Can the diffeomorphism and Gauss constraints be holonomy corrected in the deformed algebra approach to modified gravity?

TL;DR

This work assesses whether holonomy corrections in the deformed algebra approach to modified gravity can extend to the diffeomorphism and Gauss constraints beyond the Hamiltonian constraint. It shows that correcting only the diffeomorphism constraint cannot yield a first-class algebra unless the diffeomorphism correction is absorbed by a counterterm, effectively leaving the Hamiltonian correction unchanged. When all constraints are holonomy corrected, the diffeomorphism and Gauss corrections must be proportionally related, and a new differential condition on the background correction function $g$ emerges, ensuring algebraic closure; this introduces new degrees of freedom and freedom in the functional form of the corrections, notably $oldsymboleta_{13}$ and $oldsymboleta_9$–$oldsymboleta_{12}$ linked to $oldsymbol h$ and $oldsymbol l$. The Bar–Mu scheme satisfies the derived relation, indicating compatibility with known successful holonomy prescriptions, while the analysis highlights potential phenomenological implications for perturbations and casts light on fundamental aspects of the effective theory.

Abstract

Deforming the algebra of constraint is a well known approach to effective loop quantum cosmology. More generally, it is a consistent way to modify gravity from the Hamiltonian perspective. In this framework, the Hamiltonian (scalar) constraint is usually the only one to be holonomy corrected. As a heuristic hypothesis, we consider the possibility to also correct the diffeomorphism and Gauss constraints. It is shown that it is impossible to correct the diffeomorphism constraint without correcting the Gauss one, while maintaining a first-class algebra. However, if all constraints are corrected, the algebra can be closed. The resulting differential equations to be fulfilled by the corrections (of the background and of the perturbations) are derived.