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.