Infrared limit of quantum gravity

We explore the infrared limit of quantum gravity in the presence of a cosmological constant or effective potential for scalar fields. For a positive effective scalar potential, one-loop perturbation theory around flat space is divergent due to an instability of the graviton propagator. Functional renormalization solves this problem by a flow of couplings avoiding instabilities. This leads to a graviton barrier limiting the maximal growth of the effective potential for large values of scalar fields. In the presence of this barrier, variable gravity with a field dependent Planck mass can solve the cosmological constant problem by a cosmological runaway solution. We discuss the naturalness of tiny values of the cosmological constant and cosmon mass due to a strong attraction towards an infrared fixed point.