  The symmetron is a typical example of screened modified gravity, wherein the
symmetron force is dynamically suppressed in dense environments. This allows it
to hide in traditional tests of gravity. However, the past decade has seen
great experimental progress towards measuring screened forces in the laboratory
or in space. Screening relies on nonlinearities in the equation of motion,
which significantly complicates the theoretical analysis of such forces. Here,
we present a calculation of the symmetron force between a dense plate and
sphere surrounded by vacuum. This is done via semi-analytical approaches in two
limiting cases, based on the size of the sphere: large spheres are analyzed via
the proximity force approximation, whilst small spheres are treated as screened
test particles. In the intermediate regime we solve the problem numerically.
Our results allow us to make contact with Casimir force experiments, which
often employ a plate and sphere configuration for practical reasons, and may
therefore be used to constrain symmetrons. We use our results to forecast
constraints on the symmetron's parameters for a hypothetical Casimir experiment
that is based on the current state of the art. The forecasts compare favorably
to other leading laboratory tests of gravity, particularly atom interferometry
and bouncing neutrons. We thus conclude that near-future Casimir experiments
will be capable of placing tight new bounds on symmetrons. Our results for the
symmetron force are derived in a scale-invariant way, such that although we
here focus on Casimir experiments, they may be applied to any other
plate-sphere system, ranging from microscopic to astrophysical scales.
