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The coupling between a pseudo-scalar inflaton and a gauge field leads to an amount of additional density perturbations and gravitational waves (GWs) that is strongly sensitive to the inflaton speed. This naturally results in enhanced GWs at (relatively) small scales that exited the horizon well after the CMB ones, and that can be probed by a variety of GW observatories (from pulsar timing arrays, to astrometry, to space-borne and ground-based interferometers). This production occurs in a regime in which the gauge field significantly backreacts on the inflaton motion. Contrary to earlier assumptions, it was later shown that this regime is characterized by an oscillatory behavior of the inflaton speed, with a period of O ( 5 ) e-folds. Bursts of GWs are produced at the maxima of the speed, imprinting nearly periodic bumps in the frequency-dependent spectrum of GWs produced during inflation. This can potentially generate correlated peaks appearing in the same or in different GWs experiments. While recent lattice studies show that the inclusion of inflaton gradients can modify significantly the dynamics of this system in the strong backreaction regime, this is not the case for the first oscillation or two of the inflaton speed, so that we expect our results to be robust for modes that were excited during that epoch.