Extreme-mass-ratio Gravitational Wave Bursts From

The Galactic Centre

Supervisor: Jonathan Gair

Christopher Berrycplb2@cam.ac.uk

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)

Christopher Berry (IoA) EMRBs From The GC February 2012 5 / 21

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.

Christopher Berry (IoA) EMRBs From The GC February 2012 6 / 21

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.

Christopher Berry (IoA) EMRBs From The GC February 2012 7 / 21

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.

Christopher Berry (IoA) EMRBs From The GC February 2012 9 / 21

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.

Christopher Berry (IoA) EMRBs From The GC February 2012 21 / 21

Thank you

Christopher Berry (IoA) EMRBs From The GC February 2012 22 / 21

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

⊙ .

Christopher Berry (IoA) EMRBs From The GC February 2012 26 / 21

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•.

Christopher Berry (IoA) EMRBs From The GC February 2012 28 / 21

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.

Christopher Berry (IoA) EMRBs From The GC February 2012 30 / 21

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.

Christopher Berry (IoA) EMRBs From The GC February 2012 31 / 21

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.

Christopher Berry (IoA) EMRBs From The GC February 2012 32 / 21

MCMC Results

φp

Sam

ple

s

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5×10−3

0.0

0.5

1.0

1.5

2.0

2.5

Figure: Marginalised posterior for periapsis position. True value φp = 0.

Christopher Berry (IoA) EMRBs From The GC February 2012 33 / 21