| Summary: | In this study, we investigate the formation of primordial black holes (PBHs) in a scalar field inflationary model coupled to the Gauss–Bonnet term with fractional power-law potentials. The coupling function enhances the curvature perturbations, then results in the generation of PBHs and detectable secondary gravitational waves (GWs). We identify three separate sets of parameters for the potential functions of the form ϕ ^1/3 , ϕ ^2/5 , and ϕ ^2/3 . By adjusting the model parameters, we decelerate the inflaton during the ultra-slow-roll phase and enhance curvature perturbations. Our calculations predict the formation of PBHs with masses of ${ \mathcal O }(10){M}_{\odot }$ , which are compatible with LIGO-Virgo observational data. Additionally, we find PBHs with masses around ${ \mathcal O }({10}^{-6}){M}_{\odot }$ and ${ \mathcal O }({10}^{-5}){M}_{\odot }$ , which can explain ultra-short-timescale microlensing events in OGLE data. Furthermore, our proposed mechanism could lead to the formation of PBHs in mass scales around ${ \mathcal O }({10}^{-14}){M}_{\odot }$ and ${ \mathcal O }({10}^{-13}){M}_{\odot }$ , contributing to approximately 99% of the dark matter in the Universe. We also study the production of secondary GWs in our model. In all cases of the model, the density parameter of secondary GWs ${{\rm{\Omega }}}_{{\mathrm{GW}}_{0}}$ exhibits peaks that intersect the sensitivity curves of GW detectors, providing a means to verify our findings using data of these detectors. Our numerical results demonstrate a power-law behavior for the spectra of ${{\rm{\Omega }}}_{{\mathrm{GW}}_{0}}$ with respect to frequency, given by ${{\rm{\Omega }}}_{{\mathrm{GW}}_{0}}{(f)\sim (f/{f}_{c})}^{n}$ . Additionally, in the infrared regime where f ≪ f _c , the power index takes a log-dependent form, specifically $n=3-2/\mathrm{ln}({f}_{c}/f)$ .
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