Expansion history-dependent oscillations in the scalar-induced
  gravitational wave background
  Oscillations in the frequency profile of the scalar-induced contribution to
the stochastic gravitational wave background are a characteristic signal for
small-scale features during inflation. We investigate how this oscillatory
frequency profile is affected by the expansion history of the post-inflationary
universe. Our results are applicable as long as the equation of state of the
universe can be taken as constant during the period in which the gravitational
waves are produced, and we compute the spectrum of gravitational waves induced
by both sharp and resonant features, associated with oscillations in $k$ and
$\log(k)$, respectively. For resonant features, the frequencies of the
oscillatory contributions to the gravitational wave spectrum are unaffected by
the equation of state, but not their relative amplitudes, allowing one to
extract information about both inflationary physics and the post-inflationary
expansion history from the oscillatory pattern. For sharp features we find that
the gravitational wave spectrum only exhibits prominent modulations as long as
the propagation speed of density fluctuations is $c_s<1$, with a frequency
larger by a factor $c_s^{-1}$ than that of the scalar power spectrum. We find
that the stiffer the equation of state, the larger the relative amplitude of
the oscillations. In particular, a relative amplitude significantly higher than
$20 \%$ is not achievable for the 'standard' case of radiation domination, and
would be a smoking-gun signal of both nontrivial inflationary dynamics on small
scales, and a post-inflationary universe not dominated by radiation.
