  We consider the density response of a spin-imbalanced ultracold Fermi gas in
an optical lattice in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. We
calculate the collective mode spectrum of the system in the generalised random
phase approximation and find that though the collective modes are damped even
at zero tempererature, the damping is weak enough to have well-defined
collective modes. We calculate the speed of sound in the gas and show that it
is anisotropic due to the anisotropy of the FFLO pairing, which implies an
experimental signature for the FFLO state.
