclusion of bulk viscosity. The two viscous coefficients are discussed in literature: (i)shear viscosity and (ii) bulk viscosity. In early universe, Hogeveen and his team have examined the contribution ofboth shear and bulk viscosity by using kinetic theory and it is discovered that the impact of viscosity is very smallin early universe whereas the impact may be significant in future universe [45]. Later on, Brevik and his co-authorshave investigated the effect of viscosity in early universe for both homogeneous and in-homogeneous equation ofstate [46, 47]. Recently, in [48–50] the authors have investigated bulk viscous embedded hybrid universe in generalrelativity and f ( R, T ) theory of gravitation respectively.The main goal of this paper is to investigate the bulk viscous anisotropic universe in the framework f ( R, T ) theoryof gravity. The paper is organized as follows: Section II represents the explicit solutions of field equations of LRS BIuniverse. Section III deals with the confrontation of derived model with observational data. The behaviour of energyconditions are discussed in Section IV. In Section V, we present the Om(z) analysis and finally the conclusion is givenin section VI.
II. METRIC AND FIELD EQUATIONS
The spatial homogeneous and anisotropic LRS BI metric is read as ds = − dt + A ( t ) dx + B ( t )( dy + dz ) (1)where A ( t ) and B ( t ) are scale factors along the spatial direction.The energy momentum tensor for bulk-viscous fluid is read as T µν = ( ρ + ¯ p ) u µ u ν − ¯ pg µν (2)where u µ = (0 , , ,
1) is the four velocity vector in co-moving co-ordinate system satisfying u µ u ν = 1 and ¯ p = p − ξH .Here, ξ , ¯ p and p are the bulk viscous coefficient, bulk viscous pressure and normal pressure respectively.The energy-momentum tensor for barotropic bulk viscous fluid under f ( R, T ) = R + 2 ζT formalism is given by R µν − Rg µν = 8 πT µν − T µν + Θ µν ) f (cid:48) ( T ) + f ( T ) g ij (3)where f ( T ) = ζT , ζ is an arbitrary constants.Thus, the equation (3) and (1)lead to − BB − ˙ B B = (8 π + 3 ζ )¯ p − ζρ (4) − ¨ AA − ¨ BB − ˙ A ˙ BAB = (8 π + 3 ζ )¯ p − ζρ (5)2 ˙ A ˙ BAB + ˙ B B = (8 π + 3 ζ ) ρ − ζ ¯ p (6)Equation (4) -(6) are the system of three equations with four unknown variables. In general, it is impossible tosolve these equations however the exact solution is only possible by taking into account at least one physical relationamong parameters. So, we assume that the average scale factor expands in power function of time i.e. a = ( mDt ) m (7)where m , D and n are positive constants .Equation (7) gives the following expressions for directional scale factors A and BA = ( m D t ) m (8) B = ( m D t ) m (9)where m , m , D and D are constant and satisfies the following relation. m ( m + 2 m ) = 3 m m , D m = D m D m The Hubble’s parameter ( H ) and deceleration parameter ( q ) are given by H = 1 mt (10) q = m − ρ ), pressure (p) and bulk viscous pressure (¯ p ) for model (1) are read as ρ = 18 π + ζ (3 − γ ) (cid:20) m + m m m t − ξmm t (cid:21) (12) p = γ π + ζ (3 − γ ) (cid:20) m + m m m t − ξmm t (cid:21) (13)¯ p = γ π + ζ (3 − γ ) (cid:20) m + m m m t − ξmm t (cid:21) − ξmt (14) III. CONFRONTATION WITH OBSERVATIONAL DATA
The relation between scale factor a and redshift z is given by a = a z (15)where a is the present value of scale factor.The differential age of the galaxies are used for observational Hubble data (OHD) by following equation [52] H ( z ) = −
11 + z dzdt (16)We consider 36 data points of OHD in the redshift range 0 . ≤ z ≤ .
