Approximate reconstruction of inflationary potential with ACT observations

TL;DR

The paper tackles the tension between ACT measurements and universal inflationary attractors by parameterizing the scalar spectral index as $n_s = 1 - \frac{p}{N+α}$ with $1.338<p<1.746$, and reconstructs the inflationary potential within the slow-roll framework. This reconstruction yields the KKLT potential $V(φ)=V_0/[1+(M/φ)^n]$ with $n=\frac{2(p-1)}{2-p}$ and $r\approx\frac{16(p-1)}{C (N+α)^p}$, where $C$ is a function of $p$ and $M$. The KKLT form emerges as a model-independent description on CMB scales and also approximates D-brane inflation and polynomial $\alpha$-attractors, with $r$ becoming arbitrarily small for small $M$. The analysis further shows how reheating, via $N_{re}$ and $w_{re}$, shifts the allowed $p$ and $N$, linking post-inflationary dynamics to the inflationary parameter constraints. Overall, the KKLT potential provides a robust, data-compatible template for a broad class of inflationary scenarios under current ACT constraints, adaptable to future measurements.

Abstract

The Atacama Cosmology Telescope (ACT) has recently reported updated measurements of the scalar spectral index $n_s$, revealing a tension with the predictions of many conventional inflationary models. In this work, we adopt a parameterization of the spectral index in the form $n_s = 1 - p/(N + α)$ with $1.338<p<1.746$, to reconstruct an inflationary potential consistent with the latest ACT data. The resulting potential is the Kachru-Kallosh-Linde-Trivedi (KKLT) potential $V(φ) = V_0/[1 + (M/φ)^n]$, where the power index is given by $n = 2(p-1)/(2-p)$. The corresponding tensor-to-scalar ratio is approximately $r \approx 16(p-1)/[C (N+α)^p]$ with $C= 2^{2p-1} \left[\sqrt{p-1}/(2-p)\right]^{2p-2} /M^{2p-2}$. Since the reconstruction under the slow-roll approximation is model-independent, the KKLT model can serve as an effective approximation to a broad class of inflationary scenarios that are consistent with the latest ACT measurements of $n_s$, at least on large scales.

Figures (3)

• Figure 1: Comparison of the reconstructed KKLT potential \ref{['brane1']} and the Einstein-frame potential of the nonminimally coupled model. Solid curves represent the reconstructed potential, while dashed curves correspond to the Einstein-frame potential of the inflation model with the nonminimal coupling. The black, red, and blue curves denote the potentials with $n=1$, $n=2$, and $n=3$, respectively. In all cases, we set $M = 2$. The two potentials coincide in the slow-roll region, confirming the model-independent nature of the reconstructed form, at least on large scales. For the case with $n=1$, the two potentials are nearly indistinguishable throughout the entire field range. Dot-dashed curves show the slow-roll parameter $\epsilon_V$ of the reconstructed potential.
• Figure 2: The scalar spectral index $n_s$ and tensor-to-scalar ratio $r$ as functions of the parameter $n$ for various values of $M$. Arrows indicate the direction of decreasing $M$ within the range $0.001 < M < 2$. Solid curves represent the numerical results from the polynomial $\alpha$ attractors potential \ref{['rec:potential']}, while dashed curves correspond to the slow-roll approximation given by Eq. \ref{['nsr:para']}. The black, red, blue, green, and cyan curves correspond to $n = 1$, $n = 2$, $n = 3$, $n = 4$, and $n=5$, respectively.
• Figure 3: Constraints on the reheating $e$-folding number $N_{re}$ and the parameter $p$ from the observational constraint \ref{['ns:constraint']}. The left panel corresponds to a soft equation of state $w_{re}= 1/6$ and the right panel to a stiff equation of state $w_{re}=2/3$.