Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1212 https://internationalpubls.com Two Fluid Interacting and Non-Interacting Scenarios for Kaluza- Klein Dark Energy Cosmological Model in Lyra Geometry Parimal Gaidhane1, A. M. Pund2 1(Research Scholar, Department of Mathematics, S. S. E. S. Amravati’s Science College, Congress Nagar, Nagpur, India- 440012, parimalgaidhane95@gmail.com) 2(Associate Professor, Department of Mathematics, S. S. E. S. Amravati’s Science College, Congress Nagar, Nagpur, India- 440012, ashokpund64@rediffmail.com) Article History: Received: 10-10-2024 Revised: 20-11-2024 Accepted: 29-11-2024 Abstract: In this paper, we studied the evaluation of dark energy parameter in the spatial homogeneous and anisotropic five dimensional Kaluza-Klein space time filled with barotropic fluid and dark energy within the frame- work of Lyra geometry. To solve the Einstein field equation by considering the shear scalar (σ) in these models is proportional to expansion scalar (θ). Here we consider two cases; (a) when these fluids are not interacting with each other and, (b) when these fluids interact with each other. We examined the physical and geometric properties and analysed their graphical characteristics. In addition, we investigated several cosmological parameters, look-back time, proper distance, luminosity distance, angular diameter distance and distance modulus. Keywords: Dark energy, Barotropic fluid, five-dimensional Kaluza-Klein spacetime, Lyra geometry 1. Introduction Dark energy is a critical factor in understanding cosmological phenomena. Recent astronomical observations [1-4] reveal that the observable universe is currently undergoing accelerated expansion. This acceleration is widely attributed to the influence of dark energy, whose fundamental nature remains one of the central enigmas in modern cosmology. According to estimates from the Wilkinson Microwave Anisotropy Probe (WMAP), the composition of the universe comprises approximately 73% dark energy, 23% dark matter, and 4% ordinary matter. Various cosmological models, such as the cosmological constant [5], quintessence [6], equation of state parameterizations [7-11], and interactive dark energy models [12-18], have been proposed to explore the properties of dark energy and its role in driving the universe's accelerating expansion. Einstein formulated the field equations of general relativity using the principles of gravitational geometry. Building on these ideas, Weyl [19] proposed a geometric framework integrating gravity and electromagnetism. However, Weyl's