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

On 2015 September 14, the Advanced LIGO detectors picked up the first binary black hole coalescence, GW 150914, beginning a new era of observational gravitational-wave (GW) astronomy (Abbott et al. 2016). Meanwhile, it is believed that the coalescence of a binary neutron star (BNS) system is expected to produce, in addition to GWs, multiple electromagnetic (EM) signatures on different timescales (Nakar 2007; Metzger & Berger 2012). For a long time, people had been looking for the EM partners of GWs, but well-accepted results have not been obtained other than a few possible events such as the Fermi Gamma-ray Burst Monitor (GBM) transient 150914 (Connaughton et al. 2016). Then, a big breakthrough came with the detection of the GW signal GW 170817, which was recorded by the LIGO/Virgo (LIV) GW observatory network on 2017 August 17, 12:41:04 UTC. Later analysis by Abbott et al. (2017a) showed that GW 170817 was consistent with a BNS inspiral and merger. The GW 170817 sky map was then released by LIGO/Virgo, driving an intensive multimessenger campaign covering the entire EM spectrum to search for the counterparts of the event (Abbott et al. 2017b). A gamma-ray signal, classified as a short gamma-ray burst (sGRB), GRB 170817A, coincident in time and sky location with GW 170817, was independently detected by Goldstein et al. (2017) using the GBM and by Savchenko et al. (2017) using the International Gamma-Ray Astrophysics Laboratory. Beyond the sGRB, multiple independent surveys across the EM spectrum were launched in search of a counterpart. An optical counterpart (OT), Swope Supernova Survey 2017a (SSS 17a/AT 2017gfo), was first discovered by the One-Meter Two Hemisphere (1M2H) team in the optical less than 11 hr after the merger, associated by Coulter et al. (2017) with NGC 4993, a nearby early-type E/S0 galaxy. Five other teams, DLT40 (Yang et al. 2017), VISTA (Tanvir et al. 2017), MASTER (Lipunov et al. 2017), DECam (Soares-Santos et al. 2017), and Las Cumbres (Arcavi et al. 2017), made independent detections of the same optical transient and host galaxy all within about one hour and reported their results to one another within about five hours. Meanwhile, the source was reported to be offset from the center of NGC 4993 by a projected distance of about 10'' (Abbott et al. 2017c; Coulter et al. 2017; Haggard et al. 2017; Kasliwal et al. 2017; Levan et al. 2017), and the binary was determined to potentially lie in front of the bulk of the host galaxy due to the absence of interstellar medium (ISM) absorption in the counterpart spectrum in Hubble Space Telescope (HST) and Chandra imaging, combined with Very Large Telescope (VLT)/MUSE integral field spectroscopy (Levan et al. 2017). It should be mentioned that the statement above on the discovery of the EM counterpart of GW 170817 is not sufficiently convincing, and we direct the reader to the relevant reviews concerning the complete research on the counterpart of GW 170817 (e.g., Abbott et al. 2017b).

Testing fundamental physics through high-energy astronomical events (HEAE) has always been the subject of research (Will 2014, 2006). One famous scheme consists of testing the weak equivalence principle (WEP) by comparing difference waves in HEAE. The pioneering test was that between photons and neutrinos in supernova SN 1987A in the Large Magellanic Cloud by Longo (1988) and Krauss & Tremaine (1988). Recently, such schemes have sprung up in physics and astronomy, mainly focusing on cosmic transients such as GRBs (e.g., Gao et al. 2015), fast radio bursts (e.g., Wei et al. 2015; Tingay & Kaplan 2016), blazar flares (e.g., Wang et al. 2016; Wei et al. 2016), and the GW event of GW 150914 (e.g., Kahya & Desai 2016; Wu et al. 2016; Liu et al. 2017).

After the BNS merger GW 170817 and its multiple EM signatures were observed by various astronomical observatories, several pioneering works presented WEP tests and produced constraints on the parameterized post-Newtonian (PPN) parameters (Abbott et al. 2017d; Wang et al. 2017; Wei et al. 2017). Abbott et al. (2017d) constrained the deviation of the speed of gravity, and violations of Lorentz invariance and the equivalence principle were presented via the observed temporal offset, the distance to the source, and the assumed emission time difference, in which the bound on the difference of ${\gamma }_{\mathrm{GW}}-{\gamma }_{\mathrm{EM}}$ was given in the range of $[-2.6\times {10}^{-7},1.2\times {10}^{-6}]$. Then, by assuming the simultaneous emission of GWs and photons, Wang et al. (2017) presented a result of ${\rm{\Delta }}\gamma \leqslant {10}^{-7}$, which could be improved to $4\times {10}^{-9}$ using the potential fluctuations from large-scale structure, as originally proposed by Nusser (2016). Meanwhile, Wei et al. (2017) considered a Keplerian potential ${\rm{\Phi }}=-{GM}/r$ for two cases: the MW and the Virgo Cluster. A total mass of $6\times {10}^{11}{M}_{\odot }$ was adopted in the former case, and it gave the upper limits $\sim {10}^{-8}$ for GW 170817/GRB 170817A and $\sim {10}^{-3}$ for GW 170817/AT 2017gfo.

