Extreme-mass-ratio Gravitational Wave Bursts From
The Galactic Centre
Supervisor: Jonathan Gair
Christopher [email protected]
Institute of Astronomy, University of Cambridge
2nd Iberian Gravitational Wave MeetingFebruary 2012
Christopher Berry (IoA) EMRBs From The GC February 2012 1 / 21
Outline
IntroductionMassive black holesMeasurements of MBH spinGravitational wave bursts
Generating waveformsSignal analysis
DetectabilityParameter Estimation
Event ratesConclusion
Christopher Berry (IoA) EMRBs From The GC February 2012 2 / 21
Introduction Massive black holes
Galactic cores
Most galaxies are believed to have amassive black hole (MBH) at theircentre.
The masses of these MBHs are knownto correlate with the properties oftheir host bulges.
Figure: Mass-velocity dispersion relation(Graham et al. 2011)
Christopher Berry (IoA) EMRBs From The GC February 2012 3 / 21
Introduction Massive black holes
The centre of the Galaxy
The Milky Way has an MBH coincident with Sagittarius A∗ (Sgr A∗). Thishas a mass M• = 4.31 × 106M⊙ and is at a distance of only R0 = 8.33 kpc(Gillessen et al. 2009).
Figure: Infra-red Hubble/Spitzer composite image of the centre of the Milky Way(NASA/ESA)
Christopher Berry (IoA) EMRBs From The GC February 2012 4 / 21
Introduction Massive black holes
Black hole properties
Figure: The no-hair theorem
“The black holes of nature are themost perfect macroscopic objectsthere are in the universe.” (Chan-drasekhar 1998)
Astrophysical black holes are de-scribed by just their mass M• and spin
a∗ =cJ
GM 2•
. (1)
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Introduction Measurements of MBH spin
X-ray observations of active galactic nuclei
AGN a∗ Study
1H0707–495 ≥ 0.976 Zoghbi et al. (2010)
Ark 120 0.74+0.19
−0.50Nardini et al. (2011)
Fairall 9 0.60 ± 0.07 Schmoll et al. (2009)
0.44+0.04
−0.11Patrick et al. (2011a)
0.39+0.48
−0.30Emmanoulopoulos et al. (2011)
0.67+0.10
−0.11Patrick et al. (2011b)
MCG–6-30-15 0.989+0.009
−0.002Brenneman & Reynolds (2006)
0.86+0.01
−0.02de la Calle Perez et al. (2010)
0.49+0.20
−0.12Patrick et al. (2011b)
Mrk 335 0.70+0.12
−0.01Patrick et al. (2011a)
Mrk 509 0.78+0.03
−0.04de la Calle Perez et al. (2010)
NGC 3783 ≥ 0.88 Brenneman et al. (2011)< 0.32 Patrick et al. (2011b)
NGC 7469 0.69+0.09
−0.09Patrick et al. (2011a)
SWIFT J2127.4+5654 0.6 ± 0.2 Miniutti et al. (2009)
0.70+0.10
−0.14Patrick et al. (2011a)
Table: Estimates of MBH spin from Fe K emission lines.
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Introduction Measurements of MBH spin
Sgr A∗: quasi-periodic oscillations
101
102
103
104
105
106
107
-1.0
-0.5
0.0
0.5
1.0
Sp
in P
ara
mete
r
M [M ]
GRS 1915+105
XTE 1550-564
GRO 1655-40
XTE J1859+226
XTE J1650-500
a* = 0.44±0.08
(4.31±0.66) 106M (Gillessen+ 09)
(4.5±0.4) 106M (Ghez+ 08)
(3.7±1.5) 106M (Schodel+ 02)
Figure: Discseismic model fit (Kato et al. 2010)
Estimates of the spin of theMBH have been made usingquasi-periodic oscillations inflare intensity.
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Introduction Measurements of MBH spin
Sgr A∗: VLBI
It may soon be possible to im-age objects on the scale ofthe event horizon using inter-ferometry.
Black holes cast a shadow, sur-rounded by a bright photonring, the shape of which is de-termined by their spin.
For Sgr A∗ this has a angulardiameter of about 50 µas.