theory was ultimately dismissed due to the inability to integrate vector lengths under parallel displacement. In response, Lyra introduced a modification to Riemannian geometry that incorporates gauge functions in manifolds, offering an alternative that diverges from the structure of Weyl geometry. This development spurred further research into scalar-tensor theories and cosmology within the context of Lyra geometry. Sen [20] and Sen and Dunn [21] extended this work, formulating a novel scalar- tensor theory of gravity and deriving an analogue of Einstein’s field equations based on Lyra geometry. Halford [22] demonstrated that scalar-tensor theories derived from Lyra geometry yield predictions consistent with those of general relativity. Subsequent studies have focused on cosmological models grounded in Lyra geometry, with contributions from several mailto:parimalgaidhane95@gmail.com mailto:ashokpund64@rediffmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1213 https://internationalpubls.com researchers. Recent investigations by V. G. Mete et al. [23], P. M. Lambat et al. [24], Y. Aditya et al. [25], J. K. Singh et al. [26], and Bishi et al. [27] have explored diverse aspects of these models, further enriching the understanding of Lyra-based cosmological frameworks. High-dimensional cosmology is pivotal in understanding the early stages of the universe's evolution immediately following the Big Bang. Over time, the universe condensed into its present four-dimensional form. Prominent researchers such as Witten [28] and Appelquist et al. [29] have extensively studied the implications of higher-dimensional cosmology. In the context of Kaluza-Klein five-dimensional geometry (proposed by Kaluza [30] and Klein [31]), additional spatial dimensions are employed to unify gravity and electromagnetism. This framework introduced the concept of the "cosmological reduction process," where the extra dimensions contract to scales so minute that they become undetectable. The dynamic contraction of extra dimensions was further explored by Chodos and Detweller [32] and Freund and Hawking [33], who demonstrated that these dimensions are rendered imperceptible due to their diminishing scale. Kaluza-Klein cosmological models have been examined extensively within the framework of Lyra geometry. Studies by Pawar et al. [34], Y. Aditya et al. [35], and Mishra et al. [36] have delved into these models in Lyra geometry, while research by V. G. Mete et al. [37], D. Trivedi et al. [38], and M. V. Santhi et al. [39] has focused on incorporating dark energy