v 1 1 4 Se p 20 04 Cosmological constant problem and long - distance modifications of Einstein theory

We construct the covariant nonlocal action for recently suggested long-distance modifications of gravity theory motivated by the cos-mological constant and cosmological acceleration problems. This construction is based on the special nonlocal form of the Einstein-Hilbert action explicitly revealing the fact that this action within the covariant curvature expansion begins with curvature-squared terms. New approach to the solution of the cosmological constant problem consists in the assumption that, instead of adjusting the vacuum energy of quantum matter to zero (or to small value of the cosmological acceleration), one should modify the purely gravitational sector of the theory in far in-frared region. Matter sources with wavelengths comparable with the horizon size of the present Universe L ∼ 1/H 0 ∼ 10 28 cm gravitate with the long-distance gravitational constant G LD which is much smaller than the conventional Planckian value G P. Therefore, the vacuum energy E, T µν = Eg µν , of TeV or even Planckian scale (necessarily arising in all conceivable models with spontaneously broken SUSY or in quantum gravity) will not generate a catastrophically big spacetime curvature incompatible with the tiny observable Hubble constant H 2 0 ∼ G LD E ≪ G P E. This mechanism is drastically different from the old suggestions of supersymmetric cancellation of E [1], because it relies on the fact that the nearly homogeneous vacuum energy gravitates very little, rather than it is itself very small. This idea implies that the gravitational coupling constant should be promoted to the level of the operator, G P ⇒ G(2), which for sake of covariance 1