The second RIT binary black hole simulations catalog and its application to gravitational waves parameter estimation

TL;DR

This paper introduces a comprehensive catalog of 320 numerical relativity simulations of binary black hole mergers, which aids in gravitational wave analysis and parameter estimation, including reanalysis of GW150914.

Contribution

The paper presents the second public catalog of black-hole-binary waveforms with extensive simulations and tools for waveform analysis and parameter estimation.

Findings

Catalog includes 320 simulations covering a range of mass ratios and spins.

Waveforms are extrapolated for accurate gravitational wave interpretation.

Application to GW150914 demonstrates the catalog's utility in parameter estimation.

Abstract

The RIT numerical relativity group is releasing the second public catalog of black-hole-binary waveforms \url{http://ccrg.rit.edu/~RITCatalog}. This release consists of 320 accurate simulations that include 46 precessing and 274 nonprecessing binary systems with mass ratios $q=m_1/m_2$ in the range $1/6\leq q\leq1$ and individual spins up to $s/m^2=0.95$. The new catalog contains search and ordering tools for the waveforms based on initial parameters of the binary, trajectory information, peak radiation, and final remnant black hole properties. The final black hole remnant properties provided here can be used to model the merger of black-hole binaries from its initial configurations. The waveforms are extrapolated to infinite observer location and can be used to independently interpret gravitational wave signals from laser interferometric detectors. As an application of this waveform catalog we reanalyze the signal of GW150914 implementing parameter estimation techniques that make use of only numerical waveforms without any reference to information from phenomenological models.

Figures

Tables

Table 1. Table 1: Highest ln ⁡ ℒ ℒ \ln{\cal L} nonprecessing and precessing simulations. The nonprecessing simulation has highest overall ln ⁡ ℒ ℒ \ln{\cal L} , and the precessing simulation has 13th highest.

Config.	$q$	${\overrightarrow{\chi }}_1$	${\overrightarrow{\chi }}_2$	${\overrightarrow{S}}_{hu}/m^2$	$M_{total}/M_{\odot }$	$\ln ℒ$
RIT:BBH:0113	0.85	(0, 0, 0)	(0, 0, 0)	(0, 0, 0)	73.6	261.8
RIT:BBH:0126	0.75	(-0.46, -0.48, -0.44)	(0.06, -0.38, 0.12)	(-0.15, -0.42, -0.11)	72.5	260.5

Table 2. Table 2: Initial data parameters for the quasi-circular configurations with a smaller mass black hole (labeled 1), and a larger mass spinning black hole (labeled 2). The punctures are located at r → 1 = ( x 1 , 0 , 0 ) subscript → 𝑟 1 subscript 𝑥 1 0 0 \vec{r}_{1}=(x_{1},0,0) and r → 2 = ( x 2 , 0 , 0 ) subscript → 𝑟 2 subscript 𝑥 2 0 0 \vec{r}_{2}=(x_{2},0,0) , with individual linear momenta P = ± ( P r , P t , 0 ) 𝑃 plus-or-minus subscript 𝑃 𝑟 subscript 𝑃 𝑡 0 P=\pm(P_{r},P_{t},0) , spin magnitudes | S i | subscript 𝑆 𝑖 |S_{i}| , puncture mass parameters m p / m superscript 𝑚 𝑝 𝑚 m^{p}/m , horizon (Christodoulou) masses m H / m superscript 𝑚 𝐻 𝑚 m^{H}/m , total ADM mass M ADM subscript 𝑀 ADM M_{\rm ADM} , and dimensionless spins | a / m H | = | S / m H 2 | 𝑎 subscript 𝑚 𝐻 𝑆 superscript subscript 𝑚 𝐻 2 |a/m_{H}|=|S/m_{H}^{2}| . The spin directions for the precessing simulations are given in the catalog and in Tab.  LABEL:tab:IDr_prec .