and barotropic fluid into various cosmological scenarios. In this study, we explore the estimation of dark energy parameters within a five-dimensional homogeneous and anisotropic Kaluza-Klein spacetime, incorporating a barotropic fluid and dark energy. Section 2 outlines the metric and field equations that form the basis of the analysis. Section 3 focuses on deriving solutions to the field equations and determining relevant physical and geometric parameters. In Section 4, we examine two scenarios involving fluid interactions: interacting and non-interacting fluids. Section 5 is dedicated to the computation of various cosmological parameters, while Section 6 presents the conclusions drawn from the study. 2. Metric and Field Equation We consider the spatially homogeneous and anisotropic five-dimensional Kaluza-Klein space-time given by, 𝑑𝑠2 = 𝑑𝑡2 − 𝐴2(𝑑𝑥2 + 𝑑𝑦2 + 𝑑𝑧2) − 𝐵2𝑑𝜓2 (1) where, 𝐴 and 𝐵 are functions of cosmic time 𝑡 only and the fifth coordinate 𝜓 is assumed to be space-like co- ordinate Einstein’s field equation is given by, 𝑅𝑖 𝑗 − 1 2 𝑅𝑔𝑖 𝑗 + 3 2 𝜙𝑖𝜙 𝑗 − 3 4 𝑔𝑖 𝑗 𝜙𝑎𝜙𝑎 = −8𝜋𝑇𝑖 𝑗 (2) where 𝜙𝑖 a displacement field vector is defined 𝜙𝑖 = (0,0,0,0, 𝛽(𝑡)) , and other symbols have their usual meaning, as in Riemannian geometry. The energy momentum tensor for two fluid is given by, 𝑇𝑖 𝑗 = 𝑇𝑖 𝑗(𝐵) + 𝑇𝑖 𝑗(𝐷) (3) where 𝑇𝑖 𝑗 (𝐵) and 𝑇𝑖 𝑗 (𝐷) represents energy momentum tensor for Barotropic fluid and Dark energy respectively. Also, 𝑇𝑖 𝑗(𝐵) = (𝜌𝐵 + 𝑝𝐵)𝑢𝑖𝑢 𝑗 − 𝑝𝐵𝑔𝑖 𝑗 (4) 𝑇𝑖 𝑗(𝐷) = (𝜌𝐷 + 𝑝𝐷)𝑢𝑖𝑢 𝑗 − 𝑝𝐷𝑔𝑖 𝑗 (5) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1214 https://internationalpubls.com Where 𝜌𝐵 and 𝜌𝐷 represents the energy density of Barotropic fluid and Dark energy, respectively. Also 𝑝𝐵 and 𝑝𝐷 represents the pressure of Barotropic fluid and Dark energy, respectively. The equation of state parameter (𝜔), which is considered an important quantity for describing the dynamics of the universe, is the ratio of the pressure (𝑝) to the energy density (𝜌) and is given by 𝜔𝐵 = 𝑝𝐵 𝜌𝐵 (6) 𝜔𝐷 = 𝑝𝐷 𝜌𝐷 (7) The Einstein field equation (2) and (3) for the metric (1), it follows that 2 �̈� 𝐴 + �̈� 𝐵 + �̇�2 𝐴2 + 2 �̇��̇� 𝐴𝐵 = −8𝜋(𝑝𝐵 + 𝑝𝐷) − 3 4 𝛽2 (8) 3 �̈� 𝐴 + 3 �̇�2 𝐴2 = −8𝜋(𝑝𝐵 + 𝑝𝐷) − 3 4 𝛽2 (9) 3 �̇�2 𝐴2 + 3 �̇��̇� 𝐴𝐵 = −8𝜋(𝜌𝐵 + 𝜌𝐷) + 3 4 𝛽2 (10) The energy conservation equation 𝑇𝑖 ;𝑗 𝑗 = 0 gives, (𝜌�̇� + 𝜌�̇�) + (𝜌𝐵 + 𝜌𝐷 + 𝑝𝐵 + 𝑝𝐷) (3 �̇� 𝐴 + �̇� 𝐵 ) = 0 (11) And (𝑅𝑖 𝑗 − 1 2 𝑅𝑔𝑖 𝑗 ) ;𝑗 + 3 2 (𝜙𝑖𝜙 𝑗);𝑗 − 3 4 (𝜙𝑎𝜙𝑎𝑔𝑖 𝑗 ) ;𝑗 = 0 (12) Equation (10) gives, 3 2 𝛽�̇� + 3 2 𝛽2 (3 �̇� 𝐴 + �̇� 𝐵 ) = 0 (13) 3. Solution of the field equations: Equations (6)-(9) and (11) are five equations with seven unknowns 𝐴, 𝐵, 𝑝𝐵 , 𝑝𝐷 , 𝜌𝐵 , 𝜌𝐷 and 𝛽. To obtain a solution of the field equations, we assume that the shear scalar (𝜎) in these models is proportional to expansion scalar (𝜃). This condition leads to, 𝐴 = 𝐵𝑛 (14) Where 𝐴 and 𝐵 are the metric potentials and 𝑛 is an arbitrary constant. Then, from the field equations, we get 𝐴 = 𝑐1𝑡 𝑛 3𝑛−1 (15) 𝐵 = 𝑐2𝑡 1 3𝑛−1 (16) Where 𝑐1 and 𝑐2 are constants. Using equation (15) and (16) in equation (1), the line element can now be written as 𝑑𝑠2 = 𝑑𝑡2 − 𝑐1 2𝑡 2𝑛 3𝑛−1 (𝑑𝑥2 + 𝑑𝑦2 + 𝑑𝑧2) − 𝑐2 2𝑡 2 3𝑛−1𝑑𝜓2 (17) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1215 https://internationalpubls.com The physical parameters of the model, like spatial volume (𝑉), Hubble parameter (𝐻), expansion scalar (𝜃), shear scalar (𝜎2), anisotropic parameter (∆) and deceleration parameter (𝑞), are given by, 𝑉 = 𝑐𝑡 𝑛 3𝑛−1 (18) 𝐻 = 1 4 3𝑛 + 1 3𝑛 − 1 1 𝑡 (19) 𝜃 = 4𝐻 = 3𝑛 + 1 3𝑛 − 1 1 𝑡 (20) ∆= 3(𝑛 − 1)2 (3𝑛 − 1)2 (21) 𝜎2 = 3 8 (𝑛 − 1)2 (3𝑛 − 1)2 1 𝑡2 (22) 𝑞 = 9𝑛 − 5 3𝑛 + 1 (23) 𝛽 = 1 𝐴3𝐵 = 𝑐𝑡− 3𝑛+1 3𝑛−1 (24) lim 𝑡→∞ 𝜎2 𝜃2 = 3(𝑛 − 1)2 8(3𝑛 + 1)2 ≠ 0 (26) Graphs of physical parameters are drawn below. Fig. 1. Graph of spatial volume (𝑉) versus time (𝑡) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1216 https://internationalpubls.com Fig. 2. Graph of Hubble parameter (𝐻) versus time (𝑡) Fig. 3. Graph of expansion scalar (𝜃) versus cosmic time (𝑡) Fig. 4. Graph of shear scalar (𝜎2) versus cosmic time (𝑡) It can be seen from Fig. 1 that the Spatial volume 𝑉 = 0 𝑡 = 0 and it increases with time and tends to infinity as 𝑡 → ∞. Fig. 2 shows that the Hubble parameter (𝐻) diverges for the initial time (𝑡), and it decreases with increasing time and tends to zero as 𝑡 → ∞. From Fig. 3, the expansion scalar (𝜃) diverges for initial time (𝑡) and decreases with time tending to zero as 𝑡 → ∞. From Fig. 4, the shear scalar (𝜎2) diverges through initial time (𝑡) and decreases over time, approaching zero as 𝑡 → ∞. Further we consider two conditions of the fluids: a) Non-interacting two fluid model and b) Interacting two fluid model. 4.1 Non-interacting two fluid model Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1217 https://internationalpubls.com The conservation equation for the barotropic fluid and dark energy separately gives, 𝜌�̇� + 4𝐻(𝑝𝐵 + 𝜌𝐵) = 0 (26) 𝜌�̇� + 4𝐻(𝑝𝐷 + 𝜌𝐷) = 0 (27) Integrating equation (26), we get 𝜌𝐵 = 𝑐1𝑡 − (𝜔𝐵+1)(3𝑛+1) (3𝑛−1) (28) Substituting value of 𝜌𝐵 from equation (28) in equation (10), we get 𝜌𝐷 = 3𝑛(𝑛 + 1) (3𝑛 − 1)28𝜋 1 𝑡2 − 𝑐𝑡 − (𝜔𝐵+1)(3𝑛+1) (3𝑛−1) + 3 