The curvature perturbation generated by thermal fluctuations during thermal inflation

During thermal inflation, the temperature determines the number of e-folds of expansion of the universe and so thermal fluctuations are magnified into curvature perturbations. We use classical thermodynamics to calculate the subhorizon thermal fluctuations and trace their evolution into superhorizon temperature perturbations. We convert the temperature perturbations into curvature perturbations using the $\delta N$-formalism, or equivalently the junction condition of curvature perturbations at the end of thermal inflation, denoted by subscript c, and show that the late-time power spectrum is $P_\mathcal{R} = \frac{15}{4\pi^4} \frac{H^3_\mathrm{c}}{g_* T^3_\mathrm{c}} \frac{k^3}{k^3_\mathrm{c}}$.