week ending PRL 119, 161102 (2017) PHYSICAL REVIEW LETTERS 20 OCTOBER 2017 Bounding the Speed of Gravity with Gravitational Wave Observations Neil Cornish eXtreme Gravity Institute, Department of Physics, Montana State University, Bozeman, Montana 59717, USA Diego Blas Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland Germano Nardini AEC, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland (Received 19 July 2017; revised manuscript received 24 August 2017; published 18 October 2017) The time delay between gravitational wave signals arriving at widely separated detectors can be used to place upper and lower bounds on the speed of gravitational wave propagation. Using a Bayesian approach that combines the first three gravitational wave detections reported by the LIGO Scientific and Virgo Collaborations we constrain the gravitational waves propagation speed cgw to the 90% credible interval 0 55 1 42 . c<cgw < . c, where c is the speed of light in vacuum. These bounds will improve as more detections are made and as more detectors join the worldwide network. Of order 20 detections by the two LIGO detectors will constrain the speed of gravity to within 20% of the speed of light, while just five detections by the LIGO- Virgo-Kagra network will constrain the speed of gravity to within 1% of the speed of light. DOI: 10.1103/PhysRevLett.119.161102 k4 The first detections of gravitational waves from merging ω2 ¼ m2 þ c2 k2 þ a þÁÁÁ; ð1Þ black hole binaries [1–3] have been used to test many g gw Λ2 fundamental properties of gravity [3–6], and have been used where mg refers to the mass of the graviton, cgw is what we to place the first observational upper limit on the speed of call “speed” of gravitational waves, and the rest of operators gravitational wave propagation [7]. In this Letter we set a are wave-number-dependent modifications suppressed by a more stringent upper limit on the gravitational waves high-energy scale Λ (for a parametrization in scenarios propagation speed cgw by combining all the detections breaking rotation invariance, see, e.g., Ref. [10]). Both m announced to date, and by applying a full Bayesian analysis. g and Λ can be constrained by the absence of dispersion of the We also provide the first direct lower bound on the propa- Λ 0 55 waves traveling cosmological distances. The scale is gation speed: cgw > . c at 95% confidence. While there ≥ already constrained to be very large [9], making it very are strong theoretical arguments that demand cgw c to difficult to constrain the operator a. For the graviton mass the prevent gravitational Cherenkov radiation [8], the LIGO LIGO Scientific and Virgo Collaborations put the strong detections provide the first direct observational constraints. bound m < 7.7 × 10−23 eV=c2 [3]. However, the parameter Gravitational waves generically propagate at a speed g c cannot be tested by dispersion measurements; other different from c and with frequency dependence dispersion gw methods are required [7]. relations in theories of modified gravity, see, e.g., Measuring c .—In the following we focus on possible Refs. [6,7,9–12]. Thus, a precise determination of c is gw gw ways to directly measure c . Since the signals measured a test of gravitation complementary to other observations. gw by LIGO are dominated by the signal-to-noise accumulated To quantify what “precise” tests mean for general relativity, in a narrow band between 50–200 Hz, our time delay let us recall that some post-Newtonian parameters are bounds can be interpreted as constraints on the speed of known to Oð10−4Þ [13], while cosmological or other gravity at a frequency f ∼ 100 Hz. Since the LIGO bounds astrophysical observations typically constrain modifica- ð10−2Þ constrain dispersion effects to be small over hundreds of tions to general relativity at the O level [14,15]. Mpc, they can safely be ignored on the terrestrial distance A convenient parametrization for theories preserving scales we are considering. Note that the inference that the rotation invariance is to write the dispersion relation as observed signals come from hundreds of Mpc away relies on waveform models derived from general relativity, and ≠ may not apply to a theory that predicts cgw c. Published by the American Physical Society under the terms of The most obvious way to measure the speed of gravi- the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to tational wave propagation is to observe the same astro- the author(s) and the published article’s title, journal citation, physical source using both gravity and light. However, for and DOI. the three gravitational wave detections that