  We consider a rapidly rotating two-component Bose-Einstein condensate (BEC)
containing a vortex lattice. We calculate the dispersion relation for small
oscillations of vortex positions (Tkachenko modes) in the mean-field quantum
Hall regime, taking into account the coupling of these modes with density
excitations. Using an analytic form for the density of the vortex lattice, we
numerically calculate the elastic constants for different lattice geometries.
We also apply this method to calculate the elastic constant for the
single-component triangular lattice. For a two-component BEC, there are two
kinds of Tkachenko modes, which we call acoustic and optical in analogy with
phonons. For all lattice types, acoustic Tkachenko mode frequencies have
quadratic wave-number dependence at long-wavelengths, while the optical
Tkachenko modes have linear dependence. For triangular lattices the dispersion
of the Tkachenko modes are isotropic, while for other lattice types the
dispersion relations show directional dependence consistent with the symmetry
of the lattice. Depending on the intercomponent interaction there are five
distinct lattice types, and four structural phase transitions between them. Two
of these transitions are second-order and are accompanied by the softening of
an acoustic Tkachenko mode. The remaining two transitions are first-order and
while one of them is accompanied by the softening of an optical mode, the other
does not have any dramatic effect on the Tkachenko spectrum. We also find an
instability of the vortex lattice when the intercomponent repulsion becomes
stronger than the repulsion within components.