Meanwhile, we noticed that according to the K-band luminosity in the 2MASS Redshift Survey (see Huchra et al. 2012), the stellar mass of NGC 4993 ($\sim 6.2\times {10}^{10}{M}_{\odot }$) is almost equal to that of the MW ($\sim 6.4\times {10}^{10}{M}_{\odot }$). Therefore, it can be expected that the gravitational effect from the host galaxy will largely enhance the WEP test when comparing with the tests that only consider the MW. As far as we know, no works refer to tests involving NGC 4993. Motivated by this situation, we restudy the WEP test of GW 170817 but consider a joint gravitational potential that consists of the host galaxy and the MW. In this work, we focus on three aspects of the tests: the observed angle offset from the source, the possible large natal kick on the BNS, and the typical uncertainty of 0.2 dex on the halo mass of NGC 4993.

The outline of this paper is as follows. In Section 2, we present a computable galactic model by considering the observed angle offset. In Section 3, we obtain the constraints on the WEP test via the joint potential. We then explore the impacts of the large natal kick and the halo mass of NGC 4993 on the tests in Sections 4 and 5, respectively. Section 6 presents the conclusion.

2.1. Travel Path of Waves from the Merge Position

2.2. Gravitational Potential from the Joint Effect of the MW and NGC 4993

In this work, the enclosed masses consist of the stellar mass and the dark matter (DM) halo with spherically symmetric profiles. The former is described by a Hernquist profile (Hernquist 1990), and the latter is described by a Navarro–Frenk–White (NFW) profile (Navarro et al. 1996). The density distribution of the stellar component was given by Hernquist (1990) as follows:

$$\begin{eqnarray}&&{\rho }_{s}(r)=\displaystyle \frac{{M}_{s}{a}_{b}}{2\pi r{\left(r+{a}_{b}\right)}^{3}},\end{eqnarray} \tag{ 1 }$$

where M_s is the total stellar mass and a_b is a scale length. The potential is thus given by

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{stellar}}(r)=-\displaystyle \frac{{{GM}}_{s}}{r+{a}_{b}}.\end{eqnarray} \tag{ 2 }$$

The stellar mass of the MW is $6.4\times {10}^{10}{M}_{\odot }$, given by McMillan (2011), and the stellar mass of NGC 4993 is $6.2\times {10}^{10}{M}_{\odot }$, provided by Lim et al. (2017). The bulge scale length is 0.5 kpc for the MW, given by Sofue et al. (2009). The bulge scale length of NGC 4993 is about 0.55 times the half-light radius ${R}_{\mathrm{eff}}$ (Hernquist 1990), which was recently observed to be 155 ± 15, which corresponds to a 3.0 kpc offset for a distance of 40 Mpc, using HST measurements (Hjorth et al. 2017).

The density distribution of the DM halo component was given by Navarro et al. (1996) as

$$\begin{eqnarray}&&{\rho }_{\mathrm{DM}}(r)=\displaystyle \frac{{\rho }_{0}{R}_{s}}{r}{\left(1+\displaystyle \frac{r}{{R}_{s}}\right)}^{-2},\end{eqnarray} \tag{ 3 }$$

where ${\rho }_{0}$ is the density parameter, and R_s is the scale radius defined by ${R}_{s}={R}_{200}/{c}_{200}$. R₂₀₀ is the position at which the enclosed density is 200 times the universe's critical density. c₂₀₀ is the concentration parameter obtained via the empirical expression given by Duffy et al. (2008):

$$\begin{eqnarray}&&{\mathrm{log}}_{10}{c}_{200}=({\mathrm{0.76\pm }}_{0.01}^{0.01})+(-{\mathrm{0.10\pm }}_{0.01}^{0.01}){\mathrm{log}}_{10}\left(\displaystyle \frac{{M}_{200}}{{M}_{\mathrm{pivot}}}\right),\end{eqnarray} \tag{ 4 }$$

where the median halo mass ${M}_{\mathrm{pivot}}=2\times {10}^{12}{h}^{-1}\,{M}_{\odot }$. Based on the report by the Planck Collaboration (2016), the median value for the Hubble parameter is h = 0.679. The adopted halo mass of the MW is 2.5 ± 1.5 × 10¹² ${M}_{\odot }$, obtained from the numerical action method by Phelps et al. (2013). The adopted halo mass of NGC 4993 is $({10}^{12.2}{h}^{-1}){M}_{\odot }$, obtained from the 2MASS Redshift Survey (2MRS) in the low-redshift universe by Lim et al. (2017). Therefore, the parameters (ρ₀, R_s, R₂₀₀, and c₂₀₀) can be obtained by modeling the NFW halo, and they are listed in Table 2. The potential of the NFW halo is thus given by

