Instantaneous-reheating horizon matching ($e^N\gtrsim (H_i/H_0)(a_{\mathrm{reh}}/a_0)$)

Start with an inflationary patch of physical radius $H_i^{-1}$ in units $c=1$. After $N$ e-folds and subsequent expansion, its present radius is $H_i^{-1}{e}^N{a}_0/a_{\mathrm{reh}}$. Requiring it to cover the present Hubble radius gives the displayed inequality. With instantaneous conversion to radiation, fixed relativistic degrees of freedom and a fixed ratio $H_i/H_{\mathrm{end}}$, energy conservation gives $H_i\propto T_{\mathrm{reh}}^2$, whereas adiabatic subsequent expansion gives $a_0/a_{\mathrm{reh}}\propto T_{\mathrm{reh}}$. Thus the required $N$ increases approximately as $\log T_{\mathrm{reh}}$. Holding $H_i$ fixed while varying the end energy independently is a different comparison and reverses that simple temperature-only trend. The inflationary energy history and post-inflationary entropy history must be specified.^[ref]