= Instantaneous-reheating horizon matching
{title2=$e^N\gtrsim(H_i/H_0)(a_{\rm reh}/a_0)$}
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 $H_i^{-1}e^Na_0/a_{\rm reh}$. 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 $H_i/H_{\rm end}$, <energy> conservation gives $H_i\propto T_{\rm reh}^2$, whereas adiabatic subsequent expansion gives $a_0/a_{\rm reh}\propto T_{\rm reh}$. Thus the required $N$ increases approximately as $\log T_{\rm reh}$. 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.<How long before the end of inflation were observable perturbations produced? by Andrew R. Liddle and Samuel M. Leach>[Section II, equations 2 and 6–8]{ref}
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