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{DM}}(r)=-\displaystyle \frac{4\pi G{\rho }_{0}{R}_{s}^{3}}{r}\mathrm{ln}\left(1+\displaystyle \frac{r}{{R}_{s}}\right).\end{eqnarray} \tag{ 5 }$$

Table 2.  NFW DM Halo Parameters

NFW Parameters	MW	NGC 4993
Median r200 (kpc)	288	282
Concentration parameter c200	5.8	5.9
Density parameter ρ0 (${10}^{-3}{M}_{\odot }\,{\mathrm{pc}}^{-3}$)	1.6	1.6
Scale radius Rs (kpc)	49	48

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Based on the main components of the stellar and DM halos, the total potential Φ_total can be given as

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{total}}={{\rm{\Phi }}}_{\mathrm{mw}}+{{\rm{\Phi }}}_{\mathrm{host}},\end{eqnarray} \tag{ 6 }$$

where the potential ${{\rm{\Phi }}}_{\mathrm{mw}}$ of the MW (or the ${{\rm{\Phi }}}_{\mathrm{host}}$ of NGC 4993) is composed of the Hernquist stellar sector ${{\rm{\Phi }}}_{{\rm{s}}1}$ (or ${{\rm{\Phi }}}_{{\rm{s}}2}$ of NGC 4993) from Equation (2) and the NFW halo sector ${{\rm{\Phi }}}_{{\rm{D}}1}$ (or ${{\rm{\Phi }}}_{{\rm{D}}2}$ of NGC 4993) from Equation (5). Therefore, ${{\rm{\Phi }}}_{\mathrm{mw}}$ and ${{\rm{\Phi }}}_{\mathrm{host}}$ are shown by

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{mw}}={{\rm{\Phi }}}_{{\rm{s}}1}(r)+{{\rm{\Phi }}}_{{\rm{D}}1}(r),\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{host}}={{\rm{\Phi }}}_{{\rm{s}}2}(\chi )+{{\rm{\Phi }}}_{{\rm{D}}2}(\chi ),\end{eqnarray} \tag{ 8 }$$

where $\chi (r,\theta )$ is given in Equation (A12) and refers to the dynamical distance from the center of NGC 4993 to a point on the travel path of the waves. The total potential Φ_total is illustrated in Figure 1. The left panel shows the profile, and the median magnitude of the luminosity distance d = 40 Mpc is adopted. The right panel shows the path considering the condition given by Equation (A12), where the observed luminosity distance of $d={40}_{-14}^{+8}$ Mpc is adopted. The two panels strongly suggest that the impacts of the host galaxy on the total potential should not be ignored. The model parameters for the Hernquist stellar profile both in the MW and NGC 4993 are given by Equation (2), and the parameters of the NFW halo are listed in Table 2.

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The Shapiro time delay ${\rm{\Delta }}{t}_{\mathrm{gra}}$ can be obtained by integrating the potential along the path (e.g., Shapiro 1964; Krauss & Tremaine 1988; Longo 1988):

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{\mathrm{gra}}=-\displaystyle \frac{1+\gamma }{{c}^{3}}{\int }_{{r}_{e}}^{{r}_{o}}{\rm{\Phi }}(r){dr},\end{eqnarray} \tag{ 9 }$$

where r_e = r_S and r_o = r_G denote the positions of the sender and receiver, respectively. Meanwhile, in order to define the waves traveling along the path from the merge position to Earth, the condition given by Equation (B1) must be upheld. Due to the possible large natal kick of the binary (see Abbott et al. 2017c), the source is possibly kicked outside of the gravitational grasp of NGC 4993. The angle offset and the large natal kick thus become the major factors affecting the position of the transient in the WEP test.

