Hamiltonian structure of bi-gravity, problem of ghost and bifurcation

We analyze the Hamiltonian structure of a general theory of bi-gravity where the interaction term is a scalar function of the form $V(\mathcal{X}^{n})$ where $\mathcal{X}$ may be $\sqrt{g^{-1}f}$ or $g^{-1}f$. We give necessary conditions for the interaction term of such a theory to be ghost free. We give a precise constraint analysis of the bi-gravity theory of Hassan- Rosen and show that the additional constraint which omit the ghost is just one possibility at the bifurcation point.