= Solution
Assume the present universe is expanding, $\dot a(t_0)>0$. The stated energy inequality and the <Raychaudhuri equation> imply $\ddot a\leq0$ wherever $a>0$. Therefore for $t<t_0$ in that positive branch, $\dot a(t)\geq\dot a(t_0)$, and
$$
a(t_0)-a(t)=\int_t^{t_0}\dot a(s)\,ds\geq(t_0-t)\dot a(t_0).
$$
Nonnegativity of $a(t)$ forces $t_0-t\leq a(t_0)/\dot a(t_0)=H_0^{-1}$. If the solution remains valid towards the past, its positive branch must thus end in $a=0$ within that time; a positive <scale factor> extending further would contradict the inequality. Hence
$$
\boxed{t_0-t_i\leq H_0^{-1}.}
$$
This argument presumes an expanding branch and the stated energy condition throughout it, rather than proving either assumption.
There are two relevant failures of the naive numerical bound. The supplied ten-billion-year <Hubble time> is an approximate observational input: a smaller inferred present expansion rate gives a larger <Hubble time>, and comparison with stellar ages must include these uncertainties. More decisively, a positive <cosmological constant>, or other <dark energy> with sufficiently negative <pressure>, has $\rho+3P/c^2<0$ and permits accelerated expansion, so the concavity assumption fails. Observations of acceleration therefore remove this particular upper-bound argument. This is a discussion of the supplied historical estimate, not an assertion that its numerical value is a current measurement.
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