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We perform a thorough study of the theoretical consistency of recentlyproposed, viable, quadratic modifications of gravity that are functions of thethe Gauss-Bonnet invariant, regarding the stability of their perturbationsaround vacuum, maximally symmetric spaces of constant curvature. We pay specialattention, in particular, to the investigation of pathological instabilitiesassociated with the occurrence of propagating spin-0 tachyon modes, and withthe development of a graviton ghost. The latter effect is associated with theknown "Ricci stability" issue, well studied in $f(R)$-theories. Withinquadratic modifications of gravity it is discussed for the first time. Specialattention is paid to the requirement of non-negativity of the effectivegravitational coupling, which warrants that the graviton is not a negative-normstate. It is demonstrated that, several theories that pass the cosmological aswell as the solar system tests, have to be rule out on the basis of theunavoidable character of these pathological instabilities.

Vacuum Structure of Cosmologically Viable Quadratic Modifications of Gravity that are Functions of the Gauss-Bonnet Invariant