Anomaly freedom in effective Loop Quantum Cosmology refined: extended functional dependence of the counter-terms

TL;DR

This work probes whether counterterms in the deformed algebra approach to effective loop quantum cosmology can depend on the full phase-space $(K^i_a,E^a_i)$ rather than solely on background variables $(\mathfrak{c},\mathfrak{p})$. By deriving the extended constraint brackets and anomalies, the authors show that the usual holonomy replacement $\mathfrak{c}\rightarrow \sin(\delta\mathfrak{c})/\delta$ cannot, in general, be consistently implemented under this extension. They compute the full anomaly structure and obtain two explicit, nontrivial scenarios for the background/perturbation holonomy corrections, leading to two principal differential relations between $g$ and $\tilde{g}$, as well as further restrictions on the counterterms. A key outcome is that a consistent first-class algebra demands strong constraints on holonomy corrections, and in a simple illustrative case even forces $g=\mathfrak{c}$ (i.e., no correction at the background level). The results imply that extended functional dependence of counterterms significantly reshapes the space of viable effective LQC models and motivate further study of functional dependencies and their phenomenological consequences for cosmological perturbations.

Abstract

Instead of assuming that they depend only on the background variables, we investigate the hypothesis that counter-terms appearing in the deformed algebra approach to loop quantum cosmology depend on the full phase-space variables. We derive the associated anomalies and solve the entire system in several specific cases. New restrictions on the generalized holonomy corrections are obtained.