Constraining the parameters of GW150914 and GW170104 with numerical relativity surrogates

Gravitational-wave (GW) detectors have begun to observe coalescences of heavy black hole binaries ($M\gtrsim 50M_{\odot }$) at a consistent pace for the past few years. Accurate models of gravitational waveforms are essential for unbiased and precise estimation of source parameters, such as masses and spins of component black holes. Recently developed surrogate models based on high-accuracy numerical relativity (NR) simulations provide ideal models for constraining physical parameters describing these heavy black hole merger events. In this paper, we first demonstrate the viability of these multi-modal surrogate models as reliable parameter estimation tools. We show that within a fully Bayesian framework, NR surrogates can help extract additional information from GW observations that is inaccessible to traditional models. We demonstrate this by analyzing a set of synthetic signals with NR surrogate templates and comparing against contemporary approximate models. We then consider the case of two of the earliest binary black holes detected by the LIGO observatories, GW150914 and GW170104. We reanalyze their data with the generically precessing NR-based surrogate model and freely provide the resulting posterior samples as supplemental material. We find that our refined analysis is able to extract information from sub-dominant GW harmonics in data, and therefore better resolve the degeneracy in measuring source luminosity distance and orbital inclination for both events. Our analysis estimates the sources of both events to be 20%–25% further away than was previously estimated. Our analysis also constrains their orbital orientations more tightly around face-on or face-off configurations than before. Additionally, for GW150914 we constrain the effective inspiral spin ${\chi }_{\mathrm{eff}}$ more tightly around zero. This work is one of the first to unambiguously extract sub-dominant GW mode information from real events. It is also a first step toward eliminating the approximations used in semi-analytic waveform models from GW parameter estimation. It strongly motivates that NR surrogates be extended to cover more of the binary black hole parameter space.

Figure 1

Parameter recovery for select synthetic injections with NRSur7dq2HM templates. Here we show the mass (top) and source distance/inclination angle (bottom) recovery for four injections, labeled #0,8,16,32 in Table 1. Each of the top and bottom graphs has three panels: One shows two-dimensional 90%-credible regions for the joint measurement of both parameters (for that graph), while the other two show one-dimensional marginalized probability distributions measured for each of the same two source parameters (for that graph). All four injections have identical source mass and spin parameters. The first three have identical source inclination angles as well, but they differ in the distance at which their source is located: 1500 Mpc, 1000 Mpc, and 500 Mpc, respectively. The fourth injection (#32) is similar to injection #8, except that its orbital inclination is much closer to edge-on; i.e., ${\theta }_{\mathrm{JN}}=75°$ for #32. In the bottom graph, we additionally show results from analyses with NRSur7dq2L2 templates as dashed 2D contours. These can be directly contrasted with solid contours to read the effect of including $l>2$ modes in templates. In both the 2D and 1D panels, solid colored lines mark the true injected parameter value, and dashed vertical lines show the limits of 90%-credible regions for the relevant parameter. In all 1D panels, black curves show the sampling prior for that parameter. See text for further discussion.

Figure 2

Estimated masses, spins, and other physical parameters for a set of synthetic GW signals, analyzed using different approximants. In different colors, we show the results for three template models: IMRPhenomPv2 (blue, labeled PP), NRSur7dq2L2 (labeled NR22) and NRSur7dq2HM (labeled NRHM), each restricted to the domain of validity of the NRSur7dq2 surrogate model. In addition, we show results for IMRPhenomPv2 with the model allowed to explore its entire domain of support [22, 35, 36] (yellow, labeled PP-FullP). Parameters of injected signals are given in Table 1 and described in text. True values of injection parameters are marked by black triangles. Vertical line segments show measured 90%-credible intervals, and colored circles show the corresponding median estimates. Panel letters are indicated in the top-right corners. See text for discussion.

Figure 3

(a)–(d) Shown are the mean systematic biases $\overline{\delta {\theta }_{\mathrm{syst}}}$ and statistical uncertainties $\overline{\delta {\theta }_{\mathrm{stat}}}$ for various binary parameters averaged over all injections with ${\theta }_{\mathrm{JN}}=30°$. Four distinct template configurations are considered: NRSur7dq2L2 and NRSur7dq2HM models, and IMRPhenomPv2 with and without being artificially restricted to the domain of validity of NRSur7dq2. The full prior for IMRPhenomPv2 extends over $1\le q\le 8$ and spin magnitudes $|{\overrightarrow{\chi }}_{1,2}|\le 0.89$. Since the signal model is NRSur7dq2HM, when the recovery model is also NRSur7dq2HM, the mean systematic biases and statistical uncertainties reflect the shape of the posterior itself rather than modeling error. (e)–(j): Similar to the four left panels, except (i) the averaging of $\overline{\delta {\theta }_{\mathrm{syst}}}$ and $\overline{\delta {\theta }_{\mathrm{stat}}}$ is performed over all injections with ${\theta }_{\mathrm{JN}}=75^{\circ }$, and (ii) additional panels (i)–(j) show results for the orbital inclination angle.

