Perturbations of ultralight vector field dark matter
  We study the dynamics of cosmological perturbations in models of dark matter
based on ultralight coherent vector fields. Very much as for scalar field dark
matter, we find two different regimes in the evolution: for modes with $k^2\ll
{\cal H}ma$, we have a particle-like behaviour indistinguishable from cold dark
matter, whereas for modes with $k^2\gg {\cal H}ma$, we get a wave-like
behaviour in which the sound speed is non-vanishing and of order $c_s^2\simeq
k^2/m^2a^2$. This implies that, also in these models, structure formation could
be suppressed on small scales. However, unlike the scalar case, the fact that
the background evolution contains a non-vanishing homogeneous vector field
implies that, in general, the evolution of the three kinds of perturbations
(scalar, vector and tensor) can no longer be decoupled at the linear level.
More specifically, in the particle regime, the three types of perturbations are
actually decoupled, whereas in the wave regime, the three vector field
perturbations generate one scalar-tensor and two vector-tensor perturbations in
the metric. Also in the wave regime, we find that a non-vanishing anisotropic
stress is present in the perturbed energy-momentum tensor giving rise to a
gravitational slip of order $(\Phi-\Psi)/\Phi\sim c_s^2$. Moreover in this
regime the amplitude of the tensor to scalar ratio of the scalar-tensor modes
is also $h/\Phi\sim c_s^2$. This implies that small-scale density perturbations
are necessarily associated to the presence of gravity waves in this model. We
compare their spectrum with the sensitivity of present and future gravity waves
detectors.
