Age comparison for an initial vacuum-dominated phase ($\Delta t=H_1^{-1}\{\log (1/ϵ)-(1-{ϵ}^2)/2\}>0$)

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 $ϵR_1$ to $R_1$ the former takes longer by $H_1^{-1}{\int }_{ϵ}^1(1/x-x)dx>0$. Its duration diverges as $ϵ\to 0$; 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.