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

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 ($\alpha_s$) of the scalar spectral index $n_s$ imposes significant restrictions on the parameter space for a large class of ultra-slow-roll (USR) models. 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 and the Einstein Telescope. However, this discrepancy is minimal for asteroid-mass PBHs, which are still capable of feasibly constituting the entirety of dark matter. To numerically probe the six-dimensional parameter space of the models, we adapted a Markov Chain Monte Carlo approach to efficiently scan over the viable configurations. 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.