36 with their corresponding standard deviation σ H , recently compiled in Refs. [33, 51].We define matter energy density parameter (Ω m ), anisotropy curvature parameter (Ω σ ) and bulk viscous parameter(Ω b ) as following Ω m = 8 πρ H , Ω σ = σ H a , Ω b = ξ H (17)and Ω m + Ω σ + Ω b = 1 (18)where σ = σ ij σ ij is the shear scalar and ξ = 3 ζ ( ρ + ξH ).Solving equations (6), (15), (17) and (18), we obtain H = H [(Ω m ) (1 + z ) + (Ω σ ) (1 + z ) + (Ω b ) ] (19)where subscript 0 denotes the present value of parameters. H is the present value of Hubble constant in kms − M pc − .Here we find constraints on H and (Ω b ) by using 36 H(z) data points, recently published in Table II of the paper ofAkarsu et al [51]. For this sake, we define χ as following χ OHD ( H , (Ω b ) ) = (cid:88) i =1 (cid:20) H ( z i , H , (Ω b ) ) − H obs ( z i ) σ i (cid:21) (20)where H obs ( z i ) is the observed value of Hubble parameter with standard deviation σ i and H ( z i , H , (Ω b ) ) isthe theoretical values obtained from the derived model. We find that the best fit values of the parameters are H = 65 . ± . b ) = 0 . ± .
06 together with reduced χ OHD = 7 .
61. The likelihood contour at 68.3% ,95.4% and 99.7% confidence level around the best fit values in the H − (Ω b ) plane is shown in Fig. 1. SimilarlyFig. 2 depicts the likelihood contour at 68.3%, 95.4% and 99.7% confidence level around the best fit values as q = − . ± .
03 & m = 0 . ± .
04 in the m − q plane obtained by fitting derived model with 36 observationHubble data points.
55 60 65 70 75 (cid:45) H (cid:87) b FIG. 1: The likelihood contour at 68.3% (inner contour), 95.4% (middle contour) and 99.7% (outer contour) confidence levelaround the best fit values as H = 65 . ± . b ) = 0 . ± .
06 in the H − (Ω b ) plane obtained by fitting derived modelwith H ( z ). (cid:45) (cid:45) (cid:45) (cid:45) m q FIG. 2: The likelihood contour at 68.3% (inner contour), 95.4% (middle contour) and 99.7% (outer contour) confidence levelaround the best fit values as q = − . ± .
03 & m = 0 . ± .
04 in the m − q plane obtained by fitting derived model with36 observation Hubble data points. A. Type Ia Supernova
We fit our model with 580 points of SN Ia data set [53] and choose the value of current Hubble constant as H = 65 . kms − M pc − to complete the data set. Thus χ SN is obtained as χ SN ( m, q ) = (cid:88) i =1 (cid:20) µ ( z i , m, q ) − µ obs ( z i ) σ µ ( z i ) (cid:21) (21)where µ ( z i , m, q ) and µ obs ( z i ) are the theoretical and observed values distance modulus for the model under consid-eration respectively. σ µ ( z i ) represents the standard error in the observed value of µ . The distance modulus is givenby µ ( z ) = m b − M = 5 log D L ( z ) + µ (22)where m b , M are the apparent magnitude and absolute magnitude of a standard candle respectively. The luminositydistance D L and nuisance parameter ( µ ) are read as D L = c (1 + z ) H (cid:90) z dzh ( z ) ; h ( z ) = HH (23)and µ = 5 log (cid:18) H − M pc (cid:19) + 25 (24)Therefore, the distance modulus and apparent magnitude are given by µ ( z ) = 5 log (cid:20) c (1 + z ) H (cid:90) z dzh ( z ) (cid:21) (25) m b ( z ) = 16 .
08 + 5 log (cid:20) z . (cid:90) z dzh ( z ) (cid:21) (26)Thus χ total is given by χ joint = χ OHD + χ SN (27)Fig. 3 shows the likelihood contour at 68.3%, 95.4% and 99.7% confidence level around the best fit values as q = − . ± .
