What Shape is the Inflationary Bispectrum?

Abstract

Non-linear interactions during inflation generate non-Gaussianities in the distribution of primordial curvature. In many theories, the physics is scale-invariant, such that the induced three-point function depends solely on a dimensionless shape function $S(x,y)\sim k^6B_ζ(kx,ky,k)$. To confront such models with observations, one typically builds specialized estimators for each shape, then applies them to cosmic microwave background datasets at significant computational expense. In this Letter, we take a different approach, directly reconstructing $S(x,y)$ from observations using an efficient logarithmically-binned estimator in primordial-space (motivated by the modal program). Applying this to temperature and polarization maps from Planck, we obtain high-resolution shape measurements across the full $(x,y)$-plane, including squeezed limits. Our approach is close-to-optimal, highly interpretable, and preserves the information content on (optimally-analyzed) standard templates within $\approx 10\%$; moreover, we can use it to assess the scale-dependence of our constraints, finding that Planck is sensitive to $\approx 6$ $e$-folds of non-Gaussian evolution with a peak sensitivity around $0.1h\,\mathrm{Mpc}^{-1}$. Since we work directly in shape-space, data and theory can be compared in milliseconds. As an example, we perform a search for massive particle exchange using a suite of over $20\,000$ theoretical templates computed with exact bootstrap methods (for the first time) across a wide range of masses, spins, and sound-speeds; the spin-two analysis yields a maximum significance of $2.6σ$. Our approach can be used to probe a wide range of scale-invariant models in orders-of-magnitude less time than with direct estimators, allowing the inflationary paradigm to be explored in new ways.

Figures (7)

• Figure 1: Two-dimensional shape functions, $S^{\rm 2D}(x,y)$, reconstructed from Planck PR4 temperature and polarization anisotropies. The top panel shows the signal-to-noise ratio in each $x,y$-bin (imposing the triangle conditions, with $x\leq y\leq 1$), whilst the bottom shows the empirical error, obtained from $400$ FFP10 simulations and normalized such that $S^{\rm 2D}(1,1)=f_{\rm NL}$. The constraints have similar magnitude across most of the $(x,y)$-plane, but are significantly degraded in the squeezed limit Visually, we find no evidence for a primordial signal with a $\chi^2$ of $175.9$ for $171$ degrees of freedom. By comparing these measurements with theoretical shape functions, we can immediately place constraints on a wealth of scale-invariant primordial models; an example is shown in Fig. \ref{['fig: bootstrap-constraints']}.
• Figure 2: Squeezed limit of the primordial shape function reconstructed from Planck temperature and polarization data, as a function of $x$ ($=k_{\rm long}/k_{\rm short}$). Shaded regions and errorbars show the theoretical and empirical errors from the inverse Fisher matrix and FFP10 simulations respectively, and we omit configurations with $x<0.005$, which cannot be meaningfully constrained using Planck data. For reference, we show the scalings of three fiducial models (with arbitrary normalization): local (orange), equilateral (green), and the weakly-mixed cosmological collider (red; with $m\approx 5H$). Notably, the shape noise scales as $1/x$; this makes it difficult to probe the collider oscillations ($\propto x^{1/2}$) in practice.
• Figure 3: Constraints on the exchange of massive particles in inflation. We show results for four choices of interaction (induced by the cubic vertices shown in the titles), as a function of mass parameter $\mu\equiv \sqrt{m^2_\sigma/H^2-9/4}$ (or $\sqrt{m^2_\sigma/H^2-(s-1/2)^2}$ for spin $s>0$), relative sound-speed $c_s=c_\pi/c_\sigma$, and spin (zero, zero, one, and two respectively). For higher-spin exchange, we restrict to $\mathrm{Im}[\mu]\leq 1/2$ according to the Higuchi bound Higuchi:1986py. The theoretical models are computed numerically using bootstrap techniques (in contrast to most previous works, which used simplified forms, often focussing on the non-oscillatory contribution). The top panel shows the signal-to-noise ratio, whilst the bottom gives the $1\sigma$ error on the $f_{\rm NL}$ amplitude. We report no detection of collider non-Gaussianity with a maximum detection significance of $2.6\sigma$ (for spin-two with $\mu \approx 3, c_s\approx 5$), and place strong constraints on the models across a broad range of masses and spins. Given the shape function measurements from Fig. \ref{['fig: shape-functions-2d']}, computation of these constraints required just $0.6$ seconds.
• Figure S1: Comparison of the empirical (left) and theoretical (right) $1\sigma$ errors on the two-dimensional bispectrum shape. The former are obtained using $400$ FFP10/npipe simulations, whilst the latter are computed from the inverse of the (mask- and beam-dependent) Fisher normalization matrix. We find excellent agreement across the full two-dimensional plane, indicating that our measurements are close-to-optimal. A comparison of the errors in the squeezed-limit is shown in Fig. \ref{['fig: error-comparison-squeezed']}.
• Figure S2: As Fig. \ref{['fig: error-comparison']} but restricting to isosceles triangles with $y=1$. We find excellent agreement between empirical (red) and theoretical (black) errorbars across three orders-of-magnitude in $x$, validating our implementation. We also show the errorbars obtained when dropping the linear term in the estimator (blue, \ref{['eq: 3d-estimator']}); as expected, this term significantly significantly reduces the variance for highly squeezed configurations. We observe that configurations with squeezing ratios beyond $x\approx 0.002$ cannot be meaningfully constrained with Planck data.