重力波

ブラックホールを探る

• 重力波の発見とブラックホール

• ブラックホールとは

• 銀河中心の巨大なブラックホール

• 重力波とは？

• 今回の重力波はどのように検出されたのか

• ブラックホールの合体ってどうしてわかるのか？

• 10億年前に起きたとは？

2015年に重力波が２度検出されました2017年1月に３度目の重力波が検出

• 1. 重力波の発見とブラックホール

中性子星合体による重力波

重力波とは？• 電波は電場と磁場の振動で空間を伝わる

• 空間は電波を伝える性質を持つ

• 重力は質量を持つ物同士に働く力，

• 重力は時間と空間を歪める

• その歪みが振動すると，波として伝わる．

• これが重力波

重力が時間と空間を歪める？

• 一般相対論で重力を考えるとそのように考えられる

9相対論とは• 光を伝える媒質検出の実験（マイケルソンモーレイの実験）の失敗から

• 光を表す電磁気学の方程式では，時間も空間座標も同時に変換される

• 時間と空間がこの変換に従うと，物体の運動法則も変更される（特殊相対論：アインシュタイン）

相対論と時間の遅れ

• ３次元空間（たて、よこ、高さ）x, y, z

• 時間軸を加えて４次元• 異なる一定速度で動くふたつの系の間で時間の進み方が異なる（動く時計はゆっくり）

+

一般相対論（重力を含めた相対論）•重力とエレベーター

11

重力 加速度

重力も加速度も同じ効果

無重力の世界

一般相対論と光•重力で光が曲がる

12

加速度

重力で空間が歪む

重力で時間の進み方も影響をうける

一般相対論によるGPSの誤差の補正

重力で空間が歪む• 質量を持つ物体の周りの空間は歪む

• 重い物体を激しく揺すると重力波が生まれて伝わる

• アインシュタインは重力波の存在を疑問視

重力波の振動の様子

ずれの相対値を図る

重力波の初観測

• 2015年9月14日

• 2台のLIGO ルイジアナ州とワシントン州

• 二つのブラックホールの合体

• 12億年前に放射された重力波LIGO

重力波の放射と検出

LIGO

LIGOとはこんな装置

the gravitational-wave signal extraction by broadening thebandwidth of the arm cavities [51,52]. The interferometeris illuminated with a 1064-nm wavelength Nd:YAG laser,stabilized in amplitude, frequency, and beam geometry[53,54]. The gravitational-wave signal is extracted at theoutput port using a homodyne readout [55].These interferometry techniques are designed to maxi-

mize the conversion of strain to optical signal, therebyminimizing the impact of photon shot noise (the principalnoise at high frequencies). High strain sensitivity alsorequires that the test masses have low displacement noise,which is achieved by isolating them from seismic noise (lowfrequencies) and designing them to have low thermal noise(intermediate frequencies). Each test mass is suspended asthe final stage of a quadruple-pendulum system [56],supported by an active seismic isolation platform [57].These systems collectively provide more than 10 ordersof magnitude of isolation from ground motion for frequen-cies above 10 Hz. Thermal noise is minimized by usinglow-mechanical-loss materials in the test masses and their

suspensions: the test masses are 40-kg fused silica substrateswith low-loss dielectric optical coatings [58,59], and aresuspended with fused silica fibers from the stage above [60].To minimize additional noise sources, all components

other than the laser source are mounted on vibrationisolation stages in ultrahigh vacuum. To reduce opticalphase fluctuations caused by Rayleigh scattering, thepressure in the 1.2-m diameter tubes containing the arm-cavity beams is maintained below 1 μPa.Servo controls are used to hold the arm cavities on

resonance [61] and maintain proper alignment of the opticalcomponents [62]. The detector output is calibrated in strainby measuring its response to test mass motion induced byphoton pressure from a modulated calibration laser beam[63]. The calibration is established to an uncertainty (1σ) ofless than 10% in amplitude and 10 degrees in phase, and iscontinuously monitored with calibration laser excitations atselected frequencies. Two alternative methods are used tovalidate the absolute calibration, one referenced to the mainlaser wavelength and the other to a radio-frequency oscillator

(a)

(b)

FIG. 3. Simplified diagram of an Advanced LIGO detector (not to scale). A gravitational wave propagating orthogonally to thedetector plane and linearly polarized parallel to the 4-km optical cavities will have the effect of lengthening one 4-km arm and shorteningthe other during one half-cycle of the wave; these length changes are reversed during the other half-cycle. The output photodetectorrecords these differential cavity length variations. While a detector’s directional response is maximal for this case, it is still significant formost other angles of incidence or polarizations (gravitational waves propagate freely through the Earth). Inset (a): Location andorientation of the LIGO detectors at Hanford, WA (H1) and Livingston, LA (L1). Inset (b): The instrument noise for each detector nearthe time of the signal detection; this is an amplitude spectral density, expressed in terms of equivalent gravitational-wave strainamplitude. The sensitivity is limited by photon shot noise at frequencies above 150 Hz, and by a superposition of other noise sources atlower frequencies [47]. Narrow-band features include calibration lines (33–38, 330, and 1080 Hz), vibrational modes of suspensionfibers (500 Hz and harmonics), and 60 Hz electric power grid harmonics.

