Friedmann–Lemaître–Robertson–Walker cosmology in scalar-vector-tensor theories of gravity

We generalize our previous theorem for Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes within the framework of generic metric gravity theories. In earlier work, we proved that, in the absence of additional matter fields, the field equations of metric gravity theories constructed from the curvature tensor and its covariant derivatives reduce in FLRW spacetime to an effective perfect-fluid form. In the present work, we extend this tensorial reduction to a class of local scalar-vector-tensor theories in which the gravitational action contains a homogeneous scalar field, a purely timelike isotropic vector field, and their covariant derivatives at arbitrary order. We prove that, under the FLRW-compatible assumptions ϕ = ϕ(t) and Aμ = ψ(t)uμ, the symmetric rank-two tensors entering the metric field equations necessarily reduce to the FLRW perfect-fluid tensorial sector. When an explicitly normalized Einstein-Hilbert term is included in the action, the metric field equations may therefore be written in Einstein form with an effective perfect-fluid source, supplemented by the corresponding scalar and vector field equations. This result should be understood as a background-level structural reduction, or as a perfect-fluid-type tensorial universality, rather than as a claim of strict metric universality. The effective density and pressure remain theory dependent, and therefore so does the resulting cosmological evolution. We illustrate the theorem using recently proposed Einstein–scalar and Einstein–Proca theories arising from regularized four-dimensional Einstein–Gauss–Bonnet gravity.