Generation and Characterization of Large Non-Gaussianities in Single Field Inflation

TL;DR

This paper addresses how large primordial non-Gaussianities can arise in single-field inflation by identifying two distinct mechanisms: horizon-scale effects from localized slow-roll violations (features in the potential or brief inflation) and sub-horizon resonance from small-scale structure in the potential. It develops factorable analytic approximations for the primordial 3-point function and introduces a boundary-regulated numerical scheme to evaluate the Maldacena integrals efficiently and accurately. The horizon-scale mechanism yields scale-dependent, oscillatory signatures tied to the feature scale, while the resonance mechanism produces a log-periodic, scale-spanning NG signal with amplitude controlled by the modulation parameters. Collectively, the work provides a practical computational framework and physical intuition for detecting or constraining such signals in CMB data, and points to future extensions to multi-field scenarios and refined estimators.

Abstract

Inflation driven by a single, minimally coupled, slowly rolling field generically yields a negligible primordial non-Gaussianity. We discuss two distinct mechanisms by which a non-trivial potential can generate large non-Gaussianities. Firstly, if the inflaton traverses a feature in the potential, or if the inflationary phase is short enough so that initial transient contributions to the background dynamics have not been erased, modes near horizon-crossing can acquire significant non-Gaussianities. Secondly, potentials with small-scale structure may induce significant non-Gaussianities while the relevant modes are deep inside the horizon. The first case includes the "step" potential we previously analyzed while the second "resonance" case is novel. We derive analytic approximations for the 3-point terms generated by both mechanisms written as products of functions of the three individual momenta, permitting the use of efficient analysis algorithms. Finally, we present a significantly improved approach to regularizing and numerically evaluating the integrals that contribute to the 3-point function.

Figures (8)

• Figure 1: This plot compares the 3-point correlation function, computed using the $\beta$ described in Chen:2006xj and the boundary regulator described in this section, in the case of the step model [Sec. \ref{['subsect:sharpfeatures']} with model parameters $(c,d,\phi_s) = (0.0018,0.022M_p,14.84M_p)$]. We plot the dimensionless variable $G/k^3$ defined in Eq. (\ref{['eqn:g']}) for the equilateral case. The solid and the dashed lines are results obtained $\beta=0.01$ and $\beta=0.005$ respectively, while the dotted line is the result obtained from using the boundary regulator. The value of $\beta$ is chosen such that it gives optimal results at around $k=1$; if $\beta$ is too small the early time oscillation will not be suppressed while if $\beta$ is too large it will over-suppress the high $k$ values. The boundary regulator does not suffer from this arbitrariness, and matches the limit found when $\beta\rightarrow0$.
• Figure 2: The evolution of $\epsilon\times \eta'$ with units $k_*\tau = -1.2$ where $k_*=1$ is set to be the scale when $\phi$ crosses the center of the feature for our two models . The step ($c=0.0018, d=0.022M_p, \phi_s = 14.84M_p$) is the solid line, and bump is the dashed line $(c=0.0005, d=0.01M_p, \phi_b = 14.84M_p)$. In both cases $\eta'$ is boosted by ${\cal O}(1000)$ boost for around one Hubble time ($\delta \tau \approx 1$ in our units).
• Figure 3: A sample result of the step (left) and bump (right) models, where for both plots $k_3=9$. We numerically computed the 3-point correlation functions of both models for $1 < k < 9$ such that $k=1/1.2$ corresponds to the scale of the feature at $\phi_s = 14.84M_p$. For the step model, we use $(c=0.0018, d=0.022)$ while for the bump model we use $(c=0.0005, d=0.01)$.
• Figure 4: A comparison of the ansatz Eq. (\ref{['SFansatz']}) (full line) to our numerical results (dashed line) for the step model $(c=0.0018,d=0.022)$ on the left and the bump model $(c=0.0005,d=0.01)$ on the right with $k_*\approx 1/1.2$. This is a reasonable match to our analytical from. As explained in the text, the drop-off at small and large $k$ is not captured by the ansatz but can be easily incorporated by adding in a window function when comparing the ansatzen with data. We have added a phase factor into the ansatz to synchronize with the numerical results; this phase factor is physically important but is not estimated analytically.
• Figure 5: This figure illustrates resonance between the total mode $K=k_1+k_2+k_3$ momentum and an oscillatory $\eta'$. The solid line shows $\zeta(k)$, the dashed line describes $\eta'$, and the dotted line is the integral Eq. (\ref{['eqn:Itype1']}) from $-\infty$ to time $\tau$ with $k_1=k_2=k_3=k$. Resonance occurs when the frequency of $\eta'$ is roughly $3k$. This figure was generated after numerically evaluating the mode functions for the parameters of Eq. (\ref{['Nparameters']}), with all the lines rescaled to arbitrary units to emphasise the effect. The universe grows by roughly one e-fold over the range of this plot, and the relevant modes are well inside the horizon.