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, $\smash{g=-dt^2+dr^2+ S\_{k(t)}^2(r) g\_{\mathbb{S}^{n-1}}}$, where $g\_{\mathbb{S}^{n-1}}$ is the metric of the standard sphere, $\smash{S\_{k(t)}(r)=\sin(\sqrt{k(t)} r)/\sqrt{k(t)}}$ when $k(t)\geq 0$ and $\smash{S\_{k(t)}(r)=\sinh(\sqrt{-k(t)} r)/\sqrt{-k(t)}}$ when $k(t)\leq 0$. In the open case, the $t$-slices are (non-compact) Cauchy hypersurfaces of curvature $k(t)\leq 0$, thus homeomorphic to $\mathbb{R}^n$; a typical example is $k(t)=-t^2$ (i.e., $\smash{S\_{k(t)}(r)=\sinh(tr)/t}$). 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.