Auto-Regressive analysis of GWTC-3 Ringdown data
MergerRingdown
BH quasi-normal modes⇦ BH perturbation in GR
Inspiral⇨ (M, a)Strong Gravity Effect⇨ Reasonable for testing GR“Ringdown part” is quite short (3.7 ms for 60 Msun, a=0.75)
AR analysis enables to ﬁnd out waveform without template

Hisaaki Shinkai (OIT)
真貝寿明（大阪工大）
http://www.oit.ac.jp/is/shinkai/
1
2022/1/23 A01
Boot CampStatus of GW observation

Observation 1/2/3a/3b & 4
2015
Sep12

LIGO

2016

2017

Jan19

Aug25

O1
60-80 Mpc

Nov30

2018

2020

Apr1̶Sep 30

O2

O3a

60-100 Mpc
Aug 1-25

Virgo

2019

O3a

O4

terminated due
to COVID-19

O3b

Apr1̶Sep 30

30 Mpc2022

Nov1̶ Mar27

120 - 130 Mpc

O2

2021

Nov1̶ Mar27

O4

O3b

50 Mpc

KAGRA

O4
1 Mpc

Gravitational Wave Transient Catalog

Planned to startfrom late Dec.

GWTC-1
2018/12/3

GWTC-2
2020/10/28
GWTC-2.1
2021/8/2
GWTC-32021/11/5

https://www.gw-openscience.org

3

重力波観測の現状

O1 (2015/9/12 - 2016/1/19)

3 BHBH
GW150914: the first ever detection of gravitational waves from the merger of two black holes more than a billion light years away
https://media.ligo.northwestern.edu/gallery/mass-plot

Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot Camp

4

重力波観測の現状

O2 (2016/11/30 - 2017/8/25)

After O2：GWTC1 (2018/12/3 released)• GW170814: the first GW signal measured by the three-detector network, also from a binary black hole

10 BHBH•

1 NSNS(BBH) merger;
GW170817: the first GW signal measured from a binary neutron star (BNS) merger — and also the firstevent observed in light, by dozens of telescopes across the entire electromagnetic spectrum.
https://media.ligo.northwestern.edu/gallery/mass-plot

Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot Camp

5

重力波観測の現状

O3a (2019/4/1 - 2019/9/30)

•
•
•
•
•
•
•
•

After O3a：GWTC2 (2020/10/28 released)

GW190412: the first BBH with definitively asymmetric component masses, which also shows evidence for higher
harmonics
GW190425: the second gravitational-wave event consistent with a BNS, following GW170817
GW190426_152155: a low-mass event consistent with either an NSBH or BBH
GW190514_065416: a BBH with the smallest effective aligned spin of all O3a events
GW190517_055101: a BBH with the largest effective aligned spin of all O3a events
GW190521: a BBH with total mass over 150 times the mass of the Sun
GW190814: a highly asymmetric system of ambiguous nature, corresponding to the merger of a 23 solar mass
black hole with a 2.6 solar mass compact object, making the latter either the lightest black hole or heaviest
neutron star observed in a compact binary
GW190924_021846: likely the lowest-mass BBH, with both black holes exceeding 3 solar masses

Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot Camp

46 BHBH
2 NSNS
2 BH+?

6

重力波観測の現状

O3b (2019/11/1 - 2020/3/27)

After O3b：GWTC3 (2021/11/7 released)

Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot CampAuto-Regressive analysis of GWTC-3 ringdown data

LV PRX 9, 031040 (2019) arXiv:1811.12907

GWTC-1 (O1/O2)

Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot Camp

7

Auto-Regressive analysis of GWTC-3 ringdown data

LV PhysRevX 11 (2021) 021053 arXiv:2010.14527

GWTC-2 (O3a)

Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot Camp

8

Auto-Regressive analysis of GWTC-3 ringdown data

GWTC-2 (O3a)

LV PhysRevX 11 (2021) 021053 arXiv:2010.14527

continued

Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot Camp

9

Auto-Regressive analysis of GWTC-3 ringdown data

LV PhysRevX 11 (2021) 021053 arXiv:2010.14527 10

GWTC-2 (O3a)

Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot CampAuto-Regressive analysis of GWTC-3 ringdown data

