A comprehensive forecast for cosmological parameter estimation using joint observations of gravitational waves and short $\gamma$-ray bursts

In the third-generation (3G) gravitational-wave (GW) detector era, the multi-messenger GW observation for binary neutron star (BNS) merger events can exert great impacts on exploring the cosmic expansion history. In this work, we comprehensively explore the potential of 3G GW standard siren observations in cosmological parameter estimations by considering the 3G GW detectors and the future short $\gamma$-ray burst (GRB) detector THESEUS-like telescope joint observations. Based on the 10-year observation of different detection strategies, we predict that the numbers of detectable GW-GRB events are 334--674 with the redshifts $z<3.5$ and the inclination angles $\iota<15^{\circ}$. For the cosmological analysis, we consider the $\Lambda$CDM, $w$CDM, $w_0w_a$CDM models, and interacting dark energy (IDE) models. We find that GW can tightly constrain the Hubble constant with precisions of $0.345\%-0.065\%$, but perform not well in constraining other cosmological parameters. Fortunately, GW could effectively break the cosmological parameter degeneracies generated by the mainstream EM observations, CMB+BAO+SN (CBS). When combining the mock GW data with the CBS data, CBS+GW can tightly constrain the equation of state of dark energy $w$ with a precision of 1.26\%, close to the standard of precision cosmology. Meanwhile, the addition of GW to CBS could improve constraints on cosmological parameters by $34.2\%-94.9\%$. In conclusion, GW standard siren observations from 3G GW detectors could play a crucial role in helping solve the Hubble tension and probe the fundamental nature of dark energy.