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Inflation with moderately sharp features in the speed of sound: GSR and in-in formalism for power spectrum and bispectrum

We continue the study of mild transient reductions in the speed of sound of the adiabatic mode during inflation, of their effect on the primordial power spectrum and bispectrum, and of their detectability in the Cosmic Microwave Background (CMB). We focus on the regime of \emph{moderately sharp} mild reductions in the speed of sound during uninterrupted slow-roll inflation, a theoretically well motivated and self-consistent regime that admits an effective single-field description. The signatures on the power spectrum and bispectrum were previously computed using a slow-roll Fourier transform (SRFT) approximation, and here we compare it with generalized slow-roll (GSR) and in-in methods, for which we derive new formulas that account for moderately sharp features. The agreement between them is excellent, and also with the power spectrum obtained from the numerical solution to the equation of motion. We show that, in this regime, the SRFT approximation correctly captures with simplicity the effect of higher derivatives of the speed of sound in the mode equation, and makes manifest the correlations between power spectrum and bispectrum features. In a previous paper we reported hints of these correlations in the Planck data and here we perform several consistency checks and further analyses of the best fits, such as polarization and local significance at different angular scales. For the data analysis, we show the excellent agreement between the CLASS and CAMB Boltzmann codes. Our results confirm that the theoretical framework is consistent, and they suggest that the predicted correlations are robust enough to be searched for in CMB and Large Scale Structure (LSS) surveys.