Bounding the number of spacetime dimensions from precessing black hole binaries with the third-generation gravitational-wave detectors

Abstract: In the theories with extra dimensions, gravitational waves can leak into extra dimensions, resulting in a reduction in the amplitude of the observed gravitational waves. Such an effect modifies the standard luminosity distance of gravitational wave sources, which enables constraining extra dimensions by measuring both the luminosity distance and redshift from gravitational wave events. The main purpose of this paper is to assess the capacity of the third-generation gravitational wave detector network, Einstein Telescope and Cosmic Explorer, to constrain the theory of extra dimensions through observations of precessing binary black hole mergers. To this end, we generate a dataset of signals of precessing binary black hole mergers detectable by the detector network with one year of observations. We then employ the Fisher information matrix method for parameter estimation, together with a dark siren approach that obtains the redshift information from the galaxy catalog within a hierarchical Bayesian framework. Our results show that the precession in the binary system can significantly improve the precision of the luminosity distances and narrow their redshift ranges, thus leading to a tighter constraint on the extra dimensions. Based on this approach, we constrain the number of spacetime dimensions $D$ and the screening scale $R_{\rm c}$ of the extra dimensions, obtaining $D = 3.99^{+0.07}_{-0.06}$ and $\log_{10}(R_c/\mathrm{Mpc}) > 3.76$ at $68\%$ credible level. This result improves previous ones from an analysis with GWTC-3 by about one order of magnitude. This work provides theoretical foundations and projected sensitivities for future third-generation gravitational wave observations in exploring the effects of extra dimensions.