Does Gravity Fall Down? Evidence for Gravitational-wave Deflection along the Line of Sight to GW170817

1.734 ± 0.054 s after the LIGO/Virgo detection of the gravitational waves from GW170817, Fermi and International Gamma-ray Astrophysics Laboratory observed the arrival of gamma-rays (Abbott et al. 2017a, 2017b). This near-simultaneous arrival of gravitational waves and photons over 130 Myr of travel time provides a strict test for modified gravity theories (Baker et al. 2017; Langlois et al. 2018). Boran et al. (2018) estimate that the Shapiro delay (Shapiro 1964) was ∼400 days (see also Abbott et al. 2017b), enabling further tests of general relativity (GR).

Recently, Mukherjee et al. (2019a, 2019b) have proposed measuring gravitational lensing of gravitational waves as a new probe of GR. They estimate that this lensing may be detectable in the future. In this work, we propose performing the same test, but using the geometric component of the lensing time delay as a test of GR. If gravitational waves and photons are both subject to the same geometric deflection, then they undergo the same lensing amplification. Likewise, if gravitational waves and photons undergo the same geometric deflection, then they should require the same travel time.⁴ Investigating the possibility that gravitational waves undergo larger deflection than photons requires constraining the intrinsic time delay between gravitational-wave emission and photon emission. For example, gravitational waves could be emitted 100 s ahead of the photons, but delayed by 100 extra s due to larger deflection for near-simultaneous arrival. Considering this possibility is beyond the scope of this work.

Section 2 shows that, for typical nearby (tens of Mpc) gravitational-wave sources, the geometric component of the time delay is of the order of tens of seconds. Section 3 shows our computation of an approximate time delay for GW170817, obtaining a 68% confidence interval of 400–2200 s. Thus, we show that the GW170817 gravitational waves must have been geometrically deflected line-of-sight halos by an amount at least comparable to photons to have arrived at nearly the same time. We summarize and conclude in Section 4.

We show an illustration of a single thin gravitational lens in Figure 1. For an axially symmetric thin lens (line-of-sight size much less than the line-of-sight distances), the lensing deflection is given by

$$\begin{eqnarray}&&\hat{\alpha }=\displaystyle \frac{4{GM}(\xi )}{{c}^{2}\xi }=1.91\times {10}^{-7}\ \mathrm{rad}\left[\displaystyle \frac{M/({10}^{12}{M}_{\odot })}{\xi /\mathrm{Mpc}}\right],\end{eqnarray} \tag{ 1 }$$

where ξ is the impact parameter, and $M(\xi )$ is the enclosed mass at radius ξ (e.g., Schneider et al. 1992). For typical impact parameters of hundreds of kiloparsecs and galaxy-scale lenses, we can approximate $M(\xi )$ as the total mass M. For nearby sources like GW170817 (redshift ∼0.01), we can neglect the expansion of the universe (setting z = 0). The extra path length due to the geometric deflection from the lens is

$$\begin{eqnarray}&&\sqrt{{D}_{{ds}}^{2}+{\left(\xi -\eta \right)}^{2}}+\sqrt{{D}_{d}^{2}+{\xi }^{2}}-\sqrt{{D}_{s}^{2}+{\eta }^{2}}.\end{eqnarray} \tag{ 2 }$$

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Assuming small angles so that we can substitute $\beta =\eta /{D}_{s}$, $\theta =\xi /{D}_{d}$, $\alpha \,{D}_{s}=\hat{\alpha }\,{D}_{{ds}}$, and also using $\alpha +\beta =\theta$, we expand Equation (2) to lowest order in $\hat{\alpha }$ and θ to find the extra path length is

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}\displaystyle \frac{{D}_{d}{D}_{{ds}}}{{D}_{s}}{\hat{\alpha }}^{2}.\end{eqnarray} \tag{ 3 }$$

