1. 1 Massive Black Hole Mergers: As Sources and Simulations John Baker Gravitational Astrophysics Laboratory NASA/GSFC CGWP Sources & Simulations February 03, 2005

2. 2 Massive Black Hole Mergers MBHs are believed to lurk at centers of all galaxies with bulges Most galaxies are believed to have at least one merger  massive black hole binary mergers Merger rates: –depend on size of “seed” black holes, accretion rates, merger efficiency... –expect ~ 10s (more or less) per year Gravitational waves from final merger are detectable by LISA to high z (eg ~20) Observations of such events by LISA… –may be used to map merger history of MBHs (and their host structures) to high z –May provide tests of strong GR dynamics

3. 3 Some relevant astrophysics… Theory (Look out to z~20, prefer small MBH) –ΛCDM models … are they correct? Small primordial density fluctuations in dark matter grow over time Matter collects in gravitational potential wells and small galaxies with small MBH form As the potential wells deepen the galaxies merge progressively Predicts many smaller (10 4 -10 5 M Sun ) binaries –Do the MBH usually merge? Dynamical friction turns off as the binary hardens Three-body interactions can take over but there must be a process which continues to bring objects near the binary Observation (Look to z>2, prefer large MBH) –Is there a shortage of smaller (<10 6 M Sun ) black holes (eg in SDSS)? –Dearth of merger candidates at z<1-2 –Do larger MBH form earlier? (possible interpretation of recent X-ray observations)

4. 4 Detecting MBH binary mergers with LISA LISA measures strain due to incoming GW signals Instrumental strain noise spectrum Characteristic strain of GW signal LISA measures redshifted frequency Expected signal-to-noise ratio (SNR) for obs of a chirping source using matched filtering For a detection, source must be within LISA’s band of sensitivity at good SNR

5. 5 MBH binary inspirals and LISA symbols at 10 yrs, 1 yr, 1 mo, & 1 d before the onset of merger, and at the onset of merger (the merger & subsequent ringdown occurs at higher frequencies)

6. 6 LISA …observation is in the motion… Joint NASA/ESA mission all-sky monitor, measures both GW polarizations  2 time series 3 spacecraft –equilateral triangle –arm length L = 5 x 10 6 km –orbits Sun at 1 AU –20 o behind Earth in its orbit Detector motions –orbital motion around the Sun –yearly rotation of the triangular spacecraft constellation around the normal to the detector plane tilted at 60 o to ecliptic These motions induce modulations of incident GW signal that encode sky position and orientation of source

7. 7 Observing MBH binary mergers with LISA For a good observation, want masses, spins, sky position, z… This information must be extracted from LISA’s data stream Source parameters are entangled: –Track phase of inspiral waveform  measure (1+z)m 1 and (1+z)m 2 –Overall amplitudes of waveforms depend on Luminosity distance D(z) (Knowing cosmology, invert D(z) to get redshift) chirp mass M = M 2/5  3/5 (M = total mass,  = reduced mass) orientation and sky position Info on orientation and sky position (and thus z) is encoded in modulations of LISA’s signal due to its yearly motion  need source to be within band of sensitivity for significant fraction of LISA’s yearly orbit

8. 8 Information extraction vs observation time Courtesy A Vecchio The signal needs to be visible for 6 months before coalescence in order to preserve information extraction m 1 = 10 6 M sun m 2 = 10 5 M sun z = 1

9. 9 Observing – MBH inspirals 1 An MBH binary can be observed by LISA for 6 months in band if it’s ‘x’ is above a given sensitivity curve. X’s label systems. Space of X’s looks like… x x x x x x x

10. 10 Science Reach – MBH inspirals An MBH binary with chirp mass M at redshift z can be observed by LISA for 6 months in band if it is above a given sensitivity curve

11. 11 Observing MBH binary mergers with LISA Detection: LISA data stream contains source signal at good SNR Observation: source parameters can be extracted accurately How to extract source parameters? –Use motion-induced modulations  Source must be w/in LISA’s band for ~ 6 months Can we relax the 6-month rule….and demands on low frequency sensitivity? Other options… –Tolerate incomplete information…e.g. just get (1+z) M ….many more systems accessible…. –Include multipole components higher than quadrupole –May get useful source info from merger/ringdown phase (But we have to understand mergers first)