05 & m = 0 . ± .
02 in the m − q plane obtained by fitting derived model with H ( z ) + SN Ia data. (cid:45) (cid:45) (cid:45) m q FIG. 3: The likelihood contour at 68.3% (inner contour), 95.4% (middle contour) and 99.7% (outer contour) confidence levelaround the best fit values as q = − . ± .
05 & m = 0 . ± .
02 in the m − q plane obtained by fitting derived model with H ( z ) + SN Ia data. The density plot of H(z) and SN Ia data are shown in Figures 4 and 5 respectively. -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 . . . . . . z D en s i t y
40 50 60 70 80 . . . . . H0 D en s i t y FIG. 4: Density plot of red-shift and Hubble’s parameter of observational Hubble data points . . . . z D en s i t y
35 40 45 . . . . Distance modulus D en s i t y FIG. 5: Density plot of red-shift and distance modulus of SN Ia data z D i s t an c e . m odu l u s FIG. 6: The observational 580 SN Ia data points are shown with error bar (black color). The best fit model distance modulus( µ ( z )) curve (solid red points online) based on theoretical values is shown versus z. TABLE I: Summary of the numerical result.Source/Data Model parameters Values at presentH(z) m 0 . ± . q − . ± . . ± . q − . ± . H . ± . b ) . ± . z A ppa r en t. m agn i t ude Apparent magnitude (Observed)Apparent magnitude (Our model)
FIG. 7: Apparent magnitude versus red-shift best fit curve.
In order to get the best fit curve of derived model with 580 SN Ia data points for distance modulus and apparentmagnitude, we compute R test by using the following statistical formula R SN ( µ ) = 1 − (cid:80) i [( µ i ) obs − ( µ i ) th ] (cid:80) i [( µ i ) obs − ( µ i ) mean ] (28)where ( µ i ) obs and ( µ i ) th are the observed and theoretical values of distance modulus. R = 1 corresponds to the idealcase when the observed data and corresponding values of theoretical function merge exactly. We find R SN ( µ ) = 0 . µ ( z ) curve(solid red points online) is shown versus z. Similarly we compute R test for apparent magnitude and find that R SN ( m b ) = 0 .
979 with root mean square error (RMSE)= 6.21 which also shows appreciable consistency of derivedmodel with observations. This result is graphed in Fig. 7. We have compiled the numerical result in Table 1.
IV. CONSEQUENCES OF ENERGY CONDITIONS
Some important issues in theoretical cosmology have been discussed with help of energy conditions (ECs) namelyweak energy condition (WEC), null energy condition (NEC), dominant energy condition (DEC) and strong energycondition (SEC). In particular the violation of SEC implies that the important problem of accelerated expansion ofuniverse is supported by anisotropic universe [54]. The ECs in f ( R, T ) gravity theory by incorporating conservationof energy-momentum tensor is presented in the literature [55] and it has been analyzed that T sector can not bechosen arbitrarily but it has special form. The ECs in modified gravitational field equations are given as follows [56] W EC ⇐⇒ ρ ≥ ρ + ¯ p ≥ ,N EC ⇐⇒ ρ + ¯ p ≥ ,DEC ⇐⇒ ρ ≥ | ¯ p | ,SEC ⇐⇒ ρ + 3¯ p ≥ . (29)The behaviors of the above ECs are depicted in Fig. 10 with m = 0 . γ = 0 .
