Measuring the mass and angular momentum of black hole binaries

1. Measuring the mass and angular momentum of black hole binaries Bernd Brügmann (University of Jena) José González (University of Jena) Mark Hannam (University of Jena) Sascha Husa (University of Jena) Pedro Marronetti (Florida Atlantic University) Ulrich Sperhake (University of Jena) Wolfgang Tichy (Florida Atlantic University)

2. Defining M & J • Surface integral at infinity • Non-local • Essential quality control monitors (in particular J, in the case of circular orbits)

3. Ways of estimating M & J • Surface integrals at infinity (or as far away as you can go from the sources) • Relatively easy to code • Sensitive to boundary effects • Limited Quality Control use • Only the few grid points close to the integration surface are used • Estimation from the Black Hole (Apparent, Event, or Isolated) Horizon • Conceptually robust (isolated horizons) • Orbital J is left out • Requires AH to be found (slow) • Resolution sensitive • Estimation using the emission of M and J by gravitational waves • Largely independent estimation • Indirect estimation • Sensitive to boundary effects • Only the few grid points close to the integration surface are used • Converting Surface to Volume integrals • Method used successfully in Neutron Star Binaries (Duez et. al. PRD 67 (2003), PM et al. PRL 92 (2004), PM Class. Quant. Grav. 22 (2005), PM Class. Quant. Grav. 23 (2006).) • Used for Single Black Hole evolutions: Yo et al. PRD 66 (2002) (Appendix A).

4.  inner outer Single non-spinning BH: From Surface to Volume Integrals Conversion from surface at infinity to volume integral using Gauss (Yo, Shapiro and Baumgarte 2002)

5. Inner Surface inner Volume  BAM grid structure

6. Varying Grid Volume convergence Single spinning BH with initial values M0 = 0.542 J0 = 0.125 J/M2 = 0.425

7. Varying Resolution convergence Single spinning BH with initial values M0 = 0.542 J0 = 0.125 J/M2 = 0.425

8. Varying Inner Surface Position convergence Single spinning BH with initial values M0 = 0.542 J0 = 0.125 J/M2 = 0.425

9. Single spinning BH Single High-Spin BH with initial values M0 = 0.4921 J0 = 0.1875 J/M2 = 0.77 Vol. = 65M3 Max. Res. = M/128 Inn. Surf. = 0.4M Volume Method Surface Method (radius = 20M)  Monopole

10. BBH Example • Run details (comparable to ‘++’ run from Campanelli, Lousto and Zlochower 2006) • m1 = m2 = 0.332 M • s1/m12 = s2/m22 = 0.77 (aligned with orbital angular momentum) • y = 3.06 M • M = 0.980, J/M2 = 1.16 • Max. Res. M/32 • BBox = 130M Run performed in two processors, lasting a week Punctures merge at about 360M

11. BBH: M vs. time Volume Method ΔM = -4.75% Surface Method (r=20M) ΔM ~ 0% Surface Method (r=100M) ΔM = 1% Mass loss due to GW emission: recently implemented

12. BBH: J vs. time Volume Method ΔJ = -40% Surface Method (r=20M) ΔJ = -30% Surface Method (r=100M): too noisy

13. (Very preliminary)Conclusions • More work on the estimation of the mass and (particularly) the angular momentum is needed • Pros and Cons of the Volume Method • Provides tracking of Total (orbital + spin) angular momentum • Real-time Quality Control: Fairly computationally inexpensive (compared with horizon methods) • Relatively easy to code (compared with horizon methods) • Largely independent of the grid size • Possibly not strongly affected by outer boundary effects • Resolution and Inner Surface position dependent (do your convergence homework before use) • It must be used in combination with other methods