A significant abundance of primordial black hole (PBH) dark matter can be produced by curvature perturbations with power spectrum $Delta_zeta^2(k_{mathrm{peak}})sim mathcal{O}(10^{-2})$ at small scales, associated with the generation of observable sc
alar induced gravitational waves (SIGWs). However, the primordial non-Gaussianity may play a non-negligible role, which is not usually considered. We propose two inflation models that predict double peaks of order $mathcal{O}(10^{-2})$ in the power spectrum and study the effects of primordial non-Gaussianity on PBHs and SIGWs. This model is driven by a power-law potential, and has a noncanonical kinetic term whose coupling function admits two peaks. By field-redefinition, it can be recast into a canonical inflation model with two quasi-inflection points in the potential. We find that the PBH abundance will be altered saliently if non-Gaussianity parameter satisfies $|f_{mathrm{NL}}(k_{text{peak}},k_{text{peak}},k_{text{peak}})|gtrsim Delta^2_{zeta}(k_{mathrm{peak}})/(23delta^3_c) sim mathcal{O}(10^{-2})$. Whether the PBH abundance is suppressed or enhanced depends on the $f_{mathrm{NL}}$ being positive or negative, respectively. In our model, non-Gaussianity parameter $f_{mathrm{NL}}(k_{mathrm{peak}},k_{mathrm{peak}},k_{mathrm{peak}})sim mathcal{O}(1)$ takes positive sign, thus PBH abundance is suppressed dramatically. On the contrary, SIGWs are insensitive to primordial non-Gaussianity and hardly affected, so they are still within the sensitivities of space-based GWs observatories and Square Kilometer Array.
Enormous information about interactions is contained in the non-Gaussianities of the primordial curvature perturbations, which are essential to break the degeneracy of inflationary models. We study the primordial bispectra for G-inflation models pred
icting both sharp and broad peaks in the primordial scalar power spectrum. We calculate the non-Gaussianity parameter $f_{mathrm{NL}}$ in the equilateral limit and squeezed limit numerically, and confirm that the consistency relation holds in these models. Even though $f_{mathrm{NL}}$ becomes large at the scales before the power spectrum reaches the peak and the scales where there are wiggles in the power spectrum, it remains to be small at the peak scales. Therefore, the contributions of non-Gaussianity to the scalar induced secondary gravitational waves and primordial black hole abundance are expected to be negligible.
The scalar induced gravitational waves (SIGWs) is a useful tool to probe the physics in the early universe. To study inflationary models with this tool, we need to know how the waveform of SIGWs is related to the shape of the scalar power spectrum. W
e propose two parameterizations to approximate the scalar power spectrum with either a sharp or a broad spike at small scales, and then use these two parameterizations to study the relation between the shapes of $Omega_{GW}$ and the scalar power spectrum. We find that the waveform of SIGWs has a similar shape to the power spectrum. Away from the peak of the spike, the frequency relation $Omega_{GW}(k)sim mathcal{P}_zeta^2(k)$ holds independent of the functional form of the scalar power spectrum. We also give a physical explanation for this general relationship. The general relation is useful for determining the scalar power spectrum and probing inflationary physics with the waveform of SIGWs.