5. From Fig. 10, we observe thatall the energy conditions are violated for the fixed values of the free parameters. The values of free parameters areobtained by bounding the derived model with observational data and positive energy density. r FIG. 8: Energy density and pressure vs. t with γ = 0 . m = . p FIG. 9: Bulk viscous pressure vs. t with γ = 0 . m = . rr+ p r- p r+ FIG. 10: Energy conditions vs. t with γ = 0 . m = . V. OM(Z) DIAGNOSTIC ANALYSIS
In literature, the state finder parameters r − s and analysis of Om diagnostic are used to study dark energy models[57]. The Om(z), is a combination of the Hubble parameter H and the cosmological redshift z . The Om(z) parameter0in modified gravity is given as [58] Om ( z ) = (cid:104) H ( z ) H (cid:105) − z ) − H is the present Hubble parameter. The negative, zero and positive values of Om( z ) represents the quintessence( ω > − ω < −
1) DE models respectively [59]. In the present model the Om(z) parameter isobtained as Om ( z ) = (1 + z ) m − z ) − Fig. 11 . O m ( z ) FIG. 11: Variation of Om(z) against z with m = . VI. CONCLUDING REMARKS
In this paper, we have estimated the numerical values of model parameters of derived model by bounding it withobservational data. The numerical results are tabulated in Table I.Some key observations of present study are as follows: • The computed value of deceleration parameter for derived model is q = − . ± .
05 at 1 σ confidence level. • The distance modulus ( µ ( z )) and apparent distance ( m b ( z )) of derived best fit model fit well to the observationaldata points from astronomical observations (see Fig. 6 & 7). • The derived model is LRS BI anisotropic universe which tends to isotropic at t → ∞ i.e. the anisotropy is null forlarger time. • In the derived model, we find (Ω b ) = 0 . ± .
06 at 1 σ level of 36 OHD points. It declares the significantcontribution of bulk viscosity in the present universe. • The NEC, WEC, DEC and SEC are violated in our present model (Fig. 10). It is worth to mention here that dueto the current accelerated expansion of the universe, SEC must be violated [60]. Hence, we can conclude that ourderived model is a spar with the current accelerated expansion of the universe. • The Om(z) is plotted with respect to redshift in the range 0 ≤ z ≤ ACKNOWLEDGMENT
PKS acknowledges CSIR, New Delhi, India for financial support to carry out the Research project[No.03(1454)/19/EMR-II Dt.02/08/2019]. We are very much grateful to the honourable referee and the editor for theilluminating suggestions that have significantly improved our work in terms of research quality and presentation. [1] S. Perlmutter et al., Nature , 51 (1998). [2] S. Perlmutter et al., Astrophys. J. , 5 (1999).[3] A. G. Riess et al., Astron. J. , 1009 (1998).[4] J. L. Tonry et al., Astrophys. J. , 1 (2003).[5] A. Clocchiatti et al., Astrophys. J. , 1 (2006).[6] P. de Bernardis et al., Nature , 955 (2000).[7] A. Clocchiatti et al., Astrophys. J. , L5 (2000).