Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 911 https://internationalpubls.com LRS Bianchi Type-I Magnetized Dark Energy Cosmological Model in Bimetric Theory of Relativity with Cosmological Term – 𝜦 Swapnil S. Charjan1, N. P. Gaikwad2, S. S. Charjan3, P. V. Gaykwad4 1 Research Scholar, Post Graduate Teaching Department of Mathematics, Rashtrasant Tukadoji Maharaj Nagpur University, Nagpur-440033 Email: charjan91@gmail.com 2Associate Professor, Department of Mathematics, Dharampeth M.P. Deo Memorial Science College, Nagpur-440033, Maharashtra, India. Email: np_gaikwad@rediffmail.com 3Assistant Professor, Department of Mathematics, Late K.Z.S. Science College Bramhani, Kalmeshwar- 441502, Maharashtra, India. Email: sscharjan1967@gmail.com 4Assistant Professor, Department of Mathematics, A. D. College, Bharshingi-441305, Maharashtra, India. Email: pvgayakwad07@gmail.com Article History: Received: 1-11-2024 Revised: 17-12-2024 Accepted: 25-12-2024 Abstract: In this paper, to study Locally Rotationally Symmetric (LRS) Bianchi type-I magnetized dark energy cosmological models with cosmological constant -𝛬, we have calculated Rosen,s field equations for LRS Bianchi type -I spacetimes in the presence of dark energy and discuss their geometrical as well as physical significances using two cases (i) 𝜌 + πœ† = 0 (ii) 𝜌 βˆ’ πœ† = 0. To obtain the determinate solution of the model, we considered i) scalar expansion πœƒ is directly proportional to share 𝜎 ii) 𝛬 = 𝛼 𝛒𝛣2 . In the presence of a magnetic field for the conditions 𝜌 + πœ† = 0, scalar expansion πœƒ attains its minimum and increases with increase in 𝑇 indicating that the universe is expanding with increment in 𝑇, for the case 𝜌 βˆ’ πœ† = 0 scalar expansion πœƒ attains its maximum and decreases with increase in 𝑇 indicate that slows down the rate of expansion. Keywords: LRS Bianchi Type I; Bimetric; Gravitation; Dark Energy; Dark Matter. 1. Introduction It presently appears that the cosmos is composed of 4.9% baryonic (ordinary) matter, 26.8% dark matter, and 68.3% dark energy in the form of energy densities. Recently, various cosmological theories including dark matter as well as dark energy have been investigated. Due to the development of the dark energy problem, the constraints of general relativity have once more become apparent. It has been assumed that dark energy will explain how the universe is clearly expanding at an acceleration rate. Dark energy has been suggested to be responsible for of the universe's observed acceleration in expansion. Dark energy (DE) is a speculative type of energy acting appallingly which allows all of space and will in general speed up the extension of the cosmos. This sped-up extension of the cosmos was affirmed by the perceptions like Supernovae type I, LSS, SDS Survey, WMA Probe, BAO and CMBR Some authors have discussed this aspect viz; Borkar et al.(1), Mamon et al.(2), Riess et al.(3), Perlmutter et al.(4), Spergel et al.(5), Tegmark et al.(6), Hinshaw et al.(7), Nolta et al.(8), Hinshaw et al.(9), mailto:charjan91@gmail.com mailto:np_gaikwad@rediffmail.com mailto:sscharjan1967@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 912 https://internationalpubls.com Anderson et al.