FLRW-compatible tensors and perfect-fluid universality in modified gravity

We study FLRW-compatible tensor fields on Friedmann–Lemaître–Robertson–Walker spacetimes in $$(D+1)$$ ( D + 1 ) dimensions equipped with Levi-Civita connection. We prove that any covariant rank- N tensor field invariant under the spatial homogeneity and isotropy group, with $$D>N$$ D > N , admits a finite decomposition into monomials constructed from the metric tensor gμ ν g μ ν and the unit timelike vector u_μ u μ , with scalar coefficients depending only on cosmic time. As a direct consequence, any rank-two correction term appearing in an Einstein-like rearrangement of the FLRW background field equations has the effective perfect-fluid form for $$D>2$$ D > 2 . Thus, the result provides an arbitrary-rank tensorial classification underlying the perfect-fluid structure of modified gravity equations on FLRW backgrounds, complementing existing rank-two universality results. Applications to generalized Friedmann equations and to the geodesic deviation equation in modified gravity are also discussed.