Modeling Cosmological Perturbations of Thermal Inflation
  We consider a simple system consisting of matter, radiation and vacuum
components to model the impact of thermal inflation on the evolution of
primordial perturbations. The vacuum energy magnifies the modes entering the
horizon before its domination, making them potentially observable, and the
resulting transfer function reflects the phase changes and energy contents. To
determine the transfer function, we follow the curvature perturbation from well
outside the horizon during radiation domination to well outside the horizon
during vacuum domination and evaluate it on a constant radiation density
hypersurface, as is appropriate for the case of thermal inflation. The shape of
the transfer function is determined by the ratio of vacuum energy to radiation
at matter-radiation equality, which we denote by $\upsilon$, and has two
characteristic scales, $k_{\rm a}$ and $k_{\rm b}$, corresponding to the
horizon sizes at matter radiation equality and the beginning of the inflation,
respectively. If $\upsilon \ll 1$, the universe experiences radiation, matter
and vacuum domination eras and the transfer function is flat for $k \ll k_{\rm
b}$, oscillates with amplitude $1/5$ for $ k_{\rm b} \ll k \ll k_{\rm a}$ and
oscillates with amplitude $1$ for $k \gg k_{\rm a}$. For $\upsilon \gg 1$, the
matter domination era disappears, and the transfer function reduces to being
flat for $k \ll k_{\rm b}$ and oscillating with amplitude $1$ for $k \gg k_{\rm
b}$.