[8] C. Blake et al., Mon. Not. R. Astron. Soc. , 1707 (2011)[9] N. Padmanabhan et al., Mon. Not. R. Astron. Soc. , 2132 (2012)[10] L. Anderson et al., Mon. Not. R. Astron. Soc. , 3435 (2013)[11] C. L. Bennett et al. [WMAP Collaboration], Astrophys. J. Suppl. , 20 (2013)[12] N. A. Hamed et al., Phys. Rev. Lett. , 4434 (2000)[13] C. Brans, R. H. Dicke, Phys. Rev. D , 925 (1961)[14] G. Lyra, Math. Z. , 52 (1951)[15] S. M. Carroll, V. Duvvuri, M. Trodden, M. S. Turner, Phys. Rev. D , 043528 (2004).[16] S. Nojiri, S. D. Odintsov, Int. J. Geom. Methods. Mod. Phys. , 115 (2007).[17] S. Nojiri, S. D. Odintsov, Phys. Lett. B , 1 (2005).[18] E. V. Linder, Phys. Rev. D , 127301 (2010).[19] R. Myrzakulov, Eur. Phys. J. C , 1752 (2011)[20] T. Harko et al., Phys. Rev. D , 024020 (2011)[21] S. Nojiri & S. D. Odintsov, Phys. Rep. , 59 (2011)[22] H. Shabani & M. Farhoudi, Phys. Rev. D , 044048 (2013)[23] S. Nojiri, S. D. Odintsov, V. K. Oikonomou, Phys. Rept. , 1 (2017)[24] Z. Yousaf, K. Bamba, and M. Z.-ul-H. Bhatti, Phys. Rev. D , 124048 (2016).[25] Z. Yousaf, K. Bamba, M. Z. Bhatti, and U. Ghafoor, Phys. Rev D , 024062 (2019)[26] Z. Yousaf, Eur. Phys. J. Plus , 276 (2017)[27] T.R. Jaffe et al., Astrophys. J. Lett. , L1 (2005)[28] G. Hinshaw et al., Astrophys. J. Suppl. Ser. , 225 (2009)[29] P. Gran, Astrophys. Space Sci. , 191 (1990)[30] C. W. Misner, Astrophys. J. , 431 (1968)[31] W. Zimdahl et al. Phys. Rev. D , 063501 (2001)[32] C. P. Singh, Pankaj Kumar, Eur. Phys. J. C. , 3070 (2014)[33] H. Amishashchi, S. Amirhashchi, Phys. Rev. D , 023516 (2019)[34] A. K. Yadav et al., EPJ Plus , 127 (2012)[35] G. K. Goswami et al., Int. J. Theor. Phys. , 8 (2016)[36] G. K. Goswami et al., Astrophys Space Sc. , 47 (2016)[37] L. K. Sharma, B. K. Singh, A. K. Yadav, arXiv: 1907.03552 [physics.gen-ph][38] A. K. Yadav, L. Yadav, Int. J. Theor. Phys. , 218 (2011)[39] A. K. Yadav, Astrophys. Space Sc. , 565 (2011)[40] A. K. Yadav, Astrophys. Space Sc. , 276 (2016)[41] A. K. Yadav, Braz. J. Phys. , 262 (2019)[42] S. Kumar, C. P. Singh, Gen. Relativ. Grav. , 1427 (2011)[43] B. Mishra et al. Mod. Phys. Lett. A , 1850170 (2018)[44] B. Mishra et al., Astrophys. Space Sc. , 86 (2018)[45] F. Hogeveen et al. Phys. A Stat. Mech. Appl. , 458 (1986)[46] I. Brevik et al., arXiv:1706.02543v1 [gr-qc], (2017)[47] I. Brevik et al., arXiv:1708.06244v1 [gr-qc], (2017)[48] B. Mishra et al., Eur. Phys. J. C , 34 (2019)[49] A. K. Yadav et al., Mod. Phys. Lett. A , 1950145 (2019)[50] P. K. Sahoo, Parbati Sahoo, B. K. Bishi, Int. J. of Geom. Meth. in Mod. Phys. , 1750097 (2017)[51] O. Akarsu, S. Kumar, S. Sharma, L. Tedesco, Phys. Rev. D , 023532 (2019)[52] O. Akarsu et al. Eur. Phys. J Plus , 22 (2014).[53] N. Suzuki et al., Astrophys. J. , 85 (2012)[54] M. Visser, C. Barcelo. arXiv:gr-qc/0001099[55] S. Chakraborty, Gen. Relativ. Gravit. , 2039 (2013)[56] P. H. R. S. Moraes, P. K. Sahoo, Eur. Phys. J. C , 480 (2017)[57] V. Sahni, A. Shafieloo, A. A. Starobinsky, Phys. Rev. D , 103502 (2008)[58] P. K. Sahoo, P. H. R. S. Moraes, P. Sahoo, B. K. Bishi, Eur Phys J C , 736 (2018)[59] M. Shahalam, S. Sami, A. Agarwal, Mon. Not. Roy. Astron. Soc. , 2948 (2015)[60] C. Barcelo, M. Visser, Int. J. Mod. Phys. D11