Nonminimal coupling, quartic potential and perfect fluid cosmologies

Perfect-fluid matter, satisfying the equation of state p = (gamma - 1)rho, is considered in cosmologies where the geometry is nonminimally coupled with a scalar field phi and the potential of phi is lambda phi(4) + Lambda. Exact solutions are found when gamma is a constant describing the ordinary forms of matter (gamma = 1, dust, gamma = 4/3, radiation, gamma = 2, stiff matter and gamma = 0, scalar field matter) and a discussion is done in order to recover Einstein gravity and the Newton constant observed today. The various solutions can be classified according to the different values of gamma, lambda and Lambda.