ska-japanシンポジウム2019@国立天文台ska-jp.org/skajpws2019/day3/3-14_yonemaru.pdf · 10...

SKA-Japanシンポジウム2019@国立天文台 !

宇宙ひもからの重力波バーストの探索

熊本大学　米丸直之

共同研究者：　George Hobbs (CSIRO)、黒柳幸子(名古屋大学)、　高橋慶太郎、他Parkes Pulsar Timing Arrayメンバー

構成

• 宇宙ひも起源の重力波バースト

• パルサー・タイミング・アレイによる　　　　　　　　　　　　　　　　　　　　　　　　重力波バーストの探索

• 宇宙ひものパラメーターへの制限

宇宙ひも起源の重力波バースト

宇宙ひも• 宇宙ひも … 初期宇宙に形成される一次元位相欠陥 - 非常に重く、長く、細く、速く動くひも状の天体 - 宇宙ひもの綿密度(質量)を決めるGμ= ひもの張力により特徴づけられる。 - 形成メカニズム : 初期宇宙の真空の相転移 or 超ひも由来

VQRQNQIKECN�FGHGEV

ᆅࣱƕᄊǕǔƱƖݣVQRQNQIKECN�FGHGEVƕưƖǔƜƱƕƋǔŵ

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* *�

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8

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H+H- μ

Horizon size

ループの形成• 宇宙ひもどうしが交差すると、ひもどうしが繋ぎ替わりループを形成する。 - 形成時のループサイズ : L ~α/ H (H : ハッブルスケール)

!

!

• ループ上を逆方向に伝播する波が衝突する際に”cusp”という特異点構造ができ、重力波が放射される。

Cosmic string？

loop

infinite string

infinite string becomes a loop by reconnection

strings emit gravitational waves especially from singular structures

kink cusp

loops lose energy and shrink by emitting gravitational waves and eventually evaporate

重力波バーストの波形• 波形(Damour & Vilenkin (2000))

!

!

• 波形の例 GW bursts from single cosmic strings 5

-1x10-15

-8x10-16

-6x10-16

-4x10-16

-2x10-16

0

53000 54000 55000 56000

MJD [days]

MJD [days]

Waveform

W = 1,000 days

[ns]

53000 54000 55000 56000

W = 4,000 days

53000 54000 55000 56000

W = 10,000 days

53000 54000 55000 56000 57000

W = 100,000 days

-15

-10

-5

0

5

10

15

53000 54000 55000 56000

Post-fit residual

W = 1,000 days

53000 54000 55000 56000

W = 4,000 days

53000 54000 55000 56000

W = 10,000 days

53000 54000 55000 56000 57000

W = 100,000 days

Figure 1. Waveforms (top panels) and post-fit timing residuals (bottom panels) induced by a GW burst from a cosmic string cusp withamplitude A

peak

= 10

�12 at MJD 55200. In the bottom panel, pulsar parameters such as the pulse period and spin-down rate are fitted.

each sky position, epoch and width, we perform a global fitfor A

1

and A2

while fitting for the parameters specific toeach pulsar simultaneously.

Firstly, we fit for the cosmic string amplitudes (A1

andA

2

) over 1034 sky positions, nine epochs (MJD 53000, 53500,54000, 54500, 55000, 55500, 56000, 56500 and 57000) and10 widths (20, 50, 100, 200, 500, 1000, 2000, 4000 and 8000days). Fig. 4 shows the maximum detection statistics in thesky as a function of widths for nine epochs, which only thestatistics for the epoch of MJD 53500 is described as the redline. As seen from this figure the detection statistics tendto be larger as width becomes narrower, and we find max-imum D of 31.2 for the width of 100 days at MJD 53500.As discussed in section ??, however, the detection statisticcaused by noise is typically less than ⇠10. We check the tim-ing residuals of all pulsars at that epoch and determined themeasured quadrupole pattern in the sky. Consequently, wefind a “bump” in the timing residuals of PSR J1939+2134shown in Fig. 5, and the measured quadruploe pattern isrepresented in Fig. 6. As can be seen from Fig. 6, D

max

of31.2 is mostly contributed by PSR J1939+2134 that im-plies there two probabilities: this detection statistic is gen-erated by a GW burst, or bump-like noise appearing in onlyPSR J1939+2134 is causing a false detection of the GWsignal. The timing residuals for PSR J1939+2134 is knownto be significantly a↵ected by red noise and the ToA preci-

sions are high meaning that it is challenging to get a precisered and white noise model for this pulsar. We therefore ex-pect that this is a“false-detection”caused by imperfect mod-elling of the PSR J1939+2139 residuals, but note that thisquadrupolar signature will imply a clear signal in the North-ern hemisphere IPTA pulsars that are not observed by thePPTA (we consider this further in the discussion section).As we assume that this is not a GW detection, we continueour analyse without the inclusion of PSR J1939+2134 in ourdata set.

After we removed PSR J1939+2134 from the sample, wereprocessed the data using a finer grid with 1034 sky posi-tions, 18 epochs and 23 widths. Fig. 7 shows the maximumdetection statistics in the sky as a function of widths. Inanalogy with Fig. 4, the detection statistics increase as thewidth decreases, but the maximum one is 19.8 in this case.Here, we discuss the statistical significance of the observeddetection statistics. Although there is no signal, the detec-tion statistic could be significant in the case the number oftrials m is large. The probability of judging an observableas a signal at least once in m trials in the absence of sig-nals, called the family-wise error rate (FWER), is given by1� (1� s)m, where s is the significance level. In this work, thenumber of trials m is 18 epochs ⇥ 23 widths = 414 and thecorresponding FWER is 0.9999999994 for s = 0.05, whichmeans that we “detect” a signal almost certainly by simply

MNRAS 000, 1–12 (2018)

h+(t) =

(Afit

h|t� t0|1/3 �

�12W

�1/3i(t0 � 1

2W t < t0 +12W )

0 (otherwise)

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W<latexit sha1_base64="ZOSgdiqMuEAh/8/KpdlZQSzgqpY=">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</latexit><latexit sha1_base64="ZOSgdiqMuEAh/8/KpdlZQSzgqpY=">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</latexit><latexit sha1_base64="ZOSgdiqMuEAh/8/KpdlZQSzgqpY=">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</latexit><latexit sha1_base64="ZOSgdiqMuEAh/8/KpdlZQSzgqpY=">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</latexit>: バーストの到来時刻 : バーストの継続時間 → ループサイズにより決まる