PRL 116, 061102 (2016) P HY S I CA L R EV I EW LE T T ER S week ending12 FEBRUARY 2016

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長さ４kmの間隔で取り付けた４個の重りと 鏡，レーザー源からできている

重りレーザー源

6200億ドル

properties of space-time in the strong-field, high-velocityregime and confirm predictions of general relativity for thenonlinear dynamics of highly disturbed black holes.

II. OBSERVATION

On September 14, 2015 at 09:50:45 UTC, the LIGOHanford, WA, and Livingston, LA, observatories detected

the coincident signal GW150914 shown in Fig. 1. The initialdetection was made by low-latency searches for genericgravitational-wave transients [41] and was reported withinthree minutes of data acquisition [43]. Subsequently,matched-filter analyses that use relativistic models of com-pact binary waveforms [44] recovered GW150914 as themost significant event from each detector for the observa-tions reported here. Occurring within the 10-ms intersite

FIG. 1. The gravitational-wave event GW150914 observed by the LIGO Hanford (H1, left column panels) and Livingston (L1, rightcolumn panels) detectors. Times are shown relative to September 14, 2015 at 09:50:45 UTC. For visualization, all time series are filteredwith a 35–350 Hz bandpass filter to suppress large fluctuations outside the detectors’ most sensitive frequency band, and band-rejectfilters to remove the strong instrumental spectral lines seen in the Fig. 3 spectra. Top row, left: H1 strain. Top row, right: L1 strain.GW150914 arrived first at L1 and 6.9þ0.5

−0.4 ms later at H1; for a visual comparison, the H1 data are also shown, shifted in time by thisamount and inverted (to account for the detectors’ relative orientations). Second row: Gravitational-wave strain projected onto eachdetector in the 35–350 Hz band. Solid lines show a numerical relativity waveform for a system with parameters consistent with thoserecovered from GW150914 [37,38] confirmed to 99.9% by an independent calculation based on [15]. Shaded areas show 90% credibleregions for two independent waveform reconstructions. One (dark gray) models the signal using binary black hole template waveforms[39]. The other (light gray) does not use an astrophysical model, but instead calculates the strain signal as a linear combination ofsine-Gaussian wavelets [40,41]. These reconstructions have a 94% overlap, as shown in [39]. Third row: Residuals after subtracting thefiltered numerical relativity waveform from the filtered detector time series. Bottom row:A time-frequency representation [42] of thestrain data, showing the signal frequency increasing over time.

PRL 116, 061102 (2016) P HY S I CA L R EV I EW LE T T ER S week ending12 FEBRUARY 2016

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観測された重力波の波形

検出した重力波の特徴

• 振幅が大きくなりながら周期が短くなり，最後に急に振幅が小さくなり消えた

• 最も短い周期は0.01秒くらい

• 波の振幅の最大値は10の２１乗分の１

検出した重力波の説明には

• 合体直前の重力波の周期から，太陽の１００倍くらいの天体の合体

• 太陽の百倍くらいの天体の合体による発生した重力波を検出したので，距離は１０億光年程度

ブラックホールの合体propagation time, the events have a combined signal-to-noise ratio (SNR) of 24 [45].Only the LIGO detectors were observing at the time of

GW150914. The Virgo detector was being upgraded,and GEO 600, though not sufficiently sensitive to detectthis event, was operating but not in observationalmode. With only two detectors the source position isprimarily determined by the relative arrival time andlocalized to an area of approximately 600 deg2 (90%credible region) [39,46].The basic features of GW150914 point to it being

produced by the coalescence of two black holes—i.e.,their orbital inspiral and merger, and subsequent final blackhole ringdown. Over 0.2 s, the signal increases in frequencyand amplitude in about 8 cycles from 35 to 150 Hz, wherethe amplitude reaches a maximum. The most plausibleexplanation for this evolution is the inspiral of two orbitingmasses, m1 and m2, due to gravitational-wave emission. Atthe lower frequencies, such evolution is characterized bythe chirp mass [11]

M ¼ ðm1m2Þ3=5

ðm1 þm2Þ1=5¼ c3

G

!5

96π−8=3f−11=3 _f

"3=5

;

where f and _f are the observed frequency and its timederivative and G and c are the gravitational constant andspeed of light. Estimating f and _f from the data in Fig. 1,we obtain a chirp mass of M≃ 30M⊙, implying that thetotal mass M ¼ m1 þm2 is ≳70M⊙ in the detector frame.This bounds the sum of the Schwarzschild radii of thebinary components to 2GM=c2 ≳ 210 km. To reach anorbital frequency of 75 Hz (half the gravitational-wavefrequency) the objects must have been very close and verycompact; equal Newtonian point masses orbiting at thisfrequency would be only ≃350 km apart. A pair ofneutron stars, while compact, would not have the requiredmass, while a black hole neutron star binary with thededuced chirp mass would have a very large total mass,and would thus merge at much lower frequency. Thisleaves black holes as the only known objects compactenough to reach an orbital frequency of 75 Hz withoutcontact. Furthermore, the decay of the waveform after itpeaks is consistent with the damped oscillations of a blackhole relaxing to a final stationary Kerr configuration.Below, we present a general-relativistic analysis ofGW150914; Fig. 2 shows the calculated waveform usingthe resulting source parameters.

