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

Abstract

We construct globally hyperbolic spacetimes such that each slice {t=t0​} of the universal time t is a model space of constant curvature k(t0​) which may not only vary with t0​∈R but also change its sign. The metric is smooth and slightly different to FLRW spacetimes, namely, g=−dt2+dr2+Sk(t)2​(r)gSn−1​, where gSn−1​ is the metric of the standard sphere, Sk(t)​(r)=sin(k(t) ​r)/k(t) ​ when k(t)≥0 and Sk(t)​(r)=sinh(−k(t) ​r)/−k(t)

​ when k(t)≤0.

In the open case, the t-slices are (non-compact) Cauchy hypersurfaces of curvature k(t)≤0, thus homeomorphic to Rn; a typical example is k(t)=−t2 (i.e., Sk(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.