Large Non-Gaussianities in Single Field Inflation

TL;DR

This paper develops a numerical method to compute the 3-point (bispectrum) non-Gaussianity for general single-field inflation without relying on slow-roll, enabling analysis of models with brief features in the potential. It shows that a localized step in the inflaton potential can dramatically amplify non-Gaussianities and imprint a distinctive, scale- and shape-dependent bispectrum that correlates with the 2-point function. Using a step inspired by CMB glitches, the authors demonstrate that the resulting non-Gaussian signal could be within Planck's reach, providing a powerful consistency check on inflationary potentials with features. The work establishes a framework for linking power-spectrum features and higher-order statistics, and it lays the groundwork for extending to multifield and higher-point correlators.

Abstract

We compute the 3-point correlation function for a general model of inflation driven by a single, minimally coupled scalar field. Our approach is based on the numerical evaluation of both the perturbation equations and the integrals which contribute to the 3-point function. Consequently, we can analyze models where the potential has a "feature", in the vicinity of which the slow roll parameters may take on large, transient values. This introduces both scale and shape dependent non-Gaussianities into the primordial perturbations. As an example of our methodology, we examine the ``step'' potentials which have been invoked to improve the fit to the glitch in the $<TT>$ $C_l$ for $l \sim 30$, present in both the one and three year WMAP data sets. We show that for the typical parameter values, the non-Gaussianities associated with the step are far larger than those in standard slow roll inflation, and may even be within reach of a next generation CMB experiment such as Planck. More generally, we use this example to explain that while adding features to potential can improve the fit to the 2-point function, these are generically associated with a greatly enhanced signal at the 3-point level. Moreover, this 3-point signal will have a very nontrivial shape and scale dependence, which is correlated with the form of the 2-point function, and may thus lead to a consistency check on the models of inflation with non-smooth potentials.

Figures (7)

• Figure 1: The $\eta'$ (left) and $\epsilon$ (right) evolution over the step for the model $c=0.0018,~d=0.022$, with its amplitude momentarily $\eta'\approx 10c^2/(\epsilon d^2)$. This is the primary source of the large non-Gaussianities in the step potential, as the leading term in the 3 point expansion is of order $\eta' \epsilon$. On the other hand, since the height of the step is small, so the ratio of the kinetic energy to the potential energy remains tiny and $\epsilon \ll 1$.
• Figure 2: Comparison of ${\cal A}/k^3$ between the analytical slow roll results of equation (\ref{['eqn:ca_sr']}) and numerical results from our code, for the equilateral case where $k_1=k_2=k_3=k$ run from $0.5<k<6.5$ and $\beta = 0.05$. The plot shows the discrepancy between the two sets of values. The rapid oscillation at small $k$ indicates the breakdown of the current value of $\beta$ to suppress the early time oscillations. Since the $k$ space spans a few efolds, we have included the contribution from the $\epsilon$ running when computing the analytic estimate but ignored the ${\cal O}(\epsilon^2)$ terms. As we can see, the two results agree to within a few percent.
• Figure 3: The power spectrum for the standard $m^2\phi^2$ model and the step potential (\ref{['eqn:potential']}) for a decade of $k$. In this plot, $m=10^{-6}M_p$ for the $m^2\phi^2$ model while for the step potential we have used the parameters $(c,\phi_s,d) = (0.0018,14.81 M_p,0.022)$.
• Figure 4: The running of non-Gaussianity ${\cal G}/k_1 k_2 k_3$ (in the equilateral case $k_1=k_2=k_3\equiv k$) for the large scale step potential model of $(c,\phi_s,d)=(0.0018,14.81M_p,0.022)$. For comparison, the standard slow roll model will yield ${\cal G}/k_1 k_2 k_3 \approx {\cal O}(\epsilon)$.
• Figure 5: The shape of non-Gaussianities ${\cal G}/k_1 k_2 k_3$ for the large scale step potential model of $(c,\phi_s,d)=(0.0018,14.81M_p,0.022)$, with $k_1$ ranging from $6.5$ to $2.5$ going from top left to bottom right. We have set the forbidden triangle regions outside of $k_i\leq k_j+k_l~,~i\neq j,l$ to $-10$ for visualization purposes.