Primordial black holes and scalar induced gravitational waves from the $E$ model with a Gauss-Bonnet term

We study an inflationary $E$ model with the Gauss-Bonnet coupling, which can enhance the curvature perturbation at small scales and thus produce a significant abundance of primordial black holes (PBHs) and detectable scalar induced gravitational waves (SIGWs). PBHs from the $E$ model with mass $30M_{\odot}$, $10^{-5}M_{\odot}$, and $10^{-12}M_{\odot}$ can explain the LIGO-Virgo events, the ultrashort-timescale microlensing events in the OGLE data, and all dark matter, respectively. SIGWs produced by the $E$ model can account for the recent NANOGrav signal. We also compute the primordial non-Gaussianity and discuss its impact on PBHs and SIGWs. The probability distribution of density contrast is modified to be right-tailed, which we find prompts the formation of PBHs, so that the abundance of PBHs is underestimated with Gaussian approximation. On the contrary, the fractional energy density of SIGWs is hardly affected.