  In canonical scalar field inflation, the Starobinsky model (with a linear
potential but discontinuous slope) is remarkable in that though slow-roll is
violated, both the power-spectrum and bi-spectrum can be calculated exactly
analytically. The two-point function is characterised by different power on
large and small scales, and a burst of small amplitude superimposed
oscillations in between. Motivated by string-theory models with stuck branes,
we extend this analysis to Dirac Born Infeld (DBI) inflation, for which
generalised slow-roll is violated at the discontinuity and a rapid variation in
the speed of sound c_s occurs. In an attempt to characterise the effect of
non-linear kinetic terms on the oscillatory features of the primordial
power-spectrum, we show that the resulting power spectrum has a shape and
features which differ significantly from those of the standard Starobinsky
model. In particular, the power-spectrum now takes very similar scale invariant
values on large and small scales, while on intermediate scales it is
characterised by much larger amplitude and higher frequency superimposed
oscillations. We also show that calculating non-Gaussianities in this model is
a complicated but interesting task since all terms in the cubic action now
contribute. Investigating whether the superimposed oscillations could fit to
the Planck Cosmic Microwave Background (CMB) data (for instance by explaining
the large scale Planck anomalies) with, at the same time, small
non-Gaussianities remains an intriguing and open possibility.