GWTC-2.1 (O3a)

Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot CampLV arXiv:2108.01045

11

Auto-Regressive analysis of GWTC-3 ringdown data

GWTC-3 (O3b)

Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot CampLVK arXiv:2111.03606

12

Auto-Regressive analysis of GWTC-3 ringdown data

GWTC-3 (O3b)

LVK arXiv:2111.03606

continued

Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot Camp

13

Auto-Regressive analysis of GWTC-3 ringdown data

GWTC-3 (O3b)

Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot CampLVK arXiv:2111.03606

14

Auto-Regressive analysis of GWTC-3 ringdown data

Independent Ring-down Search (Mockdata challenge)

ringdown search
60 mockdata

matched filtering
Hilbert-Huan Transformation
Auto-Regression Method
Neural Network method
Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot Camp

15

Auto-Regressive analysis of GWTC-3 ringdown data

Auto-Regressive model (Method, general) I
Fitting data with linear func.

rn t
x
=
Ae
cos(!n t)e.g.
nZ1 = e (r j!) t
Z2 = e (r+j!) t
1.0

0.8

0.6

0.4

0.5

0.2

5

10

15

20

25

30
5

10

15

-0.2

-0.4
-0.5

-0.6

can be applied also to noisy data by adjusting M
Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot Camp

16

Auto-Regressive analysis of GWTC-3 ringdown data

Auto-Regressive model (Method, general) II
Fitting data with linear func.

• find aj (Burg method)• find M (FPE final prediction error method)• re-construct wave signal from fitted function• apply FFT with arbitrary precision.
power spectrum

・・・・・

・・・・・

Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot Camp17

Auto-Regressive analysis of GWTC-3 ringdown data

Auto-Regressive model vs Short FFT
sampling rate=4096

segment = 1/64 sec = 64 points

h(t)shift = 1/512 sec = 8 points
The order M can be fixed at 2〜8.

P(f)
FFT

Even for short segment,
AR model shows precise powerspectrum.

AR
freq.

[mock data, SNR=40, inspiral part]
Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot Camp18

Auto-Regressive analysis of GWTC-3 ringdown data

Auto-Regressive model (Method, general) III
Fitting data with linear func.

• find aj (Burg method)• find M (FPE final prediction error method)• re-construct wave signal from fitted function• apply FFT with arbitrary precision.
power spectrum

characteristic eq.

Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot Camp

19

20

Auto-Regressive analysis of GWTC-3 ringdown data

Procedures
Data ̶> Ringdown partSpecify (freq., damping rate) by AR method
for each segment, a couple of combinations are available.

would be possible to ﬁnd multi-modes

Look for the segments with constant frequencies.
Convert (freq., damping rate) to (M̲ﬁnal, a̲ﬁnal) using GR formula.
(Berti-Cardoso-Will). (Uses the redshift z by LVK catalog.)
check if three data (Hanford, Livingston, Virgo)
Spin aconverges

GW190828_063405

Kerr parameter a
1.0

0.8

Compare with the value (M̲ﬁnal, a̲ﬁnal) by LVK catalog.

LVK paper

0.6(AR) Hanford
0.4

check if the analysis is consistent with GRor having a shift
Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot Camp(AR) Livingston(AR) Virgo

0.2

20

40

60

Mass M

80

100

Mass

21

Auto-Regressive analysis of GWTC-3 ringdown data

GW150914

LIGO paper

freq [Hz]after whitening

before whitening▲mergertime

AR modelHanford4096 sampling rate
150-450 Hz filter
1 segment = 1/64 sec= 64 points
1 shift = 1/512 sec = 8 points

freq.