(10). Cosmologists such as Kamenshchik et al.(11), Katore et al.(12), Borkar et al.(13) presented several contenders for dark energy based on these senses and their theories about it. The cosmological steady up-and-comer experiences two notable issues, specifically, the tweaking and the grandiose incidental problems studied by Copeland et al.(14). Positive energy thickness and negative tension are combined in the pith dynamical dark energy model. The basic scenario is further inspired by the tracker field this study was done by Saha et al.(15) and Sing et al.(16). The Chaplygin gas model, which is seen as the combined effect of dark matter along with dark energy, is another option to the dynamical dark energy model studied by Dev et al.(17), Bento et al.(18), Abdusattar et al.(19). Rosen(20),(21) Investigated the importance of attractiveness in the cosmos is important, given that the attractive field exists in the space between universes and plays a very important role in cosmology. Melvin proposes that matter was severely ionized during the creation of the cosmos also, because of the expansion of the cosmos, this matter is once more flawlessly united with the attractive field and builds unbiased matter. Therefore, it makes sense that there would be an attracting field throughout the cosmos. The incredible result of general relativity did not stop any other explanations for the universe's expanding pace and the existence of dark matter from being put forth. In actuality, a short while after Einstein's theory was published, numerous elective speculations have arisen telling the best way to expand, how to connect with other theories, and how to consolidate. The bimetric theory of gravity proposed by Rosen is one of the other alternatives, which is free of singularities as well as black holes which where appear in the Big Bang theory of the cosmological model. The theory is consistent with the most recent observational evidence on GR, and it substantially agrees with the covariance and equivalence principles of GR. Therefore, Rosen's BTG is significant and is dependent on two matrices namely fundamental metric and flat metric. The metric tensor 𝑔𝑖𝑗 is the Riemannian metric tensor depicts the gravitational potential whereas the metric tensor 𝛾𝑖𝑗 is background metric tensor depicts the inertial forces related to the speed increase of the reference frame and it has no direct actual importance except for shows up in the field equation. The gravitational potential represented by the metric tensor 𝑔𝑖𝑗is not fixed by the field equation or by interactions between matter and gravity. In this way, the flat space- time metric collaborates with the Riemannian metric yet not straightforwardly with matter. Without a trace of matter, one would have 𝑔𝑖𝑗= 𝛾𝑖𝑗 fulfilling the standards covariance as well as the equivalence of the development of GR. In this manner at each point of space-time in Bimetric relativity, there are two matrices as follows. 𝑑𝑠2 = 𝑔𝑖𝑗𝑑π‘₯𝑖𝑑π‘₯𝑗 (1) 𝑑𝑛2 = 𝛾𝑖𝑗𝑑π‘₯𝑖𝑑π‘₯𝑗 (2) The bimetric theory of gravitation via Rosen's field equations is 𝑁𝑖 𝑗 βˆ’ 1 2 𝑁𝛿𝑖 𝑗 βˆ’ 𝛬𝑔𝑖 𝑗 = βˆ’8πœ‹π‘˜π‘‡π‘– 𝑗 (3) ( ) rpsi sjprj i ggN  2 1 = , Here ( )| denotes  –covariant derivative and j iT represents energy– momentum tensor. i iNN = , π‘˜ = βˆšπ‘” 𝛾⁄ together with𝑔 = π‘‘π‘’π‘‘π‘’π‘Ÿπ‘šπ‘–π‘›π‘Žπ‘›π‘‘(𝑔𝑖𝑗) and 𝛾 = π‘‘π‘’π‘‘π‘’π‘Ÿπ‘šπ‘–π‘›π‘Žπ‘›π‘‘( 𝛾𝑖𝑗). