The thermal evolution of neuron stars depends on the elementary excitations affecting the stellar matter. In particular, the low-energy excitations, whose energy is proportional to the transfered momentum, can play a major role in the emission and propagation of neutrinos. In this paper, we focus on the density modes associated with the proton component in the homogeneous matter of the outer core of neutron stars (at density between one and three times the nuclear saturation density, where the baryonic constituants are expected to be neutrons and protons). In this region, it is predicted that the protons are superconductor. We study the respective roles of the proton pairing and Coulomb interaction in determining the properties of the modes associated with the proton component. This study is performed in the framework of the Random Phase Approximation, generalized in order to describe the response of a superfluid system.The formalism we use ensures that the Generalized Wards Identities are satisfied. An important conclusion of this work is the presence of a pseudo-Goldstone mode associated with the proton superconductor in neutron-star matter. Indeed, the Goldstone mode, which characterizes a pure superfluid, is suppressed in usual superconductors due to the long-range Coulomb interaction, which only allows a plasmon mode. However, for the proton component of stellar matter, the Coulomb field is screened by the electrons and a pseudo-Goldstone mode occurs, with a velocity increased by the Coulomb interaction.
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