Lemma: Global hyperbolicity of the open slab

Every surface $t=t_0$, $t_0\in I$, is a Cauchy surface of $(\mathcal{M},\bar{g})$.

Proof. Conformal time $\eta$, defined by $\mathrm{d}\eta =\mathrm{d}t/a(t)$, identifies the metric conformally with Minkowski space on an open time interval. Along a causal curve, $\text{abs}\mathrm{d}\mathbf{x}/\mathrm{d}t\le a(t{)}^{-1}$. If an inextendible causal curve had a time endpoint strictly inside $I$, this estimate would give a finite spatial endpoint, and the curve could be extended. Its time range therefore crosses every $t_0\in I$, exactly once for a future-directed causal curve after the usual harmless removal of constant parameter intervals.