α/M•
β/M•
2 4 6 8−2−4−6−8
2
4
6
8
−2
−4
−6
−8
(a) a∗ = −0.2
α/M•
β/M•
2 4 6 8−2−4−6−8
2
4
6
8
−2
−4
−6
−8
(b) a∗ = 0.4
α/M•
β/M•
2 4 6 8−2−4−6−8
2
4
6
8
−2
−4
−6
−8
(c) a∗ = 0.9
α/M•
β/M•
2 4 6 8−2−4−6−8
2
4
6
8
−2
−4
−6
−8
(d) a∗ = 0.998
Figure: BH shadow in equatorial plane.
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Introduction Gravitational wave bursts
Extreme-mass-ratio bursts
Figure: Extreme-mass-ratio black holespacetimes (NASA).
Extreme-mass-ratio inspirals (EMRIs)are well studied. Objects undergomany orbits allowing a high signal-to-noise ratio (SNR) to accumulate.
Extreme-mass-ratio bursts (EMRBs)are produced by higher eccentric or-bits. Only one burst of radiation isemitted per orbit.
An EMRB orbit may evolve into anEMRI.
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Generating waveforms
Numerical kludge
Figure: A kludge trajectory.
Use semi-relativistic approximation toproduce waveforms: assume particlefollows exact Kerr geodesic, but useflat space radiation formula.
This is inconsistent, but waveformsare accurate to a few percent.
Also ignores radiative effects.
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Generating waveforms
Parabolic orbits
xflat/rg
yflat/rg
z flat
/rg
−300−200
−1000
100200
300
−50
0
50
100
150
200
-40
-30
-20
-10
0
10
Figure: Trajectory plotted in flat spacetime about Kerr black hole.
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Generating waveforms
Quadrupole-octupole formula
At periapsis, velocities can be highly relativistic: include higher order termsin radiation formula (e.g. Yunes et al. 2008)
hjk(t, x) = −2G
c6r
(I jk − 2ni S
ijk + ni
...M
ijk)
t′ = t−r/c, (2)
where
I jk = c2µx jxk ; (3)
S ijk = cµvix jxk ; (4)
M ijk = cµx ix jxk . (5)
Maximal difference is about 10%.
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Generating waveforms
Example waveform
f /Hz
h(f
)/H
z−1
10−6 10−5 10−4 10−3 10−2 10−1 10010−22
10−21
10−20
10−19
10−18
10−17
10−16
10−15
10−14
10−13
Figure: Frequency domain burst waveform. Periapse rp ≃ 11.8M•.
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Signal analysis Detectability
Signal-to-noise ratio
log10 (rp/M•)
log
10(ρ
)
1.0 1.2 1.4 1.6 1.8 2.0−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Figure: SNR, ρ =√
(h|h), vs periapsis. Approximate scaling ρ ∝ (rp/M•)−2.56
.
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Signal analysis Parameter Estimation
Parameter set
λa Description
M• MBH massa∗ MBH spinζ = R0/µ Distance to Galactic centre divided by compact object massLz Orbital angular momentumQ Carter constantΘK, ΦK Orientation of MBH spintp Time of periapsisχp, φp Orbital phase at periapsis
Table: Waveform input parameters.
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Signal analysis Parameter Estimation
Likelihoods
The likelihood of the parameters λ0 is
p(s(t)|λ0) ∝ exp
[−
1
2(s − h0|s − h0)
]. (6)
In the high SNR limit this may be approximated as
p(s(t)|λ) ∝ exp
[−
1
2(∂ahML|∂bhML) (λa − λa
ML)(λb − λb
ML
)]. (7)
The Fisher information matrix is
Γab = (∂ah|∂bh) . (8)
Check the high SNR approximation using the maximum-mismatch criterion(Vallisneri 2008)
log r = −1
2
(λa∂ah − ∆h
∣∣∣λb∂bh − ∆h
). (9)
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Signal analysis Parameter Estimation
MCMC Results
ΘK
ΦK
Sam
ple
s
1.042 1.044 1.046 1.048 1.0500.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5 × 104
−0.792
−0.790
−0.788
−0.786
−0.784
−0.782
−0.780
−0.778
−0.776
−0.774
Figure: Posterior for MBH orientation with rp ≃ 3.29M•, ρ ≃ 28800.
