How will our knowledge on short gamma-ray burst affect the distance measurement of binary neutron star?

Binary neutron star associated with short gamma-ray burst has drawn wide attention ever since the observation of GW170817, due to its prospect in cosmology. While further application of this sort of event suffers from the problem of degeneracy between luminosity distance and inclination angle. To address this issue, proposed in previous research is the utilization of information from short gamma-ray burst. In this paper, we aim to investigate the effect of different Gaussian priors of the inclination angle. To test the property of resulting posterior distribution, we generate four catalogues of 1000 events by varying the number of third-generation detectors and the scale of prior. It turns out that a network of detectors tends to recognize more and farther events than a single detector. Besides, adopting tighter prior and employing multiple detectors both lead to lower error at given redshifts. Also considered is the validity of a widely adopted formula $\sigma_{\Delta d_{\rm L}0} = 2d_{\rm L} / \rho$, which undergoes the change from overestimating to underestimating with the increase of redshift. In the case of $\Lambda$CDM cosmology, 800, 300, 800 and 600 events are required for the four configurations to achieve $1\%$ $H_0$ accuracy. With all 1000 events in each catalogue, $H_0$ can be constrained to $0.90\%$, $0.60\%$, $0.91\%$ and $0.77\%$, while the errors of $\Omega_m$ are 0.014, 0.010, 0.018 and 0.016 respectively. Besides, adopting $\sigma_{\Delta d_{\rm L}0}$ leads to underestimation on the errors of cosmological parameters for tighter prior and overestimation in the opposite case. Results of Gaussian process also show that gravitational wave standard siren can reach higher redshift than traditional standard candles, especially when a network of detector is available, while alteration of the prior only has moderate impact.