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We investigate the impacts of the gravitational-wave (GW) standard sirenobservation of the Einstein Telescope (ET) on constraining the total neutrinomass. We simulate 1000 GW events that would be observed by the ET in its10-year observation by taking the standard $\Lambda$CDM cosmology as a fiducialmodel. We combine the simulated GW data with other cosmological observationsincluding cosmic microwave background (CMB), baryon acoustic oscillations(BAO), and type Ia supernovae (SN). We consider three mass hierarchy cases forthe neutrino mass, i.e., normal hierarchy (NH), inverted hierarchy (IH), anddegenerate hierarchy (DH). Using Planck+BAO+SN, we obtain $\sum m_\nu<0.175$ eVfor the NH case, $\sum m_\nu<0.200$ eV for the IH case, and $\sum m_\nu<0.136$eV for the DH case. After considering the GW data, i.e., usingPlanck+BAO+SN+GW, the constraint results become $\sum m_\nu<0.151$ eV for theNH case, $\sum m_\nu<0.185$ eV for the IH case, and $\sum m_\nu<0.122$ eV forthe DH case. We find that the GW data can help reduce the upper limits of $\summ_\nu$ by 13.7%, 7.5%, and 10.3% for the NH, IH, and DH cases, respectively. Inaddition, we find that the GW data can also help break the degeneracies between$\sum m_{\nu}$ and other parameters. We show that the GW data of the ET couldgreatly improve the constraint accuracies of cosmological parameters.

Impacts of gravitational-wave standard siren observation of the Einstein Telescope on weighing neutrinos in cosmology