Scalar-tensor theory of gravity carrying a conserved current

A general scalar-tensor theory of gravity carries a conserved current for a trace-free minimally coupled scalar field, under the condition that the potential V(ϕ) of the non-minimally coupled scalar field is proportional to the square of the parameter f(ϕ) that is coupled with the scalar curvature R. The conserved current relates the pair of arbitrary coupling parameters f(ϕ) and ω(ϕ) with the gravitational field variable, where ω(ϕ) is the Brans-Dicke coupling parameter. Thus fixing up the two arbitrary parameters by hand, it is possible to explore the symmetries and the form of conserved currents corresponding to standard and many different nonstandard models of gravity. To demonstrate how the presence of conserved current makes it easier to tackle the field equations a couple of homogeneous cosmological solutions are presented. Modification of such a general scalar-tensor theory of gravity with the introduction of higher-order curvature-invariant terms also admits the same conserved current, though under a different condition.