  We determine the Tomita-Takesaki modular data for CFTs in double cone and
light cone regions in conformally flat spacetimes. This includes in particular
the modular Hamiltonian for diamonds in the de Sitter spacetime. In the limit
where the diamonds become large, we show that the modular automorphisms become
time translations in the static patch. As preparation, we also provide a
pedagogical rederivation of the known results for Minkowski spacetime. With our
results and using the Araki formula, it becomes possible to compute relative
entanglement entropies for CFTs in these regions.
