  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.
