Dynamical Barbero--Immirzi field coupled to quintessence: gravitational-wave propagation constraints and next-generation forecasts

Abstract

We investigate the imprints of a dynamical Barbero--Immirzi (BI) field $\gamma(x)$ coupled to a quintessence scalar field $\phi$ on gravitational-wave (GW) propagation. In the framework of Einstein--Cartan--Holst gravity, promoting $\gamma$ to a dynamical scalar introduces a stress--energy that back-reacts on the metric, modifying the GW friction term. A minimal coupling $\propto\beta\,\phi^2\gamma^2$ between the BI field and quintessence leads to a two-parameter extension of the Belgacem--Maggiore parametrization, characterized by $\xBI$ (from the isolated BI field) and $\xcp$ (from the coupling). Using the LIGO--Virgo--KAGRA GWTC-3 dark-siren constraint $\Xi_0=1.2^{+0.7}_{-0.7}$, we obtain the first simultaneous constraints: $|\xBI|\lesssim0.7$ and $|\xcp|\lesssim0.13$ at 90\% credibility. We then forecast the sensitivity of next-generation detectors Einstein Telescope (ET) and Cosmic Explorer (CE), showing that a 10-year observation campaign can improve these bounds by roughly one to two orders of magnitude depending on the parameter---a factor of $\sim\!20$ for $\xBI$ and $\sim\!20$ for $\xcp$---reaching $\sigma(\xBI)\sim3\times10^{-2}$ and $\sigma(\xcp)\sim1.2\times10^{-2}$. Translated into microscopic parameters, this corresponds to $\gamma_{\rm dyn}\lesssim10^{-12}$ and $\beta\lesssim10^{-3}$, providing a powerful new observational window into the interplay between quantum-gravity phenomenology and dark energy.