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

Abstract

We study FLRW-compatible tensor fields on Friedmann–Lemaître–Robertson–Walker spacetimes in (D+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(D+1)$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D>N$$\end{document}, admits a finite decomposition into monomials constructed from the metric tensor gμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\mu \nu }$$\end{document} and the unit timelike vector uμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\mu$$\end{document}, 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D>2$$\end{document}. 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.