Observer horizons in two-dimensional de Sitter spacetime ($d(\chi ,0)<\pi /2-T\text{or}d(\chi ,0)<T+\pi /2$)

For the complete observer $\chi =0$ in two-dimensional de Sitter spacetime, let $d(\chi ,0)\in [0,\pi ]$ be shortest angular distance. An event can signal to the observer exactly when $d(\chi ,0)<\pi /2-T$, and can receive its signals exactly when $d(\chi ,0)<T+\pi /2$. Equality describes its future and past observer event horizons. Their intersection is the static patch of de Sitter spacetime containing the observer. These are observer-dependent cosmological horizons; constant-$T$ circles are Cauchy hypersurfaces, so there are no Cauchy horizons in the full spacetime.