Mild Non-Gaussianities under Perturbative Control from Rapid-Turn
  Inflation Models
  Inflation can be supported in very steep potentials if it is generated by
rapidly turning fields, which can be natural in negatively curved field spaces.
The curvature perturbation, $\zeta$, of these models undergoes an exponential,
transient amplification around the time of horizon crossing, but can still be
compatible with observations at the level of the power spectrum. However, a
recent analysis (based on a proposed single-field effective theory with an
imaginary speed of sound) found that the trispectrum and other higher-order,
non-Gaussian correlators also undergo similar exponential enhancements. This
arguably leads to `hyper-large' non-Gaussianities in stark conflict with
observations, and even to the loss of perturbative control of the calculations.
In this paper, we provide the first analytic solution of the growth of the
perturbations in two-field rapid-turn models, and find it in good agreement
with previous numerical and single-field EFT estimates. We also show that the
nested structure of commutators of the in-in formalism has subtle and crucial
consequences: accounting for these commutators, we show analytically that the
naively leading-order piece (which indeed is exponentially large) cancels
exactly in all relevant correlators. The remaining non-Gaussianities of these
models are modest, and there is no problem with perturbative control from the
exponential enhancement of $\zeta$. Thus, rapid-turn inflation with negatively
curved field spaces remains a viable and interesting class of candidate
theories of the early universe.
