  In cosmological scenarios with thermal inflation, extra eras of moduli matter
domination, thermal inflation and flaton matter domination exist between
primordial inflation and the radiation domination of Big Bang nucleosynthesis.
During these eras, cosmological perturbations on small scales can enter and
re-exit the horizon, modifying the power spectrum on those scales. The largest
modified scale, $k_\mathrm{b}$, touches the horizon size when the expansion
changes from deflation to inflation at the transition from moduli domination to
thermal inflation. We analytically calculate the evolution of perturbations
from moduli domination through thermal inflation and evaluate the curvature
perturbation on the constant radiation density hypersurface at the end of
thermal inflation to determine the late time curvature perturbation. Our
resulting transfer function suppresses the power spectrum by a factor $\sim 50$
at $k \gg k_\mathrm{b}$, with $k_\mathrm{b}$ corresponding to anywhere from
megaparsec to subparsec scales depending on the parameters of thermal
inflation. Thus, thermal inflation might be constrained or detected by small
scale observations such as CMB distortions or 21cm hydrogen line observations.
