Symmetric Yang--Mills theory in FLRW universes

In this work, we set up the theoretical framework and indicate future applications of symmetric Yang--Mills fields to cosmology. We analyze the coset space dimensional reduction scheme to construct pure Yang--Mills fields on spacetimes given as cylinders over cosets. Particular cases of foliations using $H^n$, dS$_n$ and AdS$_n$ slices as non-compact symmetric spaces are solved, compared to previous results in the literature, and generalized in a structured fashion. Coupling to general relativity in FLRW-type universes is introduced via the cosmological scale factor. For the hyperbolic slicing in 4D, the dynamics of the Einstein--Yang--Mills system is analytically solved and discussed. Finally, we generalize the analysis to warped foliations of the cylinders, which enlarge the range of possible spacetimes while also introducing a Hubble friction-like term in the equation of motion for the Yang--Mills field.