3.1. Constraints on the WEP Test between GW 170817 and GRB 170817A

In our tests, GRB 170817A and AT 2017gfo are substituted into the calculations, acting as the counterparts of GW 170817. The temporal offset between the BNS merger and the GRB is 1.734 ± 0.054 s, which can be treated as the maximum time delay caused by the gravitational potential. Substituting the Shapiro time into Equation (9), we obtain the upper limit of the PPN parameter difference between GW 170817 and GRB 170817A in the potentials of the two galaxies, denoted by $| {\rm{\Delta }}{\gamma }_{1}| \equiv | {\gamma }_{\mathrm{GW}}-{\gamma }_{\mathrm{EM}}|$. Here, two extreme positions of the transient are considered: one is located at the projected point of the center of NGC 4993 and the other is located near the edge of the galaxy. Appendix B gives the details of the geometry of the angle offset and the extreme positions of the transient.

The results are listed in Table 3 (see ${\gamma }_{1}$) and primarily show that the difference $| {\rm{\Delta }}{\gamma }_{1}|$ between GW 170817 and GRB 170817A is under 10⁻⁹ due to the joint effect of the MW and NGC 4993. When comparing with previous results, which only accounted for the MW potential, our result is more stringent by two orders of magnitude than the result of 10⁻⁷, using the method of the impact parameter from Wang et al. (2017). Additionally, it is also more stringent by one order of magnitude than the result of 10⁻⁸ obtained via the Keplerian potential method from Wei et al. (2017). The total mass of the MW was adopted as $6\times {10}^{11}{M}_{\odot }$ in both methods. When the gravitational contribution of NGC 4993 was added to the WEP test, the mass of the galaxies ($\sim {10}^{12}{M}_{\odot }$) is larger than that of the MW ($\sim {10}^{11}{M}_{\odot }$) adopted before. Therefore, our results are enhanced significantly by one to two orders of magnitude compared to those only considering the MW when adding the contribution of the host galaxy to the tests.

Table 3.  Upper Limits of the PPN Parameter Differences for Three Kinds of Enclosed Mass

Comparison Type	${r}_{S}={GN}^{\prime}$ a	rS = GRNb	δ1c	Enclosed Mass
GW 170817/GRB 170817A ($| {\rm{\Delta }}{\gamma }_{1}| \lesssim$)	${6.1}_{-0.3}^{+0.8}\times {10}^{-9}$	${6.1}_{-0.3}^{+0.8}\times {10}^{-9}$	0.0%	MWd
	${6.5}_{-0.3}^{+0.8}\times {10}^{-9}$	${6.5}_{-0.3}^{+0.9}\times {10}^{-9}$	1.3%	NGC 4993
	${3.1}_{-0.2}^{+0.4}\times {10}^{-9}$	${3.2}_{-0.2}^{+0.4}\times {10}^{-9}$	0.6%	MW + NGC 4993e
GW 170817/AT 2017gfo ($| {\rm{\Delta }}{\gamma }_{2}| \lesssim$)	${1.4}_{-0.1}^{+0.2}\times {10}^{-4}$	${1.4}_{-0.1}^{+0.2}\times {10}^{-4}$	0.0%	MWd
	${1.5}_{-0.1}^{+0.2}\times {10}^{-4}$	${1.5}_{-0.1}^{+0.2}\times {10}^{-4}$	1.3%	NGC 4993
	${7.1}_{-0.3}^{+0.9}\times {10}^{-5}$	${7.1}_{-0.4}^{+0.9}\times {10}^{-5}$	0.6%	MW + NGC 4993e

Notes. The constraints of the WEP tests are calculated through two possible source locations r_S = GN' (or GR_N) by taking into account the observed angle offset (Abbott et al. 2017c; Coulter et al. 2017) and the absence of ISM absorption in the counterpart's spectrum (Levan et al. 2017).

^aIt corresponds to the maximum propagation distance, and the source is located at the projected point N' from the center of NGC 4993 (see Figure A1). ^bIt corresponds to the minimum propagation distance, and the source is located at the point R_N near the edge of NGC 4993. ^cThe influence of the change of the source position on the WEP test is quantified through the deviation δ₁ defined by ${\delta }_{1}=\left[{\rm{\Delta }}\gamma ({{GR}}_{N})-{\rm{\Delta }}\gamma ({GN}^{\prime} )\right]/{\rm{\Delta }}\gamma ({GN}^{\prime} )$. A positive value of δ₁ means that the constraint on the PPN parameter becomes tighter when the source position changes from the edge to the center of NGC 4993. ^dThe test of only the MW is calculated via the potential Φ_MW, Equation (7), and the deviation δ₁ that disappeared indicates that the impact of the source location in NGC 4993 on the test is almost entirely suppressed. ^eThe test of the maximum enclosed mass is calculated via the total potential Φ_total, Equation (6).

