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We analyse the classical stability of the model proposed by Maggiore andMancarella, where gravity is modified by a term $\sim m^2 R \Box^{-2} R$ toproduce the late-time acceleration of the expansion of the universe. Our studytakes into account all excitations of the metric that can potentially drive aninstability. There are some subtleties in identifying these modes, as anon-local field theory contains dynamical fields which yet do not correspond todegrees of freedom. Since some of them are ghost-like, we clarify the impact ofsuch modes on the stability of the solutions of interest that are the flatspace-time and cosmological solutions. We then find that flat space-time isunstable under scalar perturbations, but the instability manifests itself onlyat cosmological scales, i.e. out of the region of validity of this solution. Itis therefore the stability of the FLRW solution which is relevant there, inwhich case the scalar perturbations are known to be well-behaved by numericalstudies. By finding the analytic solution for the late-time behaviour of thescale factor, which leads to a big rip singularity, we argue that the linearperturbations are bounded in the future because of the domination of Hubblefriction. In particular, this effect damps the scalar ghost perturbations whichwere responsible for destabilizing Minkowski space-time. Thus, the modelremains phenomenologically viable.

Stability analysis and future singularity of the $m^2 R \Box^{-2} R$ model of non-local gravity