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

TL;DR

This paper constructs a new class of cosmological models with spatially constant curvature that can change sign over time, leading to novel spacetime topologies and behaviors resembling inflation.

Contribution

It introduces a smooth metric for cosmological models where the spatial curvature varies and can change sign, extending FLRW models to include more general curvature dynamics.

Findings

Models include curvature changing sign over time.

Spacetimes have Cauchy hypersurfaces homeomorphic to spheres or Euclidean space.

Some models exhibit finite-time disappearance of observers.

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, $g=-dt^2+dr^2+ S_{k(t)}^2(r) g_{\mathbb{S}^{n-1}}$, where $g_{\mathbb{S}^{n-1}}$ is the metric of the standard sphere, $S_{k(t)}(r)=\sin(\sqrt{k(t)}\, r)/\sqrt{k(t)}$ when $k(t)\geq 0$ and $S_{k(t)}(r)=\sinh(\sqrt{-k(t)}\, r)/\sqrt{-k(t)}$ 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., $S_{k(t)}(r)=\sinh(tr)/t$). 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.