Casimir effect in extended theories of gravity

We study the Casimir vacuum energy density and the Casimir pressure for a massless scalar field confined between two nearby parallel plates in a slightly curved, static spacetime background, employing the weak-field approximation in the framework of extended theories of gravity (ETG). Following a perturbative approach, we find the gravity corrections to the Casimir vacuum energy density and pressure. The corrections to the vacuum energy density in the framework of general relativity (GR) are small, and today they are still undetected with current technology. However, future sensitivity improvements in gravitational interferometer experiments will give a useful tool to detect such effects induced by gravity. For these reasons, which are interesting from a theoretical point of view, we generalize the outcomes of GR in the context of ETG. Finally, we find the general relation to constrain the free parameters of the ETG.