W = 1, 000 days<latexit sha1_base64="lr3mYOdwEbS3rGrSfrFhVm2ArAQ=">AAACeHichVHLSsNAFD2N7/qquhHcBIsvkHJTBUUQRDcufdUKrZQkTmswL5K0UIt+gD/gwoUoiIqf4cYfcOEniEsFQVx4mwZERb3DzJw5c8+dMzOaaxp+QPQQk5qaW1rb2jvinV3dPb2Jvv5N3yl7usjojul4W5rqC9OwRSYwAlNsuZ5QLc0UWW1vqb6frQjPNxx7I6i6YttSS7ZRNHQ1YKqQGMjK87IySUSHec+q7ahV/6CQSFKKwpB/AiUCSUSx4iQukccOHOgow4KAjYCxCRU+txwUEFzmtlFjzmNkhPsCB4iztsxZgjNUZvd4LPEqF7E2r+s1/VCt8ykmd4+VMkbonq7pme7ohh7p/ddatbBG3UuVZ62hFW6h92hw/fVflcVzgN1P1Z+eAxQxG3o12LsbMvVb6A19Zf/4eX1ubaQ2Suf0xP7P6IFu+QZ25UW/WBVrJ4jzByjfn/sn2EynlKlUenU6ubAYfUU7hjCMcX7vGSxgGSvI8LlVnOIK17E3SZbGpIlGqhSLNAP4ElL6A6QjkIw=</latexit>

W = 4, 000 days<latexit sha1_base64="NfpmlqIyui2EPNzHp+Q3z3E4k8o=">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</latexit>

W = 10, 000 days<latexit sha1_base64="RebY8IYQLSOV/Qxx5cC0hGnOlRc=">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</latexit>

W = 100, 000 days<latexit sha1_base64="BD+v4f3YZrJ267nBlLCHvYp4U9A=">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</latexit>

MJD [day]<latexit sha1_base64="8uU7clIthRjfseHjdFJXI6uITio=">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</latexit>

Timing Residual• 宇宙ひもからの重力波バーストによるpre-fit timing residual

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!

!• 重力波の解析の前にパルサーのパラメーター補正が行われる → post-fit timing residual = pre-fit timing residual 　　- best-fitの二次曲線

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residual [time]

パルサー・タイミング・アレイによる重力波バーストの探索

Parkes Pulsar Timing Array Data Release 2

• Parkes Pulsar Timing Array (Parkes PTA) … Parkes 64m電波望遠鏡を用いて2004年から多数のミリ秒パルサーを観測

• Parkes PTA Data Release 2 - 26個のミリ秒パルサー - 観測頻度：1回/2 ~ 3週間 - 観測期間：2004 ~ 2016 - 周波数：700 MHz, 1.4 GHz, 　　　　　3.1GHz

Detection Statistic• Detection Statistic (DS): - : 振幅の最尤値（+, ×モード） - : 共分散行列

• 各 - バーストの到来時刻 - バーストの継続時間 - 天球面上での位置に対して振幅をフィット、 DSを得る。

• 右図：PPTA DR2より得た天球面上で最大のDS

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A<latexit sha1_base64="BbOCOMXG4VWrtzOx/DSOOR1FIB0=">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</latexit><latexit sha1_base64="BbOCOMXG4VWrtzOx/DSOOR1FIB0=">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</latexit><latexit sha1_base64="BbOCOMXG4VWrtzOx/DSOOR1FIB0=">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</latexit><latexit sha1_base64="BbOCOMXG4VWrtzOx/DSOOR1FIB0=">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</latexit>

C<latexit sha1_base64="izycfcaM71nlqbF1p+/OueibW0w=">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</latexit><latexit sha1_base64="izycfcaM71nlqbF1p+/OueibW0w=">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</latexit><latexit sha1_base64="izycfcaM71nlqbF1p+/OueibW0w=">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</latexit><latexit sha1_base64="izycfcaM71nlqbF1p+/OueibW0w=">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</latexit>

6 Yonemaru et al.

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Detection Statistics

Pro

bab

ilit

y

MJD 53500

W = 100 days

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Pro

bab

ilit

y

MJD 53500

W = 100 daysW = 50 days

MJD 54000MJD 54000 MJD 54500MJD 54500 MJD 55000MJD 55000

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20

MJD 55500

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20

MJD 55500

5 10 15 20

MJD 56000

5 10 15 20

MJD 56000

5 10 15 20

MJD 56500

5 10 15 20

MJD 56500

5 10 15 20

MJD 57000

5 10 15 20

MJD 57000

Figure 2. Probability distributions of maximum D in the sky obtained from 100 realizations of simulated timing residuals. The red andblue lines show the distributions for the width of 100 and 50 days. (upper panels) epoch = MJD 53500, 54000, 54500 and 55000. (lowerpanels) epoch = MJD 55500, 56000, 56500 and 57000.

0

0.05

0.1

0.15

0.2

0 5 10 15 20 25 30 35 40

Pro

bab

ilit

y


t0 = MJD 54750, W = 75 dayst0 = MJD 53750, W = 150 days

Figure 3. Probability distributions of maximum D in the skyobtained from 1,000 realizations of simulated timing residuals.Red and blue lines represent the distributions for the epoch of54750 and 53750, and width of 75 and 150 days.

testing the significance for each detection statistic even ifthere is no signal. Then, we perform the Fisher’s producttest to combine each significance and test the global signif-icance of a set of the observed detection statistics. In theabsence of signals, the test statistics

F = �2

m’

i

ln pi (12)

follows a �2 distribution with 2m degrees of freedom underan assumption that the tests are independent, where pi isthe p-value. Although the observed detection statistics seemhave correlation with respect to the width, we assume that

0

8

16

24

32

20 100 1000 8000

Det

ecti

on

Sta

tist

ics

Width (day)

t0 = 53500 (MJD)other epoch

Figure 4. Detection statistics obtained from the PPTA DR2against widths for nine epochs. Solid and dashed ones show thestatistics for the epoch of MJD 53500 and other eight epochs.

the detection statistics are independent of the epoch andwidth, and all of their probability distributions are same ex-cept the epochs of MJD 54750 and 53750 since the simulateddistributions look similar as can be seen from Fig. 2. Underthese assumptions, we calculate p-values for the epochs ofMJD 54750 and 53750 using the simulated probability dis-tributions in Fig. 3 and for other epochs using the summeddistribution of Fig. 2. As a result, we obtain F = 751.9 andits p-value to the �2 distribution is 0.97. Consequently, weconclude that there is no significant event in our data. Notethat this conclusion is conservative because we utilize theprobability distributions for the widths of 100 and 50 daysto determine the p-values for other longer widths.

