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

We present a generic form of the covariant nonlocal action for infrared modifications of Einstein theory recently suggested within the weak-field curvature expansion. In the lowest order it is determined by two nonlocal operators-kernels of Ricci tensor and scalar quadratic forms. In models with a low strong-coupling scale, this action also incorporates the strongly coupled mode which cannot be perturbatively integrated out in terms of the metric field. This mode enters the action as a Lagrange multiplier for the constraint on metric variables which reduces to the curvature scalar and enforces the latter to vanish on shell. Generic structure of the action is demonstrated on the examples of the Fierz-Pauli and Dvali-Gabadadze-Porrati models and their extensions. The gauge-dependence status of the strong-coupling and van Dam-Veltman-Zakharov problems is briefly discussed along with the manifestly gauge-invariant formalism of handling the braneworld gravitational models at classical and quantum levels.