Inflationary Magnetogenesis without the Strong Coupling Problem II:
  Constraints from CMB anisotropies and B-modes
  Recent observational claims of magnetic fields stronger than $10^{-16}$ G in
the extragalactic medium motivate a new look for their origin in the
inflationary magnetogenesis models. In this work we shall review the
constraints on the simplest gauge invariant model $f^2(\phi)F_{\mu \nu}F^{\mu
\nu}$ of inflationary magnetogenesis, and show that in the optimal region of
parameter space the anisotropic constraints coming from the induced bispectrum,
due to the generated electromagnetic fields, yield the strongest constraints.
In this model, only a very fine tuned scenario at an energy scale of inflation
as low as $10^{-2}$ GeV can explain the observations of void magnetic fields.
These findings are consistent with the recently derived model independent
constraints. However, if the detection of primordial tensor modes by BICEP2 is
confirmed, the possibility of low scale inflation is excluded. Assuming the
validity of the BICEP2 claim of a tensor-to-scalar ratio
$r=0.2^{+0.07}_{-0.05}$, we provide the updated constraints on inflationary
magnetogenesis. On the Mpc scale, the maximal allowed magnetic field strength
from inflation is less than $10^{-30}$ G.