Run	$x_1/m$	$x_2/m$	$P_r/m$	$P_t/m$	$m_1^p/m$	$m_2^p/m$	$\left|S_1/m^2\right|$	$\left|S_2/m^2\right|$	$m_1^H/m$	$m_2^H/m$	$M_{\mathrm{ADM}}/m$	$\left|a_1/m_1^H\right|$	$\left|a_2/m_2^H\right|$
RIT:BBH:0127	-5.71	4.29	0	0.09386	0.2578	0.347	0.1469	0.2612	0.4286	0.5714	0.9901	0.8	0.8
RIT:BBH:0128	-6.67	3.33	0	0.08542	0.1992	0.4072	0.08889	0.3556	0.3333	0.6667	0.9911	0.8	0.8
RIT:BBH:0129	-7.50	2.50	0	0.07226	0.1485	0.4604	0.05	0.45	0.25	0.75	0.9925	0.8	0.8
RIT:BBH:0130	-5.71	4.29	0	0.09362	0.2578	0.3469	0.1469	0.2612	0.4286	0.5714	0.9901	0.8	0.8
RIT:BBH:0131	-6.67	3.33	0	0.08502	0.1992	0.4072	0.08889	0.3556	0.3333	0.6667	0.9911	0.8	0.8
RIT:BBH:0132	-7.50	2.50	0	0.07189	0.1485	0.4603	0.05	0.45	0.25	0.75	0.9925	0.8	0.8
RIT:BBH:0136	-10.00	5.00	0	0.06464	0.3248	0.546	0	0.2667	0.3333	0.6667	0.9934	0	0.6
RIT:BBH:0220	-7.50	5.00	-4.96e-04	0.08177	0.3889	0.3666	0	0.288	0.4	0.6	0.9921	0	0.8
RIT:BBH:0221	-10.83	2.17	-1.51e-04	0.04638	0.09955	0.7352	0.02222	0.3472	0.1667	0.8333	0.9957	0.8	0.5
RIT:BBH:0222	-10.83	2.17	-1.20e-04	0.04418	0.09964	0.7352	0.02222	0.3472	0.1667	0.8333	0.9954	0.8	0.5
RIT:BBH:0223	-7.03	5.97	-4.30e-04	0.08115	0.279	0.5162	0.1689	0.07305	0.4595	0.5405	0.992	0.8	0.25
RIT:BBH:0224	-10.83	2.17	-1.09e-04	0.04312	0.09967	0.515	0.02222	0.5556	0.1667	0.8333	0.9954	0.8	0.8
RIT:BBH:0226	-10.83	2.17	-1.27e-04	0.0447	0.09961	0.8278	0.02222	0	0.1667	0.8333	0.9955	0.8	0
RIT:BBH:0227	-7.03	5.97	-3.89e-04	0.07931	0.279	0.5163	0.1689	0.07305	0.4595	0.5405	0.9918	0.8	0.25
RIT:BBH:0228	-6.50	6.50	-3.94e-04	0.07983	0.2579	0.4557	0.2125	0.1	0.5	0.5	0.9918	0.85	0.4
RIT:BBH:0230	-7.50	5.00	-4.73e-04	0.08102	0.3889	0.3666	0	0.288	0.4	0.6	0.992	0	0.8
RIT:BBH:0231	-7.50	5.00	-4.74e-04	0.08104	0.3889	0.3666	0	0.288	0.4	0.6	0.992	0	0.8
RIT:BBH:0232	-7.03	5.97	-5.02e-04	0.08372	0.2789	0.3295	0.1689	0.2337	0.4595	0.5405	0.9924	0.8	0.8
RIT:BBH:0233	-7.50	5.00	-4.76e-04	0.0811	0.3889	0.3666	0	0.288	0.4	0.6	0.9921	0	0.8
RIT:BBH:0234	-7.50	5.00	-4.77e-04	0.08113	0.3889	0.3666	0	0.288	0.4	0.6	0.9921	0	0.8
RIT:BBH:0235	-7.50	5.00	-4.76e-04	0.0811	0.3889	0.3666	0	0.288	0.4	0.6	0.9921	0	0.8
RIT:BBH:0236	-7.50	5.00	-4.74e-04	0.08104	0.3889	0.3666	0	0.288	0.4	0.6	0.992	0	0.8
RIT:BBH:0237	-6.86	5.14	-5.90e-04	0.08569	0.311	0.5391	0.1286	0.09796	0.4286	0.5714	0.9917	0.7	0.3
RIT:BBH:0238	-6.86	5.14	-6.35e-04	0.08695	0.3701	0.2953	0.09184	0.2776	0.4286	0.5714	0.9919	0.5	0.85
RIT:BBH:0239	-6.86	5.14	-5.61e-04	0.08478	0.3702	0.2953	0.09184	0.2776	0.4286	0.5714	0.9916	0.5	0.85
RIT:BBH:0240	-7.03	5.97	-4.87e-04	0.08322	0.3983	0.2793	0.1056	0.2484	0.4595	0.5405	0.9922	0.5	0.85
RIT:BBH:0241	-5.62	4.19	-9.95e-04	0.09488	0.1605	0.3708	0.1634	0.2534	0.4271	0.5729	0.9901	0.8957	0.7719
RIT:BBH:0242	-7.50	5.00	-4.74e-04	0.08104	0.3889	0.3666	0	0.288	0.4	0.6	0.992	0	0.8
RIT:BBH:0243	-7.50	5.00	-4.76e-04	0.0811	0.3889	0.3666	0	0.288	0.4	0.6	0.9921	0	0.8
RIT:BBH:0244	-7.50	5.00	-4.74e-04	0.08104	0.3889	0.3666	0	0.288	0.4	0.6	0.992	0	0.8
RIT:BBH:0245	-8.67	4.33	-3.67e-04	0.0735	0.2871	0.3467	0.05556	0.3778	0.3333	0.6667	0.9929	0.5	0.85
RIT:BBH:0246	-10.83	2.17	-1.34e-04	0.04524	0.0996	0.8277	0.02222	0	0.1667	0.8333	0.9955	0.8	0
RIT:BBH:0247	-8.67	4.33	-2.97e-04	0.07024	0.2873	0.3467	0.05556	0.3778	0.3333	0.6667	0.9926	0.5	0.85
RIT:BBH:0248	-7.50	5.00	-4.73e-04	0.08102	0.3889	0.3666	0	0.288	0.4	0.6	0.992	0	0.8
RIT:BBH:0249	-7.03	5.97	-3.85e-04	0.07913	0.3718	0.5299	0.1267	0	0.4595	0.5405	0.9917	0.6	0
RIT:BBH:0250	-7.50	5.00	-4.77e-04	0.08113	0.3889	0.3666	0	0.288	0.4	0.6	0.9921	0	0.8
RIT:BBH:0251	-7.50	5.00	-4.76e-04	0.0811	0.3889	0.3666	0	0.288	0.4	0.6	0.9921	0	0.8
RIT:BBH:0252	-6.86	5.14	-4.43e-04	0.08022	0.3704	0.2954	0.09184	0.2776	0.4286	0.5714	0.9912	0.5	0.85
RIT:BBH:0253	-4.00	4.00	0	0.104	0.5	0.5	0.2375	0.2375	0.5156	0.5156	0.9883	0.95	0.95
RIT:BBH:0254	-7.03	5.97	-4.35e-04	0.08135	0.3718	0.5298	0.1267	0	0.4595	0.5405	0.9919	0.6	0
RIT:BBH:0255	-6.69	4.46	-5.65e-04	0.08352	0.3878	0.3659	0	0.288	0.4	0.6	0.9908	0	0.8
RIT:BBH:0256	-6.69	4.46	-5.66e-04	0.08354	0.3878	0.3659	0	0.288	0.4	0.6	0.9909	0	0.8
RIT:BBH:0257	-6.69	4.46	-5.67e-04	0.08357	0.3878	0.3658	0	0.288	0.4	0.6	0.9909	0	0.8
RIT:BBH:0258	-6.69	4.46	-5.68e-04	0.08359	0.3878	0.3658	0	0.288	0.4	0.6	0.9909	0	0.8
RIT:BBH:0259	-6.69	4.46	-5.67e-04	0.08357	0.3878	0.3658	0	0.288	0.4	0.6	0.9909	0	0.8
RIT:BBH:0260	-6.69	4.46	-5.66e-04	0.08354	0.3878	0.3659	0	0.288	0.4	0.6	0.9909	0	0.8
RIT:BBH:0261	-6.50	6.50	-4.36e-04	0.08168	0.2579	0.4556	0.2125	0.1	0.5	0.5	0.992	0.85	0.4
RIT:BBH:0262	-6.86	5.14	-4.64e-04	0.08124	0.3111	0.5392	0.1286	0.09796	0.4286	0.5714	0.9912	0.7	0.3
RIT:BBH:0263	-7.03	5.97	-3.56e-04	0.07755	0.3985	0.2794	0.1056	0.2484	0.4595	0.5405	0.9917	0.5	0.85
RIT:BBH:0264	-7.05	4.70	-5.25e-04	0.08267	0.3883	0.3662	0	0.288	0.4	0.6	0.9914	0	0.8
RIT:BBH:0265	-7.03	5.97	-4.32e-04	0.08121	0.3983	0.2794	0.1056	0.2484	0.4595	0.5405	0.992	0.5	0.85
RIT:BBH:0266	-7.05	4.70	-5.28e-04	0.08274	0.3883	0.3662	0	0.288	0.4	0.6	0.9915	0	0.8
RIT:BBH:0267	-7.05	4.70	-5.33e-04	0.08287	0.3883	0.3661	0	0.288	0.4	0.6	0.9915	0	0.8
RIT:BBH:0268	-7.05	4.70	-5.27e-04	0.08274	0.3883	0.3662	0	0.288	0.4	0.6	0.9915	0	0.8
RIT:BBH:0269	-6.84	4.56	-5.45e-04	0.08314	0.388	0.366	0	0.288	0.4	0.6	0.9911	0	0.8
RIT:BBH:0270	-6.84	4.56	-5.48e-04	0.08319	0.388	0.366	0	0.288	0.4	0.6	0.9911	0	0.8
RIT:BBH:0271	-6.84	4.56	-5.52e-04	0.08329	0.388	0.3659	0	0.288	0.4	0.6	0.9912	0	0.8
RIT:BBH:0272	-6.84	4.56	-5.54e-04	0.08334	0.388	0.3659	0	0.288	0.4	0.6	0.9912	0	0.8
RIT:BBH:0273	-6.84	4.56	-5.52e-04	0.08329	0.388	0.3659	0	0.288	0.4	0.6	0.9912	0	0.8
RIT:BBH:0274	-6.84	4.56	-5.47e-04	0.08319	0.388	0.366	0	0.288	0.4	0.6	0.9911	0	0.8
RIT:BBH:0276	-7.03	5.97	-4.62e-04	0.08235	0.279	0.5162	0.1689	0.07305	0.4595	0.5405	0.9921	0.8	0.25
RIT:BBH:0277	-7.03	5.97	-3.87e-04	0.07925	0.3985	0.2794	0.1056	0.2484	0.4595	0.5405	0.9918	0.5	0.85
RIT:BBH:0278	-10.83	2.17	-1.15e-04	0.04369	0.09965	0.7352	0.02222	0.3472	0.1667	0.8333	0.9954	0.8	0.5
RIT:BBH:0279	-7.03	5.97	-4.62e-04	0.08235	0.3983	0.4705	0.1056	0.1461	0.4595	0.5405	0.992	0.5	0.5
RIT:BBH:0280	-7.05	4.70	-5.34e-04	0.08287	0.3883	0.3661	0	0.288	0.4	0.6	0.9915	0	0.8
RIT:BBH:0281	-7.05	4.70	-5.36e-04	0.08293	0.3883	0.3661	0	0.288	0.4	0.6	0.9916	0	0.8
RIT:BBH:0283	-7.20	4.80	-6.30e-04	0.08574	0.2415	0.3663	0.128	0.288	0.4	0.6	0.9922	0.8	0.8
RIT:BBH:0284	-9.00	3.00	-3.91e-04	0.0671	0.2136	0.3911	0.03125	0.4781	0.25	0.75	0.9938	0.5	0.85
RIT:BBH:0285	-8.25	2.75	-3.49e-04	0.06534	0.2132	0.3908	0.03125	0.4781	0.25	0.75	0.9928	0.5	0.85
RIT:BBH:0286	-9.00	3.00	-3.64e-04	0.06616	0.2137	0.3912	0.03125	0.4781	0.25	0.75	0.9937	0.5	0.85
RIT:BBH:0287	-7.20	4.80	-5.25e-04	0.0826	0.2415	0.3664	0.128	0.288	0.4	0.6	0.9918	0.8	0.8
RIT:BBH:0288	-8.25	2.75	-3.35e-04	0.06449	0.2132	0.3908	0.03125	0.4781	0.25	0.75	0.9927	0.5	0.85
RIT:BBH:0289	-7.20	4.80	-4.73e-04	0.08083	0.2416	0.3664	0.128	0.288	0.4	0.6	0.9916	0.8	0.8
RIT:BBH:0290	-5.71	4.29	-1.40e-03	0.1082	0.4286	0.5714	0.1745	0.3102	0.4286	0.5714	0.9894	0.95	0.95
RIT:BBH:0291	-7.20	4.80	-4.34e-04	0.07908	0.3451	0.4661	0.08	0.234	0.4	0.6	0.9913	0.5	0.65
RIT:BBH:0292	-7.20	4.80	-4.18e-04	0.07823	0.2417	0.3664	0.128	0.288	0.4	0.6	0.9914	0.8	0.8
RIT:BBH:0293	-7.20	4.80	-4.71e-04	0.08074	0.3451	0.466	0.08	0.234	0.4	0.6	0.9915	0.5	0.65
RIT:BBH:0294	-7.20	4.80	-5.41e-04	0.08319	0.2416	0.5889	0.128	0	0.4	0.6	0.9918	0.8	0
RIT:BBH:0295	-7.20	4.80	-4.33e-04	0.07899	0.2417	0.5232	0.128	0.18	0.4	0.6	0.9914	0.8	0.5
RIT:BBH:0296	-8.67	4.33	-4.00e-04	0.07477	0.287	0.3466	0.05556	0.3778	0.3333	0.6667	0.9931	0.5	0.85
RIT:BBH:0297	-7.20	4.80	-4.96e-04	0.08169	0.2416	0.5231	0.128	0.18	0.4	0.6	0.9917	0.8	0.5
RIT:BBH:0298	-7.20	4.80	-4.58e-04	0.08014	0.3451	0.5739	0.08	0.09	0.4	0.6	0.9914	0.5	0.25
RIT:BBH:0299	-9.19	3.81	-3.51e-04	0.0698	0.2517	0.3684	0.04289	0.425	0.2929	0.7071	0.9936	0.5	0.85
RIT:BBH:0300	-9.19	3.81	-3.26e-04	0.06877	0.2517	0.3684	0.04289	0.425	0.2929	0.7071	0.9934	0.5	0.85
RIT:BBH:0301	-7.20	4.80	-5.94e-04	0.08476	0.2415	0.523	0.128	0.18	0.4	0.6	0.992	0.8	0.5
RIT:BBH:0302	-7.20	4.80	-5.47e-04	0.08339	0.345	0.5738	0.08	0.09	0.4	0.6	0.9917	0.5	0.25
RIT:BBH:0303	-7.29	5.21	-4.14e-04	0.07919	0.2522	0.5726	0.1389	0	0.4167	0.5833	0.9917	0.8	0
RIT:BBH:0304	-7.20	4.80	-5.28e-04	0.08271	0.3449	0.466	0.08	0.234	0.4	0.6	0.9917	0.5	0.65
RIT:BBH:0305	-7.20	4.80	-5.02e-04	0.08188	0.345	0.5738	0.08	0.09	0.4	0.6	0.9916	0.5	0.25
RIT:BBH:0306	-7.20	4.80	-5.90e-04	0.08465	0.3449	0.4659	0.08	0.234	0.4	0.6	0.9919	0.5	0.65
RIT:BBH:0307	-7.20	4.80	-4.63e-04	0.08032	0.2416	0.589	0.128	0	0.4	0.6	0.9915	0.8	0
RIT:BBH:0308	-7.20	4.80	-4.99e-04	0.08172	0.2416	0.5231	0.128	0.18	0.4	0.6	0.9917	0.8	0.5
RIT:BBH:0309	-7.20	4.80	-4.94e-04	0.08154	0.345	0.5739	0.08	0.09	0.4	0.6	0.9915	0.5	0.25
RIT:BBH:0311	-7.20	4.80	-4.61e-04	0.0803	0.3886	0.5231	0	0.18	0.4	0.6	0.9914	0	0.5
RIT:BBH:0312	-7.20	4.80	-5.42e-04	0.0832	0.3884	0.5231	0	0.18	0.4	0.6	0.9917	0	0.5
RIT:BBH:0314	-8.67	4.33	-2.82e-04	0.06922	0.2873	0.3467	0.05556	0.3778	0.3333	0.6667	0.9925	0.5	0.85
RIT:BBH:0316	-9.19	3.81	-2.55e-04	0.06527	0.2519	0.3684	0.04289	0.425	0.2929	0.7071	0.9931	0.5	0.85
RIT:BBH:0317	-5.71	4.29	-7.48e-04	0.09202	0.4286	0.5714	0.1745	0.3102	0.4286	0.5714	0.988	0.95	0.95
RIT:BBH:0318	-10.83	2.17	-1.64e-04	0.04711	0.1422	0.4366	0.01389	0.5903	0.1667	0.8333	0.9957	0.5	0.85
RIT:BBH:0319	-7.29	5.21	-4.83e-04	0.08199	0.2522	0.5725	0.1389	0	0.4167	0.5833	0.992	0.8	0
RIT:BBH:0321	-10.83	2.17	-1.58e-04	0.04672	0.1422	0.4366	0.01389	0.5903	0.1667	0.8333	0.9957	0.5	0.85
RIT:BBH:0322	-9.19	3.81	-2.45e-04	0.06445	0.2519	0.3684	0.04289	0.425	0.2929	0.7071	0.993	0.5	0.85
RIT:BBH:0324	-5.50	5.50	-5.85e-04	0.08541	0.1802	0.1802	0.225	0.225	0.5	0.5	0.9904	0.9	0.9
RIT:BBH:0336	-8.67	4.33	-3.48e-04	0.07276	0.2011	0.6573	0.08889	0	0.3333	0.6667	0.9929	0.8	0
RIT:BBH:0337	-10.83	2.17	-1.11e-04	0.0434	0.1424	0.4366	0.01389	0.5903	0.1667	0.8333	0.9953	0.5	0.85
RIT:BBH:0338	-8.67	4.33	-3.11e-04	0.07092	0.2011	0.6574	0.08889	0	0.3333	0.6667	0.9927	0.8	0
RIT:BBH:0339	-9.75	3.25	-2.79e-04	0.06278	0.2413	0.4618	0	0.45	0.25	0.75	0.9941	0	0.8
RIT:BBH:0344	-7.81	4.69	-5.07e-04	0.08096	0.3233	0.3241	0.07031	0.332	0.375	0.625	0.9925	0.5	0.85
RIT:BBH:0345	-10.83	2.17	-1.09e-04	0.04311	0.1424	0.4366	0.01389	0.5903	0.1667	0.8333	0.9953	0.5	0.85
RIT:BBH:0348	-7.81	4.69	-4.52e-04	0.0791	0.3641	0.5459	0	0.1953	0.375	0.625	0.9921	0	0.5
RIT:BBH:0350	-7.81	4.69	-4.88e-04	0.08035	0.3233	0.4863	0.07031	0.2539	0.375	0.625	0.9923	0.5	0.65
RIT:BBH:0352	-7.81	4.69	-4.82e-04	0.08013	0.364	0.3241	0	0.332	0.375	0.625	0.9923	0	0.85

Table 3. Table 3: The mass and spin of the nonprecessing BHBs in Table  LABEL:tab:ID after the BHs had time to equilibrate ( t / m = 200 𝑡 𝑚 200 t/m=200 ). Also provided are the difference in the masses, δ ​ m = ( q − 1 ) / ( q + 1 ) 𝛿 𝑚 𝑞 1 𝑞 1 \delta m=(q-1)/(q+1) , sum of the spins S / m 2 = ( χ 2 ​ z + q 2 ​ χ 1 ​ z ) / ( q + 1 ) 2 𝑆 superscript 𝑚 2 subscript 𝜒 2 𝑧 superscript 𝑞 2 subscript 𝜒 1 𝑧 superscript 𝑞 1 2 S/m^{2}=(\chi_{2z}+q^{2}\chi_{1z})/(q+1)^{2} , and spin difference, Δ / m 2 = ( χ 2 ​ z − q ​ χ 1 ​ z ) / ( 1 + q ) Δ superscript 𝑚 2 subscript 𝜒 2 𝑧 𝑞 subscript 𝜒 1 𝑧 1 𝑞 \Delta/m^{2}=(\chi_{2z}-q\chi_{1z})/(1+q) .