22

Auto-Regressive analysis of GWTC-3 ringdown data

GW150914

ASD [1/√Hz]

Livingston (SNR=14.2)

ASD [1/√Hz]

Hanford (SNR=20.6)

f [Hz]

f [Hz]

f [Hz]

f [Hz]

time

time

23

Auto-Regressive analysis of GWTC-3 ringdown data

GW150914
Hanford (SNR=20.6)

LV paper ▶︎+0.05(M, a) = (63.1+3.4
,
0.69
3.0
0.04 )
f220 = 271.8 Hz, f221 = 266.0 Hz, f222 = 254.7 Hz

H100_SpectrogramAR

fQNM▶︎

f210 = 380.7 Hz, f211 = 225.7 Hz, f200 = 252.8 Hz
f330 = 430.9 Hz, f331 = 427.4 Hz, f332 = 421.1 Hz
f320 = 387.9 Hz, f310 = 351.1 Hz, f300 = 320.3 Hz
GW150914

f_imag (Hz)100f̲imag

LV paper(AR) Hanford(AR) Livingston

80

60

40

20

Livingston (SNR=14.2) L100_SpectrogramARf̲real
100

200

300

400

500

f_real (Hz)GW150914
Kerr parameter a
1.0

0.8

Kerr
param

0.6

0.4

0.2

LV paper(AR) Hanford(AR) Livingston

Mass
20

40

60

80

100

Mass

24

Auto-Regressive analysis of GWTC-3 ringdown data

GW150914
GW150914

Hanford (SNR=20.6)
Livingston (SNR=14.2)

Kerr parameter a
1.0

Kerr
param

0.8

0.6

0.4

0.2

20

LV paper(AR) Hanford(AR) Livingston
40

60

80

100

Mass

GW170809

Mass
Kerr
param

LV paper(AR) Hanford(AR) Livingston(AR) Virgo

Hanford (SNR=5.9)
Livingston (SNR=10.7)
Virgo (SNR=1.1)

Mass

25

Auto-Regressive analysis of GWTC-3 ringdown data

Event List ordered by SNR

AR filtering N/A (due to small M)

AR filtering N/A (due to small M)
AR filtering N/A (due to large M)

tried in this study
list overlapped

https://www.gw-openscience.org/eventapi/html/allevents/

26

Auto-Regressive analysis of GWTC-3 ringdown data

Event List ordered by SNR (continued)

AR filtering N/A (due to small M)

AR filtering N/A (due to small M)
AR filtering N/A (due to small M)

GW190925
GW190910
GW190707

AR filtering N/A (due to small M)

https://www.gw-openscience.org/eventapi/html/allevents/

27

Auto-Regressive analysis of GWTC-3 ringdown data

+0.06
+0.05
(M, a, z) = (60.3+4.
,
0.73
,
0.18
3.3
0.06
0.07 )<latexit sha1_base64="nam+QkRHLWt55DoaOJ80ia2jOco=">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</latexit>

GW200129̲065458

LVK paper

Hanford

Network SNR=26.8

Expected fQNM
(detector frame)

f220 = 293.8 Hz, f221 = 288.4 Hz, f222 = 277.8 Hz<latexit sha1_base64="PnT9zr/BDcRs3ktAuor2Tmw1kBU=">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</latexit>

f210 = 394.1 Hz, f211 = 240.9 Hz, f200 = 263.3 Hz
Livingston

Virgo

f330 = 464.9 Hz, f331 = 461.7 Hz, f332 = 455.8 Hz
f320 = 414.4 Hz, f310 = 371.8 Hz, f300 = 336.8 Hz

28

Auto-Regressive analysis of GWTC-3 ringdown data

+0.06
+0.05
(M, a, z) = (60.3+4.
,
0.73
,
0.18
3.3
0.06
0.07 )<latexit sha1_base64="nam+QkRHLWt55DoaOJ80ia2jOco=">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</latexit>

GW200129̲065458
Hanford

H1_SpectrogramAR

LVK paper

Network SNR=26.8

355Hz◀323Hz
300Hz

Livingston

L1_SpectrogramAR

303 Hz◀296 Hz
291 HzVirgo

V1_SpectrogramAR

224 Hz◀206 Hz
189 Hz29

Auto-Regressive analysis of GWTC-3 ringdown data

+0.06
+0.05
(M, a, z) = (60.3+4.
,
0.73
,
0.18
3.3
0.06
0.07 )<latexit sha1_base64="nam+QkRHLWt55DoaOJ80ia2jOco=">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</latexit>

