The Expansion Behavior of the Hyperbolic Solution in f(R, G) Gravity

In this study, our primary focus lies in meticulously exploring the intricacies of the universe by employing a flat Friedmann–Lemaître–Robertson–Walker (FLRW) model within the framework of Formula: see text gravity. In our analysis, the Formula: see text function is delineated as the sum of two distinct components, commencing with a quadratic correction of the geometric term denoted by Formula: see text, structured as Formula: see text, alongside a matter term denoted by Formula: see text, where Formula: see text and Formula: see text symbolize the Ricci scalar and Gauss–Bonnet invariant, respectively. In the pursuit of solutions to the gravitational field equations within the Formula: see text formalism, we embrace a specific expression for the scale factor, represented as Formula: see text D. Rabha and R. R. Baruah, The dynamics of a hyperbolic solution in [Formula: see text gravity, Astron. Comput. 45 (2023) 100761]. In this context, the parameters Formula: see text and Formula: see text intricately shape the scale factor’s behavior. The model posits the intriguing prospect of perpetual cosmic acceleration when Formula: see text, signifying a continuous expansion of the universe. Conversely, for Formula: see text, the model proposes a pivotal transition from an early deceleration phase to the present accelerated epoch, a transformative shift in line with our understanding of cosmic evolution. Furthermore, the model demonstrates its credibility by satisfying Jean’s instability condition during the shift from a radiation-dominated era to a matter-dominated era, substantiating the formation of cosmic structures. In our analysis, a central focus is directed toward scrutinizing the equation of state parameter Formula: see text within our model. We delve into a comprehensive examination of the scalar field and meticulously assess the energy conditions surrounding the derived solution. To establish the robustness of our model, we deploy an array of diagnostic tools, including the Jerk, Snap, and Lerk parameters, along with the Om diagnostic, Classical stability of the model, and statefinder diagnostic tools, Observational Constraints on the Model Parameters. The outcomes, intertwined with a detailed analysis of both the results and the inherent intricacies of the model, are diligently clarified and presented.