Using our value of Dmax

for each width, we can de-

MNRAS 000, 1–12 (2018)

32

16

24

8

020 100 80001000

D <latexit sha1_base64="8eSQWchmR3f/b2eGJyudXC45MVg=">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</latexit>

W<latexit sha1_base64="XpFfJ/6r8YH8aAQmDZAhQE8XhDE=">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</latexit>

t0 = MJD 53500<latexit sha1_base64="un/TdqFv+Ugl0791nCbyX1x0FXs=">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</latexit>

Detection Statistic• Detection Statistic (DS): - : 振幅の最尤値（+, ×モード） - : 共分散行列

• 各 - バーストの到来時刻 - バーストの継続時間 - 天球面上での位置に対して振幅をフィット、 DSを得る。

• 右図：PPTA DR2より得た天球面上で最大のDS

D = AC�1A<latexit sha1_base64="tcIxoS4jjNsEPeT94+7uj137oi4=">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</latexit><latexit sha1_base64="tcIxoS4jjNsEPeT94+7uj137oi4=">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</latexit><latexit sha1_base64="tcIxoS4jjNsEPeT94+7uj137oi4=">AAACgHichVHLSsNAFD2Nr1pfVTeCm2JR3FhvRFAKQn0sXPqqClZLEqc1mCYhSQs1dOHWH3DhSkGKuNFvcOMPuPATxKWCGxfepgFRUe8wM2fO3HPnzIxqG7rrET1GpJbWtvaOaGesq7unty/eP7DpWmVHE1nNMixnW1VcYeimyHq6Z4ht2xFKSTXElnq42NjfqgjH1S1zw6vaYrekFE29oGuKx1Q+PryUmEvk1II/X8u5BX+xtudPyLUmkY8nKUVBJH4COQRJhLFixevIYR8WNJRRgoAJj7EBBS63Hcgg2MztwmfOYaQH+wI1xFhb5izBGQqzhzwWebUTsiavGzXdQK3xKQZ3h5UJjNIDXdEL3dM1PdH7r7X8oEbDS5VntakVdr7vZGj97V9ViWcPB5+qPz17KGA28KqzdztgGrfQmvrK0enLenpt1B+jC3pm/+f0SHd8A7Pyql2uirUzxPgD5O/P/RNsTqVkSsmr08nMQvgVUQxjBOP83jPIYBkryPK5x6jjBreSJI1Lk5LcTJUioWYQX0JKfwDqCpNV</latexit><latexit sha1_base64="tcIxoS4jjNsEPeT94+7uj137oi4=">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</latexit>

A<latexit sha1_base64="BbOCOMXG4VWrtzOx/DSOOR1FIB0=">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</latexit><latexit sha1_base64="BbOCOMXG4VWrtzOx/DSOOR1FIB0=">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</latexit><latexit sha1_base64="BbOCOMXG4VWrtzOx/DSOOR1FIB0=">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</latexit><latexit sha1_base64="BbOCOMXG4VWrtzOx/DSOOR1FIB0=">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</latexit>

C<latexit sha1_base64="izycfcaM71nlqbF1p+/OueibW0w=">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</latexit><latexit sha1_base64="izycfcaM71nlqbF1p+/OueibW0w=">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</latexit><latexit sha1_base64="izycfcaM71nlqbF1p+/OueibW0w=">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</latexit><latexit sha1_base64="izycfcaM71nlqbF1p+/OueibW0w=">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</latexit>

6 Yonemaru et al.

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Pro

bab

ilit

y

MJD 53500

W = 100 days

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Pro

bab

ilit

y

MJD 53500



0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20

MJD 55500

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20

MJD 55500

5 10 15 20

MJD 56000

5 10 15 20

MJD 56000

5 10 15 20

MJD 56500

5 10 15 20

MJD 56500

5 10 15 20

MJD 57000

5 10 15 20

MJD 57000


0

0.05

0.1

0.15

0.2

0 5 10 15 20 25 30 35 40

Pro

bab

ilit

y





F = �2

m’

i

ln pi (12)


0

8

16

24

32

20 100 1000 8000

Det

ecti

on

Sta

tist

ics

Width (day)






MNRAS 000, 1–12 (2018)

DS = 31.2は有意なシグナル？

32

16

24

8

020 100 80001000


W<latexit sha1_base64="XpFfJ/6r8YH8aAQmDZAhQE8XhDE=">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</latexit>

t0 = MJD 53500<latexit sha1_base64="un/TdqFv+Ugl0791nCbyX1x0FXs=">AAACeXichVHLLgRBFD3T3uM1HguJzTAhYjG5Y0yIRCJYiETiNUiQSXcrdKZf6a6ZhAkf4AcsbJCI4DNs/ICFTxBLEgsW7vR0IghupapOnbrn1qkqzTUNXxI9RJSq6prauvqGaGNTc0trrK19xXcKni6yumM63pqm+sI0bJGVhjTFmusJ1dJMsarlp8r7q0Xh+YZjL8s9V2xa6o5tbBu6KpnKxTpljuLj8Q3PKs3NTh8cZtIZolwsQUkKIv4TpEKQQBjzTuwSG9iCAx0FWBCwIRmbUOFzW0cKBJe5TZSY8xgZwb7AAaKsLXCW4AyV2TyPO7xaD1mb1+WafqDW+RSTu8fKOPronq7ome7ohh7p/ddapaBG2csez1pFK9xc61HX0uu/Kotnid1P1Z+eJbYxGng12LsbMOVb6BV9cf/4eWlssa/UT+f0xP7P6IFu+QZ28UW/WBCLJ4jyB6S+P/dPsDKUTKWTQwvDiYnJ8Cvq0Y1eDPB7j2ACM5hHls/dxymucB15U3qUAWWwkqpEQk0HvoSS/gCk35B8</latexit>

DSのシミュレーション• PPTA DR2から得られるノイズモデルをもとにtiming residualをシミュレーション → 宇宙ひもからの重力波バーストをサーチ6 Yonemaru et al.

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Pro

bab

ilit

y

MJD 53500

W = 100 days

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Pro

bab

ilit

y

MJD 53500



0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20

MJD 55500

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20

MJD 55500

5 10 15 20

MJD 56000

5 10 15 20

MJD 56000

5 10 15 20

MJD 56500

5 10 15 20

MJD 56500

5 10 15 20

MJD 57000

5 10 15 20

MJD 57000


0

0.05

0.1

0.15

0.2

0 5 10 15 20 25 30 35 40

Pro

bab

ilit

y





F = �2

m’

i

ln pi (12)


0

8

16

24

32

20 100 1000 8000

Det

ecti

on

Sta

tist

ics

Width (day)






MNRAS 000, 1–12 (2018)

0 102010 10 102020 20Detection Statistic

DSのシミュレーション• PPTA DR2から得られるノイズモデルをもとにtiming residualをシミュレーション → 宇宙ひもからの重力波バーストをサーチ6 Yonemaru et al.

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Pro

bab

ilit

y

MJD 53500

W = 100 days

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Pro

bab

ilit

y

MJD 53500



0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20

MJD 55500

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20

MJD 55500

5 10 15 20

MJD 56000

5 10 15 20

MJD 56000

5 10 15 20

MJD 56500

5 10 15 20

MJD 56500

5 10 15 20

MJD 57000

5 10 15 20

MJD 57000


0

0.05

0.1

0.15

0.2

0 5 10 15 20 25 30 35 40

Pro

bab

ilit

y





F = �2

m’

i

ln pi (12)


0

8

16

24

32

20 100 1000 8000

Det

ecti

on

Sta

tist

ics

Width (day)






MNRAS 000, 1–12 (2018)

せいぜいDS = 15程度 → DS = 31.2はシグナル？

0 102010 10 102020 20Detection Statistic

J1939+2134の除外• 左図：J1939+2134のtiming residual … “くぼみ”の存在

• 右図：検出された四重極パターン … J1939+2134(+マーク)のみが強いシグナルを示す → 北天のパルサーを調べるしかない ⇒ International PTAへの予言

GW bursts from single cosmic strings 7

Figure 5. Timing residuals of J1939+2134 at around MJD 53500in the PPTA DR2. A bump-like structure can be seen.

