Non-Gaussianities in models of inflation with large and negative
  entropic masses
  Models of inflation where the entropic directions have large and negative
masses $|m_s| \gg H$ can be well described by a single-field EFT with an
imaginary sound speed $c_s$. Among other features, they predict an exponential
enhancement of the spectrum of scalar perturbations which however is not
inherited by non-Gaussianities. In this work, I complete the calculation of the
trispectrum in this EFT by considering the contributions from the contact
interaction and the exchange diagram. While for most shapes the trispectrum is
approximately constant, I find that for certain configurations where all the
momenta collapse to a line the trispectrum is proportional to $(m_s/H)^5$ for
the contact interaction and to $(m_s/H)^6$ for the exchange diagram, as
anticipated by previous work. I also discuss the UV sensitivity of the results
and argue why the EFT provides a good order of magnitude estimate. In the end,
I confront the different predictions for the scalar spectrum against
observations. In models where the entropic mass is proportional to a positive
power of the slow-roll parameter $\epsilon$, like in hyperinflation, the
spectrum grows on small scales and becomes constrained by the overproduction of
primordial black holes. Imposing such constraint jointly with the correct
amplitude and spectral tilt at CMB scales excludes a large set of potentials.
Only those where the spectral tilt is controlled by $m_s \delta/H \sim {\cal
O}(-0.01)$, where $\delta=\dot{\epsilon}/(\epsilon H)$ is the second slow-roll
parameter, are likely observationally viable. Finally, the constraints on the
bispectrum generically impose $|c_s m_s|/H \lesssim 10-20$ while those on the
trispectrum give a weaker bound when using the constraints on
$g_\text{NL}^{\dot{\sigma}^4}$ as a proxy. For hyperinflation the bispectrum
bound translates into $\omega \lesssim 11$ where $\omega$ is the turning rate
in field-space.
