Gravitational-wave propagation constraints on the dynamical Barbero--Immirzi field from GWTC-3 dark sirens

The Barbero–Immirzi (BI) parameter, originally a constant in the Holst term of first-order gravity, can be promoted to a dynamical scalar field $\gamma (x)$ coupled to torsion in Einstein–Cartan theory. We formulate the leading gravitational-wave (GW) propagation effect of this dynamical BI sector as an effective friction term, ${\mathrm{\Gamma }}_{\mathrm{B}\mathrm{I}}={\xi }_{\mathrm{B}\mathrm{I}}H$, and relate it to the GW luminosity distance. We then compare the resulting distance-redshift relation with the published LIGO–Virgo–KAGRA GWTC-3 dark-siren constraint on the Belgacem–Maggiore parameter ${\mathrm{\Xi }}_0=1.2_{-0.7}^{+0.7}$. Without repeating the LVK hierarchical Bayesian analysis, and using a representative-redshift mapping together with a Gaussian approximation to the quoted posterior, we obtain the conservative effective constraint $\mathrm{\mid }{\xi }_{\mathrm{B}\mathrm{I}}\mathrm{\mid }\lesssim 2.1$ at approximately 90% credibility. For the specific BI realization considered here, ${\xi }_{\mathrm{B}\mathrm{I}}$ is non-negative at the leading order adopted, so the corresponding one-sided interpretation is $0\le {\xi }_{\mathrm{B}\mathrm{I}}\lesssim 2.1$. The observable constraint is therefore a statement about the effective BI-induced GW friction. Translating it into the microscopic fluctuation amplitude requires an independent normalization of the BI–torsion sector; we do not impose such a normalization phenomenologically. This GW-propagation test probes a sector complementary to torsion-wave and matter-sector constraints and does not rely on an assumption that astrophysical sources excite propagating torsion modes.