-1

-0.5

0

0.5

1

0°

30°

-30°

60°

-60°

8h 4h 0° 20h 16h

Figure 6. Distribution of the measured quadrupole pattern in thesky and source position represented as the black circle in the equa-torial coordinates. Red and black crosses describe J1939+2134and the other PPTA pulsars.

termine the sensitivity of the data set to GW bursts as afunction of sky position. For a GW burst with amplitude,hCS,fit, but unknown polarization, the expected values arehA2

1

i = hA2

2

i = (hCS,fit)2/2 and hA1

A2

i = 0. Thus,

hDi =⇣

hCS,fit

⌘

2 (S11

+ S22

) /2 (13)

where S = C�1

0

. The GW burst amplitude corresponding toa detection threshold D⇤

max

is

hCS,fit =�

2D⇤max

/[S11

+ S22

]�0.5 . (14)

As mentioned in section 3, this has a dimension of sec�1/3,so that we make it dimensionless which is given by

hCS,peak =

✓

1

2

W◆

1/3hCS,fit. (15)

Here, C�1

0

can be obtained directly from fitting with usingthe PPTA DR2. As an example. we show the sensitivity map

0

2

4

6

8

10

12

14

16

18

20

20 100 1000 10000 100000 1x106

Det

ecti

on

Sta

tist

ics

Width (day)

t0 = 54750 (MJD)t0 = 53750 (MJD)

other epoch

Figure 7. Detection statistics obtained from the PPTA DR2without J1939+2134 against widths for 18 epochs. Red solid andblue solid and dashed black lines show the statistics for the epochof MJD 54750 and 53750, and other 16 epochs. [SK: Maybe we

do not need to show W >Tobs

.]

Figure 8. Sensitivity of the PPTA DR 2 for hCS,peak for the epochof 54750 and width of 75 days. Black crosses represent the posi-tions of the pulsars without J1939+2134.

of the PPTA DR 2 to GW bursts for the epoch of 54750 andwidth of 75 days with using Eq.14 in Fig. 8. We find that thearray of the PPTA pulsars is more sensitive for GW burstsin the southern area where the pulsars are concentrated inthan in the northern area.

Here, we obtain constraints on hCS,peak as a function ofthe epoch, width and sky position, but only the width hasa physical meaning, which corresponds to the loop size ofthe cosmic string. Therefore, we show constraint on hCS,peak

by changing the width W . In Fig. 9, we show the maximumvalue of hCS,peak as a function of the width, searched over

di↵erent epoch and sky position. We find that hCS,peak

max

theconstraint is the strongest at a few thousand days. The rea-son for this is that timing residuals induced by the GWburst become larger as the width gets larger since a timingresidual is the cumulative waveform distribution. Note thatwe also show constraints for bursts whose duration is longerthan the observation time, W > T

obs

= 12 years ' 4400 days.We would be able to detect such bursts if the spiky shapelies within the observation period. In the figure, we see thatthe upper bound increases as / W1/3. This is because weplace the upper limit on A

lim

= Apeak

(W)(Tobs

/W)1/3, which

MNRAS 000, 1–12 (2018)



-1

-0.5

0

0.5

1

0°

30°

-30°

60°

-60°

8h 4h 0° 20h 16h



1

i = hA2

2


A2

i = 0. Thus,

hDi =⇣

hCS,fit

⌘

2 (S11

+ S22

) /2 (13)

where S = C�1

0


max

is

hCS,fit =�

2D⇤max

/[S11

+ S22

]�0.5 . (14)


hCS,peak =

✓

1

2

W◆

1/3hCS,fit. (15)

Here, C�1

0


0

2

4

6

8

10

12

14

16

18

20

20 100 1000 10000 100000 1x106

Det

ecti

on S

tati

stic

s

Width (day)

t0 = 54750 (MJD)t0 = 53750 (MJD)

other epoch



.]






max


obs


lim

= Apeak

(W)(Tobs

/W)1/3, which

MNRAS 000, 1–12 (2018)

53250 53350 536505355053450

J1939+2134

MJD [day]

J1939+2134の除外• 左図：J1939+2134のtiming residual … “くぼみ”の存在

• 右図：検出された四重極パターン … J1939+2134(+マーク)のみが強いシグナルを示す → 北天のパルサーを調べるしかない ⇒ International PTAへの予言



-1

-0.5

0

0.5

1

0°

30°

-30°

60°

-60°

8h 4h 0° 20h 16h



1

i = hA2

2


A2

i = 0. Thus,

hDi =⇣

hCS,fit

⌘

2 (S11

+ S22

) /2 (13)

where S = C�1

0


max

is

hCS,fit =�

2D⇤max

/[S11

+ S22

]�0.5 . (14)


hCS,peak =

✓

1

2

W◆

1/3hCS,fit. (15)

Here, C�1

0


0

2

4

6

8

10

12

14

16

18

20

20 100 1000 10000 100000 1x106

Det

ecti

on

Sta

tist

ics

Width (day)

t0 = 54750 (MJD)t0 = 53750 (MJD)

other epoch



.]






max


obs


lim

= Apeak

(W)(Tobs

/W)1/3, which

MNRAS 000, 1–12 (2018)



-1

-0.5

0

0.5

1

0°

30°

-30°

60°

-60°

8h 4h 0° 20h 16h



1

i = hA2

2


A2

i = 0. Thus,

hDi =⇣

hCS,fit

⌘

2 (S11

+ S22

) /2 (13)

where S = C�1

0


max

is

hCS,fit =�

2D⇤max

/[S11

+ S22

]�0.5 . (14)


hCS,peak =

✓

1

2

W◆

1/3hCS,fit. (15)

Here, C�1

0


0

2

4

6

8

10

12

14

16

18

20

20 100 1000 10000 100000 1x106

Det

ecti

on S

tati

stic

s

Width (day)

t0 = 54750 (MJD)t0 = 53750 (MJD)

other epoch



.]






max


obs


lim

= Apeak

(W)(Tobs

/W)1/3, which

MNRAS 000, 1–12 (2018)

53250 53350 536505355053450

J1939+2134

結局、シグナルかノイズか分からない

MJD [day]

再解析の結果• 統計的有意性試行回数が多いと大きいDSも生じうる → Fisher’s test !!