Run	$q^r$	$m_1^r/m$	$m_2^r/m$	${\chi }_{1z}^r$	${\chi }_{2z}^r$	$\delta m_r$	$S_r/m_r^2$	${\Delta }_r/m_r^2$
RIT:BBH:0136	$0.5001$	$0.3333$	$0.6666$	$0.0000$	$0.4244$	$-0.3333$	$0.2829$	$0.1886$
RIT:BBH:0220	$0.6670$	$0.4000$	$0.5997$	$0.0000$	$-0.8008$	$-0.1998$	$-0.4801$	$-0.2880$
RIT:BBH:0221	$0.2000$	$0.1666$	$0.8333$	$-0.8005$	$-0.5000$	$-0.6667$	$-0.2833$	$-0.3694$
RIT:BBH:0222	$0.2000$	$0.1666$	$0.8333$	$-0.8006$	$0.5000$	$-0.6667$	$0.5500$	$0.3250$
RIT:BBH:0223	$0.8496$	$0.4593$	$0.5405$	$-0.8008$	$0.2500$	$-0.0813$	$0.5028$	$-0.0958$
RIT:BBH:0224	$0.2001$	$0.1666$	$0.8329$	$0.8006$	$0.8009$	$-0.6666$	$0.5334$	$0.5778$
RIT:BBH:0226	$0.2000$	$0.1666$	$0.8333$	$0.8006$	$0.0000$	$-0.6667$	$-0.1334$	$0.0222$
RIT:BBH:0227	$0.8496$	$0.4593$	$0.5405$	$0.8007$	$-0.2500$	$-0.0813$	$-0.5028$	$0.0958$
RIT:BBH:0228	$0.9993$	$0.4997$	$0.5000$	$0.8512$	$-0.4000$	$-0.0003$	$-0.6251$	$0.1125$
RIT:BBH:0232	$0.8500$	$0.4593$	$0.5403$	$-0.8007$	$-0.8008$	$-0.0811$	$-0.0649$	$-0.4027$
RIT:BBH:0237	$0.7499$	$0.4285$	$0.5714$	$-0.7003$	$-0.3000$	$-0.1429$	$0.1286$	$-0.2265$
RIT:BBH:0238	$0.7505$	$0.4286$	$0.5710$	$-0.5000$	$-0.8512$	$-0.1425$	$-0.2717$	$-0.3694$
RIT:BBH:0239	$0.7505$	$0.4286$	$0.5710$	$0.5000$	$-0.8512$	$-0.1425$	$-0.7001$	$-0.1857$
RIT:BBH:0240	$0.8506$	$0.4595$	$0.5402$	$-0.5000$	$-0.8508$	$-0.0807$	$-0.2297$	$-0.3538$
RIT:BBH:0241	$0.7450$	$0.4267$	$0.5728$	$-0.8972$	$0.7723$	$-0.1461$	$0.8248$	$0.0900$
RIT:BBH:0245	$0.5004$	$0.3333$	$0.6662$	$0.5000$	$-0.8514$	$-0.3330$	$-0.7335$	$-0.3223$
RIT:BBH:0246	$0.2000$	$0.1666$	$0.8333$	$-0.8004$	$0.0000$	$-0.6667$	$0.1334$	$-0.0222$
RIT:BBH:0247	$0.5004$	$0.3333$	$0.6662$	$-0.5000$	$0.8513$	$-0.3330$	$0.7334$	$0.3222$
RIT:BBH:0249	$0.8499$	$0.4594$	$0.5405$	$0.6001$	$0.0000$	$-0.0811$	$-0.2757$	$0.1267$
RIT:BBH:0252	$0.7505$	$0.4286$	$0.5710$	$0.5000$	$0.8512$	$-0.1425$	$0.2717$	$0.3694$
RIT:BBH:0253	$1.0000$	$0.5010$	$0.5010$	$0.9485$	$0.9485$	$0.0000$	$0.0000$	$0.4761$
RIT:BBH:0254	$0.8499$	$0.4594$	$0.5405$	$-0.6001$	$0.0000$	$-0.0811$	$0.2757$	$-0.1267$
RIT:BBH:0261	$0.9993$	$0.4997$	$0.5000$	$-0.8512$	$0.4000$	$-0.0003$	$0.6251$	$-0.1125$
RIT:BBH:0262	$0.7499$	$0.4285$	$0.5714$	$0.7003$	$0.3000$	$-0.1429$	$-0.1286$	$0.2265$
RIT:BBH:0263	$0.8506$	$0.4595$	$0.5401$	$0.5000$	$0.8505$	$-0.0807$	$0.2296$	$0.3537$
RIT:BBH:0265	$0.8506$	$0.4595$	$0.5402$	$0.5000$	$-0.8507$	$-0.0807$	$-0.6890$	$-0.1427$
RIT:BBH:0276	$0.8496$	$0.4593$	$0.5405$	$-0.8008$	$-0.2500$	$-0.0813$	$0.2326$	$-0.2419$
RIT:BBH:0277	$0.8506$	$0.4595$	$0.5401$	$-0.5000$	$0.8506$	$-0.0807$	$0.6889$	$0.1426$
RIT:BBH:0278	$0.2000$	$0.1666$	$0.8333$	$0.8006$	$0.5001$	$-0.6667$	$0.2833$	$0.3694$
RIT:BBH:0279	$0.8500$	$0.4595$	$0.5405$	$-0.5001$	$-0.5001$	$-0.0811$	$-0.0406$	$-0.2517$
RIT:BBH:0283	$0.6667$	$0.3999$	$0.5997$	$-0.8007$	$-0.8006$	$-0.2000$	$-0.1599$	$-0.4160$
RIT:BBH:0284	$0.3336$	$0.2500$	$0.7494$	$-0.5001$	$-0.8513$	$-0.4997$	$-0.5127$	$-0.5094$
RIT:BBH:0285	$0.3336$	$0.2500$	$0.7494$	$-0.5001$	$0.8513$	$-0.4997$	$0.7626$	$0.4469$
RIT:BBH:0286	$0.3336$	$0.2500$	$0.7494$	$0.5000$	$-0.8513$	$-0.4997$	$-0.7626$	$-0.4469$
RIT:BBH:0287	$0.6667$	$0.3998$	$0.5997$	$0.8007$	$-0.8008$	$-0.2000$	$-0.8001$	$-0.1600$
RIT:BBH:0288	$0.3336$	$0.2500$	$0.7494$	$0.5001$	$0.8513$	$-0.4997$	$0.5127$	$0.5094$
RIT:BBH:0289	$0.6667$	$0.3998$	$0.5997$	$-0.8007$	$0.8008$	$-0.2000$	$0.8000$	$0.1600$
RIT:BBH:0290	$0.7509$	$0.4282$	$0.5703$	$-0.9510$	$-0.9531$	$-0.1422$	$-0.1361$	$-0.4844$
RIT:BBH:0291	$0.6668$	$0.4000$	$0.5999$	$0.5001$	$0.6503$	$-0.1999$	$0.1901$	$0.3140$
RIT:BBH:0292	$0.6667$	$0.3998$	$0.5997$	$0.8007$	$0.8007$	$-0.2000$	$0.1600$	$0.4160$
RIT:BBH:0293	$0.6668$	$0.4000$	$0.5999$	$-0.5001$	$0.6503$	$-0.1999$	$0.5901$	$0.1540$
RIT:BBH:0294	$0.6664$	$0.3998$	$0.6000$	$-0.8007$	$0.0000$	$-0.2002$	$0.3201$	$-0.1280$
RIT:BBH:0295	$0.6664$	$0.3998$	$0.6000$	$0.8007$	$0.5001$	$-0.2002$	$-0.0201$	$0.3080$
RIT:BBH:0296	$0.5004$	$0.3333$	$0.6662$	$-0.5000$	$-0.8513$	$-0.3330$	$-0.4003$	$-0.4334$
RIT:BBH:0297	$0.6664$	$0.3998$	$0.6000$	$-0.8008$	$0.5001$	$-0.2002$	$0.6201$	$0.0520$
RIT:BBH:0298	$0.6667$	$0.4000$	$0.6000$	$0.5001$	$0.2500$	$-0.2000$	$-0.0500$	$0.1700$
RIT:BBH:0299	$0.4145$	$0.2929$	$0.7066$	$-0.5001$	$-0.8513$	$-0.4139$	$-0.4548$	$-0.4679$
RIT:BBH:0300	$0.4145$	$0.2929$	$0.7066$	$0.5001$	$-0.8513$	$-0.4139$	$-0.7476$	$-0.3821$
RIT:BBH:0301	$0.6664$	$0.3998$	$0.6000$	$-0.8007$	$-0.5001$	$-0.2002$	$0.0201$	$-0.3080$
RIT:BBH:0302	$0.6667$	$0.4000$	$0.6000$	$-0.5001$	$-0.2500$	$-0.2000$	$0.0500$	$-0.1700$
RIT:BBH:0303	$0.7140$	$0.4165$	$0.5833$	$0.8007$	$0.0000$	$-0.1669$	$-0.3334$	$0.1389$
RIT:BBH:0304	$0.6668$	$0.4000$	$0.5999$	$0.5000$	$-0.6503$	$-0.1999$	$-0.5901$	$-0.1540$
RIT:BBH:0305	$0.6667$	$0.4000$	$0.6000$	$-0.5001$	$0.2500$	$-0.2000$	$0.3500$	$0.0100$
RIT:BBH:0306	$0.6668$	$0.4000$	$0.5999$	$-0.5001$	$-0.6503$	$-0.1999$	$-0.1901$	$-0.3140$
RIT:BBH:0307	$0.6664$	$0.3998$	$0.6000$	$0.8007$	$0.0000$	$-0.2002$	$-0.3201$	$0.1280$
RIT:BBH:0308	$0.6664$	$0.3998$	$0.6000$	$0.8007$	$-0.5001$	$-0.2002$	$-0.6201$	$-0.0520$
RIT:BBH:0309	$0.6667$	$0.4000$	$0.6000$	$0.5000$	$-0.2500$	$-0.2000$	$-0.3500$	$-0.0100$
RIT:BBH:0311	$0.6667$	$0.4000$	$0.6000$	$0.0000$	$0.5001$	$-0.2000$	$0.3000$	$0.1800$
RIT:BBH:0312	$0.6667$	$0.4000$	$0.6000$	$-0.0000$	$-0.5001$	$-0.2000$	$-0.3000$	$-0.1800$
RIT:BBH:0314	$0.5004$	$0.3333$	$0.6662$	$0.5000$	$0.8513$	$-0.3330$	$0.4003$	$0.4334$
RIT:BBH:0316	$0.4145$	$0.2929$	$0.7066$	$-0.5001$	$0.8513$	$-0.4139$	$0.7476$	$0.3821$
RIT:BBH:0317	$0.7509$	$0.4283$	$0.5704$	$0.9510$	$0.9532$	$-0.1422$	$0.1361$	$0.4845$
RIT:BBH:0318	$0.2002$	$0.1667$	$0.8327$	$-0.5000$	$-0.8514$	$-0.6664$	$-0.6252$	$-0.6042$
RIT:BBH:0319	$0.7140$	$0.4165$	$0.5833$	$-0.8007$	$0.0000$	$-0.1669$	$0.3334$	$-0.1389$
RIT:BBH:0321	$0.2002$	$0.1667$	$0.8327$	$0.5000$	$-0.8514$	$-0.6664$	$-0.7918$	$-0.5764$
RIT:BBH:0322	$0.4145$	$0.2929$	$0.7066$	$0.5001$	$0.8513$	$-0.4139$	$0.4548$	$0.4679$
RIT:BBH:0324	$1.0000$	$0.4995$	$0.4995$	$0.9017$	$0.9017$	$0.0000$	$-0.0000$	$0.4500$
RIT:BBH:0336	$0.4998$	$0.3332$	$0.6667$	$-0.8007$	$0.0000$	$-0.3335$	$0.2668$	$-0.0889$
RIT:BBH:0337	$0.2002$	$0.1667$	$0.8327$	$-0.5001$	$0.8514$	$-0.6664$	$0.7918$	$0.5764$
RIT:BBH:0338	$0.4998$	$0.3332$	$0.6667$	$0.8006$	$0.0000$	$-0.3335$	$-0.2667$	$0.0889$
RIT:BBH:0339	$0.3335$	$0.2500$	$0.7496$	$-0.0000$	$-0.8009$	$-0.4998$	$-0.6001$	$-0.4500$
RIT:BBH:0344	$0.6004$	$0.3750$	$0.6246$	$-0.5000$	$-0.8513$	$-0.2497$	$-0.3440$	$-0.4024$
RIT:BBH:0345	$0.2002$	$0.1667$	$0.8327$	$0.5001$	$0.8514$	$-0.6664$	$0.6252$	$0.6042$
RIT:BBH:0348	$0.6000$	$0.3750$	$0.6250$	$-0.0000$	$-0.5001$	$-0.2500$	$-0.3125$	$-0.1953$
RIT:BBH:0350	$0.6001$	$0.3750$	$0.6249$	$-0.5000$	$-0.6502$	$-0.2499$	$-0.2188$	$-0.3242$
RIT:BBH:0352	$0.6004$	$0.3750$	$0.6245$	$-0.0000$	$-0.8513$	$-0.2497$	$-0.5314$	$-0.3320$

Table 4. Table 4: The mass and spin of the precessing BHBs in Table  LABEL:tab:ID after the BHs had time to equilibrate ( t / m = 200 𝑡 𝑚 200 t/m=200 ).

