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 .