Inferring the gravitational wave memory for binary coalescence events

Full, nonlinear general relativity predicts a memory effect for gravitational waves. For compact binary coalescence, the total gravitational memory serves as an inferred observable, conceptually on the same footing as the mass and the spin of the final black hole. Given candidate waveforms for any LIGO-Virgo event, then one can calculate the posterior probability distribution functions for the total gravitational memory and use them to compare and contrast the waveforms. In this paper, we present these posterior distributions for the binary black hole merger events reported in the first Gravitational Wave Transient Catalog, using the phenomenological and effective-one-body waveforms. On the whole, the two sets of posterior distributions agree with each other quite well though we find larger discrepancies for the $\ell =2$, $m=1$ mode of the memory. This signals a possible source of systematic errors that was not captured by the posterior distributions of other inferred observables. Thus, the posterior distributions of various angular modes of total memory can serve as diagnostic tools to further improve the waveforms. Analyses such as this would be valuable especially for future events as the sensitivity of ground-based detectors improves, and for LISA which could measure the total gravitational memory directly.

Figure 1

GW150914: distribution of the final mass using the IMRPhenomPv2 and SEOBNRv3 models, compared with the numerical relativity fits applied to each of the posterior distributions. Unlike the kick-velocity results shown in Fig. 3, the results for the final mass using the two models are mostly consistent with the numerical relativity results. The reason is that the energy flux is dominated by the $\ell =2$ modes which are accurately modeled.

Figure 2

GW150914: posterior distributions for the perpendicular dimensionless spin components ${\chi }_{1,2}^{\perp }$. The phenom model uses a single effective spin while the EOB is parametrized by the individual spins. This gives rise to the bimodal distribution for ${\chi }_{1,2}^{\perp }$ in the EOB model, which in turn leads to the double hump in the posterior distribution of the final mass in Fig. 1.

Figure 3

GW150914: posterior distribution for the recoil velocity (in units with $c=1$) using the IMRPhenomPv2 (left panel) and SEOBNRv3 (middle panel) models, and also accurate fits to numerical relativity calculations (right panel). We note large discrepancies between both the models and the accurate numerical relativity results; the distribution in the left panel has the bulk of its support for $v/c\sim \mathcal{O}({10}^{-16})$ (making it consistent with being purely numerical noise), the middle panel for $\mathcal{O}({10}^{-6})$, while the correct answer in the right panel has $\mathcal{O}({10}^{-3})$. As discussed in the text, this is not surprising because neither of the models attempts to model the higher modes, or the asymmetry between the $(\ell ,\pm m)$ modes, that are important for the kick velocity. However, the discrepancy between the phenom and EOB models is also interesting to note.

Figure 4

The posterior distributions for the total memory in the (2, 0), (2, 1), and (4, 0) modes for the three O1 events. The best fit values of the individual black hole masses for GW150914, GW151012, and GW151226 are, respectively, $(35.6M_{\odot },30.6M_{\odot })$ $(13.6M_{\odot },15.2M_{\odot })$, and $(7.7M_{\odot },8.9M_{\odot })$. As in the main text, we emphasize again that this not a direct measurement of the memory, but these are instead histograms of the inferred values of the memory relying on waveform models and standard general relativity. The first two events are consistent with nonspinning initial black holes while there is some evidence for moderate spins for GW151226. The red and green vertical dashed lines show the 90% credible intervals (centered around the median) for the corresponding distribution.

Figure 5

Posterior distributions of the inferred values of the memory for some selected modes. This figure presents results for five of the O2 events, namely, GW170104, GW170608, GW170809, GW170814, and GW170818, in each case for both the EOB and phenom models. Each row of figures corresponds to a particular event, while the first, second, and third columns refer to the (2,0),(2,1), and (4,0) modes, respectively. These are all systems with moderate mass ratios, with the best fit individual masses being, respectively, $(20.0M_{\odot },21.4M_{\odot })$, $(11.0M_{\odot },7.6M_{\odot })$, $(35.0M_{\odot },23.8M_{\odot })$, $(30.6M_{\odot },25.2M_{\odot })$, and $(35.4M_{\odot },26.7M_{\odot })$. All of these are consistent with small individual spins. Note that while the posteriors of EOB and phenom generally agree, there is a large difference for the (2,1) mode for GW170814.

Figure 6

Posterior distributions of the inferred memory for the (2, 0), (2, 1), and (4, 0) modes. The respective modes are shown in the first, second, and third columns for two of the O2 events GW170729 and GW170823, and for the IMRPhenomPv2 model. The individual masses for these events are, respectively, $(50.2M_{\odot },34.0M_{\odot })$ and $(39.5M_{\odot },29.0M_{\odot })$. GW170729 has moderately strong evidence of non-negligible spins.

Figure 7

GW150914: posterior distributions of the final mass using the flux from SEOBNRv3 and IMRPhenomPv2 models with various levels of PN corrections, compared with the numerical relativity fits applied to each distribution. EOB shows significant improvement with the addition of the 0PN term for early time. However, adding higher order terms, till 3.5PN, shows negligible further change. On the other hand, for phenom, we use waveforms with a much lower frequency cutoff. Therefore, the truncation error is small and corrections are negligible.