Run	$q^r$	$m_1^r/m$	$m_2^r/m$	${\chi }_{1x}^r$	${\chi }_{1y}^r$	${\chi }_{1z}^r$	${\chi }_{2x}^r$	${\chi }_{2y}^r$	${\chi }_{2z}^r$
RIT:BBH:0127	$0.7501$	$0.4284$	$0.5712$	-	-	0.8005	0.5115	0.1815	-0.5887
RIT:BBH:0128	$0.5001$	$0.3332$	$0.6663$	0.0124	0.0059	0.8004	0.6819	0.1550	-0.3901
RIT:BBH:0129	$0.3334$	$0.2499$	$0.7496$	-	-	0.8004	0.7473	0.1250	-0.2596
RIT:BBH:0130	$0.7500$	$0.4284$	$0.5712$	-	-	-0.8007	0.5104	0.1674	0.5937
RIT:BBH:0131	$0.5001$	$0.3332$	$0.6663$	0.0003	-0.0055	-0.8006	0.6686	0.1790	0.4025
RIT:BBH:0132	$0.3334$	$0.2499$	$0.7496$	0.0097	-0.0299	-0.7999	0.7457	0.1224	0.2650
RIT:BBH:0230	$0.6670$	$0.4000$	$0.5997$	-	-	-	-0.5691	-0.0579	-0.5605
RIT:BBH:0231	$0.6670$	$0.4000$	$0.5997$	-	-	-	-0.4404	-0.3553	-0.5667
RIT:BBH:0233	$0.6670$	$0.4000$	$0.5997$	-	-	-	-0.2023	-0.5351	-0.5604
RIT:BBH:0234	$0.6670$	$0.4000$	$0.5997$	-	-	-	0.0675	-0.5711	-0.5573
RIT:BBH:0235	$0.6670$	$0.4000$	$0.5997$	-	-	-	0.3372	-0.4621	-0.5604
RIT:BBH:0236	$0.6670$	$0.4000$	$0.5997$	-	-	-	0.5192	-0.2284	-0.5653
RIT:BBH:0242	$0.6670$	$0.4000$	$0.5997$	-	-	-	0.4585	0.3360	-0.5641
RIT:BBH:0243	$0.6670$	$0.4000$	$0.5997$	-	-	-	0.1980	0.5253	-0.5711
RIT:BBH:0244	$0.6670$	$0.4000$	$0.5997$	-	-	-	-0.5181	0.2330	-0.5645
RIT:BBH:0248	$0.6670$	$0.4000$	$0.5997$	-	-	-	0.5545	0.0630	-0.5744
RIT:BBH:0250	$0.6670$	$0.4000$	$0.5997$	-	-	-	-0.0916	0.5607	-0.5643
RIT:BBH:0251	$0.6670$	$0.4000$	$0.5997$	-	-	-	-0.3355	0.4590	-0.5639
RIT:BBH:0255	$0.6670$	$0.4000$	$0.5997$	-	-	-	0.3986	0.0379	0.6935
RIT:BBH:0256	$0.6670$	$0.4000$	$0.5997$	-	-	-	0.3377	0.2299	0.6887
RIT:BBH:0257	$0.6670$	$0.4000$	$0.5997$	-	-	-	0.1446	0.3677	0.6965
RIT:BBH:0258	$0.6670$	$0.4000$	$0.5997$	-	-	-	-0.0647	0.3934	0.6945
RIT:BBH:0259	$0.6670$	$0.4000$	$0.5997$	-	-	-	-0.2532	0.3151	0.6913
RIT:BBH:0260	$0.6670$	$0.4000$	$0.5997$	-	-	-	-0.3785	0.1469	0.6903
RIT:BBH:0264	$0.6670$	$0.4000$	$0.5997$	-	-	-	0.7956	0.0910	0.0005
RIT:BBH:0266	$0.6670$	$0.4000$	$0.5997$	-	-	-	0.6415	0.4794	-0.0006
RIT:BBH:0267	$0.6669$	$0.4000$	$0.5998$	-	-	-	-0.5020	0.6238	-0.0004
RIT:BBH:0268	$0.6670$	$0.4000$	$0.5997$	-	-	-	-0.7467	0.2894	-0.0027
RIT:BBH:0269	$0.6670$	$0.4000$	$0.5997$	-	-	-	0.6911	0.0897	0.3946
RIT:BBH:0270	$0.6670$	$0.4000$	$0.5997$	-	-	-	0.5501	0.4193	0.4036
RIT:BBH:0271	$0.6670$	$0.4000$	$0.5997$	-	-	-	0.2643	0.6419	0.3992
RIT:BBH:0272	$0.6669$	$0.4000$	$0.5997$	-	-	-	-0.1192	0.6815	0.4031
RIT:BBH:0273	$0.6670$	$0.4000$	$0.5997$	-	-	-	-0.4388	0.5414	0.3943
RIT:BBH:0274	$0.6670$	$0.4000$	$0.5997$	-	-	-	-0.6429	0.2683	0.3949
RIT:BBH:0280	$0.6670$	$0.4000$	$0.5997$	-	-	-	0.2865	0.7477	0.0043
RIT:BBH:0281	$0.6669$	$0.4000$	$0.5998$	-	-	-	-0.1417	0.7880	0.0055

Table 5. Table 5: Table of the initial orbital frequency m ​ ω i 𝑚 subscript 𝜔 𝑖 m\omega_{i} , number of orbits to merger, N 𝑁 N , and the initial and final eccentricities, e i subscript 𝑒 𝑖 e_{i} and e f subscript 𝑒 𝑓 e_{f} for the spinning cases.

Run	$m{\omega }_i$	$N$	$e_i$	$e_f$
RIT:BBH:0127	$0.0317$	$5.9$	$0.0117$	$0.0022$
RIT:BBH:0128	$0.0309$	$6.3$	$0.0108$	$0.0033$
RIT:BBH:0129	$0.0300$	$7.3$	$0.0092$	$0.0010$
RIT:BBH:0130	$0.0315$	$6.3$	$0.0117$	$0.0013$
RIT:BBH:0131	$0.0306$	$7.0$	$0.0110$	$0.0039$
RIT:BBH:0132	$0.0296$	$8.2$	$0.0097$	$0.0044$
RIT:BBH:0136	$0.0156$	$22.0$	$0.0242$	$0.0009$
RIT:BBH:0220	$0.0211$	$8.8$	$0.0064$	$0.0024$
RIT:BBH:0221	$0.0199$	$14.1$	$0.0036$	$0.0015$
RIT:BBH:0222	$0.0192$	$23.6$	$0.0026$	$0.0006$
RIT:BBH:0223	$0.0196$	$11.2$	$0.0042$	$0.0011$
RIT:BBH:0224	$0.0189$	$29.6$	$0.0024$	$0.0006$
RIT:BBH:0226	$0.0194$	$20.5$	$0.0024$	$0.0009$
RIT:BBH:0227	$0.0194$	$13.6$	$0.0041$	$0.0011$
RIT:BBH:0228	$0.0194$	$13.6$	$0.0041$	$0.0012$
RIT:BBH:0230	$0.0210$	$9.6$	$0.0063$	$0.0012$
RIT:BBH:0231	$0.0210$	$9.6$	$0.0064$	$0.0010$
RIT:BBH:0232	$0.0201$	$8.3$	$0.0045$	$0.0016$
RIT:BBH:0233	$0.0209$	$9.7$	$0.0043$	$0.0015$
RIT:BBH:0234	$0.0209$	$9.7$	$0.0043$	$0.0011$
RIT:BBH:0235	$0.0209$	$9.6$	$0.0041$	$0.0017$
RIT:BBH:0236	$0.0210$	$9.6$	$0.0039$	$0.0014$
RIT:BBH:0237	$0.0226$	$7.9$	$0.0042$	$0.0019$
RIT:BBH:0238	$0.0229$	$6.8$	$0.0039$	$0.0020$
RIT:BBH:0239	$0.0224$	$8.6$	$0.0040$	$0.0017$
RIT:BBH:0240	$0.0199$	$8.9$	$0.0042$	$0.0014$
RIT:BBH:0241	$0.0306$	$6.4$	$0.0062$	$0.0029$
RIT:BBH:0242	$0.0210$	$9.6$	$0.0063$	$0.0012$
RIT:BBH:0243	$0.0209$	$9.7$	$0.0043$	$0.0015$
RIT:BBH:0244	$0.0210$	$9.6$	$0.0039$	$0.0019$
RIT:BBH:0245	$0.0198$	$10.7$	$0.0036$	$0.0001$
RIT:BBH:0246	$0.0195$	$18.5$	$0.0032$	$0.0009$
RIT:BBH:0247	$0.0192$	$16.7$	$0.0039$	$0.0008$
RIT:BBH:0248	$0.0210$	$9.6$	$0.0038$	$0.0012$
RIT:BBH:0249	$0.0193$	$13.9$	$0.0043$	$0.0010$
RIT:BBH:0250	$0.0209$	$9.7$	$0.0043$	$0.0017$
RIT:BBH:0251	$0.0209$	$9.6$	$0.0041$	$0.0017$
RIT:BBH:0252	$0.0216$	$14.0$	$0.0041$	$0.0013$
RIT:BBH:0253	$0.0419$	$6.7$	$0.0248$	$0.0038$
RIT:BBH:0254	$0.0196$	$10.9$	$0.0045$	$0.0012$
RIT:BBH:0255	$0.0245$	$10.8$	$0.0026$	$0.0023$
RIT:BBH:0256	$0.0244$	$10.8$	$0.0027$	$0.0016$
RIT:BBH:0257	$0.0244$	$10.8$	$0.0024$	$0.0016$
RIT:BBH:0258	$0.0244$	$10.8$	$0.0024$	$0.0016$
RIT:BBH:0259	$0.0244$	$10.8$	$0.0024$	$0.0014$
RIT:BBH:0260	$0.0244$	$10.8$	$0.0025$	$0.0014$
RIT:BBH:0261	$0.0196$	$11.1$	$0.0042$	$0.0011$
RIT:BBH:0262	$0.0218$	$12.6$	$0.0028$	$0.0009$
RIT:BBH:0263	$0.0191$	$16.6$	$0.0045$	$0.0010$
RIT:BBH:0264	$0.0230$	$9.9$	$0.0038$	$0.0013$
RIT:BBH:0265	$0.0196$	$11.2$	$0.0068$	$0.0010$
RIT:BBH:0266	$0.0229$	$9.9$	$0.0036$	$0.0019$
RIT:BBH:0267	$0.0228$	$9.9$	$0.0041$	$0.0017$
RIT:BBH:0268	$0.0229$	$9.9$	$0.0032$	$0.0013$
RIT:BBH:0269	$0.0239$	$10.4$	$0.0021$	$0.0016$
RIT:BBH:0270	$0.0238$	$10.4$	$0.0039$	$0.0019$
RIT:BBH:0271	$0.0237$	$10.4$	$0.0045$	$0.0020$
RIT:BBH:0272	$0.0236$	$10.4$	$0.0036$	$0.0018$
RIT:BBH:0273	$0.0237$	$10.4$	$0.0041$	$0.0016$
RIT:BBH:0274	$0.0238$	$10.4$	$0.0036$	$0.0013$
RIT:BBH:0276	$0.0198$	$9.7$	$0.0044$	$0.0018$
RIT:BBH:0277	$0.0193$	$14.0$	$0.0067$	$0.0011$
RIT:BBH:0278	$0.0191$	$25.8$	$0.0019$	$0.0008$
RIT:BBH:0279	$0.0198$	$9.7$	$0.0045$	$0.0019$
RIT:BBH:0280	$0.0227$	$10.0$	$0.0044$	$0.0018$
RIT:BBH:0281	$0.0227$	$10.0$	$0.0044$	$0.0024$
RIT:BBH:0283	$0.0231$	$6.5$	$0.0042$	$0.0033$
RIT:BBH:0284	$0.0229$	$7.5$	$0.0029$	$0.0020$
RIT:BBH:0285	$0.0245$	$14.0$	$0.0029$	$0.0011$
RIT:BBH:0286	$0.0226$	$8.5$	$0.0026$	$0.0012$
RIT:BBH:0287	$0.0224$	$9.1$	$0.0034$	$0.0018$
RIT:BBH:0288	$0.0243$	$15.3$	$0.0035$	$0.0010$
RIT:BBH:0289	$0.0220$	$11.4$	$0.0036$	$0.0009$
RIT:BBH:0290	$0.0314$	$3.4$	$0.0192$	$0.0083$
RIT:BBH:0291	$0.0217$	$13.6$	$0.0040$	$0.0010$
RIT:BBH:0292	$0.0216$	$14.8$	$0.0040$	$0.0012$
RIT:BBH:0293	$0.0219$	$11.6$	$0.0041$	$0.0011$
RIT:BBH:0294	$0.0224$	$8.8$	$0.0042$	$0.0015$
RIT:BBH:0295	$0.0217$	$13.6$	$0.0024$	$0.0011$
RIT:BBH:0296	$0.0200$	$9.1$	$0.0037$	$0.0014$
RIT:BBH:0297	$0.0221$	$10.4$	$0.0059$	$0.0011$
RIT:BBH:0298	$0.0218$	$12.1$	$0.0042$	$0.0011$
RIT:BBH:0299	$0.0200$	$9.3$	$0.0056$	$0.0012$
RIT:BBH:0300	$0.0198$	$10.7$	$0.0057$	$0.0010$
RIT:BBH:0301	$0.0228$	$7.3$	$0.0043$	$0.0026$
RIT:BBH:0302	$0.0224$	$8.6$	$0.0069$	$0.0018$
RIT:BBH:0303	$0.0205$	$13.1$	$0.0039$	$0.0013$
RIT:BBH:0304	$0.0223$	$9.1$	$0.0067$	$0.0014$
RIT:BBH:0305	$0.0221$	$10.2$	$0.0066$	$0.0015$
RIT:BBH:0306	$0.0227$	$7.4$	$0.0066$	$0.0023$
RIT:BBH:0307	$0.0219$	$11.8$	$0.0068$	$0.0014$
RIT:BBH:0308	$0.0222$	$10.1$	$0.0069$	$0.0016$
RIT:BBH:0309	$0.0221$	$10.4$	$0.0068$	$0.0014$
RIT:BBH:0311	$0.0218$	$12.0$	$0.0042$	$0.0011$
RIT:BBH:0312	$0.0224$	$8.7$	$0.0069$	$0.0018$
RIT:BBH:0314	$0.0190$	$18.6$	$0.0040$	$0.0008$
RIT:BBH:0316	$0.0191$	$18.2$	$0.0036$	$0.0007$
RIT:BBH:0317	$0.0281$	$10.8$	$0.0066$	$0.0010$
RIT:BBH:0318	$0.0201$	$11.7$	$0.0030$	$0.0007$
RIT:BBH:0319	$0.0210$	$9.7$	$0.0042$	$0.0014$
RIT:BBH:0321	$0.0200$	$12.7$	$0.0029$	$0.0011$
RIT:BBH:0322	$0.0190$	$20.0$	$0.0039$	$0.0008$
RIT:BBH:0324	$0.0247$	$12.4$	$0.0037$	$0.0012$
RIT:BBH:0336	$0.0196$	$12.0$	$0.0060$	$0.0009$
RIT:BBH:0337	$0.0189$	$28.3$	$0.0026$	$0.0004$
RIT:BBH:0338	$0.0194$	$14.9$	$0.0035$	$0.0012$
RIT:BBH:0339	$0.0199$	$10.7$	$0.0052$	$0.0008$
RIT:BBH:0344	$0.0213$	$7.8$	$0.0062$	$0.0011$
RIT:BBH:0345	$0.0188$	$29.7$	$0.0030$	$0.0005$
RIT:BBH:0348	$0.0209$	$9.8$	$0.0066$	$0.0014$
RIT:BBH:0350	$0.0212$	$8.5$	$0.0065$	$0.0014$
RIT:BBH:0352	$0.0212$	$8.7$	$0.0062$	$0.0016$

