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

V. K. Oikonomou

TL;DR

This work addresses the tension between ACT constraints and traditional inflationary models by adopting a top-down, model-agnostic approach: specify the scalar spectral index as a function of the tensor-to-scalar ratio, $n_s=f(r)$, and reconstruct the underlying potential $V(φ)$ by solving the resulting differential equation. Building on plateau-like attractors that yield a universal relation $n_s(r) \approx 1 - α r^{1/2}$, the authors analyze several simple yet tractable forms of $f(r)$, deriving explicit analytic potentials and slow-roll predictions. Four models (Models I–IV) are developed, with three main classes of $n_s(r)$ attractors based on $n_s(r)=γ ± β r ± \sqrt{r}$, each yielding closed-form $V(φ)$ and confronting ACT and Planck/BICEP constraints; viable parameter ranges are identified for ACT compatibility. The results demonstrate a viable pathway to ACT-compatible, plateau-like inflation within a top-down framework and point to further exploration, including potential embedding in no-scale supergravity theories.

Abstract

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.

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

TL;DR

This work addresses the tension between ACT constraints and traditional inflationary models by adopting a top-down, model-agnostic approach: specify the scalar spectral index as a function of the tensor-to-scalar ratio, , and reconstruct the underlying potential by solving the resulting differential equation. Building on plateau-like attractors that yield a universal relation , the authors analyze several simple yet tractable forms of , deriving explicit analytic potentials and slow-roll predictions. Four models (Models I–IV) are developed, with three main classes of attractors based on , each yielding closed-form and confronting ACT and Planck/BICEP constraints; viable parameter ranges are identified for ACT compatibility. The results demonstrate a viable pathway to ACT-compatible, plateau-like inflation within a top-down framework and point to further exploration, including potential embedding in no-scale supergravity theories.

Abstract

The ACT data have severely constrained the single scalar field models. Known models of inflation, like the Starobinsky model, the Higgs model and the -attractors are at least 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 -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 , in this work we seek for attractors of the form that may lead to ACT-compatible inflation. Specifically, we fix the function to have a specific desirable form and then solve the differential equation to find the potential which results to the relation . We discovered analytically three classes of potentials which are variants of the general form and all these models are found to be compatible with the ACT data.
Paper Structure (9 sections, 86 equations, 3 figures)

This paper contains 9 sections, 86 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Known Single Scalar Field Attractors and the $n_s(r)$ Relation
  3. Methodology for General $n_s(r)$ Attractors
  4. Examples of Simple $n_s(r)$ Attractors and Their Phenomenology
  5. Model I: $n_s(r)$ Attractors of the Form $n_s(r)=1-\frac{1}{3 \alpha}r^{1/2}$ and Phenomenology
  6. Model II: $n_s(r)$ Attractors of the Form $n_s(r)=1-\alpha r$ and Phenomenology
  7. Model III: $n_s(r)$ Attractors of the Form $n_s(r)=1-\beta r-\gamma \sqrt{r}$ and Phenomenology: Scenario I $s(\phi)<0$
  8. Model IV: $n_s(r)$ Attractors of the Form $n_s(r)=1-\beta r-\gamma \sqrt{r}$ and Phenomenology: Scenario II $s(\phi)>0$
  9. Conclusions

Figures (3)

  • Figure 1: Marginalized curves of the Planck 2018 data and the model (\ref{['potentialsolutionmodel12']}), confronted with the ACT data, the Planck 2018 data, and the updated Planck constraints on the tensor-to-scalar ratio for $N=50$ and $\alpha$ in the range $\alpha=[0.68,1.5]$.
  • Figure 2: Marginalized curves of the Planck 2018 data and the model (\ref{['potentialsolutionmodel123']}), confronted with the ACT data, the Planck 2018 data, and the updated Planck constraints on the tensor-to-scalar ratio for $N=50$, $K=10$, $\gamma=-0.0000001$ and $\beta$ in the range $\beta=[0.68,0.98]$.
  • Figure 3: Marginalized curves of the Planck 2018 data and the model (\ref{['potentialsolutionmodel1234']}), confronted with the ACT data, the Planck 2018 data, and the updated Planck constraints on the tensor-to-scalar ratio for $N=50$, $K=-10$, $\gamma=-0.0000001$ and $\beta$ in the range $\beta=[0.68,0.98]$.