Autoregressive Approach to Extract
Ring-down Gravitational Wave of Black-hole Merger
Hisa-aki Shinkai (Osaka Inst. Technology)
真貝寿明（大阪工業大学）

MergerRingdownInspiral

BH quasi-normal modes⇦ BH perturbation theory⇨ (M, a)strongest gravity we can observe⇨ test of gravity theories

We propose new method, which does not use any templates.

Autoregressive Approach to Extract Ring-down GW of BH Merger

Ideal vs Reality(Theory vs Data Analysis)
GW150914(S/N=23.7)

h(t)

16.25

16.45

challenging for data analysis
GW data is with noisesignal quickly decays(M=60Msun, a=0.75 ̶> 300Hz, tau = 3 ms)
Hisaaki Shinkai (Osaka Inst. Tech.)

2018/07/03 Marcel Grossmann 15 @ Rome2

Autoregressive Approach to Extract Ring-down GW of BH Merger

Mock data example (0) : QNM extraction contest

modiﬁed ringdown signals from GR
with LIGO detectorʼs noise

Hisaaki Shinkai (Osaka Inst. Tech.)

1. Standard Matched-ﬁltering method
2. Improved Matched-ﬁltering method
3. Hilbert-Huang transformation method
4. Auto-Regressive method
5. Neural network method

2018/07/03 Marcel Grossmann 15 @ Rome3

Autoregressive Approach to Extract Ring-down GW of BH Merger

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

e.g. xn = Ae rn t cos(!n t)
Z1 = 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 (Osaka Inst. Tech.)

2018/07/03 Marcel Grossmann 15 @ Rome!4

Autoregressive Approach to Extract Ring-down GW of BH Merger

1. 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 (Osaka Inst. Tech.)

・・・・・

・・・・・

2018/07/03 Marcel Grossmann 15 @ Rome!5

Autoregressive Approach to Extract Ring-down GW of BH Merger

Auto-Regressive modelvs Short FFTsegment = 1/128 sec = 32 points

h(t)shift = 1/1024 sec = 4 points
The order M can be ﬁxed at 2〜8.

P(f)
FFT

Even for short segment,AR model shows precise
power-spectrum.

ARfreq.
Hisaaki Shinkai (Osaka Inst. Tech.)[mock data, SNR=40, inspiral part]2018/07/03 Marcel Grossmann 15 @ Rome

6

Autoregressive Approach to Extract Ring-down GW of BH Merger

1. 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 spectrumHisaaki Shinkai (Osaka Inst. Tech.)characteristic eq.

2018/07/03 Marcel Grossmann 15 @ Rome!7

Autoregressive Approach to Extract Ring-down GW of BH Merger

Mock data example (1) ﬁtting well
Mock Data (Nakano02 p)

freq.

h(t)

spectrogramx original data, ̶ ﬁtted

8"mockN02p_segmentH.txt""mockN02p_segmentR.txt"

64
20-2-4-6
1.593 1.594 1.595 1.596 1.597 1.598 1.599

1.6

1.601 1.602

Fitting data with linear func.

a.1a.2a.3
Hisaaki Shinkai (Osaka Inst. Tech.)

2018/07/03 Marcel Grossmann 15 @ Rome

= -2.235e+00= 1.869e+00= -5.545e-01
8Autoregressive Approach to Extract Ring-down GW of BH Merger

Mock data example (2) spectrogram
Mock Data (Nakano02 p)

freq.

spectrogram

power spectrum

Hisaaki Shinkai (Osaka Inst. Tech.)

2018/07/03 Marcel Grossmann 15 @ Rome

9

Autoregressive Approach to Extract Ring-down GW of BH Merger

Mock data example (3) characteristic eq.
Mock Data (Nakano02 p)

Fitting data with linear func.

a.1
a.2
a.3

characteristic eq.

= -2.235e+00= 1.869e+00= -5.545e-01

xn 1 = zxn<latexit sha1_base64="N6OULxLyV4n+YDLj4gzs3pFIM1g=">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</latexit>z = exp[ 2⇡if t]<latexit sha1_base64="qLlUL+TuUzXdGJgBPG/gNGBc+b0=">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</latexit>

123x.r
0.962
0.962
1.447

Hisaaki Shinkai (Osaka Inst. Tech.)

x.i
0.566
-0.5660.000

f_R[Hz]346.800-346.800
0.000|x|f_I[Hz]8.025e-01 71.721
8.025e-01 71.721
4.775e-01 240.931

2018/07/03 Marcel Grossmann 15 @ Rome10Autoregressive Approach to Extract Ring-down GW of BH Merger

Mock data example (4) identify ring-down freq.
Mock Data (Nakano02 p)

characterestic eq.

xn 1 = zxn<latexit sha1_base64="N6OULxLyV4n+YDLj4gzs3pFIM1g=">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</latexit>z = exp[ 2⇡if t]<latexit sha1_base64="qLlUL+TuUzXdGJgBPG/gNGBc+b0=">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</latexit>

