Testing the Weak Equivalence Principle with the Binary Neutron Star Merger GW 170817: The Gravitational Contribution of the Host Galaxy

The successful detection of the binary neutron star merger GW 170817 and its electromagnetic 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 profile and the Hernquist profile, we present an analytic model to calculate the galactic potential, in which the possible locations of the source from the observed angle offset and the second supernova kick are accounted for. We show that the upper limit of Δγ is 10^-9 for the comparison between GW 170817 and a gamma-ray burst (GRB 170817A), and it is 10^-4 for the comparison between GW 170817 and a bright optical transient (SSS 17a, now with the IAU identification of AT 2017gfo). These limits are more stringent by one to two orders of magnitude than those determined solely using the measured MW potential in the literature. We demonstrate that the WEP test is strengthened by the 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% from the results with a typical kick velocity of 400 to ∼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 Δγ, with a maximum mass ${10}^{12.4}{h}^{-1}\,{M}_{\odot }$ , is nearly two times more stringent than that of the minimum mass ${10}^{12.0}{h}^{-1}\,{M}_{\odot }$ .