= Age comparison for an initial vacuum-dominated phase
{title2=$\Delta t=H_1^{-1}\{\log(1/\epsilon)-(1-\epsilon^2)/2\}>0$}
Match flat <Friedmann equations> histories at <scale factor> $R_1$ and <Hubble parameter> $H_1$, and give them identical subsequent evolution. Below $R_1$, constant <vacuum energy> gives $H=H_1$, while a <radiation-dominated universe> gives $H=H_1(R_1/R)^2$. From $\epsilon R_1$ to $R_1$ the former takes longer by $H_1^{-1}\int_\epsilon^1(1/x-x)\,dx>0$. Its duration diverges as $\epsilon\to0$; the radiation duration tends to $1/(2H_1)$. A finite period of <cosmic inflation> requires a finite initial state and a specified matching before such age comparisons are meaningful.
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