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 913 https://internationalpubls.com Numerous specialists have fostered the hypothesis and examined numerous models of the cosmos, in BTG and concentrated on their way of behaving geometrically as well as physically and this was studied by Karade et al.(22), Reddy et al.(23), Gaikwad et al.(24). In this examination note, we will focus on the Locally Rotationally Symmetric Bianchi type-I magnetized dark energy cosmological model in the bimetric theory of gravity (BTG) since it has a great deal more fidelity to the physics of spatial homogeneous cosmology. According to theories of gravitation, the main issue with the advancement of the cosmos is focusing on the singularity's situation and knowing its nature. Rosen(20),(21) joined the 𝛾𝑖𝑗 along with 𝑔𝑖𝑗 to keep away from the singularities in his BTG. Any actual hypothesis should be released from singularity since its presence suggests the breakdown of the rules propounded by the theory. This has led to the development of the bimetric theory of gravity, which is now free of singularities. Numerous experts, including Borkar et al.(22), Karade et al.(23)), Kruskal et al.(24), and so forth in GR are focused on it. There are two types of singularity one is removable and the other is non-removable. The nonremovable singularity problem has attracted the attention of experts in several gravitational theories studied by Penrose et al.(25), Karade et al.(26), Borkar et al.(27), Pradhan et al.(28). To depict the gravitational field around large things with singularities, Schwarzschild deduced an alternative arrangement for Einstein's vacuum spacetime field equation Borkar et al.(27). There is exactly one singularity at r = 0 in BTG for circularly symmetric static spacetime in the case of emptiness, and it would disappear in the unlikely event that the molecule represented by a matter tensor differs from zero in a specific area given by Rosen(20),(21). In this work, we study the LRS Bianchi type-I magnetized dark energy cosmological model in BTG by solving Rosen’s equations and also study physical as well as geometrical significance. 2. Line Element & Field Equations LRS Bianchi type - I space-time can be considered as 𝑑𝑠2 = βˆ’π‘‘π‘‘2 + 𝐴2𝑑π‘₯2 + 𝐡2(𝑑𝑦2 + 𝑑𝑧2) (4) where A and B rely upon enormous time t as it were. The flat metric relating to metric (4) is 𝑑𝑠2 = βˆ’π‘‘π‘‘2 + 𝑑π‘₯2 + 𝑑𝑦2 + 𝑑𝑧2 (5) Where, 𝑇𝑗 𝑖 is energy-momentum tensor for a cloud string with a magnetic field in the presence of perfect fluid is given by 𝑇𝑖 𝑗 = (𝜌 + 𝑝)πœπ‘–πœ 𝑗 + 𝑝𝑔𝑖 𝑗 βˆ’ πœ†π‘₯𝑖π‘₯𝑗 + 𝐸𝑖 𝑗 (6) πœπ‘–πœπ‘– = βˆ’π‘₯𝑖π‘₯𝑖 = βˆ’1 and πœπ‘–π‘₯𝑖 = 0 Here, ρ, Ξ», π‘₯𝑖, πœπ‘– is proper energy density, string tension density, unit space-like vector unit time-like vector respectively. We choose a comoving coordinate system as πœπ‘– = (0,0,0,1); π‘₯𝑖 = ( 1 𝐴 , 0,0,0) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 914 https://internationalpubls.com We consider particle density πœŒπ‘ of configuration as 𝜌 = πœŒπ‘ + πœ† 𝐸𝑖 𝑗 is the electromagnetic field defined as 𝐸𝑖 𝑗 = οΏ½Μ…οΏ½ [|β„Ž|2 (πœπ‘–πœ 𝑗 + 1 2 𝑔𝑖 𝑗 ) βˆ’ β„Žπ‘–β„Žπ‘—] (7) Where πœ‡ is the magnetic permeability and β„Žπ‘– is the magnetic flux vector characterized as β„Žπ‘– = βˆšβˆ’π‘” 2πœ‡ βˆˆπ‘–π‘—π‘˜π‘™ πΉπ‘˜π‘™πœπ‘— Here πΉπ‘˜π‘™ and βˆˆπ’Šπ’‹π’Œπ’ are electromagnetic field tensor and Levi-Cevita tensor density respectively. We can write a set of Maxwell’s equations as 𝐹[𝑖𝑗,π‘˜] = 0 yield F23 = H (say) = constant Because it's believed that electrical conductivity never ends, so we have 𝐹14 = 𝐹24 = 𝐹34 = 0 The one element that does not vanish of 𝐹𝑖𝑗 is 𝐹23. So that, β„Ž1 = 𝐴𝐻 πœ‡π΅2 and |β„Ž|2 = 𝐻2 πœ‡ 2 𝐡4 The electromagnetic field's constituent parts 𝐸𝑖 𝑗 are given by 𝐸1 1 = βˆ’πΈ2 2 = βˆ’πΈ3 3 = 𝐸4 4 = βˆ’ 𝐻2 πœ‡ 2 𝐡4 (8) Equation (6) of energy-momentum tensor yield 𝑇1 1 = (𝑝 βˆ’ πœ† βˆ’ 𝐻2 2πœ‡π΅4 ) , 𝑇2 2 = 𝑇3 3 = (𝑝 + 𝐻2 2πœ‡π΅4 ) , 𝑇4 4 = (βˆ’πœŒ βˆ’ 𝐻2 2πœ‡π΅4 ) (9) For the metric (4) and (5) using equation (9) Rosen’s field equations (3) gives, βˆ’π΄44 𝐴 + 2𝐡44 𝐡 + 𝐴4 2 𝐴2 βˆ’ 2𝐡4 2 𝐡2 = 16πœ‹π΄π΅2 (βˆ’π‘ + πœ† + 𝐻2 2πœ‡π΅4 ) βˆ’ 2𝛬 (10) 𝐴44 𝐴 βˆ’ 𝐴4 2 𝐴2 = 16πœ‹π΄π΅2 (βˆ’π‘ βˆ’ 𝐻2 2πœ‡π΅4) βˆ’ 2𝛬 (11) 𝐴44 𝐴 + 2𝐡44 𝐡 βˆ’ 𝐴4 2 𝐴2 βˆ’ 2𝐡4 2 𝐡2 = 16πœ‹π΄π΅2 (𝜌 + 𝐻2 2πœ‡π΅4) βˆ’ 2𝛬 (12) 3. Solution of Field Equations With Magnetic Field The field equation (10) (11) and (12) are the three equations along with six unknowns 𝐴, B, Ξ›, πœ†, 𝜌 and 𝑝. Therefore, the system is initially unknown, and to obtain the system's full solution, we need three more equations. Therefore, we supposed that scalar expansion πœƒ is directly proportional to share 𝜎. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 915 https://internationalpubls.com 𝛒 = 𝛣𝑛 and 𝛬 = 𝛼 𝛒𝛣2 Under the two scenarios listed above, we assume the following cases: (i) 𝜌 + πœ† = 0 and (ii) 𝜌 βˆ’ πœ† = 0 (13) Case I: 𝜌 + πœ† = 0 After adding equations (10) and (12) we get 4𝐡44 𝐡 βˆ’ 4𝐡4 2 𝐡2 = 16πœ‹π΄π΅2 (βˆ’π‘ + 𝐻2 πœ‡π΅4 ) βˆ’ 4𝛬 (14) Using equations (13) and (11) in equation (14) we get 𝐡44 𝐡 βˆ’ 𝐡4 2 𝐡2 = 48πœ‹π›«π›£π‘›βˆ’2 4βˆ’π‘› βˆ’ 2𝛼 (4βˆ’π‘›)𝛣𝑛+2 (15) On putting 𝐡4 = 𝑓(𝐡) and 𝐡44 = 𝑓𝑓′ in equation (15) we obtain 𝑑𝑓2 𝑑𝐡 βˆ’ 2𝑓2 𝐡 = 96πœ‹π›«π΅π‘›βˆ’1 4βˆ’π‘› βˆ’ 4𝛼 (4βˆ’π‘›)𝐡𝑛+1 (16) On integration, equation (16) leads to 𝑓2 = 96πœ‹π›«π΅π‘› (4βˆ’π‘›)(π‘›βˆ’2) + 4𝛼 (4βˆ’π‘›)(𝑛+2)𝐡𝑛 + 𝑀𝐡2 (17) Here 𝑀 is the integration constant. From (17), we get ∫ 𝑑𝐡 √ 96πœ‹π›«π΅π‘› (4βˆ’π‘›)(π‘›βˆ’2) + 4𝛼 (4βˆ’π‘›)(𝑛+2)𝐡𝑛+𝑀𝐡2 = 𝑑 + 𝑉 (18) Here 𝑉 is the constant of integration. The value of B can be calculated from (18). Through the proper coordinate transformation 𝐡 = 𝑇 , π‘₯ = 𝑋 , 𝑦 = π‘Œ π‘Žπ‘›π‘‘ 𝑧 = 𝑍 metric (4) becomes 𝑑𝑠2 = βˆ’ 𝑑𝑇2 [ 96πœ‹π›«π‘‡π‘› (4βˆ’π‘›)(π‘›βˆ’2) + 4𝛼 (4βˆ’π‘›)(𝑛+2)𝑇𝑛+𝑀𝑇2] + 𝑇2𝑛𝑑𝑋2 + 𝑇2(π‘‘π‘Œ2 + 𝑑𝑍2) (19) Case II: 𝜌 βˆ’ πœ† = 0 On subtracting equations (10) from (12) we get, 2 [ 𝛒44 𝛒 βˆ’ 𝛒4 2 𝛒2] = 16πœ‹π›’π›£2𝛲 (20) Using equations (13) and (11) in equation (20) we get 𝛣44 βˆ’ 𝛣4 2 𝛣 = βˆ’16πœ‹π›«π›£π‘›βˆ’1 3𝑛 βˆ’ 2𝛼 3𝑛𝛣𝑛+1 (21) On putting 𝐡4 = 𝑓(𝐡) and 𝐡44 = 𝑓𝑓′in equation (21) we obtain 𝑑𝑓2 𝑑𝐡 βˆ’ 2𝑓2 𝐡 = βˆ’32πœ‹π›«π›£π‘›βˆ’1 3𝑛 βˆ’ 4𝛼 3𝑛𝛣𝑛+1 (22) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 916 https://internationalpubls.com On integrating equation (22) we get, 𝑓2 = βˆ’32πœ‹π›«π΅π‘› 3𝑛(π‘›βˆ’2) + 4𝛼 3𝑛(𝑛+2)𝐡𝑛 + 𝑁𝐡2 (23) Here 𝑁 is the constant of