  The scalar and tensor fluctuations generated during inflation can be
correlated, if arising from the same underlying mechanism. In this paper we
investigate such correlation in the model of axion inflation, where the rolling
inflaton produces quanta of a $U(1)$ gauge field which, in turn, source scalar
and tensor fluctuations. We compute the primordial correlator of the curvature
perturbation, $\zeta$, with the gravitational energy density, $\Omega_{GW}$, at
frequencies probed by gravitational wave detectors. This two-point function
receives two contributions: one arising from the correlation of gravitational
waves with the scalar perturbations generated by the standard mechanism of
amplification of vacuum fluctuations, and the other coming from the correlation
of gravitational waves with the scalar perturbations sourced by the gauge
field. Our analysis shows that the former effect is generally dominant. For
typical values of the parameters, the correlator, normalized by the amplitude
of $\zeta$ and by the fractional energy in gravitational waves at
interferometer frequencies, turns out to be of the order of $ 10^{-4}\div
10^{-2}$.