GW200129̲065458
Hanford

H1_SpectrogramAR

LVK paper

Network SNR=26.8

355Hz◀323Hz

252Hz

300Hz

◀242Hz234Hz

Livingston

L1_SpectrogramAR

303 Hz◀296 Hz
291 Hz208 Hz◀192 Hz183 HzVirgo

V1_SpectrogramAR

224 Hz◀206 Hz
189 Hz229 Hz◀212 Hz201 Hz30

Auto-Regressive analysis of GWTC-3 ringdown data

+0.06
+0.05
(M, a, z) = (60.3+4.
,
0.73
,
0.18
3.3
0.06
0.07 )<latexit sha1_base64="nam+QkRHLWt55DoaOJ80ia2jOco=">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</latexit>

GW200129̲065458

LVK paper

Network SNR=26.8

Spin aSpin a
GW200129_065458

GW200129_065458

Kerr parameter a
1.0

Kerr parameter a
1.0

LVK paper(AR) Hanford(AR) Livingston(AR) Virgo

0.8

0.6

LVK paper(AR) Hanford(AR) Livingston(AR) Virgo

0.8

0.6

0.4

0.4

0.2

0.2

20

40

60

80

100

Mass

20

40

60

80

100

Mass

Mass M

Mass Mestimated from the merger time for 0.25 sec

estimated from 0.10 sec after the merger for 0.15 sec

31

Auto-Regressive analysis of GWTC-3 ringdown data

+0.18
+0.1
(M, a, z) = (54.9+7.2
,
0.61
,
0.38
4.3
0.19
0.15 )<latexit sha1_base64="z6CxwJ3+n6F6xV8IuM3bkvxc6ao=">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</latexit>

GW190828̲063405
Hanford

H1_SpectrogramAR

LV paper

Network SNR=15.182
369Hz◀354Hz344Hz
203Hz◀198Hz
195Hz

Livingston

L1_SpectrogramAR
425Hz◀407Hz389Hz
210Hz◀204Hz
200Hz

Virgo

V1_SpectrogramAR
379Hz◀366Hz355Hz
214Hz◀207Hz201Hz

32

Auto-Regressive analysis of GWTC-3 ringdown data

+0.18
+0.1
(M, a, z) = (54.9+7.2
,
0.61
,
0.38
4.3
0.19
0.15 )<latexit sha1_base64="z6CxwJ3+n6F6xV8IuM3bkvxc6ao=">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</latexit>

GW190828̲063405

LV paper

Network SNR=15.182
f220 = 213.6 Hz, f221 = 207.8 Hz, f222 = 196.8 Hz<latexit sha1_base64="YAkv2Gb1eLeybXah1tjQbfwzBik=">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</latexit>

Expected fQNM f210 = 323.2 Hz, f211 = 181.2 Hz, f200 = 212.2 Hz(source frame)

f330 = 339.7 Hz, f331 = 336.2 Hz, f332 = 329.7 Hz
f320 = 311.2 Hz, f310 = 286.2 Hz, f300 = 264.6 Hz
f220 = 294.8 Hz, f221 = 286.7 Hz, f222 = 271.6 Hz<latexit sha1_base64="RBcu4BfkeadqKy3pLVjOcksnkAI=">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</latexit>

Expected fQNM

f210 = 446.1 Hz, f211 = 250.1 Hz, f200 = 292.9 Hz(detector frame)

f330 = 468.8 Hz, f331 = 463.9 Hz, f332 = 455.0 Hz
f320 = 429.4 Hz, f310 = 394.9 Hz, f300 = 365.2 Hz

Spin aSpin a
GW190828_063405

Kerr parameter a
1.0

379Hz

GW190828_063405
Kerr parameter a
1.0

210Hz

◀366Hz

204Hz

355Hz

200Hz

0.8

0.8

369Hz

0.6

LVK paper(AR) Hanford(AR) Livingston(AR) Virgo

0.4

203Hz
0.2

214Hz

198Hz 207Hz
195Hz

20

◀354Hz

0.6

201Hz
40

60

Mass M

80

100

Mass

425Hz

344Hz

◀407Hz

0.4

389Hz
0.2

20

40

60

Mass M

80

100

Mass

33

Auto-Regressive analysis of GWTC-3 ringdown data

+0.07
+0.05
(M, a, z) = (59.0+4.8
,
0.69
,
0.23
3.9
0.07
0.07 )<latexit sha1_base64="FB8lBfafZBhUa73bX1+b3X0Nenc=">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</latexit>