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Signal analysis Parameter Estimation
MCMC Results
M•/M⊙
a∗
Sam
ple
s
4.285 4.290 4.295 4.300 4.305 4.310 4.315×106
0
2
4
6
8
10 × 104
0.693
0.694
0.695
0.696
0.697
0.698
0.699
0.700
0.701
0.702
Figure: Posterior for mass and spin.
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Signal analysis Parameter Estimation
MCMC Results
M•/M⊙
(R0/µ
)/(k
pc
M−
1⊙
)
Sam
ple
s
4.285 4.290 4.295 4.300 4.305 4.310 4.315×106
0
1
2
3
4
5 × 104
0.8300
0.8305
0.8310
0.8315
0.8320
0.8325
0.8330
0.8335
0.8340
Figure: Posterior for mass and distance parameter.
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Event rates
Frequency
Population of black holes and neutron stars in the Galactic centre should beenhanced by mass segregation.
Event rate critically depends upon inner cut-off radius.
Rubbo, Holley-Bockelmann & Finn (2006) estimated ≈ 15 yr−1.
Hopman, Freitag & Larson (2007) estimated ≈ 1 yr−1.
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Conclusion
Summary
EMRBs from the Galactic centre could be detectable with a space-bornedetector.
If detected a single burst could give an accurate measure of the MBH’sproperties. More work is required to characterise this.
Further work is required to calculate how likely such an event would be.
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Energy fluxes
rp/rg
Ener
gy
rati
o
Eoct/Epert
Equad/Epert
Equad/Eoct
EKepler/Equad
0 5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
1.2
2π/∆φ
Ener
gy
rati
o
Eoct/Epert
Equad/Epert
Equad/Eoct
EKepler/Equad
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.2
0.4
0.6
0.8
1.0
1.2
Figure: Ratios of energies calculated in different ways versus periapse and amountof rotation. 2π/∆φ = 1 for a Keplerian orbit, 2π/∆φ ≪ 1 for a zoom-whirl orbit.
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MCMC Results
M•/M⊙
Sam
ple
s
4.280 4.285 4.290 4.295 4.300 4.305 4.310 4.315 4.320×106
0
2
4
6
8
10
12
14
16
Figure: Marginalised posterior for mass. True value M• = 4.31 × 106M⊙.
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MCMC Results
a∗
Sam
ple
s
0.692 0.694 0.696 0.698 0.700 0.702 0.704 0.7060
2
4
6
8
10
12
14
16
Figure: Marginalised posterior for spin. True value a∗ = 0.7.
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MCMC Results
ζ/(kpc M −1⊙ )
Sam
ple
s
0.829 0.830 0.831 0.832 0.833 0.834 0.835 0.8360
5
10
15
Figure: Marginalised posterior for distance-compact object mass ratio. True valueζ = 0.833 kpc M −1
⊙ .
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MCMC Results
Lz/M•
Sam
ple
s
3.130 3.135 3.140 3.145 3.150 3.1550.0
0.5
1.0
1.5
2.0
2.5
Figure: Marginalised posterior for angular momentum. True value Lz = 3.14M•.
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MCMC Results
Q/M 2•
Sam
ple
s
0.74 0.76 0.78 0.80 0.82 0.84 0.860
2
4
6
8
10
12
14
16
18
Figure: Marginalised posterior for Carter constant. True value Q = 0.789M•.
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MCMC Results
ΘK
Sam
ple
s
1.040 1.042 1.044 1.046 1.048 1.050 1.052 1.0540.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Figure: Marginalised posterior for MBH orientation. True valueΘK = π/3 ≃ 1.047.
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MCMC Results
ΦK
Sam
ple
s
−0.795 −0.790 −0.785 −0.780 −0.775 −0.770 −0.7650
2
4
6
8
10
12
14
16
18
Figure: Marginalised posterior for MBH orientation. True valueΦK = −π/4 ≃ −0.785.
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MCMC Results
tp/yr
Sam
ple
s
−4 −3 −2 −1 0 1 2 3 4 5×10−9
0.0
0.5
1.0
1.5
2.0
2.5
Figure: Marginalised posterior for periapsis time. True value tp = 0 yr.
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MCMC Results
χp
Sam
ple
s
4.700 4.705 4.710 4.715 4.720 4.725 4.7300.0
0.5
1.0
1.5
2.0
2.5
Figure: Marginalised posterior for periapsis position. True valueχp = 3π/2 ≃ 4.712.
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