32𝜋 𝑡 − 2(3𝑛+1) (3𝑛−1) (29) Substituting value of 𝜌𝐷 in (27), we get 𝑝𝐷 = 9𝑛(𝑛2−1) 8𝜋(3𝑛−1)2(3𝑛+1) 1 𝑡2 − 𝑐𝜔𝐵𝑡 − (𝜔𝐵+1)(3𝑛+1) (3𝑛−1) + 3 32𝜋 𝑡 − 2(3𝑛+1) (3𝑛−1) (30) 𝜔𝐷 = 𝑝𝐷 𝜌𝐷 = 9𝑛(𝑛2 − 1) 8𝜋(3𝑛 − 1)2(3𝑛 + 1) 1 𝑡2 − 𝑐𝜔𝐵𝑡 − (𝜔𝐵+1)(3𝑛+1) (3𝑛−1) + 3 32𝜋 𝑡 − 2(3𝑛+1) (3𝑛−1) 3𝑛(𝑛 + 1) (3𝑛 − 1)28𝜋 1 𝑡2 − 𝑐𝑡 − (𝜔𝐵+1)(3𝑛+1) (3𝑛−1) + 3 32𝜋 𝑡 − 2(3𝑛+1) (3𝑛−1) (31) The expressions of Barotropic fluid density parameter (Ω𝐵) and Dark energy density parameter (Ω𝐷) are given by, Ω𝐵 = 𝜌𝐵 6𝐻2 = 8𝑐(3𝑛 − 1)2 3(3𝑛 + 1)2 𝑡 2− (𝜔𝐵+1)(3𝑛+1) (3𝑛−1) (32) Ω𝐷 = 𝜌𝐷 6𝐻2 = 𝑛(𝑛 + 1) (3𝑛 + 1)2 − 8𝑐(3𝑛 − 1)2 3(3𝑛 + 1)2 𝑡 2− (𝜔𝐵+1)(3𝑛+1) (3𝑛−1) + (3𝑛 − 1)2 4𝜋(3𝑛 + 1)2 𝑡 2− 2(3𝑛+1) (3𝑛−1) (33) The total energy density (Ω) is given by, Ω = Ω𝐵 + Ω𝐷 = 𝑛(𝑛 + 1) (3𝑛 + 1)2 + (3𝑛 − 1)2 4𝜋(3𝑛 + 1)2 𝑡 2− 2(3𝑛+1) (3𝑛−1) (34) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1218 https://internationalpubls.com Fig. 5. Graph of 𝜌𝐵 versus cosmic time (𝑡) for non-interacting two fluid scenario. Fig. 6. Graph of 𝜌𝐷 versus cosmic time (𝑡) for non-interacting two fluid scenario. Fig. 7. Graph of 𝑝𝐷 versus cosmic time (𝑡) for non-interacting two fluid scenario. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1219 https://internationalpubls.com Fig. 8. Graph of 𝜔𝐷 versus cosmic time (𝑡) for non-interacting two fluid scenario. For non-interacting two fluid model, Fig. 5, shows the graph of 𝜌𝐵 with cosmic time (𝑡). It can be observed that 𝜌𝐵 diverges at the initial instant (𝑡) and decreases with time, and tends to zero as 𝑡 → ∞. Fig. 6 shows that the graph of 𝜌𝐷 diverges at the initial time (𝑡) and decreases with time, tending to zero as 𝑡 → ∞. Fig. 7, shows the Dark energy pressure (𝑝𝐷) diverging at an initial time (𝑡) and decreasing with time down to zero as 𝑡 → ∞. In Fig. 8, the Dark energy EoS parameter (𝜔𝐷) starts from the quintessence region (𝜔𝐷 > −1) and remains in the quintessence region for the whole cosmic time (𝑡). 4.2 Interacting two fluid model In this section, we can write the energy conservation equation for the barotropic fluid and dark energy, 𝜌�̇� + 4𝐻(𝑝𝐵 + 𝜌𝐵) = 𝑄 (35) 𝜌�̇� + 4𝐻(𝑝𝐷 + 𝜌𝐷) = −𝑄 (36) We can write the above equation as, 𝜌�̇� + 4𝐻(1 + 𝜔𝐵)𝜌𝐵 = 𝑄 (37) 𝜌�̇� + 4𝐻(1 + 𝜔𝐷)𝜌𝐷 = −𝑄 (38) Where 𝜔 = 𝑝 𝜌 and the quantity 𝑄 represents the interaction between Barotropic fluid and Dark energy components. We consider 𝑄 > 0, this ensures that the energy being transferred from Dark energy to Barotropic fluid. Consider, 𝑄 = 4𝐻𝑘𝜌𝐵 , where k is a coupling constant, Using above value of 𝑄 in equation (37), we get 𝜌𝐵 = 𝑐𝑡 (𝑘−𝜔𝐵−1) (3𝑛+1) (3𝑛−1) (39) 𝜌𝐷 = 3𝑛(𝑛 + 1) 8𝜋(3𝑛 − 1)2 1 