GW200311̲115853
Hanford

LVK paper

H1_SpectrogramAR

363Hz◀350Hz
337Hz

206Hz

◀196Hz
191Hz

Livingston

L1_SpectrogramAR

357Hz◀341Hz
327Hz

190Hz

◀185Hz
180Hz

Virgo

V1_SpectrogramAR

34

Auto-Regressive analysis of GWTC-3 ringdown data

+0.07
+0.05
(M, a, z) = (59.0+4.8
,
0.69
,
0.23
3.9
0.07
0.07 )<latexit sha1_base64="FB8lBfafZBhUa73bX1+b3X0Nenc=">AAACrnichZHNTtRQFMf/VEQcBUbdmLhpnECGMDSnnTIfJCYkbHBhwtcMJDA0bb1AQ6dt2s4k0MwL8AIuXGlijHHBQ7jxBViw0TVhiYkbF552RgkL4DS959zfPf9zz73XClwnionOhqR7w/dHHow+zD16PDY+kX/ytBn5ndAWDdt3/XDTMiPhOp5oxE7sis0gFGbbcsWGdbCYrm90RRg5vrceHwai1Tb3PGfXsc2YkZF/XXxTMktH06+Kc3WFdpIZXan1jGS2rNR7JZmUSp0ZKVRNYeZTqpX7dO4/nZaNfIGUeq2i6RXOIKqqmpoGWlUv67LKJLUCBrbs5z9jG2/hw0YHbQh4iDl2YSLibwsqCAGzFhJmIUdOti7QQ461Hc4SnGEyPeBxj2dbA+rxPK0ZZWqbd3H5D1kpY5JO6Qtd0nf6Suf058ZaSVYj7eWQvdXXisCYOH6+9vtOVZt9jP0r1a09x9hFLevV4d6DjKSnsPv67tG7y7X51clkij7SBff/gc7oG5/A6/6yP62I1ffI8QP8u2X55qCpKWpF0Vf0wkJt8BSjeIGXKPJ9V7GAJSyjwfue4BQ/8FMiqSm1JKOfKg0NNM9wzaT9v7xNnos=</latexit>

GW200311̲115853

LVK paper

Network SNR=17.8
f220 = 236.3 Hz, f221 = 231.3 Hz, f222 = 221.5 Hz<latexit sha1_base64="D2UHzuOUtW3NI7AanN3M44ITXKA=">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</latexit>

Expected fQNM

f210 = 331.0 Hz, f211 = 196.3 Hz, f200 = 219.8 Hz(source frame)

f330 = 374.7 Hz, f331 = 371.6 Hz, f332 = 366.1 Hz
f320 = 337.3 Hz, f310 = 305.3 Hz, f300 = 278.5 Hz
f220 = 290.7 Hz, f221 = 284.5 Hz, f222 = 272.4 Hz<latexit sha1_base64="brvwrA5YlYisVl496ZCu6PtVgmg=">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</latexit>

Expected fQNM
(detector frame)

f210 = 407.2 Hz, f211 = 241.4 Hz, f200 = 270.4 Hz
f330 = 460.9 Hz, f331 = 457.1 Hz, f332 = 450.3 Hz
f320 = 414.9 Hz, f310 = 375.5 Hz, f300 = 342.6 Hz

