Large non-Gaussianity in multiple-field inflation

TL;DR

This work develops a fully non-linear, long-wavelength formalism with stochastic sources to compute the curvature perturbation three-point function in multifield inflation. It derives a general integral expression for the bispectrum, valid without assuming slow-roll, and specializes to a tractable two-field slow-roll case where the isocurvature perturbations can source large non-Gaussianity during inflation. The two-field analysis yields a potentially observable signal with $\langle \tilde{\zeta}^1 \tilde{\zeta}^1 \tilde{\zeta}^1 \rangle / P_{\zeta}^2$ reaching $\mathcal{O}(1)$–$\mathcal{O}(10)$ and a distinct momentum dependence, including a strong squeezed-limit enhancement. These results imply that Planck (and future CMB experiments) could test a broad class of multifield inflation models by measuring the shape and amplitude of non-Gaussianity arising from superhorizon isocurvature effects during inflation.

Abstract

We investigate non-Gaussianity in general multiple-field inflation using the formalism we developed in earlier papers. We use a perturbative expansion of the non-linear equations to calculate the three-point correlator of the curvature perturbation analytically. We derive a general expression that involves only a time integral over background and linear perturbation quantities. We work out this expression explicitly for the two-field slow-roll case, and find that non-Gaussianity can be orders of magnitude larger than in the single-field case. In particular, the bispectrum divided by the square of the power spectrum can easily be of O(1-10), depending on the model. Our result also shows the explicit momentum dependence of the bispectrum. This conclusion of large non-Gaussianity is confirmed in a semi-analytic slow-roll investigation of a simple quadratic two-field model.

Figures (4)

• Figure 1: The integrals $I_1(p,q,\chi,\Delta t_*)$ (small dashes), $I_2(p,q,\chi,\Delta t_*)$ (solid), $I_3(p,q,\chi,\Delta t_*)$ (large dashes), and $I_4(p,q,\chi,\Delta t_*)$ (dots) plotted as a function of $q/p$ for $\Delta t_* = 50$, both for $p=1$ (blue, darker) and for $p=\exp(10)$ (green, lighter) (i.e. 50 and 40 e-folds after horizon crossing of the mode $k$). The smoothing parameter $c=3$, although the dependence on $c$ is negligible. The different figures correspond with different values of $\chi$, as indicated. Note that $I_2$ and $I_4$ almost coincide in the first plot.
• Figure 2: (a) The integrals $J_1(p,q,\Delta t_*)$ (small dashes) and $J_2(p,q,\chi,\Delta t_*)$ for $\chi=0.05$ (solid) and $\chi=0.2$ (large dashes) plotted as a function of $q/p$, a few e-folds after horizon crossing of the mode $k$ when the dependence on $\Delta t_k = \Delta t_* - \ln p$ has become negligible. (b) The integrals $J_3(p,q,\chi,\Delta t_*)$ (solid) and $J_4(p,q,\chi,\Delta t_*)$ (small dashes) plotted as a function of $q/p$ for $p=1$ and $\Delta t_* = 50$, for both $\chi=0.01$ (blue, darker) and $\chi=0.05$ (green, lighter).
• Figure 3: The bispectrum (\ref{['3pcvertex']}) divided by the square of the power spectrum (\ref{['powerspec2f']}) (with two different momenta) in the limit where one of the momenta is much smaller than the other two, plotted as a function of ${\tilde{\eta}}^\perp$ and $\chi$ for $\Delta t = 50$, ${\tilde{\epsilon}}=-{\tilde{\eta}}^\parallel=0.05$, and $(\sqrt{2{\tilde{\epsilon}}}/\kappa)V_{111}/(3 H^2)= -{\tilde{\xi}}^\perp=-(\sqrt{2{\tilde{\epsilon}}}/\kappa)V_{221}/(3 H^2)=0.003$.
• Figure 4: (a) The bispectrum (\ref{['3pcorr']}), (\ref{['three point']}) without the overall $(2\pi)^3 \delta^3(\sum_s {\textbf{\em k}}_s)$ factor and divided by the square of the power spectrum (\ref{['powerspec2f']}) as defined in (\ref{['deffNL']}), plotted as a function of the relative size of the three momenta. The sum of the momenta is chosen as $(k_1+k_2+k_3) = 3 k_*$ with $k_*$ fixed by choosing $\Delta t_* = 50$. The values for the parameters are ${\tilde{\epsilon}} = -{\tilde{\eta}}^\parallel = 0.05$, ${\tilde{\eta}}^\perp = 0.2$, $\chi = 0.01$, $(\sqrt{2{\tilde{\epsilon}}}/\kappa)V_{111}/(3 H^2) = -{\tilde{\xi}}^\perp = -(\sqrt{2{\tilde{\epsilon}}}/\kappa)V_{221}/(3 H^2) = (\sqrt{2{\tilde{\epsilon}}}/\kappa)V_{222}/(3 H^2) = 0.003$ (as well as $c=3$, although the dependence on $c$ is negligible). (b) An explanation of the triangular domain used, defined in (\ref{['plotvars']}), with $k\equiv k_1+k_2+k_3$.