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 $ϵ$, 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 $m_s{ϵ}_2/H\sim O(-0.01)$, where ${ϵ}_2=\dot{ϵ}/\left(ϵH\right)$ 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 ${\eta }_{\perp }\lesssim 11$ where ${\eta }_{\perp }$ is the turning rate in field-space.