integration. ∫ 𝑑𝐡 √ βˆ’32πœ‹π›«π΅π‘› 3𝑛(π‘›βˆ’2) + 4𝛼 3𝑛(𝑛+2)𝐡𝑛+𝑁𝐡2 = 𝑑 + π‘ˆ (24) Where π‘ˆ is the integration constant. The value of B can be calculated from (24) Through the proper coordinate transformation 𝐡 = 𝑇, π‘₯ = 𝑋 , 𝑦 = π‘Œ π‘Žπ‘›π‘‘ 𝑧 = 𝑍 metric (1) becomes 𝑑𝑠2 = βˆ’ 𝑑𝑇2 [ βˆ’32πœ‹π›«π‘‡π‘› 3𝑛(π‘›βˆ’2) + 4𝛼 3𝑛(𝑛+2)𝑇𝑛+𝑁𝑇2] + 𝑇2𝑛𝑑𝑋2 + 𝑇2(π‘‘π‘Œ2 + 𝑑𝑍2) (25) 3.1 Geometrical and Physical Characteristics With Magnetic Field For the model (19) all the physical quantities are given by 𝜌 = βˆ’πœ† = 𝛫(4𝑛+2) (4βˆ’π‘›)𝑇4 βˆ’ 𝛼(π‘›βˆ’1) 4πœ‹(4βˆ’π‘›)𝑇2(𝑛+2) (26) πœŒπ‘ = 2𝛫(4𝑛+2) (4βˆ’π‘›)𝑇4 βˆ’ 𝛼(π‘›βˆ’1) 2πœ‹(4βˆ’π‘›)𝑇2(𝑛+2) (27) 𝛲 = βˆ’(2𝑛+4)𝛫 (4βˆ’π‘›)𝑇4 + 𝛼(π‘›βˆ’2) 4πœ‹(4βˆ’π‘›)𝑇2(𝑛+2) (28) πœƒ = (𝑛 + 2) [ 96πœ‹π›«π›΅π‘›βˆ’2 (4βˆ’π‘›)(π‘›βˆ’2) + 4𝛼 (4βˆ’π‘›)(𝑛+2)𝛡𝑛+2 + 𝑀] 1 2 (29) 𝜎 = (π‘›βˆ’1) √3 [ 96πœ‹π›«π›΅π‘›βˆ’2 (4βˆ’π‘›)(π‘›βˆ’2) + 4𝛼 (4βˆ’π‘›)(𝑛+2)𝛡𝑛+2 + 𝑀] 1 2 (30) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 917 https://internationalpubls.com From above it is clear that with magnetic field for the conditions 𝜌 + πœ† = 0 the physical quantities 𝛲, 𝜌, π‘Žπ‘›π‘‘ πœŒπ‘ Follows straight line path with slight deviation near 𝑇 = 0, also scalar expansion πœƒ attains its minimum and increases with an increase in 𝑇 and tends to ∞ as 𝑇 β†’ ∞. For the model (25) all physical quantities are given by 𝜌 = πœ† = 𝛼(π‘›βˆ’1) 12πœ‹π‘›π›΅2(𝑛+2) βˆ’ 2𝐾(2𝑛+1) 3𝑛𝑇4 (31) πœŒπ‘ = 0 (32) 𝑃 = βˆ’ 𝛼 12πœ‹π›΅2(𝑛+2) βˆ’ 2𝐾 3𝑇4 (33) πœƒ = (𝑛 + 2) [ 4𝛼 3𝑛(𝑛+2)𝑇𝑛+2 βˆ’ 32πœ‹π›«π‘‡π‘›βˆ’2 3𝑛(π‘›βˆ’2) + 𝑁] 1 2 (34) 𝜎 = (π‘›βˆ’1) √3 [ 4𝛼 3𝑛(𝑛+2)𝑇𝑛+2 βˆ’ 32πœ‹π›«π‘‡π‘›βˆ’2 3𝑛(π‘›βˆ’2) + 𝑁] 1 2 (35) From above it is clear that with magnetic field for the conditions 𝜌 βˆ’ πœ† = 0 the physical quantities 𝛲, 𝜌, π‘Žπ‘›π‘‘ πœŒπ‘ Follows a straight line path with slight deviation near 𝑇 = 0, also the scalar expansion πœƒ goes on decreasing with an increase in the value of 𝑇. 4. Solution of Field Equations Without Magnetic Field Case I: 𝜌 + πœ† = 0 When the magnetic field is not present equation (19) becomes 𝑑𝑠2 = βˆ’ 𝑑𝑇2 [ 4𝛼 (4βˆ’π‘›)(𝑛+2)𝑇𝑛+𝑀𝑇2] + 𝑇2𝑛𝑑𝑋2 + 𝑇2(π‘‘π‘Œ2 + 𝑑𝑍2) (36) Case II: 𝜌 βˆ’ πœ† = 0 When the magnetic field is not present equation (25) becomes Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 918 https://internationalpubls.com 𝑑𝑠2 = βˆ’ 𝑑𝑇2 [ 4𝛼 3𝑛(𝑛+2)𝑇𝑛+𝑁𝑇2] + 𝑇2𝑛𝑑𝑋2 + 𝑇2(π‘‘π‘Œ2 + 𝑑𝑍2) (37) 4.1 Geometrical and Physical Characteristics Without Magnetic Field For the model (36) all physical quantities are given by 𝜌 = βˆ’πœ† = βˆ’ 𝛼(π‘›βˆ’1) 4πœ‹(4βˆ’π‘›)𝑇2(𝑛+2) (38) πœŒπ‘ = βˆ’ 𝛼(π‘›βˆ’1) 2πœ‹(4βˆ’π‘›)𝑇2(𝑛+2) (39) 𝛲 = 𝛼(π‘›βˆ’2) 4πœ‹(4βˆ’π‘›)𝑇2(𝑛+2) (40) πœƒ = (𝑛 + 2) [ 4𝛼 (4βˆ’π‘›)(𝑛+2)𝛡𝑛+2 + 𝑀] 1 2 (41) 𝜎 = (π‘›βˆ’1) √3 [ 4𝛼 (4βˆ’π‘›)(𝑛+2)𝛡𝑛+2 + 𝑀] 1 2 (42) For the model (37) all physical quantities are given by 𝜌 = πœ† = 𝛼(π‘›βˆ’1) 12πœ‹π‘›π›΅2(𝑛+2) (43) 𝛲 = βˆ’ 𝛼 12πœ‹π‘‡2(𝑛+2) (44) πœŒπ‘ = 0 (45) πœƒ = (𝑛 + 2) [ 4𝛼 3𝑛(𝑛+2)𝑇𝑛+2 + 𝑁] 1 2 (46) 𝜎 = (π‘›βˆ’1) √3 [ 4𝛼 3𝑛(𝑛+2)𝑇𝑛+2 + 𝑁] 1 2 (47) When the magnetic field is not present for both the cases 𝜌 + πœ† = 0 and 𝜌 βˆ’ πœ† = 0 all the quantities show slight deviation from the straight line at 𝑇 = 0. For with and without magnetic field for both the cases, also 𝜎/πœƒ = π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ shows that the model does not approach isotropy in general studied by Pradhan et al.