  Gravitational vector degrees of freedom typically arise in many examples of
modified gravity models. We start to systematically explore their role in these
scenarios, studying the effects of coupling gravitational vector and scalar
degrees of freedom. We focus on set-ups that enjoy a Galilean symmetry in the
scalar sector and an Abelian gauge symmetry in the vector sector. These
symmetries, together with the requirement that the equations of motion contain
at most two space-time derivatives, only allow for a small number of operators
in the Lagrangian for the gravitational fields. We investigate the role of
gravitational vector fields for two broad classes of phenomena that
characterize modified gravity scenarios. The first is self-acceleration: we
analyze in general terms the behavior of vector fluctuations around
self-accelerating solutions, and show that vanishing kinetic terms of vector
fluctuations lead to instabilities on cosmological backgrounds. The second
phenomenon is the screening of long range fifth forces by means of Vainshtein
mechanism. We show that if gravitational vector fields are appropriately
coupled to a spherically symmetric source, they can play an important role for
defining the features of the background solution and the scale of the
Vainshtein radius. Our general results can be applied to any concrete model of
modified gravity, whose low-energy vector and scalar degrees of freedom satisfy
the symmetry requirements that we impose.