Spin aSpin a206Hz

GW200311_115853

GW200311_115853◀196HzKerr parameter a
1.0

Kerr parameter a
1.0

357Hz

191Hz

◀341Hz

0.8

0.8

0.6

0.6

LVK paper(AR) Hanford(AR) Livingston(AR) Virgo

190Hz◀185Hz

0.4

180Hz
0.2

20

40

60

Mass M

80

100

Mass

327Hz
363Hz

◀350Hz
337Hz

0.4

0.2

20

40

60

Mass M

80

100

Mass

35

Auto-Regressive analysis of GWTC-3 ringdown data

+0.05
+0.1
(M, a, z) = (56.4+4.4
,
0.70
,
0.18
4.6
0.07
0.07 )<latexit sha1_base64="pIf64utp7ZmKPJiaxfUVZnyEGvg=">AAACrHichZHPTttAEMY/3EIhbSFtL5W4RI2ogkitcWriFAkJlUN7qcS/AFKaRra7oRaObdlOJGLlBXgBDpxAQgihPkUvvVc9RGofAHEEiQsHxk4o4gAdyzuzv51vdnbX8GwrCIm6A9KDh4NDj4ZHUo+fPB0dSz97vhq4Td8UZdO1XX/d0ANhW44oh1Zoi3XPF3rDsMWasTkfr6+1hB9YrrMSbnmi2tA3HKtumXrIqJb+kPuU1/PtydncdFFWv0RTqqx2atEbVS528hmSNWJGMk3HkL2WUKWUUOUfnKylsyS/KxULapHXiTSloMRBQVPfqhmFSWxZ9G3BTR/iM77ChYkmGhBwEHJsQ0fAXwUKCB6zKiJmPkdWsi7QQYq1Tc4SnKEz3eRxg2eVPnV4HtcMErXJu9j8+6zMYIJ+0xGd0U86phO6vLNWlNSIe9lib/S0wquNbb9cvvivqsE+xLcb1b09h6ijlPRqce9eQuJTmD19q71ztjyzNBG9pn065f73qEs/+ARO69w8WBRLu0jxA1zfcubuYLUgK/zOi2p27n3/KYYxjlfI8X1rmMNHLKDM+37HL/zBX0mWVqSKVO2lSgN9zQvcMql+BY7SniI=</latexit>

GW190630̲185205
Hanford

LV paper

H1_SpectrogramAR

Network SNR=15.6

GW190630_185205
Kerr parameter a
1.0

0.8

0.6

0.4

0.2

20

Livingston

40

60

80

100

Mass

L1_SpectrogramAR

298 Hz◀270Hz262 Hz206 Hz◀197 Hz187 HzVirgo

V1_SpectrogramAR◀ﬁlter36

Auto-Regressive analysis of GWTC-3 ringdown data

+0.06
+0.07
(M, a, z) = (60.8+5.3
,
0.71
,
0.24
4.3
0.06
0.08 )<latexit sha1_base64="Xm4dtJIY7ttStPAYzi4I3oaBJWQ=">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</latexit>

GW200112̲155838
Hanford

LVK paper

Network SNR=19.8

H1_SpectrogramAR
GW200112_155838
Kerr parameter a
1.0

0.8

0.6

0.4

0.2

Livingston

20

40

60

80

100

Mass

L1_SpectrogramAR

210 Hz◀193 Hz183 HzVirgo

V1_SpectrogramAR◀ﬁlter190 Hz◀185 Hz◀ﬁlter182 Hz37

Auto-Regressive analysis of GWTC-3 ringdown data

+0.08
+0.16
(M, a, z) = (75.8+8.5
,
0.70
,
0.28
8.6
0.07
0.1 )
<latexit sha1_base64="RZ/7tn99kW0wBQkVNadlJ/3tA9k=">AAACrHichZHPStxQFMY/U9vqWOvYboRuhg7KSKfhJMZJWiiIXbSbgv9GhXEckvSODWaSkGQGNMwL9AW66KoFkSI+hRv3pQvBPkDp0oIbF55kRsSF9YTcc+7vnu/cc++1AteJYqKTAene4P0HD4eGcyOPRh+P5cefrEZ+O7RF1fZdP1y3zEi4jieqsRO7Yj0IhdmyXLFmbb9N19c6Iowc31uJdwJRb5lbntN0bDNm1Mi/K30om+Xd6TclfVY2NpMXhjzbbSQvDbnSLRdI1okZyWSkkL2eUdXIqFLpUaU73cgXSX5lVFStwutEuqIqaaDq2oxWUJikVkTfFvz8PjbwET5stNGCgIeYYxcmIv5qUEAImNWRMAs5crJ1gS5yrG1zluAMk+k2j1s8q/Wpx/O0ZpSpbd7F5T9kZQGT9It+0Bkd0wH9oYtbayVZjbSXHfZWTyuCxtjnieXzO1Ut9jE+Xav+23OMJoysV4d7DzKSnsLu6Tu7X86WXy9NJlP0nf5y/9/ohI74BF7nn723KJa+IscPcHXLhduDVZUfTdYWteLcfP8phvAMz1Hi+9Yxh/dYQJX3PcRPnOK3JEsrUk2q91Klgb7mKW6Y1LwEtZaeNA==</latexit>

