AIJRFANS19-

American International Journal of Research in Formal, Applied & Natural Sciences Available online at http://www.iasir.net ISSN (Print): 2328-3777, ISSN (Online): 2328-3785, ISSN (CD-ROM): 2328-3793 AIJRFANS is a refereed, indexed, peer-reviewed, multidisciplinary and open access journal published by International Association of Scientific Innovation and Research (IASIR), USA (An Association Unifying the Sciences, Engineering, and Applied Research) Shishir Cosmological model Bianchi type VI with use Barotropic in modified theory of general relativity Shishir Kumar Srivastava Department of Mathematics Ganpat Sahai P.G.College, Sultanpur (U.P.), INDIA. Abstract: We Consider Shishir cosmological model Bianchi type VI with use Barotropic in modified theory of general relativity, To solve the Einstein field equation by Shishir assuming 𝛼 = 𝛽 𝑚1 and 𝑝 = 𝜂𝜌 .The Physical Significance of the Cosmological model is discussed. Keywords: Bianchi type VI, Barotropic space time, Expansion scalar, shear, Hubble parameter, cosmology. 2010Mathematics Subject Classification: 83C05,83C15 I. Introduction Cosmology is the scientific study of large scale properties of the universe as a whole. cosmology is study motion of crystalline objects i.e.study of the early stages of universe. The model of Universe formed by massive string is initiated by letelier , Which was used as Bianchi type I and ‘’Kantowski- Sachs ‘’ type of Cosmological models [1,2] Chakraborty et.al [3] investigated string cosmological model in general relativity In LRS Bianchi-I Cosmological model with polytropic equation of state R.kdubey and shishir [4] Bianchi type I and Bianchi type III in westigated type by Bali et.al[5,6] Verma and Ram [7] investigated the solution of Bianchi type VI Bulk Vicous Fluid models with variable Gravitational and Cosmological Constants. Recently, Bali V Poonia [8] investigated Bianchi type VI. Inflationary Cosmological model in General Relativity, Bali et.al [9] and Bhogar et.al[10] has investigated Bianchi type VI. in general relativity.; In the following paper, We considered ‘Shishir Cosmological Bianchi type VI with use Barotropic in modified theory of general relativity. II. The Metric Equations We consider Bianchi type VI metric in the form 𝑑𝑠 2 = −𝑑𝑡 2 + 𝛼 2 𝑑𝑥 2 + 𝛽 2 𝑒 −2𝑥 𝑑𝑦 2 + 𝛾 2 𝑒 2𝑥 𝑑𝑧 2 _________(2.1) Where 𝛼 𝑎𝑛𝑑 𝛽are function of 𝑡 only. The energy momentum tensor for a bulk viscous fluid distribution is given by 𝑛 _____________(2.2) 𝑇𝑚𝑛 = (𝜌 + 𝑝)𝑣𝑚 𝑣 𝑛 + 𝑝𝑔𝑚 Where 𝜌 is the energy density of the cosmic matter and p is ,its pressure. 