  We analytically calculate the intra- and inter-surface dynamical
density-density linear responses of ultra-thin topological insulator films with
finite tunneling between their top and bottom surfaces in both metallic and
insulating regimes. Employing the random phase approximation we investigate the
dispersions of in-phase and out-of-phase collective density modes of this
system in the metallic regime. We find that in contrast to the bilayers of the
conventional two-dimensional electron gas, where finite tunneling gaps out the
out-of-phase mode, in topological insulator thin films, this mode remains
linear at long wavelengths. Depending on different system parameters, the
velocity of out-of-phase mode can be tuned to be larger or substantially
smaller than the Fermi velocity of electrons on the isolated surfaces of the
topological insulator. Finite tunneling generally reduces the energy of
collective modes, making them more confined in space.
