Binary Black Hole Merger Dynamics and Waveforms

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Binary Black Hole Merger Dynamics and Waveforms

• Dale Choi
• NASA/Goddard Space Flight Center
• 2006 Spring School on Numerical Methods in
  Gravitation and Astrophysics, APCTP, Seoul,
  Korea, Mar 22, 2006

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Collaborators

• J. Centrella, J. Baker, M. Koppitz, J. van Meter
  (GSFC)
• M. C. Miller (Univ. of Maryland)

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Outline

• Introduction/Motivation
• Methodology (Hahndol Ein-stein ?? code)
• Results
• Equal mass
• Non-equal mass

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Introduction/Motivation

• One of major predictions of General Relativity is
  gravitational waves (GWs).
• Indirect evidence Hulse-Taylor pulsar in a
  binary
• A network of laser interferometric GW
  observatories LIGO, VIRGO, GEO600, TAMA, and
  LISA to directly measure GW.

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Introduction

• In the next decade or two, we expect that
  observation of gravitational waves by the
  worldwide network of GW detectors
• Will deepen our understanding of Einsteins
  General Relativity as a fundamental theory of
  gravity.
• Will provide tests of GR in truly strong-field,
  highly nonlinear and highly dynamical regimes.
  Alternative theories?
• Will revolutionize our understanding of structure
  and evolution of universe by providing a
  fundamentally new observational window.
• Binary black hole systems are
  among
  the most strongest sources.

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Black Hole Systems

• Mounting observational evidence for a variety of
  kinds of black holes (stellar mass BH,
  Super-massive BH, Intermediate Mass BH).
• SMBH every galaxy looked at so far harbor SMBH
  at its center. (HST)

SMBH Mass estimate a few 109 solar mass
M87 Chanda X-ray image
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Binary Black Holes

• Binary black holes provide one of the simplest
  two body problem in GR.
• One of the strong source of gravitational waves.
• LIGO, f 101000Hz
• Stellar mass BBH, lt 100 Msol
• LISA, f 0.1mHz 1 Hz
• Super-massive BBH, 106-109 Msol
• Galactic mergers drive SMBH binary to the center.
• Final stage of dynamics driven by gravitational
  radiation reaction.

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BBH Coalescence

• Coalescence driven by GW emission can be roughly
  divided into 3 phases.
• Adiabatic Inspiral (orbits quickly circularize)
• Plunge/Merger
  (2bh ?
  1bh)
• Ring-down
  (Merged
  BH)

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BBH Merger Simulations

• Inspiral and ring-down phases can be understood
  in terms of analytic methods.
• Inspiral Post-Newtonian point particle,
  slow-moving, weak gravity.
• Ring-down Perturbative method (Perturbed Kerr).
• However, Numerical Relativity is required to
  understand merger phase accurately.
• Binary Black Hole Merger Simulations have long
  been one of major goals in Numerical Relativity.
• Very exciting developments over the past year.
  Several different groups doing multiple
  orbits/merger/ring-down using different methods
  and codes.

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Methodology Hahndol (??) Code

• 31 Numerical Relativity code
• BSSN formalism following Imbiriba et al, PRD70,
  124025 (2004), Alcubierre at al PRD67, 084023
  (2003).
• Uses finite differencing (mixed 2nd and 4th order
  FD), iterative Crank-Nicholson time integrator.
• Computational infrastructure based on PARAMESH
  (MacNiece, Olson). Scalability shown up to 864
  CPUs with 95 efficiency.
• Adaptive mesh refinement
• Initially fix mesh structure by hand with mesh
  boundaries at e.g. (2,4,8,16,32,64)M.
• Solve HCE (Hamiltonian constraint equation) using
  AMR Multi-Grid method at t0
• Change grid structure adaptively during the
  evolution
• Refinement criteria Coulomb scalar (represent
  coulomb part of potential).

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Hahndol Code

• Outer boundary conditions
• Impose outgoing Sommerfeld conditions on all BSSN
  variables.
• But, basic strategy is to push OB far away so
  that OB does not contaminate regions of
  interests.
• With typical OB400M800M, no harmful effects on
  the dynamics of black holes nor waveform
  extraction.
• Initial data solver
• Uses multi-grid method on a non-uniform grid
  based on algorithm by Brown Lowe, JCP 209,
  582-598, 2005 (gr-qc/0411112).
• Generates quasi-circular data by solving HCE
  using puncture method (Brandt Bruegmann, 1997).
• Bowen-York prescription for the extrinsic
  curvature for binary black holes.

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Hahndol Code

• Gauge conditions
• Lapse 1log plus advection term (generalized
  harmonic slicing).
• Shift Hyperbolic gamma driver shifting shift.
• Motivation
• Want to move black holes across the grid by
  adding advection terms.
• Empirically we find it works very well and
  excision is not needed (so far).
• Analysis
• Gravitational Waveforms
• Energy, Angular momentum
• Trajectories
• Fit to quasi-normal modes.

