Primordial Non-Gaussianity from Light Compact Scalars

TL;DR

This work analyzes how light compact scalars (axions) generate primordial non-Gaussianity during inflation, showing that the gauge symmetry associated with the compact field must be incorporated via gauge-invariant vertex operators. Using the EFT of inflation and a spectral (Källén-Lehmann) framework, the authors contrast the squeezed-limit bispectrum from a non-compact scalar with that from a compact scalar, finding a distinctive scaling B_ζ ∝ P_ζ P_ζ [(k_3/k_1)^β + (k_3/k_1)^2] where β = (H/(2π f))^2/2. They derive the amplitude f_NL(β) ∝ β cot(πβ/2) for β<2 and show that for β>2 the leading behavior becomes local-like, while integer β values introduce pole-overlap divergences. The results connect to late-time observables, notably galaxy clustering, enabling potential measurement of the decay constant f and offering a framework to discriminate compact-scalar scenarios from other sources of non-Gaussianity, with extensions to multiple axions and UV-complete models discussed. Overall, the paper establishes a robust method to extract the imprint of light compact scalars on the squeezed bispectrum and its observational consequences in the cosmos.

Abstract

We study the non-Gaussianities generated by light axions, or compact scalar fields, during inflation. To correctly calculate their impact on primordial statistics, we will argue that it is necessary to account for the periodicity, or gauge symmetry, of the compact scalars. We illustrate this point by comparing the predictions for the squeezed kinematic limit of the primordial bispectrum generated by two cases: a non-compact scalar $σ$ and a compact scalar $\varphi$. We demonstrate that while a light non-compact scalar predicts a bispectrum of the so-called local shape, the light compact scalar predicts a qualitatively different shape characterised by the ratio of the Hubble scale to its field-space circumference. In doing so, we show that ignoring the gauge symmetry of the compact scalar during inflation leads to spurious infrared enhancements which are softened by working with appropriate gauge-invariant operators. In addition, we connect our results for the primordial bispectrum with late-time cosmological observables and show that it is possible to measure the decay constant of the compact scalar using galaxy clustering measurements.

Figures (9)

• Figure 1: We show a contour plot illustrating the Watson-Sommerfeld integral (\ref{['eq:watson_somm']}) over the contour $\gamma$ on the left. On the right we show the result of pushing this contour onto the principal series line and thereby obtaining the Källén-Lehmann representation. The blue points represent potential simple poles of $[G_\mathcal{O}]_{{\hbox{[}1.0]{$-$}} \Delta}$ that we may encounter as we deform the contour, the residues of which can be interpreted as complementary state contributions to the spectral representation.
• Figure 2: The contour deformation to resolve the spectral integral of $\hat{b}_{\rm NA}(\Delta;u)$ in terms of a sum over residues. The blue points schematically indicate the poles of the spectral density $\Delta_*$ and the red points the poles of the tree-level seed function $\hat{b}_{\rm NA}(\Delta;u)$ at $\Delta=2,4,\cdots$.
• Figure 3: We compare the result of the numerical integral (\ref{['eq:comp_seed_na']}) for the operator $\sigma^2$ with the corresponding analytical result for scaling dimensions $\Delta_\sigma=0.2$ and $\Delta_\sigma=1.3$. We sum up to $\textcolor{blue}{N=10}$ and $\textcolor{Red}{N=40}$ residues for the solid lines and show the numerical integral in black dots.
• Figure 4: The same as Figure \ref{['fig:sig2_num_analyt_comparison']} but for the compact scalar. We sum up to $\textcolor{blue}{N=10}$ and $\textcolor{Red}{N=40}$ residues for the solid lines and show the numerical integral in black dots. On the left we show the comparison for $\beta=1.8$ and on the right for $\beta=3.4$. We see that as we increase $\beta$, we need to sum over more residues to match the exact numerical answer.
• Figure 5: Contour deformation to relate the momentum coefficients with the EFT coefficients $\mathcal{C}_n^{\mathcal{O}}$. Sending $-J \to n+2$ pinches the contour twice, picking up two residues along the way.