　- ノイズのみの場合、Fは自由度　 2mのカイ二乗分布にしたがう

• シミュレーションデータの DSの確率分布(右図)からp-値を計算 → Fのp-値 = 0.97 ⇒ 有意でない



-1

-0.5

0

0.5

1

0°

30°

-30°

60°

-60°

8h 4h 0° 20h 16h



1

i = hA2

2


A2

i = 0. Thus,

hDi =⇣

hCS,fit

⌘

2 (S11

+ S22

) /2 (13)

where S = C�1

0


max

is

hCS,fit =�

2D⇤max

/[S11

+ S22

]�0.5 . (14)


hCS,peak =

✓

1

2

W◆

1/3hCS,fit. (15)

Here, C�1

0


0

2

4

6

8

10

12

14

16

18

20

20 100 1000 10000 100000 1x106

Det

ecti

on

Sta

tist

ics

Width (day)

t0 = 54750 (MJD)t0 = 53750 (MJD)

other epoch



.]






max


obs


lim

= Apeak

(W)(Tobs

/W)1/3, which

MNRAS 000, 1–12 (2018)

F = �2mX

i

ln pi<latexit sha1_base64="DUxFZ8fTly4duspm7Ck2reFD6Wo=">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</latexit>

6 Yonemaru et al.

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Pro

bab

ilit

y

MJD 53500

W = 100 days

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Pro

bab

ilit

y

MJD 53500



0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20

MJD 55500

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20

MJD 55500

5 10 15 20

MJD 56000

5 10 15 20

MJD 56000

5 10 15 20

MJD 56500

5 10 15 20

MJD 56500

5 10 15 20

MJD 57000

5 10 15 20

MJD 57000


0

0.05

0.1

0.15

0.2

0 5 10 15 20 25 30 35 40

Pro

bab

ilit

y





F = �2

m’

i

ln pi (12)


0

8

16

24

32

20 100 1000 8000

Det

ecti

on S

tati

stic

s

Width (day)






MNRAS 000, 1–12 (2018)

0

20

15

5

10

10020 1000 104<latexit sha1_base64="Oh+6fjdqnI5n+eKRCiovjrP+Gvk=">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</latexit>

105<latexit sha1_base64="0lVyK/ell/x4bF1iulai3q9e0UI=">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</latexit>

106<latexit sha1_base64="C8PonWw0nnsoZaI/6DsYTDjW/JU=">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</latexit>

W [day]<latexit sha1_base64="+WmeRQ6r40PC8suXFJVjF/47FL4=">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</latexit>


0

0.15

0.1

0.05

0.2

0 10 3020 40D

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振幅の制限

8 Yonemaru et al.

hC,peak

●

●

●●

●●

●

●

●

● ●● ●●●●●

●●●

●●

●

10 100 1000 10 10 106× 10-

× 10-

× 10-12

× 10-12

∝W1/3

! ["#$%]

Figure 9. Constraint on hCS,peak

max

, which is the maximum hCS,peak

over the epoch and sky position, against the width.

should be the same value for all W > Tobs

. We get a result/ W1/3 when we convert the constraint on A

peak

(W).

6 DISCUSSION

6.1 Physical implications

In this section, we discuss the physical implication of theabove results. The strain amplitude of gravitational wavesfrom a cusp in the frequency domain ˜h( f ) =

Ø

dte2⇡i f t h(t) isgiven in terms of the string tension Gµ and loop size l as (8)

˜h( f ) = Gµl[(1 + z) f l]1/3r(z) f

, (16)

where r(z) is the distance to the GW source at redshift z. Thespectrum peaks at f = fl ⌘ (l/2)�1 and is cut o↵ below thisfrequency. This lowest frequency, determined by the loopsize, also gives the duration of the GW event Tl = l/2 =f �1

l⌘ W/2. Thus, searching bursts for di↵erent duration W

corresponds to searching for cosmic string loops of di↵erentloop size.

Let us define Ãf ⌘ Gµ ·l2/3(1+z)1/3r(z) so that the Fourier am-

plitude can be written in the simple form h( f ) = Ãf | f |4/3.Then, taking the inverse Fourier transform, we find that thestrain amplitude in the time domain is

h(t) =r

3

2⇡�(� 1

3

) Ãf |t |1/3 . (17)

The coe�cientq

3

2⇡ �(� 1

3

) Ãf corresponds to Afit

in Eq. (3).

This function implies h(t = 0) = 0, but the o↵set can changesince the full waveform is the sum of Eq. (17) and of a slowlyvarying component due to the low modes of the string. How-ever, only the relative di↵erence �h is important for PTA ob-servations, and the o↵set does not matter. In fact, we add theo↵set (W/2)1/3 A

fit

in Eq. (3) in order to set h(t = t0

±W/2) = 0,but it is absorbed when we fit the pulse frequency and itdoes not a↵ect the post-fit residuals. The important feature

is that the function is spiky at t = 0. (In Eq. (3), we general-ize the position of the spike by adding t

0

.) This sharp spikein h(t) leaves the feature in the timing residual as shown inFig. 1. Finally, considering that the time scale of the GWevent is given by Tl = f �1

l, the peak amplitude, �h in Eq.

(17), can be written in terms of the Fourier amplitude as

Apeak

⇠r

3

2⇡�(� 1

3

) Ãf T1/3l=

r

3

2⇡�(� 1

3

) ˜h( fl) fl . (18)

This corresponds the dimensionless amplitude in Eq. (4).Note that Eq. (16) has a high frequency cut-o↵ if the

observer does not lie exactly along the direction of the cuspvelocity. This cut-o↵ rounds o↵ the spike in the time-domainwaveform and reduces the amplitude of the timing residual.Since the timing residual is the cumulative of �h as in Eq.(5), the shape of the timing residual is not changed muchexcept for the amplitude. Thus, it does not a↵ect the analysisperformed in the previous section, while the e↵ect on theamplitude should be taken into account when we considerconstraints on string parameters.

Let us roughly estimate how much the timing residualis reduced. According to (29), the spike is smoothed by atime interval of order |t � t

0

| ⇠ ✓3Tl , where ✓ is the angle(in radians) between the direction of emission and the cuspvelocity. Thus, the peak amplitude is reduced by a factor of(1 � ✓3)1/3. In the Fourier space, it means that modes withfrequencies higher than | f | ⇠ (✓3Tl)�1 exponentially decay.Here, we find that, for ✓ & 1, even the lowest frequencyfl = T�1

lis smoothed out, so that the maximum angle we

can observe the GW is ✓ ⇠ 1. Taking an average in thesolid angle of ⌦ = 2⇡(1 � cos[1]), we find that the typical

smoothing time scale is |t � t0

| = Tl

⌦

Ø

2⇡0

d�Ø

1

0

d✓ sin ✓ · ✓3 ⇠0.39Tl = 0.2W , and the amplitude is reduced by a factor of

� = 1

⌦

Ø

2⇡0

d�Ø

1

0

d✓ sin ✓(1 � ✓3)1/3 ⇠ 0.81.In Fig. 9, we have provided constraints on the GW am-

plitude including W > Tobs

. However, such bursts are typ-ically smoothed out by this high-frequency cut-o↵ as thespiky feature is rounded o↵ for |t � t