GW190910̲112807
Hanford

H1_SpectrogramAR

Livingston

L1_SpectrogramAR

LV paper

Network SNR=13.4

324 Hz◀314 Hz304 Hz174 Hz◀172 Hz169 HzVirgo

V1_SpectrogramAR◀ﬁlter167 Hz◀165 Hz◀ﬁlter163 Hz38

Auto-Regressive analysis of GWTC-3 ringdown data

+0.08
+0.16
(M, a, z) = (75.8+8.5
,
0.70
,
0.28
8.6
0.07
0.1 )
<latexit sha1_base64="RZ/7tn99kW0wBQkVNadlJ/3tA9k=">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</latexit>

GW190910̲112807

LV paper

Network SNR=13.4

Livingston

f220 = 228.1 Hz, f221 = 223.4 Hz, f222 = 214.2 Hz<latexit sha1_base64="v/cy88F7MS13ia1fzao04708kCU=">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</latexit>

Expected fQNM
(detector frame)

f210 = 316.1 Hz, f211 = 188.8 Hz, f200 = 210.2 Hz
f330 = 361.4 Hz, f331 = 358.6 Hz, f332 = 353.4 Hz
f320 = 324.5 Hz, f310 = 293.1 Hz, f300 = 267.0 Hz

Virgo

GW190910_112807
Kerr parameter a
1.0

174 Hz0.8◀172 Hz0.6

169 Hz0.4

167 Hz◀165 Hz0.2

163 Hz
20

40

60

80

100

Mass

39

Auto-Regressive analysis of GWTC-3 ringdown data

+0.06
+0.06
(M, a, z) = (41.1+3.9
,
0.67
,
0.29
2.8
0.07
0.1 )
<latexit sha1_base64="vzyPq3382X5DgGeyJ/oe41vUnJQ=">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</latexit>

GW190408̲181802
Hanford

H1_SpectrogramAR

LV paper

Network SNR=14.7
367Hz◀349Hz
337Hz

Livingston

L1_SpectrogramAR

388 Hz◀356 Hz334 HzVirgo

V1_SpectrogramAR358 Hz◀347 Hz337 Hz40

Auto-Regressive analysis of GWTC-3 ringdown data

+0.06
+0.06
(M, a, z) = (41.1+3.9
,
0.67
,
0.29
2.8
0.07
0.1 )
<latexit sha1_base64="vzyPq3382X5DgGeyJ/oe41vUnJQ=">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</latexit>

GW190408̲181802

LV paperHanford

Network SNR=14.7
Expected fQNM f220 = 410.9 Hz, f221 = 401.6 Hz, f222 = 383.5 Hz
f210 = 587.5 Hz, f211 = 343.3 Hz, f200 = 388.9 Hz(detector frame)<latexit sha1_base64="ZfrXEqv2h2IFAst0DhSRfR3haCI=">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</latexit>

f330 = 652.1 Hz, f331 = 646.4 Hz, f332 = 636.1 Hz
f320 = 589.7 Hz, f310 = 536.0 Hz, f300 = 490.8 Hz

Livingston

358 HzGW190408_181802
Kerr parameter a
1.0

367Hz

◀347 Hz ◀349Hz
337 Hz337Hz

388 Hz0.8

Virgo◀356 Hz334 Hz

0.6

0.4

0.2

20

40

60

80

100

Mass

Auto-Regressive analysis of GWTC-3 ringdown data

Summary & Outlook
AR model x(t)Effective for finding frequencies and damping rates of short-time data (〜 60 pts)．
Data-base finding method, no templates are required.
Applied to LIGO/Virgo O1/O2/O3a & LVK O3b: Try to extract ring-down parts
Would be available for S/N >= 15, but some parameter tunings are necessary.
(Time width [merger time + X ms], Band Filterings [150-450Hz] )Analysis of GWTC-3 events are ongoing.
AR results of (M_final, a_final) are around the values of LV (LVK), i.e GR.

Hisaaki Shinkai (OIT) 2022/01/23 A01 Boot Camp41