  The rainbow chain is an inhomogenous exactly solvable local spin model that,
in its ground state, displays a half-chain entanglement entropy growing
linearly with the system size. Although many exact results about the rainbow
chain are known, the structure of the underlying quantum field theory has not
yet been unraveled. Here we show that the universal scaling features of this
model are captured by a massless Dirac fermion in a curved space-time with
constant negative curvature $R=-h^2$ ($h$ is the amplitude of the
inhomogeneity). This identification allows us to use recently developed
techniques to study inhomogeneous conformal systems and to analytically
characterise the entanglement entropies of more general bipartitions. These
results are carefully tested against exact numerical calculations. Finally, we
study the entanglement entropies of the rainbow chain in thermal states, and
find that there is a non-trivial interplay between the rainbow effective
temperature $T_R$ and the physical temperature $T$.
