Effective Theory of Inflationary Magnetogenesis and Constraints on
  Reheating
  Effective theory framework based on symmetry has recently gained widespread
interest in the field of cosmology. In this paper, we apply the same idea on
the genesis of the primordial magnetic field and its evolution throughout the
cosmological universe. Given the broken time-diffeomorphism symmetry by the
cosmological background, we considered the most general Lagrangian of
electromagnetic and metric fluctuation up to second order, which naturally
breaks conformal symmetry in the electromagnetic (EM) sector. We also include
parity violation in the electromagnetic sector with the motivation that has
potential observational significance. In such a set-up, we explore the
evolution of EM, scalar, and tensor perturbations considering different
observational constraints. In our analysis we emphasize the role played by the
intermediate reheating phase which has got limited interest in all the previous
studies. Assuming the vanishing electrical conductivity during the entire
period of reheating, the well-known Faraday electromagnetic induction has been
shown to play a crucial role in enhancing the strength of the present-day
magnetic field. We show how such physical effects combined with the PLANCK and
the large scale magnetic field observation makes a large class of models viable
and severely restricts the reheating equation of state parameter within a very
narrow range of $0.01 < \omega_\mathrm{eff} < 0.27$, which is nearly
independent of reheating scenarios we have considered.