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Another advantage of exploring the host galaxy is that it provides us with an alternative to alleviate the suppression of the WEP constraint caused by the integral of the potential far beyond the MW. The contribution to the test caused by the alteration of the source location is highly suppressed for the MW (see lines 1 and 4 in Table 3). However, this kind of suppression can be alleviated when the test contains the host galaxy. If the location of the transient changes from the edge of the galaxy to the center, the deviation δ₁ is positive, and the constraints are more stringent by about 1% (see lines 2, 3, 5, and 6 in Table 3).

3.2. Constraints on the WEP Test between GW 170817 and AT 2017gfo

The actual distance to the final merger is also strongly influenced by the second supernova (SN2) kick. According to the kinematic modeling from SN2 to the merger, the slingshot effect caused by the tangential SN2 kick is much more efficient than a purely radial kick driving the binary to the outer regions of the galaxy (see Abbott et al. 2017c). Therefore, a large natal kick to the binary could make it merge at a greater distance. The final merger position is possibly out of range of the galaxy for a larger SN2 kick, as long as the observed offset angle is respected. Therefore, the merger position S may be out of range of $| N^{\prime} {R}_{N}|$ (see Figure A1). The real distance R_real from the SN2 to the merger can be simplified as follows:

$$\begin{eqnarray}&&{R}_{\mathrm{real}}={\tau }_{\mathrm{gwr}}{V}_{\mathrm{kick}},\end{eqnarray} \tag{ 10 }$$

where ${\tau }_{\mathrm{gwr}}$ is the merger time of the BNS, and V_kick is the kick velocity along the radial direction. The merger time of the BNS ${\tau }_{\mathrm{gwr}}$ is in the range of

$$\begin{eqnarray}&&{\tau }_{\mathrm{gwr}0}\lesssim {\tau }_{\mathrm{gwr}}\lesssim {t}_{\mathrm{Hubble}},\end{eqnarray} \tag{ 11 }$$

where ${t}_{\mathrm{Hubble}}=1/{H}_{0}$ with H₀ = 100 h km s⁻¹ Mpc⁻¹ is the Hubble time, and ${\tau }_{\mathrm{gwr}0}$ is the minimum merger time, 86 Myr, from the observation of PSR J0737−3039A/B in a highly relativistic orbit (see Tauris et al. 2017). Meanwhile, the kick velocity V_kick is assumed to be constant after SN2, whereas there is ample observational evidence for large NS kicks (typically 400∼500 km s⁻¹) in observations of young radio pulsars. Therefore, the distance of the binary after the SN2 can be estimated as $(36\mbox{--}45)\,\mathrm{kpc}\lesssim {R}_{\mathrm{real}}\lesssim (2.7\mbox{--}3.4)\,\mathrm{Mpc}$. It is apparently beyond the diameter of the galaxy NGC 4993, 26 kpc (see Lauberts & Valentijn 1989). The constraints from the large natal kick on the WEP test are thus obtained and listed in Table 4, in which the case of the maximum enclosed mass, i.e., the scale of the MW + NGC 4993, is considered and the perturbation of distance comes from the kick distance R_real. Because the transient location is most possibly directly in front of NGC 4993, the travel path will be reduced after a larger natal kick compared to the calculation without a kick. The upper limit of Δγ is thus less stringent by about 2%–4% than the cases without the kick, in Section 3.

The results show that the maximum upper limit of Δγ comes from the case where the traveling waves have a maximum kick speed of ${V}_{\mathrm{kick}}\sim 500$ km s⁻¹ within the Hubble time, and the travel distance is reduced by 3.4 Mpc. In this case, the upper limit $3.7\times {10}^{-9}$ thus increases by nearly 20% compared to the result of $3.2\times {10}^{-9}$ without the kick effect in Table 3. This shows that the large natal kick has a significant impact on the WEP tests.

In Lim's catalogs (see Lim et al. 2017), the 2MRS(M) of the Low Redshift Group Catalog was given using Proxy-M to estimate the halo masses of the galaxies. In 2MRS(M), one can find that the group of NGC 4993 is a poor system that consists of only two galaxies. The group is inside the region of completeness for a given halo mass, and thus, we can assign halo mass by abundance matching. In Figure 2, we present five panels to illustrate how the related groups change in three kinds of catalogs: the 2MRS Group Catalog provided by Lu et al. (2016), the 2MRS(L), and the 2MRS(M) in the Low Redshift Group Catalog. We find that the group will become a poor system, with a decreasing number of members, and the properties of the group would be close to those of the galaxies. Therefore, it is reasonable to use the halo mass ${10}^{12.2}\,{h}^{-1}\,{M}_{\odot }$ in the poor system to identify that of galaxy NGC 4993.

