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.
