Adding gravitational memory to waveform catalogs using BMS balance laws

Accurate models of gravitational waves from merging binary black holes are crucial for detectors to measure events and extract new science. One important feature that is currently missing from the Simulating eXtreme Spacetimes (SXS) Collaboration’s catalog of waveforms for merging black holes, and other waveform catalogs, is the gravitational memory effect: a persistent, physical change to spacetime that is induced by the passage of transient radiation. We find, however, that by exploiting the Bondi-van der Burg-Metzner-Sachs (BMS) balance laws, which come from the extended BMS transformations, we can correct the strain waveforms in the SXS catalog to include the missing displacement memory. Our results show that these corrected waveforms satisfy the BMS balance laws to a much higher degree of accuracy. Furthermore, we find that these corrected strain waveforms coincide especially well with the waveforms obtained from Cauchy-characteristic extraction (CCE) that already exhibit memory effects. These corrected strain waveforms also evade the transient junk effects that are currently present in CCE waveforms. Last, we make our code for computing these contributions to the BMS balance laws and memory publicly available as a part of the python package sxs, thus enabling anyone to evaluate the expected memory effects and violation of the BMS balance laws.

Figure 1

Comparison of the extrapolated strain (2,0) mode ($h^{\mathrm{EXT}}$, black/solid) to the strain that is extracted using CCE ($h^{\mathrm{CCE}}$, gray/solid) and to what is expected according to the other modes of the extrapolated waveform using the various BMS balance laws ($J^{\mathrm{EXT}}$, red/dashed) computed by using the extrapolated waveform in Eq. (17). We take the news to be the retarded time derivative of the strain. In the middle panel, we plot the absolute error: $|h^{\mathrm{CCE}}-J^{\mathrm{EXT}}|$. In the bottom panel, we outline the individual contributions that come from the Bondi mass aspect (black/solid), the Bondi angular momentum aspect (red/dashed), the energy flux (blue/dotted), and the angular momentum flux (green/dashed/dotted). BBH merger: q1_nospin (see Table 3). CCE waveform: CCE-R0292 (with a supertranslation applied).

Figure 2

Violation of the BMS balance laws, Eq. (14), for extrapolated and CCE waveforms. In the top plot, we show the norm of Eq. (14) over the two-sphere. Here $J$ in Eq. (14) is computed by Eq. (13) and the ${\mathrm{\Psi }}_2$ used in Eq. (13) for the extrapolated or CCE waveforms is extracted using the procedures described in [29] and [30, 31]. In the bottom plot, we show the normalized time-integrated $L^2$ norm as a function of a few important spin-weight $-2$ spherical harmonic modes. Note that we integrate each of these modes by starting at one full orbit past the retarded time $u=0M$ to suppress misleading effects that the Cauchy evolution junk radiation or the CCE transient effects may induce. BBH merger: q1_nospin (see Table 3).

Figure 3

Identical to Fig. 2, but now with the violations for the memory-corrected extrapolated strain. BBH merger: q1_nospin (see Table 3).

Figure 4

Identical to the bottom plot of Fig. 3, but now for an equal mass, precessing system. BBH merger: q1_precessing (see Table 3).

Figure 5

The time-integrated norm of the BMS balance law violation as a function of the BBH system. Integration is performed from one full orbit past the retarded time $u=0M$ onward to avoid errors introduced by the Cauchy evolution junk radiation and most of the CCE transient effects. The labeling CCE-R[0,1,2,3] represents the four world tube radii (from smallest to largest). This plot demonstrates that the displacement memory can be effectively added to the extrapolated waveforms with the result satisfying the BMS balance laws to comparable accuracy as CCE. The parameters of these systems can be found in Table 3.

Figure 6

Comparison of the extrapolated strain (3,0) mode ($h^{\mathrm{EXT}}$, black/solid) to the BMS strain that we compute using the terms in Eq. (17) ($J^{\mathrm{EXT}}$, red/dashed). In the bottom panel, we display the contributions that come from the Bondi mass aspect (black/solid), the Bondi angular momentum aspect (red/dashed), the energy flux (blue/dotted), and the angular momentum flux (green/dashed/dotted). BBH merger: q1_nospin (see Table 3).

Figure 7

Mismatch between the extrapolated and CCE strains with (light/solid curves) and without (dark/dashed curves) the memory correction applied to the extrapolated strain. We compute the mismatch between the various waveforms according to Eq. (25). To improve the alignment between the extrapolated and CCE waveforms, we have applied an optimized $\ell \le 2$ supertranslation to the CCE waveforms. BBH merger: q1_nospin (see Table 3).

Figure 8

The strain (2,0) mode for the CCE strain waveform (second smallest extraction radius) and the extrapolated strain waveform with and without the memory correction. In the inset panel, we plot the mismatch between the extrapolated and CCE strain waveforms for the same mode using Eq. (22). Further, to improve the alignment between the extrapolated and CCE waveforms, we have applied an $\ell \le 2$ supertranslation to the CCE waveforms that minimizes the $L^2$ norm between the extrapolated and CCE waveforms. The negative slope in the CCE strain around $-6000M$ is an example of the junklike transient effects in CCE waveforms [37]. BBH merger: q1_nospin (see Table 3).

Figure 9

Norm of the proper Moreschi supermomentum, as defined by Eq. (27), for the extrapolated waveforms, both with and without memory corrections, and the CCE waveforms. BBH merger: q1_nospin (see Table 3).