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Because the halo masses assigned by the group finder are unbiased with respect to the true halo masses but have a typical uncertainty of ∼0.2 dex in the catalog of 2MRS(M), the halo mass will change, and its range then becomes $[{10}^{12.0}{h}^{-1}\,{M}_{\odot },{10}^{12.4}{h}^{-1}\,{M}_{\odot }]$. The related NFW halo parameters are listed in Table 5. By using these new parameters, the upper limits of ${\rm{\Delta }}{\gamma }_{1}$ and ${\rm{\Delta }}{\gamma }_{2}$ are recalculated in Table 6. It is thus clear that the tests of the maximum halo mass are nearly two times more stringent than those of the minimum halo mass. This means that the influence of the halo mass of NGC 4993 on the results of the WEP test is significant.

Table 5.  NFW Halo Parameters in NGC 4993

NFW Parameters	Upper Limit	Lower Limit
Halo mass ${M}_{\mathrm{DM}}$ (${M}_{\odot }\,{h}^{-1}$)	${10}^{12.4}$	${10}^{12.0}$
Median r200 (kpc)	328	243
Concentration parameter c200	5.6	6.2
Density parameter ρ0 (${10}^{-3}{M}_{\odot }\,{\mathrm{pc}}^{-3}$)	1.5	1.8
Scale radius Rs (kpc)	58	39

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Table 6.  Upper Limits of Δγ with the Maximum and Minimum Halo Masses in NGC 4993

Comparison Type	Maximum Halo Mass	Minimum Halo Mass	δ2	Enclosed Mass
GW 170817/GRB 170817A (${\rm{\Delta }}{\gamma }_{1}$)	${4.2}_{-0.2}^{+0.6}\times {10}^{-9}$	${9.8}_{-0.5}^{+1.2}\times {10}^{-9}$	133.3%	NGC 4993
	${2.5}_{-0.1}^{+0.3}\times {10}^{-9}$	${3.8}_{-0.2}^{+0.5}\times {10}^{-9}$	52.1%	MW + NGC 4993
GW 170817/AT 2017gfo (${\rm{\Delta }}{\gamma }_{2}$)	${9.5}_{-0.5}^{+1.3}\times {10}^{-5}$	${2.2}_{-0.1}^{+0.3}\times {10}^{-4}$	131.6%	NGC 4993
	${5.6}_{-0.3}^{+0.8}\times {10}^{-5}$	${8.5}_{-0.4}^{+1.1}\times {10}^{-5}$	51.8%	MW + NGC 4993

Note. The halo mass is in the range of $[{10}^{12.0}{h}^{-1}\,{M}_{\odot },{10}^{12.4}{h}^{-1}\,{M}_{\odot }]$ due to a typical uncertainty of 0.2 dex in NGC 4993. The influence of the change in halo mass on the WEP test is quantified through the deviation δ₂, defined as δ₂ = [Δγ(Min) − Δγ(Max)]/Δγ(Max). A positive value of δ₂ means that the constraint on the PPN becomes tighter when the halo mass increases.

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In this paper, a model was developed to describe the augmented test of a host galaxy that considers angle offset, large natal kicks, and the typical uncertainty on the halo mass of NGC 4993.

The transient could be located at any point along the line of sight in NGC 4993 due to the angle offset from the center, as long as the observed luminosity distance is guaranteed. Because the transient is most likely directly in front of the bulk of the host galaxy, the minimum distance of the source from the center of its host is then simply the projected distance, and the maximum distance extends near the edge of NGC 4993. The influence of the angle offset on the results can be quantified by the distance offset shown by SN' in Figure A1. The results of the maximum and minimum SN' are listed in Table 3.

The luminosity distance adopted in the calculation is ${40}_{-14}^{+8}$ Mpc, which was the closest observed GW source and the closest short γ-ray burst, with a distance measurement by Abbott et al. (2017a). Meanwhile, several other EM methods have given more precise values for the distance, e.g., 40.4 ± 3.4 Mpc using the MUSE/VLT measurement of the heliocentric redshift, 41.0 ± 3.1 Mpc using HST measurements of the effective radius and the MUSE/VLT measurements of the velocity dispersion (Hjorth et al. 2017), and $40.7\pm 1.4\pm 1.9$ Mpc using surface brightness fluctuations (Cantiello et al. 2018). Although the uncertainty of ${40}_{-14}^{+8}$ Mpc is slightly worse than these recent values shown above, it is accurate enough for the WEP test due to the suppression of the host galaxy.

