Transport theory for cold relativistic superfluids from an analogue model of gravity

We write a covariant transport equation for the phonon excitations of a relativistic superfluid valid at small temperatures. The hydrodynamical equations for this system are derived from the effective field theory associated to the superfluid phonons. We describe how to construct the kinetic theory for the phonon quasiparticles using a relativistic generalization of the analogue model of gravity developed by Unruh. This gravity analogy relies on the equivalence between the action of a phonon field moving in a superfluid background with that of a boson propagating in a given curved space-time. Exploiting this analogy we obtain continuity equations for the phonon current, entropy and energy-momentum tensor in a covariant form, valid in any reference frame. Our aim is to shed light on some aspects of transport phenomena of relativistic superfluidity. In particular, we are interested in the low temperature regime of the color flavor locked phase, which is a color superconducting and superfluid phase of high density QCD that may be realized in the core of neutron stars.