Sharp inflaton potentials and bi-spectra: Effects of smoothening the
  discontinuity
  Sharp shapes in the inflaton potentials often lead to short departures from
slow roll which, in turn, result in deviations from scale invariance in the
scalar power spectrum. Typically, in such situations, the scalar power spectrum
exhibits a burst of features associated with modes that leave the Hubble radius
either immediately before or during the epoch of fast roll. Moreover, one also
finds that the power spectrum turns scale invariant at smaller scales
corresponding to modes that leave the Hubble radius at later stages, when slow
roll has been restored. In other words, the imprints of brief departures from
slow roll, arising out of sharp shapes in the inflaton potential, are usually
of a finite width in the scalar power spectrum. Intuitively, one may imagine
that the scalar bi-spectrum too may exhibit a similar behavior, i.e. a
restoration of scale invariance at small scales, when slow roll has been
reestablished. However, in the case of the Starobinsky model (viz. the model
described by a linear inflaton potential with a sudden change in its slope)
involving the canonical scalar field, it has been found that, a rather sharp,
though short, departure from slow roll can leave a lasting and significant
imprint on the bi-spectrum. The bi-spectrum in this case is found to grow
linearly with the wavenumber at small scales, a behavior which is clearly
unphysical. In this work, we study the effects of smoothening the discontinuity
in the Starobinsky model on the scalar bi-spectrum. Focusing on the equilateral
limit, we analytically show that, for smoother potentials, the bi-spectrum
indeed turns scale invariant at suitably large wavenumbers. We also confirm the
analytical results numerically using our newly developed code BINGO. We
conclude with a few comments on certain related points.
