Induced Gravity and Quantum Cosmology

We study the Wheeler-DeWitt equation for a class of induced gravity models in the minisuperspace approximation. In such models a scalar field nonminimally coupled to gravity determines the effective Newton's constant. For simplicity our analysis is limited to power-law potentials for the scalar field which have exact classical solutions. We show that these models have exact solutions also when quantised. Finally the Einstein Frame form of these solutions is obtained and a classical-quantum correspondence is found. Realistic induced gravity models also must include a symmetry breaking term which is needed in order to obtain a gravitational constant, successful inflation and a subsequent standard cosmological evolution. Nonetheless the potentials considered are important as they may describe the inflationary phase when the symmetry breaking part of the potential is negligible.