𝑣𝑚 𝑖𝑠 The four velocity vector such that 𝑣𝑚 𝑣 𝑛 = −1 = −𝑢𝑚 𝑢𝑛 __________ (2.3) 𝑢𝑚 𝑣𝑚 = 0________(2.4) The vector 𝑢𝑚 𝑢𝑛 describes the direction of string or direction or anisotropy. We take the equation of state 𝑝 = 𝜂𝜌, 0 ≤ 𝜂 ≤ 1 The Einstein field equation 1 𝑅𝑚𝑛 − 𝑅𝑔𝑚𝑛 = −8𝜋𝐺𝑇𝑚𝑛 ________(2.5) 2 Where 𝑅𝑚𝑛 is Known as Ricci tensor and 𝑇𝑚𝑛 is the energy momentum tenser for matter For the line element(2.1) and field Equation (2.5) can be written as 𝛽̈ 𝛽 𝛼̈ 𝛼 𝛼̈ 𝛾̈ + + 𝛾 𝛾̈ + + 𝛾 𝛽̈ + + 𝛼 𝛼̇ 𝛽̇ 𝛼𝛽 𝛽̇ 𝛽 + 𝛼̇ 𝛾̇ 𝛼𝛾 𝛾̇ 𝛽̇ 𝛾̇ 𝛽𝛾 𝛼̇ 𝛾̇ 𝛼𝛾 𝛼̇ 𝛽̇ + − − 1 𝛼2 1 𝛼2 1 = −8𝜋𝐺𝑝____________(2.6) = −8𝜋𝐺𝑝____________(2.7) = −8𝜋𝐺𝑝____________(2.8) 𝛼2 𝛼𝛽 𝛽̇ 𝛾̇ 1 + 𝛽𝛾 − 𝛼2 = 8𝜋𝐺𝜌____________(2.9) ( − ) = 0 ____________________(2.10) 𝛽 𝛾 AIJRFANS 19-215; © 2019, AIJRFANS All Rights Reserved Page 60 Shishir Kumar Srivastava, American International Journal of Research in Formal, Applied & Natural Sciences, 26(1), March-May 2019, pp. 60-62 Dot(.) denotes the ordinary differentiation with respect to t. Integrating equation (2.10)we get 𝑙𝑜𝑔𝛽 − 𝑙𝑜𝑔𝛾 = log 𝑙1 𝛽 𝑙𝑜𝑔 = log 𝑙1 𝛾 𝛽 = 𝑙1 𝛾______________(2.11) Where 𝑙1 is constant of integration. We choose the value 𝑙1 = 1 then 𝛽 = 𝛾 ____________(2.12) The spatial volume for the model given by 𝜉 3 (𝑡) = (𝛼𝛽𝛾)______(2.13) 𝑉 = 𝜉 3 (𝑡) = (𝛼𝛽𝛾)______(2.14) 1 𝜉(𝑡) = (𝛼𝛽𝛾)3 ________(2.15) Where 𝜉(𝑡) as the average scale factor, The expression for scalar expansion, shear scalar 𝛽̇ 𝛼̇ 𝛾̇ 𝑚 = ( + + ) _________________(2.16) 𝜃 = 𝑣;𝑚 𝛼 𝛾 𝛽 1 𝛼 2̇ 𝜎 2 = 𝜎𝑚𝑛 𝜎 𝑚𝑛 = ( 3 𝛼2 + 𝛽̇ 2 𝛽2 + 𝛾̇ 2 𝛾2 − 𝛼̇ 𝛽̇ 𝛼𝛽 − 𝛽̇ 𝛾̇ 𝛽𝛾 − 𝛼̇ 𝛾̇ 𝛼𝛾 )_______________(2.17) The Hubble parameter and deceleration parameter are respectively defined as 𝛾̇ 1 𝛼̇ 𝛽̇ 𝜉̇ 𝐻 = = ( + + ) _____________(2.18) 𝜉 1 3 𝛼 𝛾 𝛽 𝐻 = (𝐻1 + 𝐻2 + 𝐻3 ) 𝑞= 3 𝑑 1 ( ) − 1 _________(2.19) 𝑑𝑡 𝐻 III. Solution of field Equation The system of five field equation (2.6) to(2.10) are in five unknown parameter (𝛼, 𝛽, 𝛾 , 𝑝, 𝜌). we assume that expansion scalar is proportional to shear scalar 𝜃 ∝ 𝜎. This condition leads to 𝛼 = 𝛽 𝑚1 ___________(2.20) Where 𝑚1 is any real constant. From equation (2.8) 𝛽̈ 𝛽̇ 2 1 (𝑚1 + 1 ) + [𝑚12 − 𝑚1 + 𝑚1 ] 2 − 2𝑚 = −8𝜋𝐺𝑝 𝛽 𝛽 𝛽 1 𝛽̈ (𝑚1 + 1) + 𝑚12 𝛽 𝛽̇ 2 − 𝛽2 1 𝛽 2𝑚1 = −8𝜋𝐺𝑝 ___________(2.21) From equation (2.9) 𝛽 2̇ 𝛽̇ 2 𝛽̇ 2 1 𝑚1 2 + 𝑚1 2 + 2 − 2𝑚 = 8𝜋𝐺𝜌 𝛽 𝛽 𝛽 𝛽 1 (2𝑚1 + 1) 𝛽̇ 2 𝛽2 − 1 𝛽 2𝑚1 = 8𝜋𝐺𝜌___________________(2.22) To get determinate solution we need extra condition we assume 𝑝 = 𝜂𝜌_________________(2.23) i.e. fluid is barotropic fluid. From equation (2.21),(2.22),(2.23) we get [𝑚2 +(2𝑚1 +1)𝜂] 𝛽̇ 2 1+𝜂 𝛽̈ + 1 =[ ] 𝛽1−2𝑚1 _____________(2.24) (𝑚1 +1) 𝑚1 +1 𝛽 we select the value 𝑚1 = 2 𝛽̇ 2 𝛽̈ + 𝑐1 = 𝑐2 𝛽 −3 ___________________(2.25) 𝛽 Where 𝑐1 = 𝑑 4+5𝜂 𝑎𝑛𝑑 𝑐2 = 3 1+𝜂 3 (𝛼12 𝛽 2𝑐1 ) = 2𝑐2 𝛽 2𝑐1−3 𝛽̇ _______________(2.26) 𝑑𝑡 Where 𝛼1 = 𝛽̇ Integrating equation (2.26) we get 𝛼1 = √𝑐3 𝛽 __________(2.27) Where 𝑐3 = 𝑐2 𝑐1 −1 = 1+𝜂 1+5𝜂 AIJRFANS 19-215; © 2019, AIJRFANS All Rights Reserved Page 61 Shishir Kumar Srivastava, American International Journal of Research in Formal, Applied & Natural