  Scalar-induced Gravitational Waves (SIGWs) represent a particular class of
primordial signals which are sourced at second-order in perturbation theory
whenever a scalar fluctuation of the metric is present. They form a guaranteed
Stochastic Gravitational Wave Background (SGWB) that, depending on the
amplification of primordial scalar fluctuations, can be detected by GW
detectors. The amplitude and the frequency shape of the scalar-induced SGWB can
be influenced by the statistical properties of the scalar density
perturbations. In this work we study the intuitive physics behind SIGWs and we
analyze the imprints of local non-Gaussianity of the primordial curvature
perturbation on the GW spectrum. We consider all the relevant non-Gaussian
contributions up to fifth-order in the scalar seeds without any hierarchy, and
we derive the related GW energy density $\Omega_{\rm GW}(f)$. We perform a
Fisher matrix analysis to understand to which accuracy non-Gaussianity can be
constrained with the LISA detector, which will be sensitive in the milli-Hertz
frequency band. We find that LISA, neglecting the impact of astrophysical
foregrounds, will be able to measure the amplitude, the width and the peak of
the spectrum with an accuracy up to $\mathcal{O}(10^{-4})$, while
non-Gaussianity can be measured up to $\mathcal{O}(10^{-3})$. Finally, we
discuss the implications of our non-Gaussianity expansion on the fraction of
Primordial Black Holes.
