Characterize dynamical inheritance of FLRW symmetries

Characterize the precise constraints on the action and the intrinsic symmetries of the dynamical fields that ensure those fields dynamically inherit the symmetries of the FLRW background spacetime, thereby establishing the exact boundaries of symmetry inheritance beyond its imposition as an independent postulate.

Background

The paper assumes FLRW compatibility of all dynamical fields in order to classify arbitrary-rank tensor fields and derive the perfect-fluid form of rank-two corrections. The authors note that, in many gravitational theories, the relevant fields may instead inherit the background symmetries dynamically rather than through an imposed restriction.

The unresolved issue is to determine precisely which properties of the action in the generalized modified-gravity framework and which intrinsic field symmetries guarantee this inheritance. Resolving it would clarify the scope of the universality theorem and identify when the FLRW-compatible sector follows from the dynamics itself.