Abstract
Gravitational waves (GWs) in general relativity are assumed to contain only tensorial polarization states, with generic metric theories of gravity also allowing for vector- and scalar-polarized GWs. Assuming an association between the gravitational-wave and electromagnetic transients GW190521 and ZTF19abanrhr, we find a preference for tensorial GWs with a log10 Bayes factor of 21.6 (with respect to vector-polarized GWs) and 38.4 (with respect to scalar-polarized GWs).
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1. Introduction
The association between the binary black hole (BBH) event GW190521 (Abbott et al. 2020a, 2020b) observed by the LIGO (Harry 2010) and Virgo (Acernese et al. 2015) gravitational wave (GW) detectors and the electromagnetic (EM) transient ZTF19abanrhr (Graham et al. 2020) creates an opportunity for novel tests of General Relativity (GR). Where previous studies have focused on cosmological (Chen et al. 2020; Gayathri et al. 2020; Mukherjee et al. 2020) or GW propagation (Mastrogiovanni et al. 2020) effects, here we investigate the GW polarization content.
There are nominally six polarization states allowed for perturbations in generic metric theories of gravity: two scalar modes (helicity 0), two vector modes (helicity ±1) and two tensor modes (helicity ±2) (Eardley et al. 1973; Will 2014). GR only allows for tensor polarization, a detection of scalar or vector modes in GWs would hence provide unambiguous proof of beyond-GR physics.
The measured strain in a GW detector can be expressed as h(t) = ∑A FA hA with A representing one of six polarization modes and FA being the detector's response to each mode's incident GW strain hA . Where each hA in principle can be separately dependent on the BBH source configuration, Abbott et al. (2020b) found no residual signal in the data containing GW190521 after subtraction of the maximum likelihood tensorial GW signal. We therefore assume all hA for GW190521 being represented by this tensorial model alone (Varma et al. 2019), with the six FA fully capturing the potential non-GR polarization content. This is justified through the detector response only being dependent on the GW helicity and the detector's orientation with respect to the line-of-sight to the GW source (Isi & Weinstein 2017). These assumptions follow previous investigations into GW polarization states for the GW150814 (Abbott et al. 2016), GW170814 (Abbott et al. 2017) and GW170817 (Abbott et al. 2019a) events.
2. Analysis and Discussion
We fix the sky-location of GW190521 to the position of ZTF19abanrhr and perform a coherent Bayesian analysis of the GW signal present in the LIGO/Virgo data (Abbott et al. 2019b; Gravitational Wave Open Science Center 2020) using using the nested sampling algorithm from LALInference (Veitch et al. 2015) and otherwise following the same analysis configuration as Isi (2020). We repeat the analysis with three separate assumptions of the GW polarization content, a signal consisting of only tensor, vector or scalar modes coherent across the three GW detectors. While mixed-polarization content in principle could be considered, this is beyond the capabilities currently implemented in LALInference so we leave this for future studies.
We report our findings in Figure 1, as the logarithm (base 10) Bayes factors between the different polarization models considered. We find overwhelming support for tensorial GWs, with a log10 Bayes factor of 21.6 and 38.4 over GWs containing purely vector or scalar polarization modes. For comparison, we also report Bayes factors with respect to a model assuming the data surrounding GW190521 containing only Gaussian noise, without the presence of a coherent GW signal, and find it to be virtually indistinguishable to the scalar-GW model. The Bayes factor reported here for the tensor/vector constraint is comparable to the Bayes factor of 20.8 measured for GW170817 (Abbott et al. 2019a). For the tensor/scalar constraint, there is a significant increased support for GW190521, with the GW170817 test reporting a Bayes factor of 23.1. A test for the presence of non-tensorial GW polarizations for GW190521 was also reported by Abbott et al. (2020c), using an analysis based on a GW null-stream approach independent of assumptions of specific waveform models (Pang et al. 2020). Combined with relaxing the assumption of a sky-location fixed to ZTF19abanrhr, the constraining power of the analysis performed in Abbott et al. (2020c) is reduced relative to the one presented here reporting a log10 Bayes factor of 0.09 and 1.16 for tensor/vector and tensor/scalar respectively.
Figure 1. The log10 Bayes factors between the evidence 
recovered for three GW polarization models and one noise-only model. The Bayes factors are reported between models on the vertical and horizontal axes.
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In conclusion, we find gravitational waves containing tensorial polarizations only being preferred with log10 Bayes factors of 21.6 and 38.4 over vector or scalar polarization GWs under the assumption of association between GW190521 and ZTF19abanrhr. As the global GW detector network expands and improves over the coming years (Abbott et al. 2018), combined with observations of a larger number of GW–EM counterparts, we expect the constraints on GW polarization states to strengthen further.
The author would like to thank Maximiliano Isi for helpful discussions. C.J.H. acknowledge support of the National Science Foundation, and the LIGO Laboratory. The author is grateful for computational resources provided by the LIGO Laboratory and supported by National Science Foundation Grants PHY-0757058 and PHY-0823459. LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation and operates under cooperative agreement PHY-1764464.
Software: LALSuite (LIGO Scientific Collaboration 2020), numpy (Harris et al. 2020), matplotlib (Hunter 2007).