Sciences, 26(1), March-May 2019, pp. 60-62 𝛽 2 = 2√𝑐3 𝑡 + 2𝑑1 ______________(2.28) 𝛽 2 = ℎ1 𝑡 + ℎ2 1 𝛽 = (ℎ1 𝑡 + ℎ2 )2 𝛽 = √ℎ1 𝑡 + ℎ2 ________________(2.29) Where ℎ2 is integrating constant From equation (2.20) 1 𝛼 = [(ℎ1 𝑡 + ℎ2 )2 ]𝑚1 𝑚1 𝛼 = (ℎ1 𝑡 + ℎ2 ) 2 ____________(2.30) using (2.29) and (2.30) the line element (2.1) becomes 𝑑𝑠 2 = −𝑑𝑡 2 + (ℎ1 𝑡 + ℎ2 )𝑚1 𝑑𝑥 2 + (ℎ1 𝑡 + ℎ2 ) 𝑒 −2𝑥 𝑑𝑦 2 + (ℎ1 𝑡 + ℎ2 )𝑒 2𝑥 𝑑𝑧 2 _________(2.31) IV. Physical and Geometrical Behaviour of the model From this model, we can find other geometrical and physical parameter. The expression for Hubble parameter, Expansion scalar𝜃, spatial volume 𝑉,shear scalar 𝜎 2 and deceleration parameter 𝑞 respectively givenby 𝑟 1 𝐻 = [ 1 ] __________(2.32) 3 ℎ1 𝑡+ℎ2 𝜃 = 3𝐻 = (ℎ 𝑟1 1 𝑡+ℎ2 ) 𝜎= 1 ____________(2.33) Where 𝑟1 = ( 𝑟2 [ 𝑚1 2 + 1) ℎ1 = constant ]__________________(2.34) √3 (ℎ1 𝑡+ℎ2 ) Where 𝑟2 = (𝑚1 −1)ℎ1 𝑚1 +2 2 2 = Constant 𝑉 = (ℎ1 𝑡 + ℎ2 ) _______________________(2.35) −(3𝑚1 +4) _____________(2.36) 𝑞= ฀ ฀ = 3(𝑚1 +2) ฀2 √3 ฀1 =____________(2.37) V. Observation 𝜎 1- = Constant, therefore model dose not approach isotropy for large value of 𝑡. 𝜃 2- When 𝑡 → ∞ then scalar expansion 𝜃, Shear scalar 𝜎 ,Hubble parameter all approaches to Zero. 3- Negative Value of deceleration parameter indicate that model is decelerating. The special volume with time is increasing an it become for sufficiently large value of 𝑡. VI. Conclusion In this paper we have studied Shishir cosmological model Bianchi type VI with use barotropic in modified theory of general relativity. We investigated that Hubble parameter H, Scalar expansion 𝜃, and shear scalar 𝜎 are infinite at 𝑡 → 0 and approaches Zero at 𝑡 → ∞. Here we see that the spatial Volume of the universe increase with time. 𝜎 = constant, Which gives us the anisotropy is maintained for all time and it can be see that the Since lim 𝑡→∞ 𝜃 model is irrotational. Therefore, the model describes a continuously expanding shearing and non rotating universe with big bang in theory modified theory of general relativity. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] P.S. Letelier, String cosmologies. Physical review D, 28 (10) ,1983,2414. P.S Letelier, Clouds of string in general relativity. Physical Review D,20 (6), 1979, 1294. S. Chakrborty, String cosmology in Bianchi VI space-time. Indian Journal of Pure and Applied Physics,29,1991,31-33. 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Shanjit, S Nerkar, Bianchi Type VI0 Inflationary Universe with Constant Deceleration parameter and Flat Potential in General Relativity,2017 S.R Bhoyar, V.R Chirde, S.H. Shaikh, Dark Energy Dominated Bianchi Type-VI0 Universe with the Hybrid Expansion Lw in f (R,T) Gravity. Prespacetime Journal 8(9),2017. AIJRFANS 19-215; © 2019, AIJRFANS All Rights Reserved Page 62