  We present a theoretical scheme to simulate quantum field theory in a
discrete curved spacetime based on the Bose-Hubbard model describing a
Bose-Einstein condensate trapped inside an optical lattice. Using the
Bose-Hubbard Hamiltonian, we first introduce a hydrodynamic presentation of the
system evolution in discrete space. We then show that the phase (density)
fluctuations of the trapped bosons inside an optical lattice in the superfluid
(Mott insulator) state obey the Klein-Gordon equation for a massless scalar
field propagating in a discrete curved spacetime. We derive the effective
metrics associated with the superfluid and Mott-insulator phases and, in
particular, we find that in the superfluid phase the metric exhibits a
singularity which can be considered as a the manifestation of an analog
acoustic black hole. The proposed approach is found to provide a suitable
platform for quantum simulation of various spacetime metrics through adjusting
the system parameters.