(28). 5. Quintessence Model Of Dark Energy Without Magnetic Field One of the models of dark energy is the Quintessence model which obeys the equation of state. π‘ƒπ‘ž = πœ”π‘žπœŒπ‘ž , βˆ’1 ≀ πœ”π‘ž ≀ 0 (48) in which πœ”π‘ž is the equation of state parameter correlated with the quintessence model. For case (i) πœ”π‘ž is given by πœ”π‘ž = βˆ’ (π‘›βˆ’2) (π‘›βˆ’1) (49) For case (ii) πœ”π‘ž is given by πœ”π‘ž = βˆ’ 𝑛 (π‘›βˆ’1) (50) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 919 https://internationalpubls.com We are interested in three values of πœ”π‘ž which are βˆ’1,0, 1 3 π‘Žπ‘›π‘‘ 1. πœ”π‘ž = βˆ’1 corresponds to the vacuum fluid, πœ”π‘ž = 0 corresponds to the dust fluid, πœ”π‘ž = 1 3 corresponds to the radiating fluid, πœ”π‘ž = 1 stiff dominated universe. 6. Conclusion In this work, we examine the Locally Rotationally Symmetric Bianchi type-I magnetized dark energy cosmological model in BTG by solving Rosen’s field equations. The study investigates the model's geometrical and physical characteristics with and without a magnetic field in two cases (i) 𝜌 + πœ† = 0 (ii) 𝜌 βˆ’ πœ† = 0. 1. In the Presence of a magnetic field for the case 𝜌 + πœ† = 0 the increasing scalar expansion πœƒ and Shear 𝜎 over time indicate (fig 1) that the expansion of the universe. This concept is similar to the Big Bang theory, which suggests that the origin of the universe from an extremely hot and dense point and has been expanding ever since. The observations of increasing shear and scalar expansion obey the idea of an expanding universe. 2. In presence of a magnetic field for the case 𝜌 βˆ’ πœ† = 0 scenario suggested that the universe in the presence of a magnetic field influences gravitational dynamics, leading to a contraction despite the presence of constant pressure and density (fig 2). The balance between matter energy density and the cosmological constant further shapes the overall evolution of the universe. 3. When the magnetic field is not present for both cases (figures 3 & 4) all the physical quantities follow a straight-line path except at 𝑇 = 0. The universe evolves predictably according to a straight- line path for physical quantities, with a notable deviation at the beginning of time. This suggests that while the absence of a magnetic field may not directly influence the overall evolution of the cosmos, there are still some initial conditions that shape its development. 4. In the absence magnetic field, the equation of state parameter πœ”π‘ž for 𝑛 > 2 for both the cases comes out to be negative and it is independent of 𝑇. This indicates that there is other than baryonic matter and the universe is completely occupied by dark energy and dark matter when the magnetic field is not present. The equation of state parameter πœ”π‘ž is negative and remains constant over time. This indicates the presence of a dominant energy component with repulsive gravitational effects, similar to dark energy, which influences the overall evolution of the universe. 7. References 1) Borkar MS, Gayakwad PV, LRS Bianchi type I magnetized cosmological model with perfect fluid and with quintessence, Chaplygin gas dark energy in bimetric theory of gravitation. International Journal of Modern Physics D. 2017;26(07). Available from: https://doi.org/10.1142/S0218271817500614. 