Inflationary Magnetogenesis with f(R) Gravity: Dynamics, Constraints, and Observational Signatures during Reheating

Inflationary magnetogenesis provides a promising mechanism for generating primordial large-scale magnetic fields, but faces challenges. In this paper, we extend the Ratra model to a non-minimal $f(R)$ gravity framework, appearing in the Lagrangian in the form $f^2(\eta )F_{\mu \nu }{F}^{\mu \nu }$, to break the conformal invariance of the standard electromagnetic action. Focusing on the post-inflationary reheating epoch, we derive analytic expressions for the magnetic and electric energy density spectra by constructing a broken-power-law evolution for the scale factor and the coupling function across the inflation-to-reheating transition. Three key theoretical constraints are imposed on the model parameter space: the strong coupling condition, the backreaction constraint, and the CMB isotropy requirement. Through numerical calculations, we obtain predictions for the present-day magnetic field strength $B_0$ and coherence length $L_{c0}$ for various combinations of the inflationary energy scale $H_f$ and the reheating temperature $T_r$. We further incorporate the nonlinear effects of magnetohydrodynamic (MHD) turbulence after reheating, including the inverse transfer process, which significantly enhances the coherence length (up to $0.1$ Mpc) while reducing the field strength by several orders of magnitude. By comparing with observational constraints from radio observations and Fermi-LAT gamma-ray data, we demonstrate that the inflationary energy scale $H_f$, the reheating temperature $T_r$, the parameter $\beta$, and the e-folds numbers $N_f$, $N_r$ must satisfy stringent joint constraints. This work provides a viable theoretical framework for inflationary magnetogenesis that simultaneously satisfies theoretical consistency conditions and current observational bounds.