General Scalar Field Inflation ACT Attractors: Utilizing the $n_s(r)$ relation

The ACT data have severely constrained the single scalar field models. Known models of inflation, like the Starobinsky model, the Higgs model and the $a$-attractors are at least $2σ$ off the ACT data. In this work we aim to provide a top-to-bottom approach in single scalar field inflationary cosmology compatible with the ACT data. Specifically, inspired by the fact that the Starobinsky model, the Higgs model and the $a$-attractors, all being plateau potentials, result to the same attractor relation between the spectral index of scalar perturbations and the tensor-to-scalar ratio, which is of the form $n_s(r)=1-αr^{1/2}$, in this work we seek for attractors of the form $n_s(r)=f(r)$ that may lead to ACT-compatible inflation. Specifically, we fix the function $f(r)$ to have a specific desirable form and then solve the differential equation $n_s(r)=f(r)$ to find the potential which results to the relation $n_s(r)=f(r)$. We discovered analytically three classes of potentials which are variants of the general form $n_s(r)=γ\pm βr \pm r^{1/2}$ and all these models are found to be compatible with the ACT data.

💡 Research Summary

The paper addresses the tension between the latest Atacama Cosmology Telescope (ACT) measurements and the predictions of the most widely studied single‑field inflationary models. ACT reports a scalar spectral index (n_s = 0.9743 \pm 0.0034) and a tensor‑to‑scalar ratio bound (r < 0.036). Classic plateau‑type models such as Starobinsky inflation, Higgs inflation and the family of (\alpha)-attractors all share the universal attractor relation (n_s(r)=1-\alpha\sqrt{r}). When evaluated for the typical number of e‑folds ((N\approx 60)), these models give (n_s\approx 0.967) and a value of (r) that sits at the edge of the ACT limit, making them at least (2\sigma) away from the new data.

Instead of starting from a specific potential and checking its phenomenology, the authors adopt a “bottom‑up” (or rather “top‑down”) strategy: they prescribe a functional form for the observable relation (n_s = f(r)) that is compatible with ACT, and then solve the resulting differential equation for the inflaton potential (V(\phi)). The key equations are the slow‑roll expressions \