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Massive Black Hole Mergers: As Sources and Simulations

John Baker

Gravitational Astrophysics Laboratory

NASA/GSFC

CGWP Sources & Simulations February 03, 2005

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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

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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 (104-105 MSun) 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 processwhich continues to bring objects near the binary

• Observation (Look to z>2, prefer large MBH)

– Is there a shortage of smaller (<106 MSun) 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)

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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

])1[()(

)1(2

Lc fz

df

dE

zD

zh

)(

)(2c

0

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ffS

fh

f

dfSNR

h

)1/( zff e

)( ffSh hc

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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)

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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 106 km– orbits Sun at 1 AU

– 20o 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

60o

to ecliptic

• These motions induce modulations of incident GW signal that encode sky position and orientation of source

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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)m1 and (1+z)m2

– Overall amplitudes of waveforms depend on

• Luminosity distance D(z) (Knowing cosmology, invert D(z) to get redshift)

• chirp mass M = M2/53/5 (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

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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

m1 = 106 Msun

m2 = 105 Msun

z = 1

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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

xx

xx

x

x

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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

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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)

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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

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Higher order finite differencing with LazEv

• Why not second order differencing?

– For 3+1D simulations: work ~ h-4;

– error ~ hn 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 4th order BSSN formulation of Einstein’s equations.

– Designed for generalization to higher-order and other formulations.

– 4th-order runs here use (Kreiss-Oliger) dissipation

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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 byfactor of 256

t (crossing)

|ρ4 CHam|L2

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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 byfactor of 256

|ρ4 CHam|L2

t (crossing)

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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 4th order shift-advection

• Mixed w/ 2nd order shift-advection

• Full 4th order with excision

– Both approaches at two resolutions compared

– Preliminary result!

Re[ψ4]l=m=2

t/M

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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/ 2nd order finite differencing (for now)

– Guard cells (ghostzones) at refinement boundaries filled by quadratic or cubic interpolation

– Thoroughly investigating interface performance

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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)

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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

20Binary BH Mesh Refinement testing with Hahndol: The movie• Same run from

last slide

• Computationaldomain

– To 256M

• Domain shown

x,y,z ≤ 64M

• Quantity shown

– ||ψ4||

• Interfaces shown

– At 2, 4, 8, 16 and 32M

• Resolution in visible regions

– M/32 to M

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Summary

• MBH-MBH systems are an exciting source for LISA observations

• Understanding of these systems based on numerical simulations will be of great valuefor LISA science

• Development toward higher-fidelity simulations is progressing on two technologies

– Higher-order differencing (Lazarus-UTB)

– Mesh refinement (GSFC)