  We study theoretically the collective mode spectrum of a strongly imbalanced
two-component unitary Fermi gas in a cigar-shaped trap, where the minority
species forms a gas of polarons. We describe the collective breathing mode of
the gas in terms of the Fermi liquid kinetic equation taking collisions into
account using the method of moments. Our results for the frequency and damping
of the longitudinal in-phase breathing mode are in good quantitative agreement
with an experiment by Nascimb\`ene et al. [Phys. Rev. Lett. 103, 170402 (2009)]
and interpolate between a hydrodynamic and a collisionless regime as the
polarization is increased. A separate out-of phase breathing mode, which for a
collisionless gas is sensitive to the effective mass of the polaron, however,
is strongly damped at finite temperature, whereas the experiment observes a
well-defined oscillation.
