Constraining axion inflation with gravitational waves from preheating

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.