Table 6. Table 6: The energy radiated, δ ​ ℳ I ​ H = M a ​ d ​ m − M r ​ e ​ m I ​ H 𝛿 superscript ℳ 𝐼 𝐻 subscript 𝑀 𝑎 𝑑 𝑚 superscript subscript 𝑀 𝑟 𝑒 𝑚 𝐼 𝐻 \delta\mathcal{M}^{IH}=M_{adm}-M_{rem}^{IH} , and final spin, χ rem I ​ H superscript subscript 𝜒 rem 𝐼 𝐻 \chi_{\mathrm{rem}}^{IH} , as measured using the IH formalism The error bars are due to variations in the measured mass and spin with time. For aligned systems with final spin antialigned to the initial orbital angular momentum the minus sign is preserved.

Run	$\delta {ℳ}^{IH}$	${\chi }_{\mathrm{rem}}^{IH}$
RIT:BBH:0127	$0.047049\pm 0.000002$	$0.642923\pm 0.000004$
RIT:BBH:0128	$0.039357\pm 0.000006$	$0.609307\pm 0.000086$
RIT:BBH:0129	$0.029840\pm 0.000001$	$0.607187\pm 0.000012$
RIT:BBH:0130	$0.051886\pm 0.000001$	$0.730998\pm 0.000003$
RIT:BBH:0131	$0.043854\pm 0.000071$	$0.768088\pm 0.000570$
RIT:BBH:0132	$0.034965\pm 0.000000$	$0.756259\pm 0.000002$
RIT:BBH:0136	$0.049301\pm 0.000002$	$0.775972\pm 0.000008$
RIT:BBH:0220	$0.034360\pm 0.000000$	$0.463421\pm 0.000032$
RIT:BBH:0221	$0.013239\pm 0.000006$	$0.104715\pm 0.000006$
RIT:BBH:0222	$0.024213\pm 0.000002$	$0.697263\pm 0.000030$
RIT:BBH:0223	$0.043177\pm 0.000003$	$0.627577\pm 0.000000$
RIT:BBH:0224	$0.039046\pm 0.000140$	$0.880312\pm 0.005699$
RIT:BBH:0226	$0.018361\pm 0.000004$	$0.427060\pm 0.000001$
RIT:BBH:0227	$0.053982\pm 0.000003$	$0.733438\pm 0.000081$
RIT:BBH:0228	$0.055817\pm 0.000003$	$0.751804\pm 0.000196$
RIT:BBH:0230	$0.038221\pm 0.000002$	$0.555080\pm 0.000038$
RIT:BBH:0231	$0.037993\pm 0.000002$	$0.556462\pm 0.000038$
RIT:BBH:0232	$0.033037\pm 0.000001$	$0.419364\pm 0.000090$
RIT:BBH:0233	$0.037722\pm 0.000002$	$0.558094\pm 0.000038$
RIT:BBH:0234	$0.037552\pm 0.000002$	$0.558906\pm 0.000037$
RIT:BBH:0235	$0.037696\pm 0.000002$	$0.557694\pm 0.000038$
RIT:BBH:0236	$0.038129\pm 0.000001$	$0.555337\pm 0.000040$
RIT:BBH:0237	$0.037119\pm 0.000001$	$0.531769\pm 0.000006$
RIT:BBH:0238	$0.033155\pm 0.000002$	$0.428593\pm 0.000007$
RIT:BBH:0239	$0.039182\pm 0.000001$	$0.542678\pm 0.000004$
RIT:BBH:0240	$0.034421\pm 0.000003$	$0.451168\pm 0.000033$
RIT:BBH:0241	$0.051719\pm 0.000007$	$0.743552\pm 0.000049$
RIT:BBH:0242	$0.037993\pm 0.000002$	$0.556462\pm 0.000038$
RIT:BBH:0243	$0.037722\pm 0.000002$	$0.558094\pm 0.000038$
RIT:BBH:0244	$0.038129\pm 0.000001$	$0.555336\pm 0.000040$
RIT:BBH:0245	$0.029791\pm 0.000003$	$0.374712\pm 0.000000$
RIT:BBH:0246	$0.016920\pm 0.000004$	$0.405685\pm 0.000002$
RIT:BBH:0247	$0.059225\pm 0.000000$	$0.854389\pm 0.000011$
RIT:BBH:0248	$0.038220\pm 0.000002$	$0.555080\pm 0.000037$
RIT:BBH:0249	$0.055860\pm 0.000004$	$0.755050\pm 0.000049$
RIT:BBH:0250	$0.037552\pm 0.000002$	$0.558906\pm 0.000037$
RIT:BBH:0251	$0.037696\pm 0.000002$	$0.557694\pm 0.000037$
RIT:BBH:0252	$0.079713\pm 0.000012$	$0.890918\pm 0.000245$
RIT:BBH:0253	$0.105833\pm 0.000053$	$0.940241\pm 0.000027$
RIT:BBH:0254	$0.041832\pm 0.000003$	$0.605990\pm 0.000026$
RIT:BBH:0255	$0.063541\pm 0.000001$	$0.832657\pm 0.000032$
RIT:BBH:0256	$0.063236\pm 0.000002$	$0.833626\pm 0.000050$
RIT:BBH:0257	$0.062808\pm 0.000003$	$0.834735\pm 0.000094$
RIT:BBH:0258	$0.062559\pm 0.000003$	$0.835152\pm 0.000087$
RIT:BBH:0259	$0.062811\pm 0.000001$	$0.834240\pm 0.000049$
RIT:BBH:0260	$0.063373\pm 0.000003$	$0.832833\pm 0.000088$
RIT:BBH:0261	$0.043225\pm 0.000002$	$0.615218\pm 0.000023$
RIT:BBH:0262	$0.062388\pm 0.000001$	$0.806245\pm 0.000047$
RIT:BBH:0263	$0.079755\pm 0.000086$	$0.885051\pm 0.001694$
RIT:BBH:0264	$0.048571\pm 0.000002$	$0.710881\pm 0.000014$
RIT:BBH:0265	$0.041759\pm 0.000006$	$0.584610\pm 0.000056$
RIT:BBH:0266	$0.047876\pm 0.000000$	$0.713698\pm 0.000001$
RIT:BBH:0267	$0.047019\pm 0.000001$	$0.715990\pm 0.000006$
RIT:BBH:0268	$0.048562\pm 0.000002$	$0.710338\pm 0.000012$
RIT:BBH:0269	$0.054878\pm 0.000001$	$0.792807\pm 0.000012$
RIT:BBH:0270	$0.055376\pm 0.000001$	$0.791199\pm 0.000013$
RIT:BBH:0271	$0.055820\pm 0.000001$	$0.790455\pm 0.000008$
RIT:BBH:0272	$0.056897\pm 0.000000$	$0.787993\pm 0.000005$
RIT:BBH:0273	$0.057247\pm 0.000001$	$0.787618\pm 0.000011$
RIT:BBH:0274	$0.055833\pm 0.000000$	$0.791237\pm 0.000007$
RIT:BBH:0276	$0.037611\pm 0.000002$	$0.530494\pm 0.000037$
RIT:BBH:0277	$0.057208\pm 0.000003$	$0.773654\pm 0.000085$
RIT:BBH:0278	$0.026856\pm 0.000002$	$0.715530\pm 0.000043$
RIT:BBH:0279	$0.037334\pm 0.000002$	$0.521499\pm 0.000062$
RIT:BBH:0280	$0.047518\pm 0.000001$	$0.715617\pm 0.000012$
RIT:BBH:0281	$0.046779\pm 0.000002$	$0.717849\pm 0.000014$
RIT:BBH:0283	$0.030805\pm 0.000002$	$0.383336\pm 0.000002$
RIT:BBH:0284	$0.019466\pm 0.000003$	$0.136838\pm 0.000011$
RIT:BBH:0285	$0.052236\pm 0.000043$	$0.883435\pm 0.001081$
RIT:BBH:0286	$0.020768\pm 0.000001$	$0.172724\pm 0.000017$
RIT:BBH:0287	$0.039013\pm 0.000001$	$0.539101\pm 0.000036$
RIT:BBH:0288	$0.060428\pm 0.000149$	$0.907461\pm 0.001717$
RIT:BBH:0289	$0.054364\pm 0.000001$	$0.781275\pm 0.000012$
RIT:BBH:0290	$0.030423\pm 0.000016$	$0.350609\pm 0.000009$
RIT:BBH:0291	$0.068688\pm 0.000005$	$0.854343\pm 0.000115$
RIT:BBH:0292	$0.083983\pm 0.000212$	$0.906306\pm 0.001585$
RIT:BBH:0293	$0.053790\pm 0.000001$	$0.773129\pm 0.000003$
RIT:BBH:0294	$0.039046\pm 0.000002$	$0.589673\pm 0.000061$
RIT:BBH:0295	$0.068155\pm 0.000005$	$0.844768\pm 0.000009$
RIT:BBH:0296	$0.026874\pm 0.000004$	$0.308819\pm 0.000001$
RIT:BBH:0297	$0.047266\pm 0.000001$	$0.711723\pm 0.000013$
RIT:BBH:0298	$0.055193\pm 0.000001$	$0.766060\pm 0.000001$
RIT:BBH:0299	$0.023398\pm 0.000003$	$0.233298\pm 0.000026$
RIT:BBH:0300	$0.025421\pm 0.000009$	$0.283357\pm 0.000034$
RIT:BBH:0301	$0.033411\pm 0.000001$	$0.462101\pm 0.000007$
RIT:BBH:0302	$0.037663\pm 0.000001$	$0.555664\pm 0.000026$
RIT:BBH:0303	$0.054631\pm 0.000001$	$0.746572\pm 0.000005$
RIT:BBH:0304	$0.038894\pm 0.000000$	$0.549270\pm 0.000030$
RIT:BBH:0305	$0.045062\pm 0.000000$	$0.679210\pm 0.000010$
RIT:BBH:0306	$0.033390\pm 0.000001$	$0.453122\pm 0.000005$
RIT:BBH:0307	$0.052619\pm 0.000000$	$0.732976\pm 0.000018$
RIT:BBH:0308	$0.043136\pm 0.000001$	$0.613587\pm 0.000039$
RIT:BBH:0309	$0.044675\pm 0.000001$	$0.648076\pm 0.000015$
RIT:BBH:0311	$0.055694\pm 0.000003$	$0.781539\pm 0.000019$
RIT:BBH:0312	$0.037596\pm 0.000001$	$0.540226\pm 0.000029$
RIT:BBH:0314	$0.073484\pm 0.000049$	$0.903065\pm 0.000334$
RIT:BBH:0316	$0.056812\pm 0.000009$	$0.871082\pm 0.000187$
RIT:BBH:0317	$0.104701\pm 0.000080$	$0.942466\pm 0.000058$
RIT:BBH:0318	$0.011757\pm 0.000002$	$-0.105807\pm 0.000008$
RIT:BBH:0319	$0.039589\pm 0.000000$	$0.589102\pm 0.000053$
RIT:BBH:0321	$0.012213\pm 0.000001$	$-0.090347\pm 0.000006$
RIT:BBH:0322	$0.068114\pm 0.000014$	$0.906331\pm 0.000200$
RIT:BBH:0324	$0.099686\pm 0.000022$	$0.927790\pm 0.001526$
RIT:BBH:0336	$0.035055\pm 0.000000$	$0.574463\pm 0.000013$
RIT:BBH:0337	$0.037908\pm 0.000002$	$0.895267\pm 0.000044$
RIT:BBH:0338	$0.043459\pm 0.000001$	$0.669149\pm 0.000004$
RIT:BBH:0339	$0.020426\pm 0.000001$	$0.178243\pm 0.000006$
RIT:BBH:0344	$0.030016\pm 0.000001$	$0.370895\pm 0.000005$
RIT:BBH:0345	$0.041385\pm 0.000010$	$0.904325\pm 0.000217$
RIT:BBH:0348	$0.035551\pm 0.000001$	$0.513676\pm 0.000012$
RIT:BBH:0350	$0.031769\pm 0.000000$	$0.428983\pm 0.000007$
RIT:BBH:0352	$0.031913\pm 0.000001$	$0.414030\pm 0.000008$

