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

TL;DR

This work links early-universe inflationary dynamics to primordial black hole (PBH) phenomenology and their associated scalar-induced gravitational waves (SIGWs). By employing a single-field mutated hilltop potential with a small step, the authors generate a transient ultra-slow-roll phase that amplifies the small-scale power spectrum, computed precisely via numerical solutions of the Mukhanov–Sasaki equation. PBH abundance and spin are predicted using peak theory with the Laplacian of the curvature perturbation, yielding a characteristic spin of order $10^{-3}$ and two benchmark mass scenarios: PBHs around $M \,\sim\ 10^{-13} M_3$ potentially accounting for all dark matter, and PBHs around $M \,\sim\ 10^{-2} M_3$ contributing a few percent of DM. The corresponding SIGW spectra lie within the reach of upcoming detectors like LISA, DECIGO, and SKA, with one parameter set also compatible with NANOGrav signals, highlighting a testable inflation–PBH–SIGW link.

Abstract

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 $\nabla^2 ζ$ treated as a Gaussian random field and parametrize the curvature profile by its amplitude $μ$ 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.

Figures (14)

• Figure 1: The plot shows the behavior of the a)Hubble parameter, b) scalar field, and c) first slow-roll parameter versus the number of e-folds for two sets of the parameters: i) $(c, \phi_s, \delta) = (7.0043 \times 10^{-4}, 3.7, 1.26 \times 10^{-2})$, and ii)$(c, \phi_s, \delta) = (2.3658 \times 10^{-4}, 4.22, 7.2548 \times 10^{-3})$. The other parameter are taken as $\alpha = 1$, $V_0 = 4.75$, and $\phi_\star = 4.75$ where $\phi_\star$ is the field at the crossing time. The parameters are chosen to have around $N=60$ e-folds of expansion for the inflationary phase.
• Figure 2: The plot displays the power spectrum as a function of wavenumber $k$ for 2 different sets of parameters: i) $(c, \phi_s, \delta) = (7.0043 \times 10^{-4}, 3.7, 1.26 \times 10^{-2})$, and ii)$(c, \phi_s, \delta) = (2.3658 \times 10^{-4}, 4.22, 7.2548 \times 10^{-3})$. The other parameter are taken as $\alpha = 1$, $\phi_\star = 4.75$, and $V_0 = 9.48 \times 10^{-11}$. The parameters are chosen to yield approximately $N=60$ e-folds of expansion for the inflationary phase and to reproduce the power spectrum $\mathcal{P}_s = 2.1 \times 10^{-9}$ at the crossing time of the pivot mode. The different constraints on the power spectrum are also plotted Planck Planck:2018jri, FIRAS Fixsen:1996nj, SKA Chluba:2019nxaInomata:2018epa, PTA Byrnes:2018txb and LISA Inomata:2018epa. Plot legends are self-explanatory. We have obtained $r=0.0024$ and $n_s=0.975$, which are consistent with the recent ACT observations ACT:2025fjuACT:2025tim.
• Figure 3: The plot displays the normalized profile versus $x = k_p r$ coordinate for different values of $K$ parameter. Here, $k_p$ is the peak mode where the power spectrum reaches the maximum value.
• Figure 4: Behavior of the compaction function versus $x = k_p r$ is illustrated: a) for different values of $\mu$ and $K = \sqrt{\gamma_3}$, and b) for different values of $K$ and $\mu = 0.5$ is shown. By increasing $\mu$, the amplitude of the compaction function increases; however, the location of the peak never changes. On the other hand, by increasing the $K$ parameter, the amplitude of the compaction function decreases and the location of the peak tends to lower values of $r$.
• Figure 5: The plot shows the compaction function, black color lines, and the average of the compaction function, blue color lines, for the numerical power spectrum obtained in Sec.\ref{['model']}. The dotted horizontal line stands for the threshold value as $\bar{\mathcal{C}(r_m)} = 2/5$.