week ending PRL 119, 161102 (2017) LETTERS 20 OCTOBER 2017

Bounding the of with Gravitational

Neil Cornish eXtreme Gravity Institute, Department of , 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 arriving at widely separated detectors can be used to place upper and lower bounds on the speed of gravitational . Using a Bayesian approach that combines the first three gravitational wave detections reported by the LIGO Scientific and Virgo Collaborations we constrain the gravitational propagation speed cgw to the 90% credible interval 0 55 1 42 . c

DOI: 10.1103/PhysRevLett.119.161102

k4 The first detections of gravitational waves from merging ω2 ¼ m2 þ c2 k2 þ a þ; ð1Þ 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 of the , 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- 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 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 [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 ; other different from c and with 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 complementary to other observations. gw by LIGO are dominated by the -to-noise accumulated To quantify what “precise” tests mean for , 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 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 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) 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 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 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 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

While the sources may be uniformly distributed on the sky, the patterns of the detectors make it more likely to detect systems above or below the plane of detectors. Assuming roughly equal sensitivity for the detectors, 3 ðθ ϕÞ2 ¼ Pthe observational bias scales as F , where F ; D ðθ ϕÞ2 D¼H;LFþ;× ; is the -averaged network antenna pattern [20]. The resulting distribution of electro- ðΔ Þ¼½1 − magnetic time delays is then well fit by p tEM ðΔ Þ2½ð3 − 2 2Þ2 3−1 − ≤ Δ ≤ tEM=tq t0=tq t0= for t0 tEM t0 with tq ¼ 10.65 ms. We use this modified distribution to define ðΔ j Þ the likelihood p t cgw . For multiple events the full like- lihood is the product of the per-event likelihoods. The FIG. 1. Posterior distributions for the gravitational wave propa- ’ posterior distribution for cgw follows from Bayes theorem, gation speed derived from each of the individual LIGO events for ð jΔ Þ¼ ðΔ j Þ ð Þ ðΔ Þ p cgw t p t cgw p cgw =p t . We consider two prior distributions uniform in cgw or ln cgw.

161102-2 week ending PRL 119, 161102 (2017) PHYSICAL REVIEW LETTERS 20 OCTOBER 2017 ≃ 0 0 66 at cgw . Note that GW151226 produces the weakest some signals if cgw < . c. On the other hand, loud single upper bound on the propagation velocity even though it has detector triggers consistent with a binary merger would not the most accurately measured time delay. This is because go un-noticed, as evidence by GW170104, which was the strongest upper limits come from events with the missed by the standard search due to the Hanford detector longest time delay; even allowing for the uncertainties in being incorrectly flagged as out of observing mode, and only the time delay measurements for GW150914 and found later in an analysis of single detector triggers from the GW170104, both are constrained to have delays that are Livingston detector [3]. It is highly unlikely that pairs of much longer than for GW151226. triggers with time delays greater than 15 ms would be Figure 2 shows the posterior distribution for cgw found overlooked, especially if they shared similar parameters by combining all three LIGO detections together for and occurred within of each other. — uniform priors in cgw or ln cgw. For the wider prior range Forecasts for more detections and more detectors. It is ¼ 100 with cL c= the combination of the three detections interesting to consider how the bounds will improve with yield an interesting lower bound on the propagation speed. additional detections and detectors, using just the gravita- The 90% credible interval for the linear and log priors are tional time delays (as mentioned earlier, combined electro- 0 55 1 42 0 41 1 39 . c8.6 ms with ten detections, and one uniform prior on cgw this gives p cgw t1; t2; t3 3 4 event with a time delay Δt>9.6 ms with 100 detections. 4c =c for c

161102-3 week ending PRL 119, 161102 (2017) PHYSICAL REVIEW LETTERS 20 OCTOBER 2017