Equations

$$\ln \mathcal{L}(\mathbf{\lambda };\theta )=-\frac{1}{2}\sum _k⟨h_k(\mathbf{\lambda },\theta )-d_k|h_k(\mathbf{\lambda },\theta )-d_k{⟩}_k-⟨d_k|d_k{⟩}_k,$$

$$⟨a|b{⟩}_k\equiv 2{\int }_{|f|>f_{\min }}df\frac{[\tilde{a}(f){]}^{\ast }\tilde{b}(f)}{S_{h,k}(|f|)}.$$

$$0.570$$

$$0.00$$

$$-0.44$$

$$66.3$$

$$0.62$$

$$0.04$$

$$0.03$$

$$-0.29$$

$$66.1$$

$$m^2{S}_{hu}=\left((1+\frac{1}{2q}){\vec{S}}_1+(1+\frac{1}{2}q){\vec{S}}_2\right)\cdot \hat{L},$$

$$m^2{\chi }_{eff}=\left((1+\frac{1}{q}){\vec{S}}_1+(1+q){\vec{S}}_2\right)\cdot \hat{L}.$$

$$0.039$$

$$0.578$$

$$0$$

$$0.041$$

$$0.60$$

Full text

I Introduction

Ten years of advances and studies since the breakthroughs Pretorius (2005); Campanelli et al. (2006a); Baker et al. (2006a) in numerical relativity led to detailed predictions of the gravitational waves from the late inspiral, plunge, merger, and ringdown of black-hole-binary systems (BHB). These predictions helped to accurately identify the first direct detection Abbott et al. (2016a) of gravitational waves with such binary black hole systems Abbott et al. (2016b, c, d, 2017) and match them to targeted supercomputer simulations Abbott et al. (2016e, f); Lovelace et al. (2016). The observed gravitational waves were remarkably consistent with the predictions of numerical general relativity Abbott et al. (2016g, d, 2017), thereby supporting the notion that general relativity is an accurate theory of gravity in the highly dynamical, strong field regime of merging binary black holes.

Numerical relativity techniques have been used to explore the late dynamics of spinning black-hole binaries, beyond the post-Newtonian regime for several years. The first generic, long-term precessing black-hole binary evolutions (i.e., without any symmetry) were performed in Ref. Campanelli et al. (2009), where a detailed comparison with post-Newtonian $\ell=2,3$ waveforms was made. More recently, the longest of such comparisons for a precessing binary was done in Lousto et al. (2016) and a full numerical simulation was performed for a nonspinning binary with 350 orbits in Ref. Szilagyi et al. (2015).

Numerical simulations have started to explore the corners of parameter space, these include near extremal 111$\chi$ denotes the spin angular momentum of a black hole in units of the square of its mass. The maximum possible spin is $\chi=1$ with $\chi=0.99$ spinning black-hole binaries in Refs. Lovelace et al. (2015); Zlochower et al. (2017), mass ratios as small as $q=1/100$ in Refs. Lousto and Zlochower (2011a); Sperhake et al. (2011), and large initial separations, $R=100M$, in Ref. Lousto and Zlochower (2013a). Similarly challenging, high energy collision of black holes were studied in Ref. Sperhake et al. (2008); Shibata et al. (2008); Healy et al. (2016) and hyperbolic black-hole encounters in Ref. Healy et al. (2009); Gold and Brügmann (2013).

Other important studies include the exploration of the hangup effect, i.e. the role individual black-hole spins play to delay or accelerate their merger Campanelli et al. (2006b); Hannam et al. (2008); Hemberger et al. (2013); Healy and Lousto (2018), the determination of the magnitude and direction of the recoil velocity of the final merged black hole Campanelli et al. (2007a, b); Herrmann et al. (2007); Pollney et al. (2007); Baker et al. (2006b); González et al. (2007); Schnittman et al. (2008); Lousto and Zlochower (2011b), and the flip-flop of individual spins during the orbital phase Lousto and Healy (2015); Lousto et al. (2016); Lousto and Healy (2016), as well as precession dynamics Schmidt et al. (2011); Lousto and Zlochower (2014a); Pekowsky et al. (2013); Ossokine et al. (2015); Lousto and Healy (2018) and the inclusion of those dynamics to construct surrogate models for gravitational waveforms Blackman et al. (2017a, b); Varma et al. (2018).

There have been several significant efforts to coordinate numerical relativity simulations to support gravitational wave observations. These include the numerical injection analysis (NINJA) project Aylott et al. (2009a, b); Ajith et al. (2012); Aasi et al. (2014), the numerical relativity and analytical relativity (NRAR) collaboration Hinder et al. (2014), and the waveform catalogs released by the SXS collaboration Mroue et al. (2013); Blackman et al. (2015); Chu et al. (2016), Georgia Tech. Jani et al. (2016), and RIT Healy et al. (2017a).

In this paper we describe a new release of the public waveform catalog by the RIT numerical relativity group that nearly triples the number of waveforms by adding a new set of 194 waveforms, with 154 aligned spins and 40 precessing binaries. The catalog has new search and ordering features and includes all modes $\ell\leq 4$ modes of $\psi_{4}$ and the strain $h$ (both extrapolated to null-infinity). The catalog can be accessed from http://ccrg.rit.edu/~RITCatalog.

This paper is organized as follows. In Section II we describe the methods and criteria for producing the numerical simulations and evaluation of their errors in order to be included in the RIT catalog. In Sec. III we describe the use of the new searching and ordering capabilities of all the relevant BHB parameters, the file format, and the full content of the data in the catalog. In Sec. IV we use the waveform catalog to estimate the binary black hole parameters that best match the first gravitational wave event GW150914. We use the Bayesian likelihood maximized over extrinsic parameters as described in Lange et al. (2017) and map it onto the grid of simulations. Use of interpolation routines lead to an estimate of the confidence intervals that are consistent with previous estimates for the aligned spin binaries. We conclude in Sec. V with a discussion of the future use of this catalog for parameter inference of new gravitational waves events and the extensions to this work to more generic precessing binaries.

II Full Numerical Evolutions

The simulations in the RIT Catalog were evolved using the LazEv code Zlochower et al. (2005) implementation of the moving puncture approach Campanelli et al. (2006a) (with the modifications suggested by Ref. Marronetti et al. (2008)). In all cases (except the very high spin where we use CCZ4 Alic et al. (2012)) we use the BSSNOK (Baumgarte-Shapiro-Shibata-Nakamura-Oohara-Kojima) family of evolutions systems Nakamura et al. (1987); Shibata and Nakamura (1995); Baumgarte and Shapiro (1998). For the runs in the catalog, we used a variety of finite-difference orders, Kreiss-Oliger dissipation orders, and Courant factors Lousto and Zlochower (2008); Zlochower et al. (2012); Healy and Lousto (2017). All of these are given in the metadata included in the catalog and the references associated with each run (where detailed studies have been performed).

The LazEv code uses the EinsteinToolkit Löffler et al. (2012); ein / Cactus cac / Carpet Schnetter et al. (2004) infrastructure. The Carpet mesh refinement driver provides a “moving boxes” style of mesh refinement. In this approach, refined grids of fixed size are arranged about the coordinate centers of both holes. The code then moves these fine grids about the computational domain by following the trajectories of the two black holes (BHs).

We use AHFinderDirect Thornburg (2004) to locate apparent horizons. We measure the magnitude of the horizon spin using the isolated horizon (IH) algorithm detailed in Ref. Dreyer et al. (2003) (as implemented in Ref. Campanelli et al. (2007c)). Once we have the horizon spin, we can calculate the horizon mass via the Christodoulou formula ${m_{H}}=\sqrt{m_{\rm irr}^{2}+S_{H}^{2}/(4m_{\rm irr}^{2})}\,,$ where $m_{\rm irr}=\sqrt{A/(16\pi)}$ and $A$ is the surface area of the horizon.

