On the equation of state of Dark Energy

The formalism in order to obtain the Dark Energy equation of state is extended to non-flat universes and we consider the inequalities that must be satisfied by Phantom Dark Energy in this case. We show that due to a non-vanishing spatial curvature satisfying the observational bounds, the uncertainty on the determination of the Dark Energy equation of state parameter $w$, when it is taken constant, can be significant and that it is minimal for some redshift $z_{cr}\sim 3$. We consider the potential of future measurements of the gravitational waves emitted by binaries at high redshifts $z>z_{cr}$ to reduce this uncertainty. Results obtained here should also be relevant for a weakly varying equation of state with $w\approx -1$.