Estimating cosmological parameters by the simulated data of gravitational waves from the Einstein Telescope

We investigate the constraint ability of the gravitational wave (GW) as the standard siren on the cosmological parameters by using the third-generation gravitational wave detector: the Einstein Telescope. We simulate the luminosity distances and redshift measurements from 100 to 1000 GW events. We use two different algorithms to constrain the cosmological parameters. For the Hubble constant $H_0$ and dark matter density parameter $\Omega_m$, we adopt the Markov chain Monte Carlo approach. We find that with about 500-600 GW events we can constrain the Hubble constant with an accuracy comparable to \textit{Planck} temperature data and \textit{Planck} lensing combined results, while for the dark matter density, GWs alone seem not able to provide the constraints as good as for the Hubble constant; the sensitivity of 1000 GW events is a little lower than that of \textit{Planck} data. It should require more than 1000 events to match the \textit{Planck} sensitivity. Yet, for analyzing the more complex dynamical property of dark energy, i.e., the equation of state $w$, we adopt a new powerful nonparametric method: the Gaussian process. We can reconstruct $w$ directly from the observational luminosity distance at every redshift. In the low redshift region, we find that about 700 GW events can give the constraints of $w(z)$ comparable to the constraints of a constant $w$ by \textit{Planck} data with type Ia supernovae. Those results show that GWs as the standard sirens to probe the cosmological parameters can provide an independent and complementary alternative to current experiments.