Narrowing the window of inflationary magnetogenesis
  We consider inflationary magnetogenesis where the conformal symmetry is
broken by the term $f^2(\phi) F_{\alpha\beta} F^{\alpha\beta}$. We assume that
the magnetic field power spectrum today between 0.1 and $10^4$ Mpc is a power
law, with upper and lower limits from observation. This fixes $f$ to be close
to a power law in conformal time in the window during inflation when the modes
observed today are generated. In contrast to previous work, we do not make any
assumptions about the form of $f$ outside these scales. We cover all possible
reheating histories, described by an average equation of state $-1/3 <\bar{w}
<1$. Requiring that strong coupling and large backreaction are avoided both at
the background and perturbative level, we find the bound $\delta_{B_0} < 5
\times10^{-15} \left( \frac{r}{0.07} \right)^{1/2} \kappa \mathrm{G}$ for the
magnetic field generated by inflation, where $r$ is the tensor-to-scalar ratio
and $\kappa$ is a constant related to the form of $f$. This estimate has an
uncertainty of one order of magnitude related to our approximations. The
parameter $\kappa$ is $<100$, and values $\gtrsim1$ require a highly fine-tuned
form of $f$; typical values are orders of magnitude smaller.