2) Mamon AA, Constraints on a generalized deceleration parameter from cosmic chronometers. Modern Physics Letters A. 2018; 33(10n11). Available from : https://doi.org/10.1142/S0217732318500566. 3) Riess AG, Filippenko AV, Challis P, Clocchiatti A, Diercks A, Garnavich PM, Gilliland RL, Hogan CJ, Jha S, Kirshner RP, et al., Observational evidence from supernovae for an https://doi.org/10.1142/S0218271817500614 https://doi.org/10.1142/S0217732318500566 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 920 https://internationalpubls.com accelerating universe and a cosmological constant. Astron. J. 1998; vol 116. Available from: DOI10.1086/300499. 4) Perlmutter S, Aldering G, Goldhaber G, Knop RA, Nugent P, Castro PG, Deustua S, Fabbro S, Goobar, Groom DE, Hook IM, et al., Measurements of Ω and κ“₯ from 42 high-redshift supernovae, Astrophysical. J. 1999; 517. Available from: DOI 10.1086/307221. 5) Spergel DN, Verde, Peiris HV, Komatsu E, Nolta MR, Bennett CL, Halpern M, Hinshaw G, Jarosik N, Kogut A, First-year wilkinson microwave anisotropy probe (wmap) 1 observations: determination of cosmological parameters. Astrophys. J. Suppl.2003;148: 175-194. Available From: DOI 10.1086/377226. 6) Tegmark M and Costa A, CMB multipole measurements in the presence of foregrounds. Physical Review D. 2006; 4(2). Available from:https://doi.org/10.1103/PhysRevD.74.023005. 7) Hinshaw G, Weiland JL, Hill RS, Odegard N, Larson D, Bennett CL, Dunkley, Gold B, Greason MR, Jarosik M, Five-year wilkinson microwave anisotropy probe observations: data processing, sky maps, and basic results. The Astrophysical Journal Supplement Series. 2009;180(2). Available from: DOI 10.1088/0067-0049/180/2/225. 8 Nolta MR, Dunkley J, Hill RS, Hinshaw G, Komatsu E, Larson D, Page L, Spergel DN, Bennett CL, Gold B, et al., Five-year wilkinson microwave anisotropy probe* observations: Angular power spectra. The Astrophysical Journal Supplement Series. 2009; 180(2). Available from: DOI 10.1088/0067-0049/180/2/296. 9) Hinshaw G, Larson D, Komatsu E, Spergel DN, Bennett CL, Dunkley JN, Nolta MR, Halpern M, Hill RS, Odegard N, et al., Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: cosmological parameter results. The Astrophysical Journal Supplement Series.2013;208(2). Available from: DOI 10.1088/0067-0049/208/2/19. 10) Anderson RL, Campagnola S, Lantoine G., Broad search for unstable resonant orbits in the planar circular restricted three-body problem. Celestial Mechanics and Dynamical Astronomy. 2016 ;124:177-99. Available from: https://doi.org/10.1007/s10569-015-9659-7. 11) Kamenshchik A, Moschella U, Pasquier V, An alternative to quintessence. Physics Letters B. 2001 ;511(2-4):265-8. Available from: https://doi.org/10.1016/S0370-2693(01)00571-8. 12) Katore SD, Kapse DV, Tayade GB, Bianchi Type VI 0 Cosmological Models with Perfect Fluid and Dark Energy. International Journal of Theoretical Physics. 2011;50:3299-312.Available from: https://doi.org/10.1007/s10773-011-0832-9. 13) Borkar MS, Charjan SS, Lepse VV, Evaluation of bianchi type VI0 cosmological models with a binary mixture of perfect fluid and dark energy in bimetric theory of gravitation. International Journal of Applied Mathematics. 