The dynamics of a hyperbolic solution in f(R,G) gravity

In this study, our focus is on exploring the dynamics of the universe using a flat FLRW model within the framework of $f(R,G)$ gravity. The specific function $f(R,G)=\xi R+\lambda R^2{G}^2$ is considered, with $R$ and $G$ representing the Ricci scalar and Gauss–Bonnet invariant, respectively. To obtain the solution to the gravitational field equations within the $f(R,G)$ formalism, we adopt a specific form for the scale factor, denoted as $a={sinh}^{\frac{1}{\alpha }}\left(\beta t\right)$ (Nagpal et al., 2019). Here, $\alpha$ and $\beta$ are parameters of the model that determine the behavior of the scale factor. The proposed model predicts the possibility of eternal cosmic acceleration when $0<\alpha <1.19$, indicating a continuous expansion of the universe. On the other hand, for $\alpha \ge 1.19$, the model suggests a transition from an early deceleration phase to the current accelerated epoch. This transition aligns with our understanding of the universe’s evolution. Additionally, the model supports the formation of structures in the universe, as it satisfies the Jeans instability condition during the transition from a radiation-dominated era to a matter-dominated era. We focus on analyzing the behavior of the equation of state parameter $\omega$ in our model. We investigate the scalar field and analyze the energy conditions about the obtained solution. To validate our model, we employ various diagnostic tools such as the Jerk, Snap, and Lerk parameters, as well as the Om diagnostic, Velocity of sound, and statefinder diagnostic tools. Additionally, we perform cosmological tests to assess the accuracy of our model. A detailed discussion of the results and the model itself is provided.