The natal kick was imparted to the binary at the same time as the SN explosions that gave rise to the neutron stars. This kind of kick should then lead to mergers at large offsets from their birth sites and host galaxy, on scales of about tens to hundreds of kiloparsecs, over a broad range of merger timescales (Berger 2014). In this work, we chose large NS kicks with V_kick typically from 400 to ∼500 km s⁻¹. The related delay time was adopted in a less stringent range from the observed minimum magnitude, 86 Myr, to the Hubble time. For more stringent constraints on the delay time, one can refer to Figure 8 and Table 2 given in Abbott et al. (2017c), where the summary statistics for output probability density functions (PDFs) and the more detailed PDFs on progenitor properties, with various delay time constraints, are presented.

We thank Dr. Jielei Zhang for helpful discussions. This work is supported by the National Natural Science Foundation of China under grant Nos. 11475143 and 11822304, and by the Nanhu Scholars Program for Young Scholars of Xinyang Normal University.

Here we present the geometry of the travel path of the waves. We use ECS to describe the localization of the merge position (see the left panel of Figure A1) and use the polar coordinate system (PCS) to describe the path of the waves (see the right panel of Figure A1). In ECS, the impact parameter b and the viewing angle α satisfy the following formula:

$$\begin{eqnarray}&&b={SG}{\left(1-\displaystyle \frac{{{SG}}^{2}}{4{r}_{G}^{2}}\right)}^{1/2},\ \ \cos \alpha =1-\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{SG}}{{r}_{G}}\right)}^{2}.\end{eqnarray} \tag{ A1 }$$

The distance $| {SG}|$ between points on the spherical surface can be given by

$$\begin{eqnarray}&&| {SG}| ={\left[{({x}_{S}-{x}_{G})}^{2}+{({y}_{S}-{y}_{G})}^{2}+{({z}_{S}-{z}_{G})}^{2}\right]}^{1/2},\end{eqnarray} \tag{ A2 }$$

where the coordinates $({x}_{S},{y}_{S},{z}_{S})$ and $({x}_{G},{y}_{G},{z}_{G})$ are shown as

$$\begin{eqnarray}&&{x}_{S}={r}_{G}\cos {\delta }_{S}\cos {\beta }_{S};\ {x}_{G}={r}_{G}\cos {\delta }_{G}\cos {\beta }_{G};\end{eqnarray} \tag{ A3 }$$

$$\begin{eqnarray}&&{y}_{S}={r}_{G}\cos {\delta }_{S}\sin {\beta }_{S};\ {y}_{G}={r}_{G}\cos {\delta }_{G}\sin {\beta }_{G};\end{eqnarray} \tag{ A4 }$$

$$\begin{eqnarray}&&{z}_{S}={r}_{G}\sin {\delta }_{S};\ \ \ \ \ \ \ \ \ {z}_{G}={r}_{G}\sin {\delta }_{G}.\end{eqnarray} \tag{ A5 }$$

Here, r_G is assumed to be the radius of the celestial sphere.

Substituting SG (A2) into Equation (A1), we get the formula below of the angle α between the line of sight and the line from Earth to the Galactic center,

$$\begin{eqnarray}&&\cos \alpha =\sin {\delta }_{S}\sin {\delta }_{G}+\cos {\delta }_{S}\cos {\delta }_{G}\cos {\rm{\Delta }}\beta ,\end{eqnarray} \tag{ A6 }$$

with ${\rm{\Delta }}\beta =| {\beta }_{S}-{\beta }_{G}|$. Then, by substituting the coordinates of points S and G into Equation (A6), we get $\alpha \approx 61\buildrel{\circ}\over{.} 28$. The impact parameter b can thus be rewritten as

$$\begin{eqnarray}&&{b}^{2}={r}_{G}^{2}\left[1-{\left(\sin {\delta }_{S}\sin {\delta }_{G}+\cos {\delta }_{S}\cos {\delta }_{G}\cos {\rm{\Delta }}\beta \right)}^{2}\right].\end{eqnarray} \tag{ A7 }$$

In PCS, at the initial time of travel, the wave is located at point S $({r}_{S},{\theta }_{S})$,

$$\begin{eqnarray}&&{r}_{S}^{2}={d}^{2}+{r}_{G}^{2}-2{r}_{G}d\cos \alpha ,\end{eqnarray} \tag{ A8 }$$

$$\begin{eqnarray}&&{d}^{2}={r}_{S}^{2}+{r}_{G}^{2}-2{r}_{G}{r}_{S}\cos {\theta }_{S},\end{eqnarray} \tag{ A9 }$$

with ${r}_{S}={GS}$ and d = OS. In this way, we obtain the angles $\psi =0\buildrel{\circ}\over{.} 01$ and ${\theta }_{S}=118\buildrel{\circ}\over{.} 71$ in $\bigtriangleup {SOG}$ at the initial time of the wave's propagation. The path of the waves from the source $({r}_{S},{\theta }_{S})$ to the final receiver $({r}_{G},0^\circ )$ is illustrated in Figure A1. For any test point P with the coordinate $(r,\theta )$, the angle α should satisfy the following formula:

$$\begin{eqnarray}&&\cos \alpha =\displaystyle \frac{{{OP}}^{2}+{r}_{G}^{2}-{r}^{2}}{2{r}_{G}\cdot {OP}}.\end{eqnarray} \tag{ A10 }$$

