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

TL;DR

This paper investigates whether holonomy corrections can be consistently applied to all constraints in the deformed algebra approach to modified gravity, revealing that all constraints must be corrected simultaneously for algebra closure.

Contribution

It demonstrates that only correcting all constraints together preserves the first-class algebra, providing new differential equations for the corrections.

Findings

Correcting only the Hamiltonian constraint is insufficient.

Diffeomorphism and Gauss constraints must be corrected together.

A set of differential equations for the corrections is derived.

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.