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 ΛCDM 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 mν < 0.175 eV for the NH case, mν < 0.200 eV for the IH case, and mν < 0.136 eV for the DH case. After considering the GW data, i.e., using Planck+BAO+SN+GW, the constraint results become mν < 0.151 eV for the NH case, mν < 0.185 eV for the IH case, and mν < 0.122 eV for the DH case. We find that the GW data can help reduce the upper limits of mν 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 mν and other parameters. We show that the GW data of the ET could greatly improve the constraint accuracies of cosmological parameters.

Detection of gravitational waves produced by merger of binary compact objects could provide an independent way for measuring the luminosity distance to the gravitational-wave burst source, indicating that gravitational-wave observation, combined with observation of electromagnetic counterparts, can provide "standard sirens" for investigating the expansion history of the universe in cosmology. In this work, we wish to investigate how the future gravitational-wave standard siren observations would break the parameter degeneracies existing in the conventional optical observations and how they help improve the parameter estimation in cosmology. We take the third-generation ground-based gravitational-wave detector, the Einstein Telescope, as an example to make an analysis. By simulating 1000 events data in the redshift range between 0 and 5 based on the ten-year observation of the Einstein Telescope, we find that the gravitational-wave data could largely break the degeneracy between the matter density and the Hubble constant, thus significantly improving the cosmological constraints. We further show that the constraint on the equation-of-state parameter of dark energy could also be significantly improved by including the gravitational-wave data in the cosmological fit.