Probing non-Gaussianities with the high frequency tail of induced gravitational waves

We investigate in detail the spectrum of gravitational waves induced by a peaked primordial curvature power spectrum generated in single field inflationary models. We argue that the $f_{\rm NL}$ parameter can be inferred by measuring the high frequency spectral tilt of the induced gravitational waves. We also show that the intrinsically non-Gaussian impact of $f_{\rm NL}$ in $\Omega_{\rm GW}$ is to broaden its peak, although at a negligible level in order not to overproduce primordial black holes. We discuss possible degeneracies in the high frequency spectral tilt between $f_{\rm NL}$ and a general equation of state of the universe $w$. Finally, we discuss the constraints on the amplitude, peak and slope (or equivalently, $f_{\rm NL}$) of the primordial power spectrum by combining current and future gravitational wave experiments with limits on $\mu$ distortions from the cosmic microwave background.