A class of cosmological models with spatially constant sign-changing curvature

Abstract

We construct globally hyperbolic spacetimes such that each slice $\{t=t_0\}$ of the universal time $t$ is a model space of constant curvature $k(t_0)$ which may not only vary with $t_0\in \mathbb{R}$ but also change its sign. The metric is smooth and slightly different to FLRW spacetimes, namely, $g=-d{t}^2+d{r}^2+S_{k(t)}^2(r)g_{{\mathbb{S}}^{n-1}}$, where $g_{{\mathbb{S}}^{n-1}}$ is the metric of the standard sphere, $S_{k(t)}(r)=\sin (\sqrt{k(t)}r)/\sqrt{k(t)}$ when $k(t)\ge 0$ and $S_{k(t)}(r)=\sinh (\sqrt{-k(t)}r)/\sqrt{-k(t)}$ when $k(t)\le 0$.

In the open case, the $t$-slices are (non-compact) Cauchy hypersurfaces of curvature $k(t)\le 0$, thus homeomorphic to ${\mathbb{R}}^n$; a typical example is $k(t)=-t^2$ (i.e., $S_{k(t)}(r)=\sinh (tr)/t$). In the closed case, $k(t)>0$ somewhere, a slight extension of the class shows how the topology of the $t$-slices changes. This makes at least one comoving observer to disappear in finite time $t$ showing some similarities with an inflationary expansion. Anyway, the spacetime is foliated by Cauchy hypersurfaces homeomorphic to spheres, not all of them $t$-slices.