Spinning Primordial Black Holes and Scalar Induced Gravitational Waves from Single Field Inflation

We investigate the formation of primordial black holes (PBHs), their spin and abundance, in a single-field inflationary model based on a mutated hilltop potential inserted with a small step-like feature. This step induces a brief phase of ultra-slow-roll inflation, producing the large enhancement of the scalar power spectrum required for an appreciable amount of PBH abundance. Instead of the commonly used analytical power spectra, we compute the primordial power spectrum accurately by numerically solving the Mukhanov-Sasaki equation. Using the obtained power spectrum, we apply peak theory with $\nabla^2 \zeta$ treated as a Gaussian random field and parametrize the curvature profile by its amplitude $\mu$ and characteristic width $K$. Confining the study to Type-I PBH, the threshold value is calculated using two robust methods: the average of the compaction function and the q-function method. Using the result, the dimensionless spin parameter of the resulting PBHs is calculated at linear order and found to be $\sqrt{\langle a_\star^2 \rangle} \sim 10^{-3}$; however, it can be higher for smaller masses. We present detailed predictions for two representative parameter sets, calculate the present-day PBH mass function $f_{\rm PBH}(M)$ and the associated scalar-induced gravitational waves (SIGW). The first produces PBHs of mass $M \simeq 10^{-13}\,M_\odot$ that can account for $100\%$ of dark matter, while the second yields $M \simeq 10^{-2}\,M_\odot$ PBHs contributing approximately $2.4\%$ of the dark matter density. The predicted signals of SIGWs lie within the sensitivity bands of future experiments such as LISA, DECIGO, BBO, and SKA. In particular, the second parameter set produces a SIGWs compatible with the recent NANOGrav evidence for a low-frequency gravitational-wave signal.