2013;28(2):2051. 14) Copeland EJ, Sami M, Tsujikawa S. Dynamics of dark energy. International Journal of Modern Physics D. 2006;15(11):1753-935. Available from: https://doi.org/10.1142/S021827180600942X. 15) Saha B, Anisotropic cosmological models with perfect fluid and dark energy. arXiv preprint gr- qc/0412078. 2004. Available from: https://doi.org/10.48550/arXiv.gr-qc/0412078. 16) Singh T, Chaubey R, Bianchi type-V cosmological models with perfect fluid and dark energy. Astrophysics and Space Science. 2009;319:149-54. Available from: https://doi.org/10.1007/s10509- 008-9959-4. https://doi.org/10.1016/S0370-2693(01)00571-8 https://doi.org/10.1142/S021827180600942X https://doi.org/10.48550/arXiv.gr-qc/0412078 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 921 https://internationalpubls.com 17) Dev A, Alcaniz JS, Jain D, Cosmological consequences of a Chaplygin gas dark energy. Physical Review D. 2003;67(2). Available from: https://doi.org/10.1103/PhysRevD.67.023515. 18) Bento MC, Bertolami O, Sen AA, Generalized Chaplygin gas, accelerated expansion, and dark- energy-matter unification. Physical Review D. 2002;66(4). Available from: https://doi.org/10.1103/PhysRevD.66.043507. 19) Prajapati S, LRS Bianchi type-I perfect fluid cosmological model with modified generalized Chaplygin gas equation of state. Astrophysics & Space Science. 2011;332(2). Available from: 10.1007/s10509-010-0508-6. 20) Rosen N, A bi-metric theory of gravitation. General Relativity and Gravitation. 1973;4:435-47. Available from: https://doi.org/10.1007/BF01215403. 21) Rosen N, A bi-metric theory of gravitation. II. General Relativity and Gravitation. 1975:259-68. Available from: https://doi.org/10.1007/BF00751570. 22) Karade TM, Spherically symmetric space-times in bimetric relativity theory I. Indian J. Pure-Appl. Math.. 1980;11:1202-9. 23) Reddy DR, Venkateswara Rao N, On some Bianchi type cosmological models in a bimetric theory of gravitation. Astrophysics and space science. 1997;257:293-8. Available from: https://doi.org/10.1023/A:1001166619709. 24) Gaikwad NP, Borkar MS, Charjan SS, Bianchi type-I massive string magnetized barotropic perfect fluid cosmological model in the bimetric theory of gravitation. Chinese Physics Letters. 2011;28(8). Available from: DOI 10.1088/0256-307X/28/8/089803. 25) Penrose R, Gravitational collapse and space-time singularities. Physical Review Letters. 1965;14(3):57. Available from: https://doi.org/10.1103/PhysRevLett.14.57. 26) Karade TM, The nature of the singularity in the gravitational field of a charged particle. Acta Physica Academiae Scientiarum Hungaricae. 1975:227-31. Available from: https://doi.org/10.1007/BF03157140. 27) Borkar MS, Karade TM, On singularity of spherically symmetric space times. Indian Journal of Pure and Applied Mathematics. 2003;34(8):1219-24. 28) Pradhan A, Amirhashchi H, Yadav MK, LRS Bianchi Type II String cosmological models for perfect fluid distribution in general relativity. Fizika B: a journal of experimental and theoretical physics. 2009;18(1):35-42. Available from: https://hrcak.srce.hr/304572. https://doi.org/10.1103/PhysRevD.66.043507 https://hrcak.srce.hr/304572