  We study gravitational wave production from gauge preheating in a variety of
inflationary models, detailing its dependence on both the energy scale and the
shape of the potential. We show that preheating into Abelian gauge fields
generically leads to a large gravitational wave background that contributes
significantly to the effective number of relativistic degrees of freedom in the
early universe, $N_\mathrm{eff}$. We demonstrate that the efficiency of
gravitational wave production is correlated with the tensor-to-scalar ratio,
$r$. In particular, we show that efficient gauge preheating in models whose
tensor-to-scalar ratio would be detected by next-generation cosmic microwave
background experiments ($r \gtrsim 10^{-3}$) will be either detected through
its contribution to $N_\mathrm{eff}$ or ruled out. Furthermore, we show that
bounds on $N_\mathrm{eff}$ provide the most sensitive probe of the possible
axial coupling of the inflaton to gauge fields regardless of the potential.
