Parametrizing modified gravities with vector degrees of freedom:
  anisotropic growth and lensing
  We consider the problem of parametrizing modified gravity theories that
include an additional vector field in the sub-Hubble regime within the
quasi-static approximation. We start from the most general set of second order
equations for metric and vector field perturbations and allow for both temporal
and spatial components of the background vector field. We find that in the case
in which dark matter obeys standard conservation equations, eight parameters
are needed to fully characterize the theory. If dark matter vorticity can be
neglected, the number of independent parameters is reduced to four. In addition
to the usual scale and redshift dependence, the effective parameters have an
additional angular dependence induced by the preferred direction set by the
background vector. In the considered sub-Hubble regime, we show that this
angular dependence appears only through even multipoles and generates
anisotropies in the growth function which translate into anisotropies in the
galaxy and lensing convergence power spectra. The angular dependence generated
by the preferred direction is different from that induced by redshift space
distortions and could be disentangled in the data collected by future galaxy
surveys.
