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

We investigate the impacts of the gravitational-wave (GW) standard siren observation of the Einstein Telescope (ET) on constraining the total neutrino mass. We simulate 1000 GW events that would be observed by the ET in its 10-year observation by taking the standard $\Lambda$CDM cosmology as a fiducial model. We combine the simulated GW data with other cosmological observations including cosmic microwave background (CMB), baryon acoustic oscillations (BAO), and type Ia supernovae (SN). We consider three mass hierarchy cases for the neutrino mass, i.e., normal hierarchy (NH), inverted hierarchy (IH), and degenerate hierarchy (DH). Using Planck+BAO+SN, we obtain $\sum m_\nu<0.175$ eV for 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., using Planck+BAO+SN+GW, the constraint results become $\sum m_\nu<0.151$ eV for the NH case, $\sum m_\nu<0.185$ eV for the IH case, and $\sum m_\nu<0.122$ eV for the DH case. We find that the GW data can help reduce the upper limits of $\sum m_\nu$ by 13.7%, 7.5%, and 10.3% for the NH, IH, and DH cases, respectively. In addition, 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 could greatly improve the constraint accuracies of cosmological parameters.