CONSTRAINTS ON GRAVITATIONAL-WAVE POLARIZATIONS FROM COMPACT-BINARY COALESCENCES
6.1 Introduction
Generic metric theories of gravity may predict up to six polarization modes for metric perturbations: two tensor (helicity±2), two vector (helicity ±1), and two scalar (helicity 0) modes [97, 98]. In contrast, one of the key predictions of general relativity (GR) is that metric perturbations possess only two tensor degrees of freedom [83, 115]. Therefore, a detection of any nontensorial mode would be unambiguous indication of physics beyond GR.
Before the beginning of LIGO’s second observation run, some evidence that gravita- tional waves (GWs) are described by the tensor metric perturbations of GR had been obtained from measurements of the rate of orbital decay of binary pulsars, in the context of specific beyond-GR theories (see e.g. [86, 87], or [88, 89] for reviews), and from the rapidly changing GW phase of binary black-hole mergers observed by LIGO, in the framework of parameterized models [1, 4, 82]. The addition of Advanced Virgo to the network in 2017 enabled another, more compelling, way of probing the nature of polarizations by studying GW geometry directly through the projection of the metric perturbation onto our detector network [18, 85, 99, 116].
The GW strain measured by a detector can be written in general ash(t)= FAhA, where hAare the 6 independent polarization modes andFArepresent the detector responses to the different modesA=(+,×,x,y,b,l). The antenna response functions depend only on the detector orientation and GW helicity, i.e. they are independent of the
Table 6.1: Polarization constraints from compact binaries (LIGO and Virgo).
GW170814 GW170817 GW170817 (fixed sky) log10B(tensor vs vector) 2.5±0.2 0.72±0.09 20.81±0.08 log10B(tensor vs scalar) 3.1±0.2 5.84±0.09 23.09±0.08
intrinsic properties of the source. We can therefore place bounds on the polarization content of a given GW by studying which combination of response functions is consistent with the signal observed [18, 79, 85, 99, 116, 117]. However, in order to break degeneracies and constrain all distinguishable polarization combinations with short compact-binary coalescences (CBCs), we would need at least 5 detectors, even if the location of the source was knowna priori—a network with fewer detectors can only make partial statements about polarizations (see Chapter 5). Furthermore, as the two LIGO instruments have similar orientations, little information about polarizations can be obtained using the LIGO detectors alone.
The first test on the polarization of GWs was performed for GW150914 [82]. The number of GR polarization modes expected was equal to the number of detectors in the network that observed the signal, rendering this test inconclusive. The addition of Virgo to the network of GW detectors allowed for the first informative test of polarization for GW170814 [5]. As described in Sec. 6.2, this analysis established that the GW data was better described by pure tensor modes than pure vector or pure scalar modes. A similar analysis was carried out for GW170817, the binary neutron star detected shortly after [6]. In that case, and as described in Sec. 6.3, the identification of an electromagnetic counterpart allowed for much stronger results, effectively rejecting the two pure-nontensorial models to any reasonable doubt [118].
Results for both events are summarized in Table 6.1.
6.2 GW170814
On August 14, 2017, GWs from the coalescence of two black holes at a luminosity distance of 540+130−210Mpc, with masses of 30.5+5−3..70Mand 25.3+2−4..82M, were observed in all three LIGO and Virgo detectors. The signal was first observed at the LIGO Livingston detector at 10:30:43 UTC, and then at the LIGO Hanford and Virgo detectors with delays of∼8 ms and∼14 ms, respectively.
The signal-to-noise ratio (SNR) time series, the time-frequency representation of the strain data and the time series data of the three detectors together with the inferred GW waveform, are shown in Fig. 6.1. The different sensitivities and responses of the
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Figure 6.1: GW170814 as observed by LIGO Hanford, LIGO Livingston and Virgo. Times are shown from August 14, 2017, 10:30:43 UTC.Top row: SNR time series produced in low latency. The time series were produced by time-shifting the best- match template from the online analysis and computing the integrated SNR at each point in time. The single-detector SNRs in Hanford, Livingston and Virgo are 7.3, 13.7 and 4.4, respectively. Second row: Time-frequency representation of the strain data around the time of GW170814. Bottom row: Time-domain detector data (in color), and 90% credible intervals for waveforms reconstructed from a wavelet analysis [25] (light gray) and GR templated models (dark gray), whitened by each instrument’s noise.(Reproduced from [5], see that Ref. for details on data manipulation for this figure.)
three detectors result in the GW producing different values of matched-filter SNR in each detector. A measurement of the signal amplitude by multiple (non-coaligned) detectors is the first ingredient needed to study polarizations.
Until Advanced Virgo became operational, typical GW position estimates were highly uncertain compared to the fields of view of most telescopes. The baseline formed by the two LIGO detectors allowed us to localize most mergers to roughly annular regions spanning hundreds to about a thousand square degrees at the 90% credible level [119–121]. Virgo adds additional independent baselines, which in cases such as GW170814 can reduce the positional uncertainty by an order of magnitude or more [120]. Good sky localization is the second ingredient needed to study polarizations.
With the addition of Advanced Virgo we can probe, for the first time, gravitational-
(a) Tensor
(b) Vector
(c) Scalar
Figure 6.2: GW170814 skymaps. Reconstructed sky location under the assumption of the three different polarization hypotheses. Color represents probability density, as a function of equatorial coordinates in a Mollweide projection.
