Hojman symmetry approach for scalar–tensor cosmology

• Cosmological models.
• Exact solutions.
• Noether's symmetries and Hojman's symmetries.
• Scalar–tensor gravity.

Scalar–tensor cosmologies can be dealt under the standard of the Hojman conservation theorem that allows to fix the form of the coupling F(ϕ)F(ϕ), of the potential V(ϕ)V(ϕ) and to find out exact solutions for related cosmological models. Specifically, the existence of a symmetry transformation vector for the equations of motion gives rise to a Hojman conserved quantity on the corresponding minisuperspace and exact solutions for the cosmic scale factor a and the scalar field ϕ can be achieved. In particular, we take advantage of the fact that minimally coupled solutions, previously obtained in the Einstein frame, can be conformally transformed in non-minimally coupled solutions in the Jordan frame. Some physically relevant examples are worked out.