Beware of the running $n_s$ when producing heavy primordial black holes

Abstract

We examine single-field inflationary models for the formation of primordial black holes (PBHs). By analyzing the latest observations from the Atacama Cosmology Telescope (ACT)~\cite{ACT:2025fju, ACT:2025tim}, we demonstrate that the observed preference for positive running ($α_s$) of the scalar spectral index $n_s$ imposes significant restrictions on the parameter space of ultra slow roll scenarios (USR). This tension becomes progressively pronounced for more massive PBHs, posing substantial challenges for USR models to yield a detectable PBH abundance, especially in the mass range probed by ongoing and future gravitational-wave experiments such as LIGO-Virgo-KAGRA (LVK) and the Einstein Telescope (ET). However, this discrepancy is minimal for asteroid-mass PBHs, which are still capable of feasibly constituting the entirety of dark matter (DM). To numerically probe the six-dimensional parameter space of polynomial models, we adapted a Markov Chain Monte Carlo (MCMC) approach to efficiently scan over the space of viable models. Our results further indicate that, in non-minimally coupled polynomial inflation, a viable cosmic microwave background (CMB) spectrum is best obtained at an inflection point for which second-order slow-roll approximation is necessary for precise CMB predictions.

Figures (9)

• Figure 1: Existing constraints (colored lines) on the primordial curvature power spectrum as a function of the scale $k$, assuming negligible primordial non-Gaussianities. While the blue constraints on the amplitude refer to the amplitude evaluated at the position of the main peak $k_{\rm pk}$, the green constraints apply to the entire spectrum. The black line corresponds to a benchmark power spectrum. The red lines show the amplitude corresponding to $f_{\rm PBH}=1$ for the TS (solid) and PT (Dashed) formalism (see Sec. \ref{['subsec:ABU']} for more details).
• Figure 2: Evolution of the parameter $\nu^2$ (green lines), the normalized curvature power spectrum $P_\zeta(N)/P_\zeta(0)$ (red lines), the field evolution (blue lines), and the second Hubble parameter $\epsilon_2$ (orange lines) as functions of the number of $e$-folds $N$. Solid lines correspond to the model including the localized bump, while dashed lines refer to the smooth potential without the bump. The two SR phases are labeled as $N_1$ and $N_3$, separated by a transient USR and CR stage whose beginning and end are given by the condition $\nu^2 \simeq 9/4$. The USR phase is indicated with a grey region.
• Figure 3: Constraints in the ($r,n_s$) plane (bottom panel) and ($\alpha_s,n_s$) plane (top panel) using the analysis of Planck data Planck:2018vyg (yellow regions) and the ACT+Planck ACT:2025tim (green regions). The plot includes 68% and 95% confidence regions. The colored lines show how the CMB observables change with varying $\beta$, while the black lines show the dependence on $N_{\rm SR}$. Prospective sensitivities on $r$ of experiments such as Spider SPIDER:2017xxz, Simons Observatory SimonsObservatory:2018koc, and LiteBIRD Matsumura:2013aja are indicated by the gray dashed lines.
• Figure 4: Summary of the inflationary predictions and associated observational consequences for the toy model using the benchmark value $\beta=0.05$. Bottom panel: The curvature power spectra for configurations with fixed $N_{\rm tot} \simeq 55$ and amplitude of the main peak $\mathcal{P}_\zeta(k_{\rm pk})\simeq10^{-2}$, but with different $N_{\rm SR}$ ( in magenta with different dashing) and with different $k_{\rm pk}$ (different colors). Top panel: The resulting SIGW spectra for the power spectrum shown in the bottom panel are color-coded accordingly.
• Figure 5: Output parameters from the NMC (blue) and MC (red) models from the MCMC scan. The dots represent model configurations that fall within the 68% or 95% confidence regions from Planck data Planck:2018vyg (yellow) and ACT+Planck ACT:2025tim (green). The corresponding $2\sigma$ bounds are indicated by black dashed lines. Configurations are further required to exhibit a power spectrum peak compatible with the NANOGrav and PBH overproduction constraints (grey regions in the top-left panel). Vivid colors indicate the $2\sigma$ contours for the configurations obtained from the Planck+ACT MCMC, while faded colors indicate those obtained from the Planck MCMC (see App. \ref{['app1']} for further details).