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On the Present Status of Inflationary Cosmology

Renata Kallosh, Andrei Linde

TL;DR

This review assesses the present status of inflationary cosmology by weighing simple, predictive models against Planck/BICEP/Keck and ACT data. It underscores the robustness of plateau-type models (Starobinsky, Higgs, and α-attractors) while highlighting viable alternatives like polynomial chaotic inflation that can fit current observables with a small parameter count. The discussion emphasizes how upcoming measurements of the scalar spectral index $n_s$ and tensor-to-scalar ratio $r$—and their interplay with reheating and dark energy—could distinguish between exponential and polynomial plateau realizations and test string-theoretic realizations of inflation. Overall, inflation remains broadly compatible with data across a spectrum of models, with observational precision now driving model discrimination more than qualitative viability.

Abstract

We give a brief review of the basic principles of inflationary theory and discuss the present status of the simplest inflationary models that can describe Planck/BICEP/Keck observational data by choice of a single model parameter. In particular, we discuss the Starobinsky model, Higgs inflation, and $α$-attractors, including the recently developed $α$-attractor models with $SL(2,\mathbb{Z})$ invariant potentials. We also describe inflationary models providing a good fit to the recent ACT data, as well as the polynomial chaotic inflation models with three parameters, which can account for any values of the three main CMB-related inflationary parameters $A_{s}$, $n_{s}$ and $r$.

On the Present Status of Inflationary Cosmology

TL;DR

This review assesses the present status of inflationary cosmology by weighing simple, predictive models against Planck/BICEP/Keck and ACT data. It underscores the robustness of plateau-type models (Starobinsky, Higgs, and α-attractors) while highlighting viable alternatives like polynomial chaotic inflation that can fit current observables with a small parameter count. The discussion emphasizes how upcoming measurements of the scalar spectral index and tensor-to-scalar ratio —and their interplay with reheating and dark energy—could distinguish between exponential and polynomial plateau realizations and test string-theoretic realizations of inflation. Overall, inflation remains broadly compatible with data across a spectrum of models, with observational precision now driving model discrimination more than qualitative viability.

Abstract

We give a brief review of the basic principles of inflationary theory and discuss the present status of the simplest inflationary models that can describe Planck/BICEP/Keck observational data by choice of a single model parameter. In particular, we discuss the Starobinsky model, Higgs inflation, and -attractors, including the recently developed -attractor models with invariant potentials. We also describe inflationary models providing a good fit to the recent ACT data, as well as the polynomial chaotic inflation models with three parameters, which can account for any values of the three main CMB-related inflationary parameters , and .
Paper Structure (22 sections, 79 equations, 20 figures)

This paper contains 22 sections, 79 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Chaotic inflation
  3. Initial conditions for inflation
  4. Can we test inflation?
  5. Constraints on inflationary models
  6. Polynomial chaotic inflation
  7. Starobinsky model
  8. Higgs inflation and $\xi$-attractors
  9. Conformal attractors
  10. $\alpha$-attractors
  11. T-models
  12. E-models
  13. Special cases: Poincare disks and $SL(2,\mathbb{Z})$ $\alpha$-attractors
  14. ACT and pre-attractor regime
  15. Quintessential $\alpha$-attractors, reheating and ACT, DES, and DESI
  16. ...and 7 more sections

Figures (20)

  • Figure 1: ACT DR6 and Planck PR3 (Planck Collaboration 2020b) combined TT (top), EE (middle), and TE (bottom) power spectra. The gray lines show the joint ACT and Planck ${\rm \Lambda CDM}$ best-fit power spectra.
  • Figure 2: Many favorite string inflation models from a decade ago, such as inflation with an inflection point, Racetrack Inflation, and D3/D7 inflation, were compatible with WMAP (red area) but have been ruled out by Planck 2018 (blue area). The yellow area corresponds to $\alpha$-attractors Kallosh:2013hoaFerrara:2013rsaKallosh:2013yoaGalante:2014ifaKallosh:2015zsaKallosh:2019eeuKallosh:2019hzo, the two dots in the lower part of this area correspond to the Starobinsky model Starobinsky:1980te and the Higgs inflation model Salopek:1988qhBezrukov:2007ep.
  • Figure 3: BICEP/Keck results for $n_{s}$ and $r$BICEP:2021xfz. The $1\sigma$ and $2\sigma$ areas are represented by dark blue and light blue colors. The purple region shows natural inflation, and the orange band corresponds to inflation driven by the scalar field with canonical kinetic terms and monomial potentials.
  • Figure 4: The figure shows the latest constraints on $n_{s}$ and $r$ according to ACT (P-ACT-LB) Louis:2025tst. The dashed line at the bottom corresponds to the Starobinsky model.
  • Figure 5: The potential $V(\phi) = {m^{2}\phi^{2}\over 2}\,\bigl(1-a\phi +a^{2}b\,\phi^{2})\bigr)^{2}$ for $a = 0.1$ and $b = 0.36$ (upper curve), 0.35 (middle) and 0.34 (lower curve). The potential is shown in units of $m$, with $\phi$ in units $M_{p}= 1$.
  • ...and 15 more figures