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In this paper we study long distance modifications of gravity obtained byconsidering actions that are singular in the limit of vanishing curvature. Inparticular, we showed in a previous publication that models that includeinverse powers of curvature invariants that diverge for r->0 in theSchwarzschild geometry, recover an acceptable weak field limit at shortdistances from sources. We study then the linearisation of generic actions ofthe form L=F[R,P,Q] where P=R_{ab}R^{ab} and Q=R_{abcd}R^{abcd}. We show thatfor the case in which F[R,P,Q]=F[R,Q-4P], the theory is ghost free. Assumingthis is the case, in the models that can explain the acceleration of theUniverse without recourse to Dark Energy there is still an extra scalar fieldin the spectrum besides the massless spin two graviton. The mass of this extraexcitation is of the order of the Hubble scale in vacuum. We neverthelessrecover Einstein gravity at short distances because the mass of this scalarfield depends on the background in such a way that it effectively decoupleswhen one gets close to any source. Remarkably, for the values of the parametersnecessary to explain the cosmic acceleration the induced modifications ofgravity are suppressed at the Solar System level but can be important forsystems like a galaxy.Comment: 27 pages, 1 figure, v2. references added, improved discussio

Consistent long distance modification of gravity from inverse powers of the curvature