𝑡2 − 𝑐𝑡 (𝑘−𝜔𝐵−1) (3𝑛+1) (3𝑛−1) − 3 32𝜋 𝑡 − 2(3𝑛+1) (3𝑛−1) (40) Substituting the value of 𝜌𝐷 in equation (36), we get 𝑝𝐷 = 9𝑛(𝑛 − 1)2 8𝜋(3𝑛 − 1)2(3𝑛 + 1) 1 𝑡2 + 𝑐(1 − 𝜔𝐵)𝑡 (𝑘−𝜔𝐵−1) (3𝑛+1) (3𝑛−1) − 3 32𝜋 𝑡 − 2(3𝑛+1) (3𝑛−1) (41) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1220 https://internationalpubls.com 𝜔𝐷 = 𝑝𝐷 𝜌𝐷 = 9𝑛(𝑛 − 1)2 8𝜋(3𝑛 − 1)2(3𝑛 + 1) 1 𝑡2 + 𝑐(1 − 𝜔𝐵)𝑡 (𝑘−𝜔𝐵−1) (3𝑛+1) (3𝑛−1) − 3 32𝜋 𝑡 − 2(3𝑛+1) (3𝑛−1) 3𝑛(𝑛 + 1) 8𝜋(3𝑛 − 1)2 1 𝑡2 − 𝑐𝑡 (𝑘−𝜔𝐵−1) (3𝑛+1) (3𝑛−1) − 3 32𝜋 𝑡 − 2(3𝑛+1) (3𝑛−1) (42) The expressions of Barotropic fluid density parameter (Ω𝐵) and Dark energy density parameter (Ω𝐷) are given as, Ω𝐵 = 𝜌𝐵 6𝐻2 = 8𝑐(3𝑛 − 1)2 3(3𝑛 + 1)2 𝑡 2+ (𝑘−𝜔𝐵−1)(3𝑛+1) (3𝑛−1) (43) Ω𝐷 = 𝜌𝐷 6𝐻2 = 𝑛(𝑛 + 1) 𝜋(3𝑛 + 1)2 − 8𝑐(3𝑛 − 1)2 3(3𝑛 + 1)2 𝑡2+ (𝑘−𝜔𝐵−1)(3𝑛+1) 3𝑛−1 − (3𝑛 − 1)2 4𝜋(3𝑛 + 1)2 𝑡 2− 2(3𝑛+1) (3𝑛−1) (44) The total energy density (Ω) is given by, Ω = Ω𝐵 + Ω𝐷 = 𝑛(𝑛 + 1) 𝜋(3𝑛 + 1)2 − (3𝑛 − 1)2 4𝜋(3𝑛 + 1)2 𝑡 2− 2(3𝑛+1) (3𝑛−1) (45) Fig. 9. Graph of 𝜌𝐵 versus cosmic time (𝑡) for interacting two fluid scenario. Fig. 10. Graph of 𝑝𝐷 versus cosmic time (𝑡) for interacting two fluid scenario. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1221 https://internationalpubls.com Fig. 11. Graph of 𝜔𝐷 versus cosmic time (𝑡) for interacting two fluid scenario. For interacting two fluid model, Fig. 9, shows the graph of 𝜌𝐵 with cosmic time (𝑡). It is observed that 𝜌𝐵 diverges for initial time (𝑡) and it decreases with time and tends to zero as 𝑡 → ∞. Figure 10, shows the graph of 𝑝𝐷 with cosmic time (𝑡). It is observed from the graph that 𝑝𝐷 diverges for 𝑡 = 0 and tends to zero as 𝑡 → ∞. Figure 11, represents the graph of EoS parameter of Dark energy (𝜔𝐷) with cosmic time (𝑡). It shows that 𝜔𝐷 = 0 when 𝑡 = 0 i.e., initially 𝜔𝐷 is in quintessence region and as 𝑡 ⟶ ∞, 𝜔𝐷 crosses PDE (𝜔𝐷 = −1) and finally stables in phantom region (𝜔𝐷 < −1). 5. Cosmological Parameters: 5.1 Look Back time: The look back time tL is given by, 𝑡𝐿 = 𝑡0 − 𝑡(𝑧) = ∫ 𝑑𝑅 𝑅 𝑅0 𝑅 (46) The relation between scale factor 𝑡𝐿 and redshift parameter (𝑧) is given below, 𝑅 𝑅0 = 1 1 + 𝑧 (47) Using equation (47) in equation (46), we get 𝑡𝐿 = 3𝑛 + 1 𝐻04(3𝑛 − 1) [1 − (1 + 𝑧) − 4(3𝑛−1) (3𝑛+1) ] (48) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1222 https://internationalpubls.com Fig. 12. Graph of 𝑡𝐿 versus Redshift (𝑧) 5.2 Proper Distance: The proper distance 𝑑(𝑧) is given by, 𝑑(𝑧) = 𝑟1𝑅0 (49) Where, 𝑟1 = ∫ 𝑑𝑡 𝑅 𝑡0 𝑡 (50) From equation (49) and (50), we get 𝑑(𝑧) = 𝑅0𝑘′ 𝑘 [1 − 3𝑛 + 1 4(3𝑛 − 1) ] [1 − (1 + 𝑧) 1− 4(3𝑛−1) (3𝑛+1) ] (51) Fig. 13. Graph of 𝑑(𝑧) versus Redshift (𝑧) 