The shape of non-Gaussianities

TL;DR

This paper analyzes how the primordial 3-point function’s dependence on triangle configuration in momentum space varies across inflationary scenarios, revealing two broadly orthogonal shape classes: equilateral-type shapes from horizon-crossing dynamics and local shapes from outside-horizon evolution. By defining a cosine-based metric and an optimal estimator, the authors quantify how distinguishable these shapes are in 3D and after projection to the CMB (2D). They show that equilateral-type models (e.g., ghost inflation, DBI) peak for equilateral triangles, while local-type models peak in the collapsed limit, and they demonstrate that translating constraints between shapes using equilateral normalization can significantly misstate the true bounds. The work underscores the importance of shape-specific analyses to extract meaningful information about early-universe physics from non-Gaussianity measurements. Overall, it provides a framework for comparing and constraining diverse non-Gaussian templates in both 3D surveys and 2D CMB data.

Abstract

We study the dependence on configuration in momentum space of the primordial 3-point function of density perturbations in several different scenarios: standard slow-roll inflation, curvaton and variable decay models, ghost inflation, models with higher derivative operators and the DBI model of inflation. We define a cosine between the distributions using a measure based on the ability of experiments to distinguish between them. We find that models fall into two broad categories with fairly orthogonal distributions. Models where non-Gaussianity is created at horizon-crossing during inflation and models in which the evolution outside the horizon dominates. In the first case the 3-point function is largest for equilateral triangles, while in the second the dominant contribution to the signal comes from the influence of long wavelength modes on small wavelength ones. We show that, because the distributions in these two cases are so different, translating constraints on parameters of one model to those of another based on the normalization of the 3-point function for equilateral triangles can be very misleading.

Figures (5)

• Figure 1: Plot of the function $F(1,x_2,x_3) \; x_2^2 x_3^2$ for the local distribution (\ref{['eq:FFNL']}). The figure is normalized to have value 1 for equilateral configurations $x_2 = x_3 =1$ and set to zero outside the region $1-x_2 \leq x_3 \leq x_2$.
• Figure 2: Plot of the function $F(1,x_2,x_3) \; x_2^2 x_3^2$ for the usual slow-roll inflation (\ref{['eq:malda']}) with $\epsilon = \eta = 1/30$. The figure is normalized to have value 1 for equilateral configurations $x_2 = x_3 =1$ and set to zero outside the region $1-x_2 \leq x_3 \leq x_2$.
• Figure 3: Plot of the function $F(1,x_2,x_3) \; x_2^2 x_3^2$ for non-Gaussianities generated by higher derivative interactions (\ref{['eq:myresult']}) and in the DBI model of inflation Silverstein:2003hfAlishahiha:2004eh. The figure is normalized to have value 1 for equilateral configurations $x_2 = x_3 =1$ and set to zero outside the region $1-x_2 \leq x_3 \leq x_2$.
• Figure 4: Plot of the function $F(1,x_2,x_3) \; x_2^2 x_3^2$ for ghost inflation (\ref{['eq:ghost']}). The figure is normalized to have value 1 for equilateral configurations $x_2 = x_3 =1$ and set to zero outside the region $1-x_2 \leq x_3 \leq x_2$.
• Figure 5: Shape dependence of the density 3-point function for $k=0.1\ h\ {\rm Mpc}^{-1}$. The signal is largest for collinear triangles, when the three wavevectors are parallel. These fluctuations are not scale invariant so both the amplitude and details of the shape are functions of the scale. We present results for a particular wavelength for illustrative purposes.