= Concavity bound on past FLRW evolution
{title2=$a(t)\le a(t_0)-\dot a(t_0)(t_0-t)$}
In an <FLRW metric> with zero separate <cosmological constant>, $\varrho+3p\ge0$ makes the <Friedmann acceleration equation> imply $\ddot a\le0$. If $a_0=a(t_0)>0$ and $v_0=\dot a(t_0)>0$, then the <scale factor> is a <concave function> and
$$
a(t)\le a_0-v_0(t_0-t)
$$
throughout its regular past interval. Thus a regular positive <scale factor> cannot persist earlier than $t_0-a_0/v_0$. If the solution continues regularly whenever $a>0$, its first past zero occurs within $a_0/v_0=H(t_0)^{-1}$, where $H$ is the <Hubble parameter>.
The continuation condition matters: without restrictions on the <equation of state>, an earlier <curvature singularity> can occur at positive $a$. For example, in fixed units $a(t)=1+\sqrt t$, $t>0$, with $k=0$, defines <energy density> $\varrho=3(\dot a/a)^2/(8\pi G)$ and <pressure> $p=-\varrho/3-\ddot a/(4\pi Ga)$. These satisfy the <Friedmann equations> and $\varrho+3p>0$, yet $a\to1$ at the singular past boundary. Conversely the <Milne model> reaches zero without divergent <curvature>. A finite duration, a zero of the <scale factor>, and a <curvature singularity> are distinct conclusions.
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