0

| < 0.2W , which meansthat we do not see the typical cusp-origin GW shape in therange of observation period if W > 5T

obs

. Note that this esti-mate is made by taking the average of the observation angleso that there would be a rare case to find a burst withoutthe cut-o↵. In the following, we focus on the case of W < T

obs

to provide constraints on cosmic string parameters.Let us translate the constraint on the peak amplitude

shown in Fig. 9 to constraints on cosmic string parameters.For string network evolution, we use the velocity-dependentone-scale model where all loops are assumed to be formedwith the same size. In this model, the string network of in-finite strings is characterized by a correlation length ⇠. Thetotal length L of infinite strings in volume V is given byL = V/⇠2, and the average string energy density is given by⇢ = µ/⇠2. Defining � ⌘ ⇠/t, the equation for energy conser-vation gives

t�

d�dt= �1 + ⌫ +

c̃pv2�+ ⌫v2, (19)

while the equations of motion for the Nambu-Goto stringyield an equation for the evolution of the typical root-mean-square velocity v of infinite strings,

dvdt= (1 � v2)H( k(v)

⌫�� 2v), (20)

MNRAS 000, 1–12 (2018)

• DSの観測値 → 振幅の制限

! はの次元をもつ !!!• 右図：各バーストの到来時刻、天球面内での最大値

hCS,fit = (2D⇤max

/[S11

+ S22

])0.5<latexit sha1_base64="MfOUxp7UR49x6UyDrFuiWd6tQQs=">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</latexit>

hCS,fit<latexit sha1_base64="RT4gjr/zw0V2yFXH2pOCnD11+R0=">AAACdXichVHLSgMxFD0d3/XRUTeCCMWquJCaqqC4Et24tK1VwUeZGdM2OC9m0oIO/QF/wIVuFFTEz3DjD7jwE8SlghsX3k4HREW9IcnJyT03J4numsKXjD3GlJbWtvaOzq54d09vX0LtH9jwnapn8ILhmI63pWs+N4XNC1JIk2+5Htcs3eSb+sFKY3+zxj1fOPa6PHT5rqWVbVEShiaJKqpqZS/Y8axgJT+VLAlZrxfVFEuzMJI/QSYCKUSx5qjX2ME+HBiowgKHDUnYhAaf2jYyYHCJ20VAnEdIhPscdcRJW6UsThkasQc0lmm1HbE2rRs1/VBt0CkmdY+USYyzB3bDXtg9u2VP7P3XWkFYo+HlkGa9qeVuMXE8lH/7V2XRLFH5VP3pWaKEhdCrIO9uyDRuYTT1taOTl/xibjyYYBfsmfyfs0d2Rzewa6/GZZbnThGnD8h8f+6fYGMmnZlNz2TnUkvL0Vd0YhijmKT3nscSVrGGAp1bwxkucRV7U0aUMWWimarEIs0gvoQy/QEaSJBh</latexit>

hCS,peak =

✓1

2W

◆1/3

hCS,fit

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sec�1/3<latexit sha1_base64="BbquXMuUOP3dYHTPSdSl8ANIXOM=">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</latexit>

S = C�10

<latexit sha1_base64="h3DhNxqQQ5GNG3AP5+I8c3jbI+g=">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</latexit>

宇宙ひものパラメーターへの制限

宇宙ひもの物理• 重力波の振幅

!

!

• エネルギー保存則 … velocity-dependent one-scale model

!

• Numbu-Goto stringの運動方程式

h̃(f) =Gµl

[(1 + z)fl]1/3r(z)f<latexit sha1_base64="MMxTcWtbE2a24he/4GoILbsttXg=">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</latexit>

Apeak ⇠r

3

2⇡�(� 1

3 )h̃(fl)fl<latexit sha1_base64="Gr6KhkF3d9u4RuykWC/dqt8U6wE=">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</latexit>

フーリエ変換f�1l ⌘ l/2 = W/2

<latexit sha1_base64="eFb2BKwKcnUTS/2/EKguxvwA4d4=">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</latexit>

最低かつピークの周波数

t

�

d�

dt= �1 + ⌫ +

c̃pv

2�+ ⌫v2

<latexit sha1_base64="JH4jCUfGWxbvbzcsX/JksTrHCoQ=">AAACs3ichVFNSxxBEH1ONNFV4yZehFyWLIaAuPRsEhICgpiLp+BHVoUdXXp6ejeN88VM74AZ5g/4Bzx4SiBIyDX/IBf/QA7mGi+So4IXD6mdGQiJJFbT3VWv6lW/7rZDV8WasZMh49bwyO07o2OV8YnJu1PVe/c34qAfCdkSgRtEWzaPpat82dJKu3IrjCT3bFdu2ruvBvnNREaxCvw3ei+U2x7v+aqrBNcEdaqvrW7ERaqz1Opxz+NZETtllDo6W5g35yy/X5srUpZWriNTkYVJljbLurwg2Wl2qnXWYLnVrjtm6dRR2kpQPYIFBwEE+vAg4UOT74IjptGGCYaQsG2khEXkqTwvkaFC3D5VSarghO7S2qOoXaI+xYOecc4WdIpLMyJmDbPsG/vEztkx+8zO2NU/e6V5j4GWPdrtgivDztT+zPrljSyPdo23v1n/1azRxYtcqyLtYY4MbiEKfvLu4Hz95dps+oh9YD9J/3t2wr7SDfzkQnxclWuHqNAHmH8/93Vno9kwnzSaq0/ri0vlV4ziAR7iMb33cyxiGSto0blf8B0/cGo8M9qGbThFqTFUcqbxhxneL/yTqf4=</latexit>

dv

dt= (1� v2)H(

k(v)

⌫�� 2v)

<latexit sha1_base64="GeNDYNM4E329WJBq/wMW8xlC5ss=">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</latexit>

� ⌘ ⇠/t, ⇠<latexit sha1_base64="LT3MO2R1EMQ1MEik9g+SCL1eu/U=">AAACe3ichVHLSgMxFD0d3/XRqiCCG7EoIlJTHyiuim5c1kdVsFJmxrSGzsuZtFirfoA/4MKVBRHRv3DjD7jwE8SlghsFb6cDoqLeYXJPTu65OUk0xxCeZOwhpDQ0NjW3tLaF2zs6uyLR7p51zy66Ok/rtmG7m5rqcUNYPC2FNPim43LV1Ay+oRUWa+sbJe56wrbWZNnh26aat0RO6KokKhvty+RV01QzfK8oSpl9MSHHjylloz EWZ34M/gSJAMQQRMqOXiKDHdjQUYQJDguSsAEVHn1bSIDBIW4bFeJcQsJf5zhCmLRFquJUoRJboDFPs62AtWhe6+n5ap12Meh3STmIYXbPrtgzu2PX7JG9/dqr4veoeSlT1upa7mQjJ/2rr/+qTMoSu5+qPz1L5DDnexXk3fGZ2in0ur50cPq8Or8yXBlhVfZE/s/ZA7ulE1ilF/1ima+cIUwPkPh+3T/B+mQ8MRWfXJ6OJReCp2jFAIYwSvc9iySWkEKa9j1EFde4Cb0rMWVMGa+XKqFA04svocx8AFaCkzM=</latexit>