Figure 4

Estimated sky location for GW150914, using different approximants: NRSur7dq2HM, NRSur7dq2L2, IMRPhenomPv2 (labeled IMRPP), and SEOBNRv4 (labeled SEOBv4). The X-FullP results correspond to an analysis with model X that allows for unrestricted mass ratios $1\le q\le 8$, and spin magnitudes up to $a_{1,2}\lesssim 0.89$ for IMRPhenomPv2 and $a_{1,2}\lesssim 0.98$ for SEOBNRv4. For all others, we a priori restrict sampling to $1\le q\le 2$ and $0\le a_{1,2}\le 0.8$—i.e., to the range where NR surrogate models are valid. In all panels showing 1D posterior distributions, the shaded region shows our prior belief. Vertical dashed lines in 1D posteriors mark 90%-credible regions. The 2D posteriors show both the 90%- (dashed line) and 68%- (solid line) credible regions.

Figure 5

Estimated source orbital orientation/luminosity distance for GW150914, using different approximants: NRSur7dq2HM, NRSur7dq2L2, IMRPhenomPv2 (labeled IMRPP), and SEOBNRv4 (labeled SEOBv4). All figure attributes are similar to those in Fig. 4.

Figure 6

Luminosity distance and mass ratio measurement for GW150914. All figure attributes are similar to those in Fig. 4. We find that the samples at large luminosity distances actually correspond to smaller mass ratios, and therefore the shifting of distance measurement to larger values when using NRSur7dq2HM is not a symptom of the model’s restricted sampling priors.

Figure 7

Estimated masses for GW150914, using different approximants: NRSur7dq2HM, NRSur7dq2L2, IMRPhenomPv2 (labeled IMRPP), and SEOBNRv4 (labeled SEOBv4). The X-FullP results correspond to an analysis with model X that allows for unrestricted mass ratios $1\le q\le 8$, and spin magnitudes up to $a_{1,2}\lesssim 0.89$ for IMRPhenomPv2 and $a_{1,2}\lesssim 0.98$ for SEOBNRv4. For all others, we a priori restrict sampling to $1\le q\le 2$ and $0\le a_{1,2}\le 0.8$—i.e., to the range where NR surrogate models are valid. In all panels showing 1D posterior distributions, the shaded region shows our prior belief. Vertical dashed lines in 1D posteriors mark 90%-credible regions. The 2D posteriors show the 90%-credible regions as a solid line.

Figure 8

Estimated total mass for GW150914, as measured in the source frame. Four different approximants are shown: NRSur7dq2HM, NRSur7dq2L2, IMRPhenomPv2 (labeled IMRPP), and SEOBNRv4 (labeled SEOBv4). Figure attributes are identical to those of Fig. 7. The shaded region shows our prior belief. Vertical dashed lines mark 90%-credible regions, and vertical solid lines show the distribution median.

Figure 9

Estimated spins for GW150914, using different approximants and different prior probability distributions. Shown are spin magnitudes for both BHs, and the tilt angles between BH spins and the orbital angular momentum at $f_{\mathrm{ref}}$. Figure attributes are identical to those of Figs. 4 and 7.

Figure 10

Estimated sky location for GW170104, using different approximants: NRSur7dq2HM, NRSur7dq2L2, IMRPhenomPv2 (labeled IMRPP), and SEOBNRv4 (labeled SEOBv4). The X-FullP results correspond to an analysis with model X that allows for unrestricted mass ratios $1\le q\le 8$, and spin magnitudes up to $a_{1,2}\lesssim 0.89$ for IMRPhenomPv2 and $a_{1,2}\lesssim 0.98$ for SEOBNRv4. For all others, we a priori restrict sampling to $1\le q\le 2$ and $0\le a_{1,2}\le 0.8$—i.e., to the range where NR surrogate models are valid. In all panels showing 1D posterior distributions, the shaded region shows our prior belief. Vertical dashed lines in 1D posteriors mark 90%-credible regions. The 2D posteriors show both the 90%- (dashed line) and 68%- (solid line) credible regions.

Figure 11

Estimated source orientation/luminosity distance for GW170104, using different approximants: NRSur7dq2HM, NRSur7dq2L2, IMRPhenomPv2 (labeled IMRPP), and SEOBNRv4 (labeled SEOBv4). All figure attributes are identical to those of Fig. 10.

Figure 12

Estimated masses for GW170104, using different approximants: NRSur7dq2HM, NRSur7dq2L2, IMRPhenomPv2 (labeled IMRPP) and SEOBNRv4 (labeled SEOBv4). The X-FullP results correspond to an analysis with model X that allows for unrestricted mass ratios $1\le q\le 8$, and spin magnitudes up to $a_{1,2}\lesssim 0.89$ for IMRPhenomPv2 and $a_{1,2}\lesssim 0.98$ for SEOBNRv4. For all others, we a priori restrict sampling to $1\le q\le 2$ and $0\le a_{1,2}\le 0.8$—i.e., to the range where NR surrogate models are valid. In all panels showing 1D posterior distributions, the shaded region shows our prior belief. Vertical dashed lines in 1D posteriors mark 90%-credible regions. The 2D posteriors show 90%-credible regions as solid contours.

Figure 13

Total mass for GW170104, as measured in the source frame. Four different approximants are shown: NRSur7dq2HM, NRSur7dq2L2, IMRPhenomPv2 (labeled IMRPP), and SEOBNRv4 (labeled SEOBv4). All figure attributes are identical to those of Fig. 12.

Figure 14

Estimated spins for GW170104, using different approximants and different prior probability distributions. Shown are spin magnitudes for both BHs and the tilt angles between BH spins and the orbital angular momentum at $f_{\mathrm{ref}}$. Figure attributes are identical to those of Figs. 10 and 12.