General covariant symmetric teleparallel cosmology

Symmetric teleparallel gravity theories, in which the gravitational interaction is attributed to nonmetricity of a flat, symmetric, but not metric-compatible affine connection, have been topic growing interest recent studies. Numerous works study cosmology symmetric assuming flat Friedmann-Lema\^itre-Robertson-Walker metric, while working so-called "coincident gauge", further that connection coefficients vanish. However, little attention has paid fact both these assumptions rely on freedom choose particular coordinate system order simplify metric or and they may, general, be achieved simultaneously. Here we construct most general geometry obeying conditions homogeneity isotropy, without making any properties coordinates, present our results usual cosmological coordinates coincident gauge. We find do agree, hold simultaneously allows only for very restricted class geometries. For case, derive energy-momentum-hypermomentum conservation relations dynamics selected theories. Our show contributes another scalar quantity into dynamics, decouples specific theories from if simple chosen, assumed systems agree. Most notably, $f(Q)$ falls this class.