We consider the application of group invariant transformations in order to constrain a flat isotropic and homogeneous cosmological model, containing a Brans-Dicke scalar field and a perfect fluid with a constant equation of state parameter $w$, where the latter is not interacting with the scalar field in the gravitational action integral. The requirement that the Wheeler-DeWitt equation be invariant under one-parameter point transformations provides us with two families of power-law potentials for the Brans-Dicke field, in which the powers are functions of the Brans-Dicke parameter ${\omega }_{\mathrm{BD}}$ and the parameter $w$. The existence of the Lie symmetry in the Wheeler-DeWitt equation is equivalent to the existence of a conserved quantity in field equations and with oscillatory terms in the wave function of the Universe. This enables us to solve the field equations. For a specific value of the conserved quantity, we find a closed-form solution for the Hubble factor, which is equivalent to a cosmological model in general relativity containing two perfect fluids. This provides us with different models for specific values of the parameters ${\omega }_{\mathrm{BD}}$, and $w$. Finally, the results hold for the specific case where the Brans-Dicke parameter ${\omega }_{\mathrm{BD}}$ is zero, that is, for the O’Hanlon massive dilaton theory and, consequently, for $f(R)$ gravity in the metric formalism.

• Received 2 November 2015

DOI:https://doi.org/10.1103/PhysRevD.93.043528