: correlation length

ループ生成によるエネルギー損失

k(v) =2p2

⇡

1� 8v6

1 + 8v6<latexit sha1_base64="g+Nto/z2auWKq01eR2T0Fh2W0qo=">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</latexit>

a(t) / t⌫<latexit sha1_base64="mGjzbHB2anr540pWeoj6iSQRKpI=">AAACeXichVG7TgJBFD2sL8QH+ChMbFCCQQsygInGymhj6YtHAkh21wE3LLub3YFEiT/gD1jYqIkx6mfY+AMWfoKxxMRCCy/LJkaJeiczc+bMPXfOzCiWrjmCsSef1NPb1z/gHwwMDY+MBkNj4xnHrNsqT6umbto5RXa4rhk8LTSh85xlc7mm6DyrVNfb+9kGtx3NNHbFocWLNbliaGVNlQVRpdCkHI6J+YJlm5Yww2KvWTDqx6 VQhMWZG+FukPBABF5smqFrFLAPEyrqqIHDgCCsQ4ZDLY8EGCziimgSZxPS3H2OYwRIW6csThkysVUaK7TKe6xB63ZNx1WrdIpO3SZlGFH2yG5Yiz2wO/bMPn6t1XRrtL0c0qx0tNwqBU+mdt7+VdVoFjj4Uv3pWaCMZderRt4tl2nfQu3oG0enrZ2V7Whzjl2yF/J/wZ7YPd3AaLyqV1t8+wwB+oDEz+fuBplkPJGKJ7cWI6tr3lf4MY1ZxOi9l7CKDWwiTece4Rw3uPW9SzNSTFropEo+TzOBbyGlPgF1tJHi</latexit>

L = V/⇠2<latexit sha1_base64="CyN29DF+2KaZUa1CRLDDSsE2Usw=">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</latexit>

重力波バーストの発生率

• ループサイズ & ループ生成率

!

!• 重力波バーストの発生率 × beaming effect, 単位時間あたりのcusp数 & comiving volume

l(t) = ↵tb � �Gµ(t� tb)<latexit sha1_base64="ZVoduTJ45bfEsoplbDc+xX0fyR0=">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</latexit>

dn

dtb=

C

↵p2t4b<latexit sha1_base64="NwtV649BIXHDspOIDm0TsmxCMLY=">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</latexit>

dn

dl(t) =

C

↵p21

(↵+ �Gµ)t4b<latexit sha1_base64="hwZ0KdxtflB3mLtYPiBZGSnWZb8=">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</latexit>

dR

dzdh̃=

3

4✓m(f, z, l)2

Nc

(1 + z)h̃

C

↵p21

(↵+ �Gµ)t4b<latexit sha1_base64="53XtMROl9xtU1Az0QPlNkjODjuU=">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</latexit>

⇥✓a(tb)

a(t)

◆3 dV

dz⇥(fl(1 + z)� 2)

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& C = 1/�2<latexit sha1_base64="Nlfp7dKFJXYtB6BCJtHASfUxfow=">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</latexit>

パラメーターの制限• 重力波バーストの個数

!

!

!• 未検出 → 95%の確率で

• α小さい(大きい) → 寿命が短い(長い) ⇒ 物質(放射)優勢期由来 … ループ数が多い(少ない)

10 Yonemaru et al.

Gμ

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50 100 500 1000 5000

10-5

0.001

0.100

10

1000

α = 10-1

10-310-5

10-7

10-9

! ["#$%]

Figure 10. Constraint on Gµ obtained for di↵erent widthsW . We show the cases for di↵erent initial loop sizes ↵ =

10

�1, 10

�3, 10

�5, 10

�7, 10

�9. Here we assume p = 1.

detection easier and the latter makes it more di�cult. Tak-ing into account both e↵ects, we find that the former ef-fect dominates and larger loops, namely larger W , are easierto detect and give better constraints. We also see that theconstraint is stronger for smaller initial loop sizes. This isbecause, since the lifetime of loops is given by ↵tb

�Gµ and is

shorter when ↵ is smaller, GWs come from loops generatedin the matter-dominated era in the case of small ↵, whileloops are long-lived in the case of large ↵ and are generatedearlier in the radiation-dominated era. Due to the di↵erencein the expansion rate, we have a larger number of loops forthose generated in the matter-dominated era, which makesthe constraint stronger.

The parameters such as Nc , C, and p2 in Eq. (29)change the overall number of GW bursts and a↵ect the up-per bound. The number of cusps per oscillation period Nc istypically considered to be of order 1. The coe�cient in theloop number density C is determined by the number of longstrings inside the horizon and by the e�ciency of energyloss to loops. This value can vary in the range O(1�10). Forexample, we use Cr = 13.7 and Cm = 2.63, while the LIGOpaper (32) uses Cr = 1.6 and Cm = 0.48 (for Model 1). Lastly,the reconnection probability p can vary a lot, as it may getsuppressed up to O(10

�3) in the case of superstrings (31),while field theoretic strings have p ⇠ 1. In order to see howthe constraints are a↵ected by those factors, we define thecombination of the parameters as 2

C0 =✓

Nc

1

◆ ✓

Cr

13.7

◆ ✓

1

p2

◆

or

✓

Nc

1

◆ ✓

Cm

2.63

◆ ✓

1

p2

◆

(32)

and, in Fig. 11, we show how the upper bound on Gµ changeswhen C0 has di↵erent values by fixing W = 4000 days, whichgives the strongest constraint in Fig. 10. We find that theupper bound on Gµ gets better for larger C0 as it becomes

2 Here, we assume that Cr and Cm are changed by the samefactor.

Gμ

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0.01 0.10 1 10 100 1000

10-7

10-6

10-5

10-4

0.001

0.010

0.100

α = 10-1

10-310-510-7

!"

Figure 11. Dependence of the upper bound of Gµ on the coe�-cient of loop number density C0 for W = 4000 days.

easier to detect GWs even for small Gµ when the numberdensity increases. The jump in the curve, most prominentfor ↵ = 10

�1, corresponds to whether the GWs are emittedin the radiation or matter era. The constraint is obtainedby considering GWs emitted in the radiation-dominated erawhen C0 is small, while GWs are from the matter-dominatedera when C0 is large.

Finally, in Fig. 12, we present the constraint in the Gµ– ↵ plane, compared with constraints by other types of ob-servations; cosmic microwave background (CMB) (34) and astochastic GW background 3. Although the stochastic back-ground provides better constraint on the string parameters,the burst GW search provides independent constraints.