Thus the geometric component of the time delay is

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{1}{2c}\displaystyle \frac{{D}_{d}{D}_{{ds}}}{{D}_{s}}{\hat{\alpha }}^{2} & = & 18.9\ {\rm{s}}\left[\displaystyle \frac{{D}_{d}{D}_{{ds}}}{{D}_{s}}\displaystyle \frac{1}{10\mathrm{Mpc}}\right]\\ & & \times \,{\left[\displaystyle \frac{M/({10}^{12}{M}_{\odot })}{\xi /\mathrm{Mpc}}\right]}^{2}.\end{array}\end{eqnarray} \tag{ 4 }$$

These time delays are of the order of the astrophysical time delay between gravitational waves and photons and will thus frequently be detectable, depending especially on M and ξ (both of which enter quadratically). It was pointed out to us by Nathan Johnson-McDaniel that a deflected ray passes farther from the lens and is thus subject to less Shapiro delay than a ray that passes straight through. Evaluating this difference in Shapiro delay in the small-angle and weak-field limits shows that it is twice the effect of the extra path length, and thus all of our computed time delays should be negative. Due to the small intrinsic time delay of ≲ 10 s (e.g., Abbott et al. 2017b), this oversight does not affect our conclusions.

To estimate the geometric time delay, we select galaxies with possible lensing contributions along the line of sight to GW170817 from the 2MASS Redshift Survey (Huchra et al. 2012). For each of the 43,533 2MASS galaxies included in the survey, we estimate the time delay up to a multiplicative constant from Equation (4). We estimate the distance from the cosmic microwave background (CMB)-centric redshift, the impact parameter from the distance and angular separation, and the mass from the absolute K-band isophotal magnitude. Coulter et al. (2017) discovered the GW170817 optical emission, necessary to obtain the coordinates on the sky and the identity of the host galaxy (NGC 4993). Cantiello et al. (2018) provide a precise distance to NGC 4993: ${40.7}_{-2.3}^{+2.5}$ Mpc. Four galaxies are the most plausible for large time delays: NGC 5084, M104, M83, and NGC 5128 (Centaurus A). The later two of these are the central galaxies in groups and all have dynamical halo mass measurements.⁵ To be conservative, we assume all dynamical mass measurements are at fixed distance. As estimated dynamical mass scales with estimated distance, in addition to the mass uncertainties discussed below, we assume additional uncertainty on mass that covaries with the distance uncertainty.

With a dynamical mass as large as ${10}^{13}{M}_{\odot }$ (Carignan et al. 1997), NGC 5084 is one of the most massive disk galaxies known. For a more conservative mass measurement, we use the Carignan et al. (1997) value excluding two possible satellites that may not be bound: $(5.2\pm 2.9)\times {10}^{12}{M}_{\odot }$, which we convert to a log ${}_{10}(M/{M}_{\odot })$ measurement of 12.72 ± 0.24. We take a distance modulus of 31.12 ± 0.54 (or 16.7 Mpc; Springob et al. 2014), giving an impact parameter of 0.84 ±0.24 Mpc. We note that for this measured distance, there is a $\sim 0.02 \%$ chance that NGC 5084 is actually behind NGC 4993 and thus NGC 5084 could not lens GW170817.

M104 has a lower mass than NGC 5084, but it is more precisely measured. Tempel & Tenjes (2006) find $2\times {10}^{12}{M}_{\odot }$, in good agreement with the Jardel et al. (2011) value for 50 kpc. Jardel et al. (2011) measure to larger scales and find a higher mass ($\sim 3\times {10}^{12}{M}_{\odot }$ with about 10% uncertainty). To be conservative, we use $(2\pm 0.2)\times {10}^{12}{M}_{\odot }$. We average two consistent tip of the red giant branch distance measurements to M104 (McQuinn et al. 2016 and the Extragalactic Distance Database; Jacobs et al. 2009) to arrive at a distance modulus of 29.88 ± 0.08 (or 9.46 Mpc), giving an impact parameter of 2.27 ± 0.08 Mpc.

