Curvature phase transition in $R^2$ quantum gravity and induction of Einstein gravity

The phase transition with respect to the curvature in the effective potential of $R^2$ quantum gravity with matter is studied. The effective potential is calculated in the framework of the renormalization-group approach up to terms linear in the curvature. A universal expression is obtained for the induced gravitational and cosmological constants. The effective potential, and also the induced cosmological and gravitational constants depend on the relationships between the coupling constants of the original theory and on the gauge parameters. When the matter is represented by a single scalar field values fixed by asymptotic freedom are chosen for the coupling constants. There is no gauge dependence for the unified parametrization- and gauge-invariant effective action.