  The observed large-scale structure in our Universe is seen as a result of
quantum fluctuations amplified by spacetime evolution. This, and related
problems in cosmology, asks for an understanding of the quantum fields of the
standard model and dark matter in curved spacetime. Even the reduced problem of
a scalar quantum field in an explicitly time-dependent spacetime metric is a
theoretical challenge and thus a quantum field simulator can lead to new
insights. Here, we demonstrate such a quantum field simulator in a
two-dimensional Bose-Einstein condensate with a configurable trap and
adjustable interaction strength to implement this model system. We explicitly
show the realisation of spacetimes with positive and negative spatial curvature
by wave packet propagation and confirm particle pair production in controlled
power-law expansion of space. We find quantitative agreement with new
analytical predictions for different curvatures in time and space. This
benchmarks and thereby establishes a quantum field simulator of a new class. In
the future, straightforward upgrades offer the possibility to enter new, so far
unexplored, regimes that give further insight into relativistic quantum field
dynamics.