(a) Tensor (b) Vector (c) Scalar
Figure 6.3: GW170814 polarization-angle posteriors. Posterior probability density in the polarization angle under the three different polarization assumptions. Note the periodicities corresponding to the different helicities.
wave polarizations geometrically by projecting the wave’s amplitude onto the three detectors. To do this, the coherent Bayesian analysis used to infer signal parameters is repeated after replacing the standard tensor antenna response functions with those appropriate for scalar or vector polarizations [116]. In our analysis, we are interested in the geometric projection of the GW onto the detector network, and therefore the details of the phase model itself are less relevant as long it is a faithful representation of the fit to the data in Fig. 6.1. Hence, we assume a GR phase model.
We find Bayes factors of more than 200 and 1000 in favor of the purely tensor polarization against purely vector and purely scalar, respectively. We also find that, as expected, the reconstructed sky location, distance and orientation change significantly depending on the polarization content of the source, with non-overlapping 90%
credible regions for tensor, vector and scalar (Fig. 6.2). In particular, the posterior on the polarization angleψreveals the symmetries intrinsic to each helicity: periodic overπ/2 for tensor, periodic overπ for vector, and totally insensitive to changes inψ for scalar (Fig. 6.3). The inferred detector-frame masses and spins are always the same, because that information is encoded in the signal phasing. An example of this is shown in Fig. 6.4 for the “chirp” mass,M = (m1m2)3/5(m1+m2)−1/5. The most probable waveforms recovered under different hypotheses confirm that Virgo data is the key differentiating factor (Fig. 6.5).
6.3 GW170817
On August 17, 2017 at 12:41:04 UTC, the Advanced LIGO and Advanced Virgo gravitational-wave (GW) detectors made their first observation of a binary neutron
(a) Vector and tensor (b) Scalar and tensor
Figure 6.4: GW170814 chirp-mass comparison. Posterior probability densities recovered for the detector-frame chirp massM= (m1m2)3/5(m1+m2)−1/5, wherem1 andm2are the two component masses. The different traces correspond to different polarization hypotheses: on the left, vector (brown) and tensor (green); on the right scalar (green) and tensor (brown). They are all equivalent.
star inspiral signal (GW170817) [6]. A representation of their data is given in Fig. 6.6.
Associated with this event, a gamma ray burst [123] was independently observed, and an optical counterpart was later discovered [124]. The source was successively associated with the galaxy NGC4993.
In terms of fundamental physics, these coincident observations led to a stringent constraint on the difference between the speed of gravity and the speed of light, allowed new bounds to be placed on local Lorentz invariance violations and enabled a new test of the equivalence principle by bounding the Shapiro delay between gravitational and electromagnetic radiation [123]. These bounds, in turn, helped to strongly constrain the allowed parameter space of alternative theories of gravity that offered gravitational explanations for the origin of dark energy [125–131] or dark matter [132].
GW170817 also offers important clues about GR polarizations. We carry out a test similar to [5] (Sec. 6.2) by performing a coherent Bayesian analysis of the signal properties using either the tensor or the vector or the scalar response functions.
We assume that the phase evolution of the GW can be described by GR templates, but the polarization content can vary [116]. The phase evolution is modeled with the GR waveform model IMRPhenomPv2 and the analysis is carried out with
(a) LIGO Hanford
(b) LIGO Livingston
(c) Virgo
Figure 6.5: GW170814 waveforms. Reconstructed waveforms with the maximum a posterioriprobability (MAP) under the assumption of fully tensor (blue), vector (orange) or scalar (green) polarizations, as seen by each of the three detectors in the network.
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Figure 6.6: Time-frequency representations [122] of data containing the gravitational- wave event GW170817 , observed by the LIGO-Hanford (top), LIGO-Livingston (middle), and Virgo (bottom) detectors. Times are shown relative to August 17, 2017 12:41:04 UTC. The amplitude scale in each detector is normalized to that detector’s noise amplitude spectral density. In the LIGO data, independently observable noise sources and a glitch that occurred in the LIGO-Livingston detector have been subtracted, as described in [6]. (Reproduced from [6].)
LALInference [20]. Tidal effects are not included in this waveform model, but this is not expected to affect the results presented below, since the polarization test is sensitive to the antenna pattern functions of the detectors and not the phase evolution of the signal, as argued above.
If the sky location of GW170817 is constrained to NGC 4993, we find overwhelming evidence in favor of pure tensor polarization modes in comparison to pure vector and pure scalar modes with a (base ten) logarithm of the Bayes factor of+20.81±0.08 and+23.09±0.08 respectively. This result is many orders of magnitudes stronger than the GW170814 case both due to the sky position of GW170817 relative to the detectors and the fact that the sky position is determined precisely by electromagnetic observations. Indeed if the sky location is unconstrained we find evidence against scalar modes with +5.84±0.09, while the test is inconclusive for vector modes with+0.72±0.09. From this analysis, we also see that only the tensor hypothesis is consistent with the location of the electromagnetic counterpart (Fig. 6.7).
The above results are to be expected given the marked difference in the network’s sensitivity to signals with different polarizations in the direction of GW170817 (with respect to the network at the time of arrival). This is demonstrated by Fig. 6.8, which shows the location of NGC 4993 (cyan star) over each detector’s effective response to tensor, vector or scalar signals (color). For the purpose of this figure, we define the “effective response” as the quadrature sum of the antenna patterns for each polarization of a given helicity evaluated at a given sky location [see Eqs. (5.23)–(5.25)]. This figure suggests that, had GW170817 been purely scalar or vector, the detectors would have measured drastically different relative amplitudes.
In particular, the signal would not have been measured loudly in the LIGO detectors and not at all in Virgo, as it was.