12. 12 Numerical Simulations for LISA science Astrophysics: –Merger kicks  ejection rates –Remnant spins  population stats. Parameter Estimation: –Better estimates  more systems accessible (larger z, larger masses) and less reliance on low-frequency band –High SNR implies small details in waveforms may be useful Improved sensitivity to other sources: –“cocktail party problem” –LISA analysis requires fitting ALL sources simultaneously How good do simulations have to be? –Any understanding may be useful –Ultimately want high-precision waveforms: eg. Run for 10000 M, with 0.1% accuracy. Moving toward more accurate waveforms: –Higher order finite differencing –Adaptive mesh refinement

13. 13 Higher order finite differencing with LazEv Why not second order differencing? –For 3+1D simulations: work ~ h -4 ; –error ~ h n  error ~ work -n/2 –To reduce error from O(1) to O(0.01) you need to work 10000 times as hard if n=2. LazEv: –A general Cactus-based evolution tool –Developed by Yosef Zlochower, J.B., and Lazarus-UTB team –Includes 4 th order BSSN formulation of Einstein’s equations. –Designed for generalization to higher-order and other formulations. –4 th -order runs here use (Kreiss-Oliger) dissipation

14. 14 Higher order finite differencing with LazEv LazEv with 1D Gowdy wave “Mexico test” (Y. Zlochower, et al) –h=0.2/ρ, ρ=2,4,8 –Evolves backward in time –Excellent convergence for 1000 crossing times –Error reduced by factor of 256 t (crossing) |ρ 4 C Ham | L2

15. 15 Higher order finite differencing with LazEv LazEv with 2D gauge wave “Mexico test”: (Y. Zlochower, et al) –h=0.2/ρ, ρ=2,4,8 –Strong wave (A=0.1) –“X”=crossing times –Excellent convergence for 60 crossing times –Error reduced by factor of 256 t (crossing)

16. 16 Higher order finite differencing with LazEv LazEv with BBH example: (Y. Zlochower, et al ) –Black holes released from rest at ISCO separation –Punctures crash with 4 th order shift-advection Mixed w/ 2 nd order shift- advection Full 4 th order with excision –Both approaches at two resolutions compared –Preliminary result! Re[ψ 4 ] l=m=2 t/M

17. 17 Mesh Refinement with Hahndol Why use AMR? –Multiple scales O(M) at black hole O(100M) for orbit wave –Can achieve higher resolution in critical regions –Can push outer boundary far away. Hahndol code for AMR: (GSFC NumRel team ) –Block-based mesh refinement using PARAMESH –BSSN formalism w/ 2 nd order finite differencing (for now) –Guard cells (ghostzones) at refinement boundaries filled by quadratic or cubic interpolation –Thoroughly investigating interface performance

18. 18 Mesh Refinement testing with Hahndol Wave propagation –Teukolsky wave tests 1BH strong field convergence studies –Geodesic slicing –1+log slicing –Quadratic or cubic guard cell filling –Gamma-driver gauge (in progress) 2BH test –Head-on collision waves (in progress) AMR studies –Brill wave collapse (in progress)

19. 19 Binary BH Mesh Refinement testing with Hahndol Brill-Lindquist data at ISCO separation –1+log slicing –cubic guard cell filling –Γ-driver shift Eight-level FMR –h=M/32 and M/16 out to x=2M –Outer boundary at 256M –Preliminary result

20. 20 Binary BH Mesh Refinement testing with Hahndol: The movie Same run from last slide Computational domain –To 256M Domain shown x,y,z ≤ 64M Quantity shown –||ψ 4 || Interfaces shown –At 2, 4, 8, 16 and 32M Resolution in visible regions –M/32 to M

21. 21 Summary MBH-MBH systems are an exciting source for LISA observations Understanding of these systems based on numerical simulations will be of great value for LISA science Development toward higher-fidelity simulations is progressing on two technologies –Higher-order differencing (Lazarus-UTB) –Mesh refinement (GSFC)