Polynomial Potential Inflation in the ACT Era: From CMB to Primordial Black Holes

TL;DR

This work tests polynomial potential inflation against ACT-era measurements, systematically analyzing $n=2$–$5$ and showing that cubic and quartic potentials can fit the CMB constraints while quintic inflation can simultaneously generate an inflection point that amplifies small-scale perturbations enough to form primordial black holes and produce scalar-induced gravitational waves. The quintic scenario yields a concrete link between large-scale $n_s$ and $r$ and small-scale PBH/SGW signatures, predicting a pronounced peak in the curvature power spectrum around $k\sim10^{13}\,\text{Mpc}^{-1}$ and a PBH mass window potentially compatible with dark matter. The analysis also highlights tensions for the quadratic case under ACT constraints and discusses the role of running and running-of-running parameters in fitting the data for higher-order potentials. Overall, higher-order polynomial inflation remains a viable framework that can connect precision CMB measurements with multi-messenger probes of early-universe physics, with future CMB and GW experiments poised to test these predictions.

Abstract

The recent measurements from the Atacama Cosmology Telescope (ACT) favor a higher value of the scalar spectral index $n_s$ compared to the Planck data, challenging many well-established inflationary models. In this work, we investigate the viability of polynomial potential inflation in light of the latest ACT data, systematically analyzing cases from $n=2$ to $n=5$. By exploring the parameter space and deriving constraints on the model coefficients, we find that the cubic to quintic models can provide a good fit to the data, while the quadratic model struggles to simultaneously accommodate the ACT data and the requirement of sufficient inflation. Notably, the quintic case ($n=5$) not only matches cosmic microwave background (CMB) observations but also produces an inflection point that simultaneously triggers primordial black hole formation and generates a scalar-induced gravitational wave. These findings establish higher-order polynomial potentials as compelling frameworks and reconcile precision CMB measurements with multi-messenger probes of early-universe physics.

Figures (6)

• Figure 1: (Left) Scalar spectral index $n_s$ and tensor-to-scalar ratio $r$ predicted by the quadratic potential for $N=50$, $N=60$ and $N=70$, compared with observational constraints. The contours are the $1\sigma$ and $2\sigma$ observational constraints from the P-ACT-LB data combined with B-mode measurements from the BICEP and Keck telescopes at the South Pole (BK18), referred to as P-ACT-LB-BK18 ACT:2025fjuACT:2025tim. (Right) Corresponding parameter values of $a_1$ and $a_2$ consistent with observational bounds. The brown and gray color regions are the constraints from $1\sigma$ and $2\sigma$ observational constraints, respectively.
• Figure 2: The $n_s-r$ trajectories for the cubic potential, evaluated at $N=50$ (red), $60$ (blue), and $70$ (green). The brown and gray color regions are the constraints from $1\sigma$ and $2\sigma$ observational constraints, respectively.
• Figure 3: The $n_s-r$ trajectories for the quartic potential, evaluated at $N=50$ (red), $60$ (blue), and $70$ (green). Each curve corresponds to a specific value of the running of the running: $\beta_s=-2.75\times10^{-4}$, $-2.60\times10^{-4}$, $-2.50\times10^{-4}$, respectively. The brown and gray color regions are the constraints from $1\sigma$ and $2\sigma$ observational constraints, respectively.
• Figure 4: Power spectrum of primordial curvature perturbations.The green shaded region is excluded by CMB observations Planck:2018jri. The orange shaded region shows the current upper bound on the power spectrum from measurements of $\mu$ distortion for COBE/FIRAS Mather:1993ijFixsen:1996nj. The forecasted constraint for the distortion experiment PIXIE Kogut:2011xw is shown as the orange dashed line. See Ref. Chluba:2019nxa for the summary of constraints on the power spectrum of curvature perturbations.
• Figure 5: Predicted PBH mass function for the parameter set given in the text. The shaded regions indicate current observational constraints on PBHs, derived from the extragalactic gamma-ray background (EG$\gamma$Carr:2009jm), the galactic 511 keV emission line ($e^{+}$DeRocco:2019fjqLaha:2019ssqDasgupta:2019cae), gravitational lensing surveys (HSC Niikura:2017zjd, EROS Tisserand:2006zx, OGLE Niikura:2019kqi), dynamical limits from stellar kinematics Koushiappas:2017chw, and CMB measurements Poulin:2017bweCarr:2020goxGreen:2020jor.