To compute the numerical initial data, we use the puncture approach Brandt and Brügmann (1997) along with the TwoPunctures Ansorg et al. (2004) code. To compute initial low eccentricity orbital parameters, we use the post-Newtonian techniques described in Healy et al. (2017b) to determine quasi-circular orbits. We then evaluate the residual eccentricity during evolution via the simple formula, as a function of the separation of the holes, $d$, $e_{d}=d^{2}\ddot{d}/M$, as given in Campanelli et al. (2009).

As discussed in Ref. Healy et al. (2017a) the main sources of numerical errors in this catalog are due to finite difference truncation, finite extraction radii, finite number of modes, and the non-zero residual initial eccentricities and displacement of the center of mass.

During the early inspiral, the irreducible masses and intrinsic spins of each black hole should be nearly constant because the levels of gravitational wave energy and momentum absorbed by the holes is 4-5 orders of magnitude smallerIsoyama and Nakano (2017) than those emitted to infinity. During a simulation, the masses and spins vary due to numerical truncation error, and we then use these variations as a measure of the size of the truncation error. For our current simulations we monitor accuracy by measuring the conservation of the individual horizon masses and spins during evolution, as well as the level of satisfaction of the Hamiltonian and momentum constraints, to ensure reaching an accuracy consistent with our main applications. Those measurements are seen to be preserved at least to one part in $10^{4}$ in the cases of the masses and one part in $10^{3}$ in the cases of the spins (see for instance Fig. 6 in Ref. Lousto and Healy (2018)).

We measure radiated energy, linear momentum, and angular momentum, in terms of the radiative Weyl Scalar $\psi_{4}$, using the formulas provided in Refs. Campanelli and Lousto (1999); Lousto and Zlochower (2007). These formulas are strictly speaking only valid on future null-infinity ($\mathscr{I}^{+}$). We therefore measure the radiated energy-momentum on a series of timelike worldtubes of finite radius and then extrapolate to $r=\infty$ using both linear and quadratic extrapolations. The difference between these two extrapolations is an estimate for the uncertainty.

Unlike the radiated energy-momentum, more care is needed to properly extrapolate the waveform itself to $\mathscr{I}^{+}$. As described in Ref. Nakano et al. (2015), we use the Teukolsky equation to obtain expressions for $r\psi_{4}$ at $\mathscr{I}^{+}$ based on its values on a timelike worldtube traced out by a fixed sphere of constant (large) areal radius $r$ [see Eq. (29), there]. The expressions there contain the corrections of order ${\cal O}(1/r)$ and ${\cal O}(1/r^{2})$ to $r\psi_{4}$. As shown in Ref. Nakano et al. (2015), this extrapolation is consistent with both the waveform and the radiated energy-momentum extrapolated using a least squares fit to a polynomial in $1/r$. Additionally, the ${\cal O}(1/r)$ perturbative corrections were shown to be consistent with a Cauchy-Characteristic extraction for an equal-mass binary in Babiuc et al. (2011).

Various simulations in this catalog were studied in detail in previous papers. In Appendix A of Ref. Healy et al. (2014), we performed a detailed error analysis of configurations with equal mass and spins aligned/antialigned with respect to the orbital angular momentum; in Appendix B of Ref. Healy and Lousto (2017), we performed convergence studies for runs with mass ratios $(q=1,3/4,1/2,1/3)$ and measured errors due to finite observer locations; and in Ref. Healy et al. (2017c), we performed convergence studies for $q\geq 1/10$ nonspinning binaries.

Finally, in addition to all the internal consistency analysis and error estimates, in Ref. Lovelace et al. (2016) we showed that for the parameter estimated for GW150914 ($q=m_{1}/m_{2}=0.82$ and spins for the small/large holes of $\chi_{1}=-0.44$ and $\chi_{2}=+0.33$), the RIT waveforms and those produced completely independently by the SXS collaboration have an excellent match Cho et al. (2013) of $\gtrsim 0.99$ overall for modes up to $\ell=5$. In Ref. Healy et al. (2018) a similar agreement between approaches has been found for five targeted precessing and nonprecessing simulations of GW170104, displaying a 4th order convergence with finite difference resolution. The comparisons were also carried up to $\ell=5$-modes. For all modes up to $l\leq 4$ we found a match of $\gtrsim 0.99$ and $\gtrsim 0.97$ for the $l=5$ modes.

In all our studies we concluded that the waveforms at the resolutions provided in this catalog are well into the convergence regime (roughly converging at 4th-order with resolution), that the horizon evaluated quantities such as the remnant final mass and spins have errors of the order of 0.1%, and that the radiatively computed quantities such as the recoil velocities and peak luminosities are evaluated at a typical error of 5%.

III The Catalog

The RIT Catalog can be found at http://ccrg.rit.edu/~RITCatalog. Figure 1 shows the distribution of the non-precessing runs in the catalog in terms of $\chi_{1,2}$ and $q$ (where $\chi_{i}$ is the component of the dimensionless spins of BH $i$ along the direction of the orbital angular momentum). The information currently in the catalog consists of the metadata describing the runs and all modes up through the $\ell=4$ modes (enough for most applications) of $Mr\psi_{4}$ extrapolated to $\mathscr{I}^{+}$ via the perturbative approach of Nakano et al. (2015). The associated metadata include the initial orbital frequencies, ADM masses, initial waveform frequencies from (2,2) mode, black hole masses, momenta, spins, separations, and eccentricities, as well the black-hole masses and spins once the initial burst of radiation has left the region around the binary. Note that we normalize our data such that the sum of the two initial horizon masses is $1M$. These relaxed quantities (at $t_{relax}=200M$ after the initial burst of radiation has mostly dissipated) are more accurate and physically relevant for modeling purposes. In addition, we also include the final remnant black hole masses, spins and recoil velocity.

The catalog is organized using an interactive table dat that includes an identification number, resolution, type of run (nonspinning, aligned spins, precessing), the initial proper length of the coordinate line joining the two BH centroids that is outside both horizons Lousto and Zlochower (2013a), the coordinate separation of the two centroids, the mass ratio of the two black holes, the components of the dimensionless spins of the two black holes, the starting waveform frequency, $Mf_{22,{\rm relax}}$, time to merger, number of gravitational wave cycles calculated from the (2,2) modes from the beginning of the inspiral signal to the amplitude peak, remnant mass, remnant spin, recoil velocity, peak luminosity, amplitude and frequency. The final column gives the appropriate bibtex keys for the relevant publications where the waveforms were first presented. The table can be sorted (ascending or descending) by any of these columns. And there is a direct search feature that runs over all table elements.

Resolutions are given in terms of the grid spacing of the refinement level where the waveform is extracted (which is typically two refinement levels below the coarsest grid) with $R_{obs}\sim 100M$. We use the notation nXYY, where the grid spacing in the wavezone is given by $h=M/X.YY$, e.g., n120 corresponds to $h=M/1.2$.

For each simulation in the catalog there are three files: one contains the metadata information in ASCII format, the other two are a tar.gz files containing ASCII files with up to and including $\ell=4$ modes of $Mr\psi_{4}$ and $h$. In the near future, data will be available in the Numerical Relativity Injection format Schmidt et al. (2017). Note that the primary data in our catalog is the Weyl scalar $Mr\psi_{4}$ extrapolated to $\mathscr{I}^{+}$ (using Eq. (29) of Ref. Nakano et al. (2015)), rather than the strain $(r/M)h$. We provide the strain but also leave it to the user to convert $Mr\psi_{4}$ to strain for most modes since this is still a sensitive process and is best handled on a mode-by-mode basis. The subtleties associated with transforming $\psi_{4}$ to $h$ arise from the two integrations required Baker et al. (2002); Campanelli et al. (2009). One of the standard techniques, developed in Ref. Reisswig and Pollney (2011), performs this integration in Fourier space with a windowing function and a low-frequency cutoff. Both of these require fine-tuning of parameters. The codes to do this are open-source and publicly available from https://svn.einsteintoolkit.org/pyGWAnalysis/trunk.

Figure 2 shows the distribution of the 274 non-precessing runs in the catalog in terms of $\chi_{1,2}$ and $q$. Those runs were motivated by systematic studies to produce a set of accurate remnant formulas to represent the final mass, spin and recoil of a merged binary black hole system and the peak Luminosity, amplitude and frequency, as a function of the parameters of the precursor binary, as reported in Healy et al. (2014); Healy and Lousto (2017, 2018). A second important motivation was to provide a grid of simulations for parameter estimation of gravitational wave signals detected by LIGO using the methods described in Abbott et al. (2016e). We will see in the next section that we have achieved a good coverage of this BHB parameter space.

The precessing runs in the catalog were motivated to study particular spin dynamics of merging BHB, such as the study of unstable spin flip-flop, as reported in Lousto and Healy (2016) and the targeted followups of gravitational wave signal from the first and second LIGO observing runs Lovelace et al. (2016); Healy et al. (2018). We have also payed special attention to the systematic study of simulations covering a 4-dimensional parameter space involving a spinning and a nonspinning black hole binary as a function of the mass ratio. Those simulations were originally performed to study remnant recoil and final masses and spins Zlochower and Lousto (2015). We have supplemented them here with additional 31 simulations to have a coverage of spin orientations (see Table LABEL:tab:ID that allows an estimation of precession as shown in Fig. 8).

Figure 3 shows the distributions of the minimal total mass of the BHB systems in the catalog given a starting gravitational wave frequency of 20 or 30 Hz in the source frame. This provides a coverage for the current events observed by LIGO (redshift effects improve this coverage by a factor of $1+z$, where $z$ is the redshift). Coverage of even lower total masses would require longer simulations or hybridization of the current waveforms with Post-Newtonian methods Ajith et al. (2012).

IV Application of the Catalog to parameter estimation of Binary Black holes

We can directly compare any of our simulations to real or synthetic gravitational wave observations by scaling that simulation and its predictions to a specific total redshifted mass $M_{z}$ and then marginalizing the likelihood for the gravitational wave data over all extrinsic parameters Pankow et al. (2015); Abbott et al. (2016e); Lange et al. (2017); O’Shaughnessy et al. (2017); Lange et al. (2018): the seven coordinates characterizing the spacetime coordinates and orientation of the binary relative to the earth. Specifically the likelihood of the data given Gaussian noise has the form (up to normalization)

[TABLE]

where $h_{k}$ are the predicted response of the kth detector due to a source with parameters ($\bm{\lambda}$, $\theta$) and $d_{k}$ are the detector data in each instrument k; $\bm{\lambda}$ denotes the combination of redshifted mass $M_{z}$ and the remaining intrinsic parameters (mass ratio and spins; with eccentricity $\approx 0$) needed to uniquely specify the binary’s dynamics; $\theta$ represents the seven extrinsic parameters (4 spacetime coordinates for the coalescence event and 3 Euler angles for the binary’s orientation relative to the Earth); and $\langle a|b\rangle_{k}\equiv\int_{-\infty}^{\infty}2df\tilde{a}(f)^{*}\tilde{b}(f)/S_{h,k}(|f|)$ is an inner product implied by the kth detector’s noise power spectrum $S_{h,k}(f)$. In practice we adopt a low-frequency cutoff fmin so all inner products are modified to

[TABLE]

For our analysis of GW150914, we adopt the same noise power spectrum employed in previous work Abbott et al. (2016e); Lange et al. (2018). After exploring a range of redshifted masses $M_{z}$ for each simulation, we estimate $\ln{\cal L}(M_{z})$ as a function of mass for that simulation and thus in particular its maximum value $\ln{\cal L}_{\rm max}$, as in Abbott et al. (2016e); Lange et al. (2017).

IV.1 Non-precessing binaries

Fig. 4 displays a likelihood map for the simulations as a function of black hole’s individual spins, with a panel for each of the eight mass ratios studied, $q=1.00,0.85,0.75,0.6667,0.4142,0.50,0.3333,0.20$ on both LIGO detectors, H1 and L1, combined. Our priors of intrinsic parameters are the discrete set of numerical relativity simulations ($q,\chi_{1},\chi_{2}$), with 100 points in uniform spacing between $40<M_{total}/M_{\odot}<120$. The heat maps are generated using a multiquadric radial basis interpolating function $\sqrt{(D/\epsilon)^{2}+1}$ between the computed likelihoods for each simulation plotted by hollow circles ($D$ being the distance of the point in parameter space and $\epsilon=0.25$). We have options for using different interpolating functions, among them Gaussian process regression. The results are all compatible and the differences decrease with the increased number of simulations. We also tested the consistency of the results by dropping randomly 10% of the simulations used to produce the interpolated maps. Note that we have restricted the grids to spin magnitudes $\leq 0.85$ in order to produce interpolation maps and avoid at this stage extrapolations to larger spins until we produce enough simulations in the $\geq 0.85$ region.