as those coming from the damping of in binary systems or . The authors would like to thank the Centro de Ciencias de Benasque Pedro Pascual for providing a wonderful venue to carry out this work. The authors appreciate the constructive feedback received from Walter Del Pozzo, Thomas Dent, John Veitch, and Brian O’Reilly during the internal LIGO review. D. B. is grateful to Víctor Planas- Bielsa and Sergey Sibiryakov for discussion. N. J. C. appreciates the support of National Foundation (NSF) Award No. PHY-1306702. G. N. is supported by the Swiss National Science Foundation (SNF) under Grant No. 200020-168988. The authors thank the LIGO Scientific Collaboration for access to the data and gratefully acknowledge the support of the United States NSF for the construction and operation of the LIGO Laboratory and Advanced LIGO as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, and the Max--Society (MPS) for support of the construction of Advanced LIGO. Additional support for Advanced FIG. 3. Slices through the joint electromagnetic time delay LIGO was provided by the Australian Research Council. distribution for the HLVK network using Hanford as the reference site. Darker colors in the two-dimensional slices indicate higher density. Note the cuplike structure of the distributions (higher at the edges than in the center). [1] B. P. Abbott et al. (Virgo and LIGO Scientific Collaborations), Phys. Rev. Lett. 116, 061102 (2016). delays between one reference detector and the other detectors [2] B. P. Abbott et al. (Virgo and LIGO Scientific Collabora- in the network. Figure 3 shows slices through the joint tions), Phys. Rev. Lett. 116, 241103 (2016). et al. electromagnetic time delay distribution for the HLVK net- [3] B. P. Abbott (Virgo and LIGO Scientific Collaborations), Phys. Rev. Lett. 118, 221101 (2017). work using Hanford as the reference site assuming a uniform [4] B. P. Abbott et al. (Virgo and LIGO Scientific Collabora- distribution of sources. Here we did not correct for the tions), Phys. Rev. Lett. 116, 221101 (2016). observational bias, since the network antenna pattern for a [5] B. P. Abbott et al. (Virgo and LIGO Scientific Collabora- four-detector network is fairly uniform. Applying the change tions), Phys. Rev. X 6, 041015 (2016). Δ ¼ð ÞΔ of variable t c=cgw tEM to this distribution as we did [6] N. Yunes, K. Yagi, and F. Pretorius, Phys. Rev. D 94, for the two-detector HL case yields the joint likelihood 084002 (2016). ðΔ Δ Δ j Þ [7] D. Blas, M. M. Ivanov, I. Sawicki, and S. Sibiryakov, Pis’ma p tHL; tHV; tHK cgw . Using simulated detections of events measured to a precision of σ ¼ 0.3 ms in each Zh. Eksp. Teor. Fiz. 103, 708 (2016) [JETP Lett. 103, 624 detector, we find that with just three detections the HLVK (2016)]. [8] G. D. Moore and A. E. Nelson, J. High Energy Phys. 09 network will typically be able to constrain cgw to the 99% ¼ 1 00 0 02 (2001) 023. credible region cgw=c . . . The constraints [9] J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, Mod. improve to better than 1% of the speed of light with five Phys. Lett. A 31, 1675001 (2016). detections. [10] V. A. Kosteleck and M. Mewes, Phys. Lett. B 757, 510 Summary.—Combining the time-delay measurements (2016). between detector sites for multiple gravitational wave [11] D. Bettoni, J. M. Ezquiaga, K. Hinterbichler, and M. events can be used to place interesting constraints on the Zumalacrregui, Phys. Rev. D 95, 084029 (2017). speed of gravity. The LIGO detections made to date already [12] C. de Rham, J. T. Deskins, A. J. Tolley, and S.-Y. Zhou, Rev. constrain the speed of gravity to within 50% of the speed of Mod. Phys. 89, 025004 (2017). light. Additional LIGO detections in the next few years [13] C. M. Will, Living Rev. Relativity 9, 3 (2006). [14] L. Amendola et al., arXiv:1606.00180. should improve the bound to of order 10%. The bounds will [15] E. Berti et al., Classical 32, 243001 improve rapidly as more detectors join the worldwide (2015). network, with just a half dozen detections by the [16] A. Nishizawa and T. Nakamura, Phys. Rev. D 90, 044048 Hanford-Livingston-Virgo-Kagra network constraining (2014). deviations to better than 1%. These bounds will allow us [17] X. Li, Y.-M. Hu, Y.-Z. Fan, and D.-M. Wei, Astrophys. J. to test general relativity to the level of other standard tests, 827, 75 (2016).

161102-4 week ending PRL 119, 161102 (2017) PHYSICAL REVIEW LETTERS 20 OCTOBER 2017

[18] X.-L. Fan, K. Liao, M. Biesiada, A. Piorkowska-Kurpas, [23] T. Tomaru (KAGRA Collaboration), KAGRA-Large-scale and Z.-H. Zhu, Phys. Rev. Lett. 118, 091102 (2017). Cryogenic Gravitational wave , in Gravitation, [19] T. E. Collett and D. Bacon, Phys. Rev. Lett. 118, 091101 , and Cosmology (World Scientific, 2016), (2017). pp. 153–159. [20] B. F. Schutz, Classical Quantum Gravity 28, 125023 (2011). [24] B. P. Abbott et al. (VIRGO and LIGO Scientific Collabo- [21] C. Cutler and E. E. Flanagan, Phys. Rev. D 49, 2658 (1994). rations), Living Rev. Relativity 19, 1 (2016). [22] B. P. Abbott et al. (Virgo and LIGO Scientific Collabora- [25] LAL Detector File Reference, http://software.ligo.org/docs/ tions), Phys. Rev. D 93, 122003 (2016). nlalsuite/lal/_l_a_l_detectors_8h.html.