Karachentsev et al. (2007) compute the tip of the red giant branch distances and the halo masses for the NGC 5128 and M83 groups. They find a mean distance of 3.76 ± 0.05 Mpc for the NGC 5128 group (for an impact parameter of 1.31 ± 0.02 Mpc) and 4.79 ± 0.1 Mpc for the M83 group (impact parameter of 0.74 ± 0.02 Mpc). Their mass for the NGC 5128 group is (6.4–8.1) $\times \,{10}^{12}{M}_{\odot }$, which we convert to a $\mathrm{log}{}_{10}(M/{M}_{\odot })$ measurement of 12.86 ± 0.05. For the the M83 group, they find (0.8–0.9) $\times \,{10}^{12}{M}_{\odot }$, which we convert to a $\mathrm{log}{}_{10}(M/{M}_{\odot })$ measurement of 11.93 ± 0.02. Of course, modeling these groups as axially symmetric masses (for the purposes of Equation (1)) is incorrect in detail. However, we show below that the implied time delays are large enough that moderate changes to our assumptions will not affect our conclusions.

3.1. Combined Result

To properly evaluate the total geometric delay due to each of these four halos along the line of sight, we wrote a simple multi-plane lensing code. This code starts a ray at the position (relative to us) of GW170817 propagating toward the origin. As the ray reaches the position of closest approach to each halo, it is deflected according to Equation (1). Due to the deflections during propagation, the ray will not pass through the origin. We thus iterate, adjusting the initial direction in the next iteration to account for the miss in the last iteration. As the deflections are $\sim {10}^{-7}$ radians, this iteration converges rapidly. The difference between an undeflected line from GW170817 and the path taken gives the time delay. As this relative difference scales as the deflection angles squared ($\sim {10}^{-14}$) and the precision limit for a double-precision (64-bit) floating point is $\sim {10}^{-16}$, we improve the accuracy of our results by using higher-precision arithmetic through the Python decimal package.

We compute a the probability density function for lensing by numerically propagating all distance and mass uncertainties. We draw 1000,000 realizations for all distances and masses and compute the geometric component of the time delay for each realization. The resulting probability density functions are shown in Figure 2. The top four panels show the time delay considering each halo in isolation; the bottom panel shows the multi-plane combination. Except for NGC 5084 (which does not have a precise enough distance measurement to securely place it along the line of sight to GW170817), we see strong ($\gt 5\sigma$) evidence of a time delay larger than the observed delay for each of the other halos. We see even stronger evidence for the combination (bottom panel of Figure 2) where the minimum time delay out of the realizations is 112 s and the 68% confidence interval is 400–2200 s. We can also interpret our result in terms of constraints on the deflection angle of gravitational waves (${\hat{\alpha }}_{\mathrm{GW}}$), assuming that the gravitational waves left no later than the photons and traveled at the same speed. Starting from the lower bound of our 99.9999% confidence interval (112 s), the gravitational waves were deflected by at least 110 s. Assuming Equation (1) applies to photons, we find

$$\begin{eqnarray}&&{\hat{\alpha }}_{\mathrm{GW}}\gt \displaystyle \frac{3.96{GM}(\xi )}{{c}^{2}\xi }.\end{eqnarray} \tag{ 5 }$$

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This work proposes a novel test of GR: detecting the deflection of gravitational waves by a gravitational potential by evaluating the geometric component of the time delay due to the deflection. Any deflection more or less than predicted by GR will lead to a different time delay with respect to the photons. We show that typical galaxy halos give detectable time delays (of order tens of seconds) for gravitational-wave sources as close as tens of Mpc. We present an initial evaluation of the deflection the gravitational waves of GW170817 underwent due to four halos along the line of sight. We see strong evidence that deflection did occur and GR passes our test.

This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. I.S. acknowledges support from the NSF, award 1616974. The authors also thank Harald Ebeling and Brent Tully for feedback on this work. B.J.S. is supported by NSF grants AST-1908952, AST-1920392, and AST-1911074. We thank the anonymous referee for valuable feedback that substantially improved this work.

Matplotlib (Hunter 2007), Numpy (van der Walt et al. 2011), Python, SciPy (Jones et al. 2001).