6.2 Prediction for the IPTA data set

As mentioned in section 5.1, if the measured D of 31.2 iscaused by the GW burst from the cosmic string, that ef-fect should appear on pulsars in the northern hemisphere.In this section, thus, we predict expected strength of theGW burst for the IPTA pulsars. Table. 2 shows the ex-pected strength of the GW burst, which causes D of 31.2with W = 75 days at MJD 53500, for IPTA pulsars whichcan not be observed from the Parkes telescope, or are ob-served by the NANOGrav and EPTA. The second and thridcolumns in the table represent the measured values of theantenna pattern and the weighted rms of the residuals (19).The forth ones is the normalized antenna pattern by �

w

de-scribing how strong the burst should be seen against timingnoise. We find that J1012+5307 has the strongest e↵ect ofthe GW burst. In order to verify the existence of the GW

3 We derived the shaded area using the upper limit on thestochastic GW background for di↵erent spectral index given in(33); A

GWB

= 1.45 ⇥ 10

�15 for � = 13/3 and it scales as / 10

�0.4� ,where the characteristic strain amplitude is parametrized as

hc ( f ) = AGWB

⇣

f

yr

�1

⌘↵and � = 3 � 2↵.

MNRAS 000, 1–12 (2018)


10-12

10-10

10-8

10-6

10-4

10-2

10-10

10-8

10-6

10-4

10-2

SKA

Gµ

α

CMB

This work

Stochastic

Figure 12. Constraint in the Gµ –↵ plane obtained by this work(orange) compared with a stochastic background search by PTA(light gray) and the CMB (dark gray). The dashed line (lightblue) is the accessible parameter space predicted for the SKA.The dotted line (dark blue) is also prediction for the SKA withthe clustering of cosmic string loops in our galaxy’s halo.

burst, IPTA data especially J1012+5307 should be investi-gated in the future.

6.3 Implications for future cosmic string burstsearches

In Fig. 12, we also show the prediction for the SKA, plottedassuming that the SKA can reach hCS,peak = 10

�16. We seethat we can improve the constraint dramatically up to Gµ ⇠10

�10 in the SKA era (the dashed light-blue line), and ifstring loops are clustered in the dark matter halo of ourgalaxy (35; 36; 37), we may even be able to reach down toGµ ⇠ 10

�12 for large ↵ (the dotted dark-blue line). Usuallystochastic background search provides better constraint onthe string parameters at the PTA frequency. However, if thestring tension is low, 10

�15 < Gµ < 10

�8, string loops liveslong and the old loops tend to cluster in our galaxy, whichcould enhance the local number density of loops up to ⇠ 10

5.Thus, we can access the parameter space of small Gµ withthe improved sensitivity of the SKA, and it may enhancethe possibility of single burst detection. The tight boundon string tension by the SKA would be extremely useful totest models of cosmic superstrings such as the KKLMMTscenario, where the tension is predicted to be in the rangeof 10

�12 < Gµ < 10

�6 (38; 39; 40).

7 CONCLUSION

A GW burst from a cusp on the cosmic string loop is de-tectable with PTA experiments as well as their GWB. Inthis work, we developed an algorithm for detecting the GWburst from a single source of the cosmic string and applied itto the most up-to-date data set from the PPTA, the PPTADR 2. We obtained the maximum detection statistic of 31.2

Table 2. Expected strength of the GW burst for IPTA pulsarswhich can not be observed from the Parkes telescope, or are ob-served by the NANOGrav and EPTA. The second and columnsrepresent the measured values of the antenna pattern and theweighted rms of the residuals (? ). The forth one is the normal-ized antenna pattern by �

w

.

Pulsar Name Value ofÕ

A F A( ˆ⌦, p̂) �w

(µs) S/N (µs�1)

J0030+0451 0.38 1.5 0.25J0034�0534 0.46 4.4 0.10J0218+4232 0.06 6.7 0.01J0610�2100 -0.41 5.2 -0.08J0621+1002 -0.80 7.2 -0.11J0751+1807 -0.62 3.3 -0.19J0900�3144 -0.41 2.8 -0.15J1012+5307 0.79 1.6 0.49J1455�3330 0.04 3.8 0.01J1640+2224 0.02 2.0 0.01J1721�2457 -0.03 25.7 0.00J1738+0333 -0.11 2.6 -0.04J1751�2857 -0.05 2.4 -0.02J1801�1417 -0.10 2.0 -0.05J1802�2124 -0.08 2.9 -0.03J1804�2717 -0.06 4.4 -0.01J1843�1113 -0.15 1.0 -0.15J1853+1303 -0.30 1.1 -0.27J1910+1256 -0.32 1.4 -0.23J1911+1347 -0.33 5.2 -0.06J1911�1114 -0.17 0.7 -0.25J1918�0642 -0.21 1.5 -0.14J1955+2908 -0.49 5.0 -0.10J2010�1323 -0.16 2.0 -0.08J2019+2425 -0.49 8.8 -0.06J2033+1734 -0.44 13.3 -0.03J2229+2643 -0.45 3.8 -0.12J2317+1439 -0.10 1.0 -0.10J2322+2057 -0.19 7.0 -0.03

which is contributed mostly by PSR J1939+2134 exhibitingbump-like structure in its timing residuals. This implies twopossibilities; this detection statistic is a signal generated by aGW burst, or just bump-like noise in only PSR J1939+2134causing a false detection. In the former case, that detectionstatistic can not be verified to be a signal or not with onlythe PPTA pulsars, thus we predicted expected strength ofthe signal for the IPTA pulsars. In the latter case, we re-moved PSR J1939+2134 from the data set and re-analyzedit. As a result of the re-analysis, no signal was detected andwe placed constraints on the cosmic string parameters. Weconstrained the string tension as a function of the widthsof the bursts describing the loop sizes by calculating theGW burst event rate using the velocity-dependent one-scalemodel. We find that the constraint on the string tension be-comes more stringent for longer width for two reasons; oneis that the sensitivity for GW amplitude is better for longerwidth when the width is shorter than the observational pe-riod, and another is that larger loops emit GW bursts withlarger amplitudes.

In the future, the SKA would allow us to access the pa-rameter space of small Gµ of ⇠ 10

�10 with the increased sen-sitivity, and accordingly probe even loop clusters of cosmicstrings in our galaxy and test models of cosmic superstrings.

MNRAS 000, 1–12 (2018)

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まとめ• Parkes PTA DR 2を用いて宇宙ひもの単一波源からの重力波バーストを探索 → DS = 31.2 (PSR J1939+2134のデータに”くぼみ”)

• - シグナルの可能性 … PPTAパルサーだけでは検証できない → International PTAへの予言 - ノイズの可能性 … PSR J1939+2134を除外 → 有意なシグナルは検出できなかった

• 宇宙ひものパラメーターへの制限 … 背景重力波やCMBによる制限よりは弱いしかし、各ループサイズごとに制限を付けられた

ska-japanシンポジウム2019@国立天文台ska-jp.org/skajpws2019/day3/3-14_yonemaru.pdf · 10...

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