have been 0031-9007=17=119(16)=161102(5) 161102-1 Published by the American Physical Society week ending PRL 119, 161102 (2017) PHYSICAL REVIEW LETTERS 20 OCTOBER 2017 ð Þ announced thus far no unambiguous electromagnetic possibilities for the prior on the speed of gravity, p cgw : ∈ ½ counterparts have been detected, and a different approach flat in cgw and flat in ln cgw in the interval cgw cL;cU .For ¼ 100 must be taken to constrain cgw. The finite distance between the results shown here we set cU c, and either ¼ 100 ¼ the Hanford and Livingston gravitational wave detectors cL c= ,orcL c. The latter limit takes into account can be used to set an absolute upper limit on the the Cherenkov radiation constraint [8]. For three or more propagation speed [7], since the observed gravitational events the choice of prior has very little impact on the upper wave signals did not arrive simultaneously in the two limit. To account for the measurement error in Δt we use a detectors. Here we show that a proper statistical treatment Markov chain Monte Carlo algorithm to marginalize over the that folds in the probability distribution of the time delays errors in the arrival times. as function of cgw also allows us to set lower bounds on the The first detections of black hole mergers by LIGO propagation velocity. It should be noted that when the first provide measurements of Δt that were quoted in terms of confirmed electromagnetic counterpart to a gravitational central values and 90% credible intervals. Since the full wave signal is finally observed, the bounds on the differ- posterior distributions for Δt were not provided, we assume j − j ence in propagation velocities, cgw c will be many that the distributions can be approximated as normal dis- orders of magnitude more stringent than what we can ever tributions with mean μ and standard deviation σ with values hope to set using gravitational wave signals alone [9,16– GW150914 (μ ¼ 6.9 ms, σ ¼ 0.30 ms) [5], GW151226 18]. Precisely for the same reason, this identification may (μ¼1.1ms, σ ¼ 0.18 ms) [5], and GW170104 (μ ¼ 3.0 ms, never happen if the speed difference is not very small, since σ ¼ 0.30 ms) [3] (for a discussion about this assumption, that would mean a significant time offset. For other possible see, e.g., Ref. [21]). The upper bound on cgw quoted in stringent model-independent bounds not relying on the Ref. [7] was found by taking the minimum time delay from detection of a counterpart see [11,19]. GW150914 as Δt ¼ μ − 2σ ¼ 6.3 ms, and demanding that — Δ ¼ 1 6 Constraints on cgw from LIGO detections. The LIGO cgw <ct0= t . c. Note that this value is lower than the gravitational wave detectors at the Hanford and Livingston bound of 1.7c quoted in Ref. [7] as they interpreted the error sites are separated by a light-travel time of t0 ¼ 10.012 ms. in Δt quoted in Ref. [1] as one-sigma errors, when in fact they The time delay for light along a propagation direction that were the bounds on the 90% credible interval. makes an angle θ with the line connecting the two sites is We compute the posterior distribution for the gravita- Δt ¼ t0 cos θ. For sources distributed isotropically on ð jΔ Þ EM tional wave propagation speed, p cgw t using a Markov the sky, there are equal numbers of sources per solid angle chain Monte Carlo algorithm that marginalizes over the θ ϕ element d cos d ; thus, the time delays for electromag- uncertainties in the time delays by drawing new values of ðΔ Þ¼ netic signals are uniformly distributed with p tEM Δt from the assumed posterior distributions at each 1 ð2 Þ − ≤ Δ ≤ = t0 for t0 tEM t0. The gravitational wave time iteration of the Markov chain. Figure 1 shows the posterior Δ ¼ð ÞΔ delay is given by t c=cgw tEM; thus, the probability distributions for c using each of the detections separately. Δ gw of observing a time delay t between gravitational wave Individually the three events yield 95% upper bounds on signals arriving at the two sites for sources uniformly the propagation velocity for the linear and (log) uni- distributed on the sky is given by the likelihood 1 37 ð1 26 Þ form priors of cgw < . c . c for GW150914, cgw < 10 1 ð8 57 Þ 3 19 ð2 94 Þ . c . c for GW151226, and cgw < . c . c for c =2ct0 for − ct0=c ≤ Δt ≤ ct0=c ; ðΔ j Þ¼ gw gw gw p t cgw GW170104. Each event also yields a 95% lower bound on 0 otherwise: the propagation velocity, but these limits are not very ð2Þ interesting since all of the distributions have some support While the sources may be uniformly distributed on the sky, the antenna patterns of the detectors make it more likely to detect systems above or below the plane of detectors.