5.3 Luminosity Distance: Luminosity distance (𝑑𝐿) is given by, 𝑑𝐿 = 𝑟1(𝑧)𝑅0(1 + 𝑧) = 𝑑(𝑧)(1 + 𝑧) (52) Substituting value of 𝑑(𝑧) from equation (51) in equation (52), we get 𝑑𝐿 = 𝑅0𝑘′ 𝑘 [1 − 3𝑛 + 1 4(3𝑛 − 1) ] [1 − (1 + 𝑧) 1− 4(3𝑛−1) (3𝑛+1) ] (1 + 𝑧) (53) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1223 https://internationalpubls.com Fig. 14. Graph of 𝑑𝐿 versus Redshift (𝑧) 5.4 Angular Diameter Distance: Angular diameter distance 𝑑𝐴 is given by, 𝑑𝐴 = 𝑑(𝑧)(1 + 𝑧)−1 (54) Substituting value of 𝑑(𝑧) from equation (51) in equation (54), we get 𝑑𝐴 = 𝑅0𝑘′ 𝑘 [1 − 3𝑛 + 1 4(3𝑛 − 1) ] [1 − (1 + 𝑧) 1− 4(3𝑛−1) (3𝑛+1) ] (1 + 𝑧)−1 (55) Fig. 15. Graph of 𝑑𝐴 versus Redshift (𝑧) 5.5 Distance Modulus: The distance modulus 𝜇(𝑧) is given by 𝜇(𝑧) = 5 log 𝑑𝐿 + 25 (56) Substituting value of dL from equation (53) in equation (56), we get Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 1224 https://internationalpubls.com 𝜇(𝑧) = 5 log 𝑅0𝑘′ 𝑘 [1 − 3𝑛 + 1 4(3𝑛 − 1) ] [1 − (1 + 𝑧) 1− 4(3𝑛−1) (3𝑛+1) ] (1 + 𝑧) + 25 (57) 6. Conclusion In the present paper, we have studied a five-dimensional Kaluza-Klein cosmological model filled with barotropic fluid and dark energy in Lyra geometry by considering the shear scalar (σ) proportional to the expansion scalar(𝜃). We have derived physical and kinematical parameters for both the interacting and non-interacting scenarios. In the physical parameters, spatial volume (𝑉) increases with cosmic time(𝑡), resulting in a model that begins to evolve at an initial instant with zero volume and an infinite expansion rate. The Hubble parameter(𝐻), expansion scalar (𝜃) and the shear scalar (𝜎2) diverges for initial time(𝑡), and it decreases as time increases and tends to zero as 𝑡 → ∞. At the same time, the displacement vector (𝛽) decreases and tends to zero as 𝑡 → ∞. The recent observations of Supernovae-Ia reveal that the present universe is accelerating, and the value of the deceleration parameter (𝑞) lie somewhere in the range −1 ≤ 𝑞 ≤ 0. Since 𝑞 < 0 for 1 3 < 𝑛 ≤ 5 9 , the inclusion of the dark energy into the system gives rise to an accelerated expansion of the universe. Equation (25) and (21) shows that lim 𝑡→∞ 𝜎2 𝜃2 ≠ 0, and the anisotropic parameter (Δ) does not vanish for 1 3 < 𝑛 ≤ 5 9 . Hence, this model does not tend to isotropy. Hence, our model is totally anisotropic and spatially homogeneous. For both interacting and non-interacting cases, the pressure and density of Dark energy and Barotropic fluid diverge for initial cosmic time (𝑡) and tends to zero as 𝑡 → ∞. 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