Using the s-wave and large-$N$ approximation the one-loop effective action for 2D dilaton coupled scalars and spinors which are obtained by spherical reduction of 4D minimal matter is found. Quantum effective equations for reduced Einstein gravity are written. Their analytical solutions corresponding to a 4D Kantowski-Sachs (KS) universe are presented. For quantum-corrected Einstein gravity we get a nonsingular KS cosmology which represents (1) a quantum-corrected KS cosmology which existed on a classical level or (2) a purely quantum solution which had no classical limit. The analogy with the Nariai BH is briefly mentioned. For purely induced gravity (no Einstein term) we found a general analytical solution but all KS cosmologies under discussion are singular. The corresponding equations of motion are reformulated as a classical mechanics problem of motion of a unit mass particle in some potential V.

• Received 4 February 1999

DOI:https://doi.org/10.1103/PhysRevD.60.024008