Probing the reheating phase through primordial magnetic field and CMB

Inflationary magnetogenesis has long been assumed to be the most promising mechanism for producing large-scale magnetic fields in our universe. However, generically, such models are plagued with either backreaction or strong coupling problems within the standard framework. This paper has shown that the reheating phase can play a crucial role in alleviating those problems along with CMB. Assuming the electrical conductivity to be negligible during the entire period of reheating, the classic Faraday electromagnetic induction changes the magnetic field's dynamics drastically. Our detailed analysis reveals that this physical phenomenon not only converts a large class of magnetogenesis model observationally viable without any theoretical problem but also can uniquely fix the perturbative average inflaton equation of state, $\omega_{\phi} = (p+2)/(p+2)$ during reheating given a specific value of the large scale magnetic field. This observation hints the inflaton to assume the potential of form $V(\phi) \sim \phi^p$ near its minimum with $p \gtrsim 3.5$ if one considers the limit of the present-day strength of the large scale magnetic field to be $\mathcal{P}_{B0}^{\frac{1}{2}} \gtrsim 10^{-18}$ G. Our analysis opens up a new avenue towards constraining the inflationary and magnetogenesis model together via reheating.