Noether analysis of scalar-tensor cosmology

A scalar-tensor theory of gravity, containing an arbitrary coupling function F(phi) and a general potential F(phi), is considered in the context of a spatially flat Friedmann-Lemaitre-Robertson-Walker model. The use of reparametrization invariance enables a particular lapse parametrization in which the minisuperspace metric completely specifies the dynamics of the system. A requirement of the existence of the maximal possible number of autonomous integrals of motion is imposed. This leads to a flat minisuperspace metric realized by a particular relation between the coupling function and the potential. The space of solutions is completely described in terms of the three autonomous integrals of motion constructed by the Killing fields of the minisupermetric and an additional rheonomous emanating from the homothetic field. The solutions contain the arbitrary function which remains after the imposition of the relation between F(phi) and F(phi). To exemplify the use of the general results, we select some particular cases and study their physical implications through an effective energy-momentum tensor, which turns out to be that of a perfect fluid.