Fig. 5 displays the error estimates of the aligned spins and mass ratio binary parameters for GW150914 at 90%, 95% ($2\sigma$), and 99.7% ($3\sigma$) confidence levels. Note that the diagonal shape of higher likelihoods the first panel is in part inherited by the symmetry of the $q=1$ case, given that the binary has comparable masses. The elongated (with bubbles) diagonal shape of the first panel translates into a vertical shapes in the last two panels.

The $90\%$ confidence level gives

[TABLE]

Where $M_{total}$ is given in solar mass $M_{\odot}$ units.

Compare these values to the GW150914 properties paper Abbott et al. (2016a)

[TABLE]

Fig. 6 displays a comparative analysis of the single spin approximations to aligned binaries using a linear interpolation. The upper panel presents our preferred variables for the spin, $S_{hu}$

[TABLE]

to describe the leading effect of hangup on the waveforms Healy and Lousto (2018). The lower panel displays a comparative heatmap using the common approximate model variable Ajith et al. (2011)

[TABLE]

The latter exhibits some “pinch” points around some simulations suggesting a remaining degeneracy by using $\chi_{eff}$. Such features are not seen using the (normalized) variable ${S}_{hu}$, which represents a better fitting to waveform phases as shown in Healy and Lousto (2018), suggesting again that it is a better (or at least a valid alternative) choice to describe aligned binaries.

The 3D interpolated results give a maximum $\ln{\cal L}$ of 261.8 at $(M_{total},q,\chi_{1},\chi_{2})=(73.6,0.8500,0.0000,0.0000)$. Values of the final mass and spin for this point are $0.952$ and $0.683$, respectively, and the recoil velocity is $44$ km/s. The mean values from the GW150914 properties paper Abbott et al. (2016a) are 0.955 and 0.67 for the final mass and spin, respectively. Converting the final mass to energy radiated and calculating the ranges in these final parameters from the simulations that fall within the 90% confidence interval as shown in Fig. 7, we find

[TABLE]

Comparing these ranges to the GW150914 properties paper Abbott et al. (2016a) (and converting from total mass and final mass to energy radiated and propagating the errors appropriately)

[TABLE]

Note that a final recoil velocity is not estimated in Abbott et al. (2016a).

IV.2 Precessing binaries

An analogous study of GW150914 using the aligned spin binaries above can be done in a completely independent way with a set of precessing binaries. We supplement the new simulations in this catalog release with those reported in Ref.  Lousto and Zlochower (2013b); Zlochower and Lousto (2015) for mass-ratio families of binaries with one spinning black hole pointing along 32 different orientations. The results of evaluation of the $\ln{\cal L}$ for a set of six different mass ratio families are displayed in Fig. 8. We find that the highest likelihood,is displayed in the $q=1$ panel and spin orientation near the equatorial (orbital) plane. This result is consistent with the low $S_{hu}$ (or low $\chi_{eff}$) displayed in Fig. 6. The $q=1$ highest likelihood is bracketed by the $q=1.4$ and $q=0.66$ panels with the former having a larger $\ln{\cal L}$ than the later, indicating that the optimal configuration should have a mass ratio between $q=1$ and $q=1.40$ (that corresponds to a case of $q=1/1.4=0.714$ with the smaller black hole spinning). This again is in agreement with the previous analysis involving only aligned simulations indicating a preference for mass ratios around $q=0.85$.

Let us note that this precessing simulations analysis is completely independent from the previous nonprecessing, aligned spins, and they do not share simulations in common and yet lead to a similar range of parameters (for the most robust mass ratios and projections of the spins onto the orbital angular momentum). While the analysis would benefit from more simulations to populate this 4D parameter space (and this will be one of the subjects of a new catalog release), it is encouraging that consistent results are already found with this minimal set of nearly 200 simulations. we populated each panel with 4 sets of initial $\theta=30,60,90,135$ degrees orientations (note that new configurations will supplant $\theta=135$ for 120 and 150 degrees) and six $\phi=0,30,60,90,120,150$ plus the poles and a few control simulations on the $\phi<0$ degrees (See Fig. 7 of Ref. Healy et al. (2018)). To extrapolate into the western hemisphere, we exploit the symmetry of the parameter space and fit a sinusoidal function to the available numerical simulations. Instead of plotting in the angles $\theta$ and $\phi$, we plot in the Hammer-Aitoff coordinates 222Weinstein, Eric W.“Hammer-Aitoff Equal-Area Projection.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Hammer-AitoffEqual-AreaProjection.html, which is a coordinate system where the whole angular space can be viewed as a 2D map. The points at the top left and bottom left are the poles, $\theta=0$ at the top, and $\theta=\pi$ at the bottom. The line connecting the two is the $\phi=0$ line. As you move from left to right from the center, $\phi$ increases from 0 to 180 degrees, and from right to left, $\phi$ decreases from 0 to -180 degrees.

IV.3 Estimation of extrinsic parameters

To complete the parameter determination from only the numerical relativity simulations we proceed to compute the distribution of the sky location, distance and binary orientation from the evaluations of the $\ln{\cal L}$ for each of our simulations on a total mass grid of 100 points between 40 and 120 $M_{\odot}$. The results for GW150914 are displayed in Fig. 9. The gray boundaries are calculated from the LIGO GWTC-1 public data 333https://dcc.ligo.org/LIGO-P1800370/public for GW150914 by first constructing a 2D kernel destiny estimation and using numpy’s percentile function. Simulations within a $\ln{\cal L}$ cut of 3.125 of the max are included in the 90% CI interval. As with the intrinsic parameters, the results are consistent but less confined than the LIGO results. Our results seem also to be compatible with the original localization in sky estimates displayed in Ref. Abbott et al. (2016h)

IV.4 Simulated versus signal waveform comparison

We use standard techniques Abbott et al. (2016e); Lange et al. (2017) to directly compare GW150914 to our simulations. For each simulation, direct comparison of our simulations to the data selects a fiducial total mass which best fits the observations, as measured by the marginalized likelihood. We can for each simulation select the binary extrinsic parameters like event time and sky location which maximize the likelihood of the data, given our simulation and mass. Then, using these extrinsic parameters, we evaluate the expected detector response in the LIGO Hanford (H1) and Livingston (L1) instruments. Figure 10 displays these reconstructions for the highest log-likelihood NR waveform of the nonprecessing and precessing simulations labeled as RIT:BBH:0113 and RIT:BBH:0126 in our catalog. The details of these simulations are provided in Table 1. They directly compare to the signals as observed by LIGO H1 and L1 and with each other. The lower panel shows the residuals of the signals with respect to the RIT simulations. A similar analysis was performed in Ref. Healy et al. (2018), Figures 4-6, for the GW170104 event.

V Conclusions and Discussion

The breakthroughs Pretorius (2005); Campanelli et al. (2006a); Baker et al. (2006a) in numerical relativity were instrumental in identifying the first detection of gravitational waves Abbott et al. (2016a) with the merger of two black holes. We have shown in this paper that the use of numerical relativity waveform catalogs provides a consistent method for parameter estimation from the observed gravitational waves from merging binary black holes (See also Refs. Abbott et al. (2016e); Lange et al. (2017); Kumar et al. (2018). It is worthwhile stressing here that this current method of direct comparison of the gravitational wave signal with numerical waveforms does not rely at all on any information from the phenomenological models Babak et al. (2017); Hannam et al. (2013) (Phenom or SEOBNR). It also shows that with the current aligned spin coverage one can successfully carry out parameter estimations with results, at least as good as with the phenomenological models Abbott et al. (2016a); see Lange et al (in prep). We also note that any new simulation produced (for instance targeted to followup any new detection or catalog expansions) will contribute to improve the binary parameter coverage, thus reducing the interpolation error. The next step will be reduce the extrapolation error at very high spins by adding more simulations with spin magnitudes above 0.90. Also the extension of the family of simulations displayed in Fig. 8 to smaller mass ratios, i.e. $q<1/3$. Coverage for low total binary masses (below $20M_{\odot}$), in turn, would require longer full numerical simulations or hybridization of the current NR waveforms with post-Newtonian waveforms.

The next area of development for the numerical relativity waveform catalogs is the coverage of precessing binaries. Those require expansions of the parameter space to seven dimensions (assuming negligible eccentricity), and is being carried out in a hierarchical approach by neglecting the effects of the spin of the secondary black holes, which is a good assumption for small mass ratios. This approach has proven also successful when applied to GW170104 Healy et al. (2018). It required an homogeneous set of simulations since the differences in $\ln{\cal L}$ are subtle. The comparison of different approaches to solve the binary black hole problem has produced an excellent agreement for the GW150914 Lovelace et al. (2016) and GW170104 Healy et al. (2018), including higher (up to $\ell=5$) modes. This leads to the possibility of regularly using multiple catalogs of numerical relativity waveforms to further improve parameter coverage as started in Abbott et al. (2016e). In a follow up paper we plan to use this upgraded catalog to evaluate the parameters of the ten binary black hole mergers reported recently by LIGO-Virgo O1-O2 observing runsAbbott et al. (2018).

Aside from the interest in producing waveforms for direct comparison with observation, the simulations of orbiting black-hole binaries produce information about the final remnant of the merger of the two holes. Numerous empirical formulas relating the initial parameters $(q,\vec{\chi}_{1},\vec{\chi}_{2})$ (individual masses and spins) of the binary to those of the final remnant $(m_{f},\vec{\chi}_{f},\vec{V}_{f})$ have been proposed. These include formulas for the final mass, spin, and recoil velocity Barausse et al. (2012); Rezzolla et al. (2008); Hofmann et al. (2016); Jiménez-Forteza et al. (2017); Lousto et al. (2010); Lousto and Zlochower (2014b); Hemberger et al. (2013); Healy et al. (2014); Zlochower and Lousto (2015), the computation of the peak frequency of the (2,2) mode $\Omega_{22}^{peak}$, peak waveform amplitude $A_{22}^{peak}$  Healy et al. (2017c); Healy and Lousto (2018) and peak luminosity Abbott et al. (2016a, d); Healy and Lousto (2017); Keitel et al. (2017). Those formulas in turn provide further tools to extract information from the observation of gravitational waves, see for instance our Fig. 7.

Acknowledgements.

The authors thank A. Williamson for assistance with Fig. 10. The authors also gratefully acknowledge the National Science Foundation (NSF) for financial support from Grants No. PHY-1607520, No. PHY-1707946, No. ACI-1550436, No. AST-1516150, No. ACI-1516125, No. PHY-1726215. This work used the Extreme Science and Engineering Discovery Environment (XSEDE) [allocation TG-PHY060027N], which is supported by NSF grant No. ACI-1548562. Computational resources were also provided by the NewHorizons, BlueSky Clusters, and Green Prairies at the Rochester Institute of Technology, which were supported by NSF grants No. PHY-0722703, No. DMS-0820923, No. AST-1028087, No. PHY-1229173, and No. PHY-1726215.

Appendix A Tables of initial data and results of the new simulations

In this appendix we provide tables with the relevant BBH configuration details. In Table LABEL:tab:ID, we provide the initial data parameters used to start the full numerical evolutions. In Tables LABEL:tab:IDr and LABEL:tab:IDr_prec, we provide the binary mass and spin parameters after they settle into a more physical value after radiating and absorbing the spurious gravitation wave content from the initial mathematical choice of conformal flatness. These relaxed values are calculated at a fiducial $t=200M$.

In Table LABEL:tab:ecc we provide the initial orbital frequency and eccentricity, as well as the number of orbits to merger and the final eccentricity. The eccentricity is expected to be reduced from its initial value by gravitational radiation, at a rate proportional to $d^{19/12}$ according to Peters (1964), with $d$, the separation of the binary (see, for instance, Fig. 6 of Ref. Mroue et al. (2010) or Fig. 9 in Ref. Lousto et al. (2016)).

Finally, In Table LABEL:tab:spinerad, we provide the values of the energy radiated during the simulation and the final black hole spin as measured through the (most accurate) isolated horizon formalism Dreyer et al. (2003).