Non-Gaussianities in Single Field Inflation and their Optimal Limits from the WMAP 5-year Data

TL;DR

This work uses the EFT of inflation to show that single-field inflationary non-Gaussianities occupy a two-dimensional shape space, captured by $f_{NL}^{\rm equil.}$ and $f_{NL}^{\rm orthog.}$, with the orthogonal shape being largely missed by earlier analyses. The authors construct two factorizable templates that span this space across relevant $c_s$ and $\tilde{c}_3$ values and perform an optimal analysis of the WMAP 5-year data, finding no evidence for primordial NG and placing joint constraints on the EFT parameters. They map the bounds on $f_{NL}$ to limits on the inflaton Lagrangian coefficients, notably constraining the speed of sound to be $c_s\ge 0.011$ (95% CL) when allowing cancellations between operators, or tighter if cancellations are not allowed, and ruling out large regions of parameter space with negative $c_s^2$. The results demonstrate how CMB bispectrum measurements probe high-energy inflationary interactions, and they set a framework for Planck-era analyses to sharpen these constraints further with the same EFT language.

Abstract

Using the recently developed effective field theory of inflation, we argue that the size and the shape of the non-Gaussianities generated by single-field inflation are generically well described by two parameters: f_NL^equil, which characterizes the size of the signal that is peaked on equilateral configurations, and f_NL^orthog, which instead characterizes the size of the signal which is peaked both on equilateral configurations and flat-triangle configurations (with opposite signs). The shape of non-Gaussianities associated with f_NL^orthog is orthogonal to the one associated to f_NL^equil, and former analysis have been mostly blind to it. We perform the optimal analysis of the WMAP 5-year data for both of these parameters. We find no evidence of non-Gaussianity, and we have the following constraints: -125 < f_NL^equil < 435, -369 < f_NL^orthog < 71 at 95% CL. We show that both of these constraints can be translated into limits on parameters of the Lagrangian of single-field inflation. For one of them, the speed of sound of the inflaton fluctuations, we find that it is either bounded to be c_s > 0.011 at 95% CL. or alternatively to be so small that the higher-derivative kinetic term dominate at horizon crossing. We are able to put similar constraints on the other operators of the inflaton Lagrangian.

Figures (12)

• Figure 1: The shape of single-field inflation. Top Left:$F_{\dot\pi({\partial}_i\pi)^2}$ (corresponding to $\tilde{c}_3=0$), which is very similar to the template Equilateral shape. Top Right: Orthogonal shape: $\tilde{c}_3=-5.4$. The cosine of this shape with the equilateral shape is approximately zero. Bottom Left: Flat shape: $\tilde{c}_3=-6$. This shape is peaked on flat triangles where the two smallest $k$'s are equal to half the larger one, instead of on equilateral triangles. Bottom Right:$F_{\dot\pi^3}$, which correponds to the case $1\ll |\tilde{c}_3|\lesssim{\cal{O}}(10)$: the contribution on flat triangles is clearly larger than in the case of $F_{\dot\pi({\partial}_i\pi)^2}$.
• Figure 2: Left: Cosine of single-field shape with the equilateral shape as we vary $\tilde{c}_3$ with $c_s\ll1$, the regime in which it is independent of $c_s$. The two horizontal lines represent when the scalar product is equal to $\pm0.7$, to give a rough measure of when the cosine becomes small. Right: Cosine with the local shape.
• Figure 3: The local shape.
• Figure 4: Cosine between $F_{\rm equil.}$ and $\tilde{F}_{\rm orthog.}$ as we let $c$ vary and as we use the 3D and 2D cosine, or as we vary the $l_{\rm max}$ in the definition of the 2D cosine. As expectable, we see that how much the two shapes are similar depends on the survey. Our choice of the template $F_{\rm orthog.}$ corresponds to $c=2/3$, which is quite close to a 2D orthogonal template with repect to $F_{\rm equil.}$ at Planck resolution.
• Figure 5: The cosine between the template shape $F_{\rm template}$ and the exact single-field shape $F$ as we vary $\tilde{c}_3$, for $c_s\ll1$ where it is independent of $c_s$. The cosine is always very close to one, reaching its minimum, equal to approximately 0.91, for $\tilde{c}_3\simeq-5.5$. To help visualization, we plot also the line corresponding to a cosine equal to 0.9.