Quantum Cosmology in Higher Derivative and Scalar-Tensor Gravity

The Bicknell theorem states that a non-linear Lagrangian can be recast in the form of a scalar-tensor theory, with a suitable potential, through a conformal transformation. In this paper, we first show that such classical equivalence remains valid at the level of the Wheeler—deWitt equation. Then, we consider a specific case, represented by a Lagrangian f(R) = R + l^–2(l²R)^4/3 whose vacuum cosmological solutions describe a non-singular Universe. The corresponding scalar-tensor theory and its cosmological solutions are written down. We find again non-singular solutions. The Wheeler—deWitt equation for this case is analyzed. The application of the Bicknell theorem leads to the interpretation of the behaviour of the scale factor in terms of the matter content, represented by the scalar field, and consequently to the energy conditions. The problem of classical and quantum regime is discussed and the classical behaviour is recovered, from the quantum solutions, near the maximum of the scale factor where the strong energy condition is satisfied.