12
3x.r
0.962
0.962
1.447

x.i
0.566
-0.5660.000

f_R[Hz]346.800-346.800
0.000|x|f_I[Hz]8.025e-01 71.721
8.025e-01 71.721
4.775e-01 240.931t
f_R (z_plane) f_I (z_plane) f_Rh(spectr) f_Rmax(spectr)
0.159375E+010.363837E+030.280414E+020.340000E+030.363000E+030.159668E+010.344258E+030.166608E+02
0.331000E+03
0.344000E+03
0.159766E+01
0.346800E+03
0.717212E+020.240000E+03
0.329000E+030.161230E+010.357677E+030.122067E+03
0.213000E+03
0.338000E+030.161328E+010.361098E+03
0.948919E+020.261000E+030.350000E+030.161523E+010.379918E+030.772796E+020.304000E+030.373000E+03
average & variance zfr =0.359E+03
0.118E+02 fr(sp) =0.350E+03
average & variance zfi =0.684E+02
0.365E+02
Hisaaki Shinkai (Osaka Inst. Tech.)

2018/07/03 Marcel Grossmann 15 @ Romef_Rh(spectr)
0.384000E+03
0.357000E+03
0.382000E+03
0.431000E+03
0.422000E+03
0.432000E+03
0.148E+02

11

Autoregressive Approach to Extract Ring-down GW of BH Merger

Application to the LIGO/Virgo data

List of Detected GW eventsref.
GW150914
LVT151012
GW151226
GW170104
GW170608
GW170814
GW170817M1+M2=Mf,

spin

Mpc

Mdiﬀ/Mtotala̲ﬁnalz

PRL116, 061102

36.2+29.1=62.3+3.0(2016/2/11)

4.59%(2016/2/11)

23+13=35+1.5
2.78%

PRL116, 241103

14.2+7.5=20.8+0.9(2016/6/15)

4.15%

PRL118, 221101

31.2+19.4=48.7+1.9(2017/6/1)

3.75%

ApJ 851, L35

12+7=18.0+1.0(2017/12/18)

5.2%

PRL119,141101

30.5+25.3=53.2+2.6(2017/10/6)

4.66%

PRL119, 161101

1.36〜1.60 + 1.17〜1.36(2017/10/16)= 2.74 + ?

Hisaaki Shinkai (Osaka Inst. Tech.)

0.680.66

0.74

0.64

0.69

0.70?

2018/07/03 Marcel Grossmann 15 @ Rome

410Mpc0.09
1000Mpc
0.20
440Mpc0.09880Mpc
0.18
340Mpc0.07540Mpc
0.1140Mpc

SNRdeg^2

23.7

600

9.7

13.0

850

13

1300

13520

1860

32.4

28!12

Autoregressive Approach to Extract Ring-down GW of BH Merger

Ringdown wave of GW150914

LIGO paper

4096 sampling rate
100-400 Hz ﬁlter
1 segment = 1/64 sec= 64 points
1 shift = 1/512 sec = 8 points

AR modelHanfordP(f)max M = 3
Hisaaki Shinkai (Osaka Inst. Tech.)

2018/07/03 Marcel Grossmann 15 @ Romefreq.

Autoregressive Approach to Extract Ring-down GW of BH Merger

Ringdown wave of GW150914

AR model16384 sampling rate

Hanford4096 sampling rate
100-400 Hz ﬁlter
1 segment = 1/64 sec= 64 points
1 shift = 1/512 sec = 8 points

AR modelHanfordP(f)max M = 3
Hisaaki Shinkai (Osaka Inst. Tech.)

2018/07/03 Marcel Grossmann 15 @ Romefreq.

Autoregressive Approach to Extract Ring-down GW of BH Merger

Ringdown wave of GW150914

AR model16384 sampling rate

Livingston

4096 sampling rate100-400 Hz ﬁlter
1 segment = 1/64 sec= 64 points
1 shift = 1/512 sec = 8 points

AR modelLivingstonmax M = 3
Hisaaki Shinkai (Osaka Inst. Tech.)

2018/07/03 Marcel Grossmann 15 @ RomeAutoregressive Approach to Extract Ring-down GW of BH Merger

Ringdown wave of GW150914
tf realf imagWe see QNM at 300Hz, 0.003s after the merger.

Berti, Cardoso & Will PRD 73, 064030 (2006).

LIGO paper says
Hisaaki Shinkai (Osaka Inst. Tech.)

2018/07/03 Marcel Grossmann 15 @ RomeAutoregressive Approach to Extract Ring-down GW of BH Merger

Ringdown wave of GW150914, GW170814, GW170104

Hisaaki Shinkai (Osaka Inst. Tech.)

2018/07/03 Marcel Grossmann 15 @ RomeAutoregressive Approach to Extract Ring-down GW of BH Merger

Summary & Outlook
New method for extracting GW signal from x(t)It works for short segment data (〜 30 pts), accurate freq.
It does not require any templates for ﬁnding signals.
Applied to 3 real events, for extracting ring-down part.
Obtained similar (M, a) with LIGO/Virgo-papers for GW150914, GW170814
But not for GW170104. (S/N=13)(S/N=23.7)(S/N=18)

By combining with other methods, we expect to test the theories of gravity
using the ring-down part of BH merger.
We hope template-independent wave extraction method will contribute toﬁnd gravitational wave from unknown sources.
Hisaaki Shinkai (Osaka Inst. Tech.)

2018/07/03 Marcel Grossmann 15 @ Rome