  We consider a model where a light scalar field (with mass $\lesssim 30\, {\rm
eV}$), conjectured to be dark matter, has a non-minimal coupling to gravity. In
the non-relativistic limit, this new coupling introduces a self-interaction
term in the scalar-field equation of motion, and modifies the source term for
the gravitational field. Moreover, in the small-coupling limit justified by the
observed dark-matter density, the system further reduces to the
Gross-Pitaevskii-Poisson equations, which remarkably also arise from a
self-gravitating and self-interacting Bose-Einstein condensate system. We
derive predictions of our model on linear and non-linear structure formation by
exploiting this unexpected connection.
