Constant roll inflationary dynamics with generalized potentials in f(R,ϕ,X) gravity

In this work, we study early-time inflation within a class of $f(R,\varphi ,X)$ gravity models under a constant-roll condition. Employing a generalized potential of the form $V{\left(\varphi \right)}^{\sigma }$, we derive expressions for the spectral index $n_s$ and tensor-to-scalar ratio $r$, demonstrating that the inflationary dynamics are primarily governed by the parameter $\sigma$ and the shape of the potential. At the upper limit of $n_s$, we obtain a de Sitter-like phase, while at the lower limit, the model transitions to a quintessence-like phase through an effective oscillating equation of state parameter (EoS). Therefore, under the tuning parameter $\sigma <1$, the model exhibits a smooth transition from a de Sitter-like phase to a quintessence-like phase via the oscillating EoS parameter. The resulting predictions are consistent with recent observations from the Atacama Cosmology Telescope (ACT), combined with CMB lensing and DESI BAO data.