Cyclic cosmology in periodic gravity within fractional action framework: Markov chain Monte Carlo analysis and constraints from the Hubble parameter

Modified gravity theories provide a compelling alternative to dark energy in explaining cosmic acceleration, and nonlinear oscillatory models like [Formula: see text] exhibit rich dynamical behavior capable of producing cyclic universes. In this work, we investigate the cosmological implications of [Formula: see text] gravity within the framework of fractional-action cosmology (FAC), focusing on early-universe dynamics, stability, and astrophysical consequences. We derive the weighted Friedmann equations for a spatially flat FRW universe, identify the conditions for curvature-driven acceleration, and analyze the oscillatory behavior of the Ricci scalar as a driver for expansion and contraction cycles. The fractional action parameter [Formula: see text] modulates the amplitude and period of these cycles, influencing both the early-universe and late-time dynamics. We show that stability conditions, such as positivity of the effective gravitational coupling and nonnegative scalaron mass squared, are satisfied for small deviations [Formula: see text], allowing a dynamically stable cyclic universe. We further explore astrophysical consequences, demonstrating that oscillations in curvature can modulate star formation rates, alter galaxy formation and evolution, and influence cluster dynamics, while also producing subtle signatures in the cosmic microwave background and primordial gravitational waves. Our analysis suggests that cyclic [Formula: see text] gravity with FAC provides a unified framework for early acceleration, long-term cosmic oscillations, and observable astrophysical effects, offering a testable alternative to standard [Formula: see text]CDM cosmology.