Testing the weak equivalence principle with the binary neutron star merger GW170817: the gravitational contribution of the host galaxy

The successful detection of the binary neutron star (BNS) merger GW170817 and its electromagnetic (EM) counterparts has provided an opportunity to explore the joint effect of the host galaxy and the Milky Way (MW) on the weak equivalence principle (WEP) test. In this paper, using the Navarro$-$Frenk$-$White (NFW) profile and the Herquist profile, we present an analytic model to calculate the galactic potential, in which the possible locations of the source by the observed angle offset and the second supernova (SN2) kick are accounted for. We show that the upper limit of $\Delta \gamma$ is $10^{-9}$ for the comparison between GW170817 and a gamma-ray burst (GRB 170817A), and it is $10^{-4}$ for the comparison between GW170817 and a bright optical transient (SSS17a, now with the IAU identification of AT 2017gfo). These limits are more stringent by one to two orders of magnitude than those determined solely by the measured MW potential in the literature. We demonstrate that the WEP test is strengthened by contribution from the host galaxy to the Shapiro time delay. Meanwhile, we also find that large natal kicks produce a maximum deviation of about $20\%$ to the results with a typical kick velocity 400$\sim$ 500 km s$^{-1}$. Finally, we analyze the impact from the halo mass of NGC 4993 with a typical 0.2 dex uncertainty, and find that the upper limit of $\Delta \gamma$, with a maximum mass $10^{12.4}h^{-1} M_{\odot}$, is nearly two times more stringent than that of the minimum mass $10^{12.0}h^{-1} M_{\odot}$.