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The Conformal Limit of Inflation in the Era of CMB Polarimetry

Enrico Pajer, Guilherme L. Pimentel, Jaap V. S. Van Wijck

TL;DR

This work shows that the absence of primordial tensor modes in single-field slow-roll inflation implies a new hierarchy $\varepsilon\ll\eta$, defining a conformal limit in which de Sitter isometries strongly constrain primordial fluctuations. By combining conformal symmetry with a soft-scalar consistency condition, the authors determine the full, leading-order power spectrum and bispectrum, predicting a small conformal (non-local) shape of non-Gaussianity tied to the running $\alpha_s$ via $f_{NL}^{\text{conf.}} = -\frac{25}{36}\alpha_s$. They validate these results through in-in calculations in flat/comoving gauges and a Wheeler–DeWitt wave-function approach, and discuss the necessary boundary terms and constraint-solution structure. The findings provide a scalar-sector consistency test for the simplest inflationary models that remains applicable even if tensor modes remain undetected, with potential implications for future CMB polarization and large-scale-structure observations.

Abstract

We argue that the non-detection of primordial tensor modes has taught us a great deal about the primordial universe. In single-field slow-roll inflation, the current upper bound on the tensor-to-scalar ratio, $r < 0.07$ $(95 \% ~CL)$, implies that the Hubble slow-roll parameters obey $\varepsilon \ll η$, and therefore establishes the existence of a new hierarchy. We dub this regime the conformal limit of (slow-roll) inflation, and show that it includes Starobinsky-like inflation as well as all viable single-field models with a sub-Planckian field excursion. In this limit, all primordial correlators are constrained by the full conformal group to leading non-trivial order in slow-roll. This fixes the power spectrum and the full bispectrum, and leads to the "conformal" shape of non-Gaussianity. The size of non-Gaussianity is related to the running of the spectral index by a consistency condition, and therefore it is expected to be small. In passing, we clarify the role of boundary terms in the $ζ$ action, the order to which constraint equations need to be solved, and re-derive our results using the Wheeler-deWitt formalism.

The Conformal Limit of Inflation in the Era of CMB Polarimetry

TL;DR

This work shows that the absence of primordial tensor modes in single-field slow-roll inflation implies a new hierarchy , defining a conformal limit in which de Sitter isometries strongly constrain primordial fluctuations. By combining conformal symmetry with a soft-scalar consistency condition, the authors determine the full, leading-order power spectrum and bispectrum, predicting a small conformal (non-local) shape of non-Gaussianity tied to the running via . They validate these results through in-in calculations in flat/comoving gauges and a Wheeler–DeWitt wave-function approach, and discuss the necessary boundary terms and constraint-solution structure. The findings provide a scalar-sector consistency test for the simplest inflationary models that remains applicable even if tensor modes remain undetected, with potential implications for future CMB polarization and large-scale-structure observations.

Abstract

We argue that the non-detection of primordial tensor modes has taught us a great deal about the primordial universe. In single-field slow-roll inflation, the current upper bound on the tensor-to-scalar ratio, , implies that the Hubble slow-roll parameters obey , and therefore establishes the existence of a new hierarchy. We dub this regime the conformal limit of (slow-roll) inflation, and show that it includes Starobinsky-like inflation as well as all viable single-field models with a sub-Planckian field excursion. In this limit, all primordial correlators are constrained by the full conformal group to leading non-trivial order in slow-roll. This fixes the power spectrum and the full bispectrum, and leads to the "conformal" shape of non-Gaussianity. The size of non-Gaussianity is related to the running of the spectral index by a consistency condition, and therefore it is expected to be small. In passing, we clarify the role of boundary terms in the action, the order to which constraint equations need to be solved, and re-derive our results using the Wheeler-deWitt formalism.

Paper Structure

This paper contains 35 sections, 184 equations, 2 figures.

Table of Contents

  1. Introduction and Summary
  2. Introduction
  3. Summary of Results
  4. The Conformal Limit of Inflation
  5. The Conformal Limit
  6. Starobinsky Inflation as an Explicit Example
  7. Constraint Equations in the Conformal Limit
  8. The Spectra from Symmetries
  9. Power Spectrum
  10. Conformally Covariant Shapes of the Bispectrum
  11. A Soft Inflaton Consistency Relation
  12. The Primordial Bispectrum
  13. Conformal non-Gaussianity
  14. The Bispectrum from the In-In Formalism
  15. Flat Gauge
  16. ...and 20 more sections

Figures (2)

  • Figure 1: We plot of the shape function $S(k_{1}, x_2, x_3)\equiv (x_2 x_3)^2 \langle \zeta(|{\bf k}_1|=1)\zeta(|{\bf k}_2|=x_2)\zeta(|{\bf k}_3|=x_3)\rangle'$ of the conformal shape for the fundamental domain $1-x_1 \leq x_2 \leq x_1 \leq 1$, with $0.5\le x_2\le 1$. We have chosen $k_{1}$ (and $\tau_{\ast})$ in such a way that the squeezed limit is positive (left hand plot) or vanishes (right hand plot). The figures are normalized to have the value 1 for the equilateral configuration $x_2 =x_3=1$.
  • Figure 2: Schematic plot of the current experimental constraints on the tensor to scalar ratio $r$ and the scalar speed of sound $c_s$. As the bound on $r$ and $c_{s}$ (from $f_{NL}^{\rm eq.}$) improves, we will exclude the region where $\varepsilon>\eta$.