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Gravitational Waveforms

• Strain amplitude h ?L/L is what is observed.
• Compute the Newman-Penrose Weyl scalar ?4 (a
  gauge invariant measure)
  
• where C is weyl tensor and (l,n,m,mbar) is a
  tetrad.
• ?4 h,tt
• Analyze harmonic decomposition using a novel
  technique due to Misner (Misner 2004 Fiske
  2005).
• Compute waveforms typically at r 30-60M.

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Initial Data

• Conformal flatness (Gij f4 fij), Maximal
  slicing (K0)
• Given Bowen-York extrinsic curvature
• That satisfy Momentum constraint equation
  analytically
• Which encode information about spin and linear
  momentum (consider here only non-spinning cases)
• solve Hamiltonian constraint equations for
  conformal factor f.
• Choose BY extrinsic curvature that approximate
  circular orbits.

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Results
gr-qc/0511103 (to appear PRL),
gr-qc/0602026 (to appear PRD)

• Initial Data Table, M total mass of space-time

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Results

• Left Merger from a roughly ISCO separation. ?4
• Right Merger from L/M 13.2 separation. Lines
  represent puncture tracks and blue surface
  represent alpha lt 0.3 (alpha0.3 being roughly
  location of apparent horizon for the gauge
  conditions used.)

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BH Trajectories

• xPUNC (t) ,t - ß (xPUNC (t) ), xPUNC is
  an location of puncture.
• Coordinate quantity, so need to be cautious. But
  the trajectory frequencies and radiation
  frequencies match extremely well.
• Tracks from 4 runs are very similar for the last
  orbit, through merger indicating black holes
  follow the same track.

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Coordinate separation

• R (coordinate separation bet. BHs) vs. t
• Remarkable agreement for the last orbit, and
  merger for the runs with different initial
  separation.

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Gravitational Radiation Waveforms

• r ?4 (t/M)
• l2, m2 mode
• Perfect agreement after t -50M with 1
  difference.
• For the preceding 500M (inset) agreement in phase
  and amplitude within 10 except for a brief
  initial transient pulse at the beginning of each
  run.

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Waveforms

• r ?4 A exp( -i f)
• A amplitude, f phase angle

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Frequencies

• Left gravitational wave frequencies, w ? fpol
  /?t, fpol
• Right Shows comparison against Post-Newtonian
  results. (1.5PN, 2PN expansion in (v/c)2 )

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Energy and angular momentum

• E radiative energy loss, J radiative angular
  momentum loss, aJ/Mf
• E, J are measured at r30M, r50M respectively.
• MQN and aQN are calcaulted independently from the
  quasi-normal fits of the ring-down waveforms by
  assuming the ring-down part of waveform ? f(t)
  A exp(-a t i w t) with A, a, w real constants.

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Summary

• Applied recently developed techniques (1) new
  gauge conditions to move black holes with no need
  for excision and (2) AMR based on curvature
  scalar.
• Simulated a set of quasi-circular initial data of
  equal mass non-spinning black holes for up to 4
  orbits with simulation up to 850M.
• Excellent agreement in dynamics and waveforms for
  the last orbit, merger, and ring-down among 4
  runs with different initial separation.
• Universal features in waveforms and trajectories
  for the last orbit, merger thru ring-down.

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• Multiple groups make rapid progress in a broad
  front. A beginning of new ear in 3D BBH research?
• Next questions
• Mass ratio--- Kicks (astro-ph/0603204)??
• Spin??
• Further separation, compare against PN results.
• Develop and/or provide feedbacks to PN models.
• BBH scattering??
• Inputs to the data analysis
• Very important for LIGO for detection
• LISA extract physics and test GR alternative
  theories of gravity?

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Gravitational Radiation Recoil
astro-ph/0603204

• Most numerical studies to date focused on highly
  symmetric situations (e.g. equal mass
  non-spinning)
• But mergers between astrophysical black holes
  will in general produce asymmetric gravitational
  radiation imparting merger remnant a kick
  (gravitational rocket).
• For SMBH mergers, kicks are important
  astrophysical input in that large kicks can eject
  merged black holes from the host structure.
• Provides a very crucial inputs to the study of
  formation and growth of supermassive black holes
  and their hosts.

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Radiation Recoil Calculations

• Need accurate estimates of higher L modes of
  waveforms.
• In general its more difficult calculations
  because we are dealing with small numbers 10-3
  vs. 1.
• Kicks are from delicate mixing of different
  harmonic modes (e.g l2 and l3 modes) of the
  waveforms.

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Kicks results

• Kicks for mass ratio 1.51
• Final value is 105 km/s with error bar of less
  than 10.

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Astrophysical implications of kicks

• Minimum halo mass to retain merged black holes as
  a function of z.

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Closing Remarks

• Using moving puncture techniques, can do
• Multiple orbit dynamics
• Extract accurate gravitational waveforms
• Study different initial data set-up covering a
  large parameter space
• Demonstrated remarkable agreements in the black
  hole dynamics and the waveforms for the last
  orbit of the equal mass non-spinning binary black
  hole coalescence starting from the different
  initial separation
• Robustness of the results provide strong evidence
  that the waveforms for the last orbit represent
  accurately gravitation radiation in GR from an
  astrophysical system of equal mass non-spinning
  binary black holes.