Therefore, the dynamic distance from Earth to any position P during the wave's travel can be obtained as

$$\begin{eqnarray}&&{OP}=\displaystyle \frac{1}{2d}\left[\zeta \pm \sqrt{{\zeta }^{2}+4{d}^{2}\left({r}^{2}-{r}_{G}^{2}\right)}\right],\end{eqnarray} \tag{ A11 }$$

where we adopt $\zeta ={d}^{2}-{r}_{S}^{2}+{r}_{G}^{2}$ and keep the sign "+" in front of the square root. When the waves propagate along the path, it requires that at the initial moment of $r\to {r}_{S}$, the condition of ${OP}\to d$ must be satisfied, and at the terminal moment of $r\to {r}_{G}$, the condition of ${OP}\to 0$ also must be satisfied. The line OP in Equation (A11) with the "+" sign is the path of the propagation of waves from the merge position to that of Earth.

In order to distinguish the potentials between the two galaxies, we use r and χ to denote the radius of the MW and NGC 4993, respectively, in Equations (7) and (8). If the waves travel along their path, the below condition should be upheld:

$$\begin{eqnarray}&&\chi (r,\theta )={\left[{r}^{2}+{{GN}}^{2}-2{GNr}\cos \left({\theta }_{S}-\theta \right)\right]}^{1/2},\end{eqnarray} \tag{ A12 }$$

where, for one cosmic source of GW 170817, we can assume ${GN}\approx d$ and ${\theta }_{S}\approx \angle {NGO}$.

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We then present the geometry of the observed angle offset. According to the projected triangle $\bigtriangleup {PNN}^{\prime}$ in Figure A1, a key relationship of the path of the waves can be given as

$$\begin{eqnarray}&&{NP}=\sqrt{N^{\prime} {N}^{2}+N^{\prime} {P}^{2}},\end{eqnarray} \tag{ B1 }$$

where N' is the projected position of the galactic center (point N) along the line of sight. The projected offset distance N'N can be used quantitatively to indicate the observed angle offset. When the angle offset disappears, i.e., N'N → 0, the distance between the two galaxies approximates the distance from the merge position to Earth, i.e., ${ON}^{\prime} \to {ON}$.

In Figure A1, the minimum of SN' (${SN}^{\prime} =0$) comes from the fact that the source is located at the projected point N'. Inversely, the maximum of SN' (${SN}^{\prime} ={13.0}_{-4.6}^{+2.6}$ kpc) appears when the source is located at the outermost front edge (the bulk denoted by the dashed line) of the galaxy along the line of sight. This also means that the source is located at the point R_N, where ${R}_{N}N={R}_{N}$ is half of the galaxy diameter. By using the NASA Extragalactic Database or the ESO-LV catalog (see Lauberts & Valentijn 1989), one can find that the diameter of NGC 4993 is about 26 kpc, which is longer than the bulge scale length (∼1.5 kpc) in the Hernquist profile. These magnitudes are consistent with each other. The reason for this is that the diameter represents the maximum range of possible source locations, and the bulge scale length represents the range of the main stellar mass producing a stellar potential, Equation (2). Therefore, we can obtain the minimum and maximum N'S shown by

$$\begin{eqnarray}&&N^{\prime} S{| }_{\min }=0,\end{eqnarray} \tag{ B2 }$$

$$\begin{eqnarray}&&N^{\prime} S{| }_{\max }=N^{\prime} {R}_{N}=\sqrt{{R}_{N}^{2}-N^{\prime} {N}^{2}}.\end{eqnarray} \tag{ B3 }$$

The range of N'S is thus determined by the distance d from the source S to the receiver O. The uncertainty in the diameter of NGC 4993, 26 kpc, is proportional to the uncertainty in the distance d = ${40}_{-14}^{+8}$ Mpc, so the uncertainty in the maximum of N'S is proportional to this as well. Therefore, we can obtain the distance N'S to be in the range of [0, ${13.0}_{-4.6}^{+2.6}$ kpc], where the maximum uncertainty comes from the luminosity distance.