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Long-wavelength phonons in the crystalline and pasta phases of neutron-star crusts

We study the long-wavelength excitations of the inner crust of neutron stars, considering three phases: cubic crystal at low densities, rods and plates near the core-crust transition. To describe the phonons, we write an effective Lagrangian density in terms of the coarse-grained phase of the neutron superfluid gap and of the average displacement field of the clusters. The kinetic energy, including the entrainment of the neutron gas by the clusters, is obtained within a superfluid hydrodynamics approach. The potential energy is determined from a model where clusters and neutron gas are considered in phase coexistence, augmented by the elasticity of the lattice due to Coulomb and surface effects. All three phases show strong anisotropy, i.e., angle dependence of the phonon velocities. Consequences for the specific heat at low temperature are discussed.