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

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