Gravitational waves from axion inflation in the gradient expansion formalism. Part I. Pure axion inflation

Axion inflation is a well-motivated model of cosmic inflation with a rich phenomenology. The abundant production of gauge fields during axion inflation notably sources a stochastic gravitational-wave (GW) background signal, which nourishes the hope that future GW searches might have a chance to probe the model. In this paper, we scrutinize GW production during axion inflation in the gradient expansion formalism (GEF), a powerful numerical technique that captures the nonlinear dynamics of the system in the limit of vanishing axion gradients. We focus on {single-field} axion inflation coupled to a pure Abelian gauge sector, i.e., pure axion inflation (PAI), and perform the first-ever {detailed} parameter scan of GW production in the Abelian PAI model close to the onset of strong backreaction. {We approximate the axion potential around its minimum by a quadratic mass term and study the tensor modes that exit the Hubble horizon as the axion rolls down this potential.} Remarkably enough, we find that GW signals within the reach of future GW interferometers can only be realized in parameter regions that also lead to strong backreaction and that are in conflict with the upper limit on $\Delta N_{\rm eff}$, i.e., the allowed energy density of dark radiation. This observation defines a clear target for future lattice studies of axion inflation that may confirm or improve the predictions of our GEF benchmark.