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We present generic form of the covariant nonlocal action for infraredmodifications of Einstein theory recently suggested within the weak-fieldcurvature expansion. In the lowest order it is determined by two nonlocaloperators -- kernels of Ricci tensor and scalar quadratic forms. In models witha low strong-coupling scale this action also incorporates the strongly coupledmode which cannot be perturbatively integrated out in terms of the metricfield. This mode enters the action as a Lagrange multiplier for the constrainton metric variables which reduces to the curvature scalar and enforces thelatter to vanish on shell. Generic structure of the action is demonstrated onthe examples of the Fierz-Pauli and Dvali-Gabadadze-Porrati models and theirextensions. The gauge-dependence status of the strong-coupling and VDVZproblems is briefly discussed along with the manifestly gauge-invariantformalism of handling the braneworld gravitational models at classical andquantum levels.Comment: 24 pages, LaTeX, discussion of the constrained version of the DGP  model (suggested in hep-th/0403161) is essentially extended, references adde

Covariant long-distance modifications of Einstein theory and strong coupling problem