Bouncing behavior in f (R, Lₘ) gravity: Phantom crossing and energy conditions

In this work, we investigate the bouncing behavior of the universe within the framework of f (R, Lₘ) gravity, using a simple form of f (R, Lₘ) =R2+Lₘ^ (where is a free model parameter) as previously studied. The model predicts a vanishing Hubble parameter in the early and late times, with the deceleration parameter approaching a specific limit at the bouncing point. The EoS parameter is observed to cross the phantom divide line (=-1) near the bouncing point, indicating a significant transition from a contracting to an expanding phase. The model satisfies the necessary energy conditions for a successful bouncing scenario, with violations indicating exotic matter near the bouncing point. Stability conditions are satisfied for certain values of near the bouncing point, but potential instabilities in late-time evolution require further investigation. Finally, we conclude that the f (R, Lₘ) gravity model is promising for understanding the universe's dynamics, especially during events like the bouncing phase.