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A linear response framework is set up for the evaluation of collectiveexcitations in a confined vapour of interacting Bose atoms at finitetemperature. Focusing on the currently relevant case of contact interactionsbetween the atoms, the theory is developed within a random phase approximationwith exchange. This approach is naturally introduced in a two-fluid descriptionby expressing the density response of both the condensate and thenon-condensate in terms of the response of a Hartree-Fock reference gas to theselfconsistent Hartree-Fock potentials. Such an approximate account ofcorrelations (i) preserves an interplay between the condensate and thenon-condensate through off-diagonal components of the response, which insteadvanish in the Hartree-Fock-Bogolubov approximation; and (ii) yields a commonresonant structure for the four partial response functions. The theory reducesto the temperature-dependent Hartree-Fock-Bogolubov-Popov approximation for thefluctuations of the condensate when its coupling with the density fluctuationsof the non-condensate is neglected. Analytic results are presented which areamenable to numerical calculations and to inclusion of damping rates.Comment: 14 pages. To appear on J. Phys. : Condens. Matte

Linear density response in the random-phase approximation for confined Bose vapours at finite temperature