III. DETECTORS

Gravitational-wave astronomy exploits multiple, widelyseparated detectors to distinguish gravitational waves fromlocal instrumental and environmental noise, to providesource sky localization, and to measure wave polarizations.The LIGO sites each operate a single Advanced LIGO

detector [33], a modified Michelson interferometer (seeFig. 3) that measures gravitational-wave strain as a differ-ence in length of its orthogonal arms. Each arm is formedby two mirrors, acting as test masses, separated byLx ¼ Ly ¼ L ¼ 4 km. A passing gravitational wave effec-tively alters the arm lengths such that the measureddifference is ΔLðtÞ ¼ δLx − δLy ¼ hðtÞL, where h is thegravitational-wave strain amplitude projected onto thedetector. This differential length variation alters the phasedifference between the two light fields returning to thebeam splitter, transmitting an optical signal proportional tothe gravitational-wave strain to the output photodetector.To achieve sufficient sensitivity to measure gravitational

waves, the detectors include several enhancements to thebasic Michelson interferometer. First, each arm contains aresonant optical cavity, formed by its two test mass mirrors,that multiplies the effect of a gravitational wave on the lightphase by a factor of 300 [48]. Second, a partially trans-missive power-recycling mirror at the input provides addi-tional resonant buildup of the laser light in the interferometeras a whole [49,50]: 20Wof laser input is increased to 700Wincident on the beam splitter, which is further increased to100 kW circulating in each arm cavity. Third, a partiallytransmissive signal-recycling mirror at the output optimizes

FIG. 2. Top: Estimated gravitational-wave strain amplitudefrom GW150914 projected onto H1. This shows the fullbandwidth of the waveforms, without the filtering used for Fig. 1.The inset images show numerical relativity models of the blackhole horizons as the black holes coalesce. Bottom: The Keplerianeffective black hole separation in units of Schwarzschild radii(RS ¼ 2GM=c2) and the effective relative velocity given by thepost-Newtonian parameter v=c ¼ ðGMπf=c3Þ1=3, where f is thegravitational-wave frequency calculated with numerical relativityand M is the total mass (value from Table I).

PRL 116, 061102 (2016) P HY S I CA L R EV I EW LE T T ER S week ending12 FEBRUARY 2016

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周期:

太陽質量の115倍で τ=0.01秒

� = 2�r/v

r � 3rg

rg � 2GM/c2

v � c

質量と周期が比例

太陽質量は10の30乗 kg

重力波の強さと距離の関係

�gµ� � 10�21

重力波の振幅は距離に反比例して弱くなる

rg ⇠ 300km ⇠ 3⇥ 107cmM = 100M�ブラックホールの質量

�gµ⌫ ⇠ rg/3r

光年r ⇠ 0.3⇥ 1021rg ⇠ 1028cm ⇠ 3⇥ 109

発生源からrでの振幅は

観測値は

レポート• 次の関係を使って回転周期が0.01秒となるブラックホールの質量を計算しなさい

� = 2�r/vr � 3rg

rg � 2GM/c2

v � c

今回の重力波検出からわかること

• 太陽の100倍の質量の天体合体で重力波を出せるのはブラックホールなので，ブラックホール合体だ

• それは10億年前に起きたらしい

• アインシュタインの一般相対論は正しい

重力波観測ネットワーク 日本の観測装置KAGRA

日本

アインシュタイン方程式

• 左辺は時空の性質 • 右辺は物質やエネルギーの分布

27

重力波の観測

• 2015年9月14日に観測

• 2台のLIGO ルイジアナ州とワシントン州

• 二つのブラックホールの合体

• 12億年前に放射された重力波

重力波の放射と検出

LIGO

LIGO

the gravitational-wave signal extraction by broadening thebandwidth of the arm cavities [51,52]. The interferometeris illuminated with a 1064-nm wavelength Nd:YAG laser,stabilized in amplitude, frequency, and beam geometry[53,54]. The gravitational-wave signal is extracted at theoutput port using a homodyne readout [55].These interferometry techniques are designed to maxi-

mize the conversion of strain to optical signal, therebyminimizing the impact of photon shot noise (the principalnoise at high frequencies). High strain sensitivity alsorequires that the test masses have low displacement noise,which is achieved by isolating them from seismic noise (lowfrequencies) and designing them to have low thermal noise(intermediate frequencies). Each test mass is suspended asthe final stage of a quadruple-pendulum system [56],supported by an active seismic isolation platform [57].These systems collectively provide more than 10 ordersof magnitude of isolation from ground motion for frequen-cies above 10 Hz. Thermal noise is minimized by usinglow-mechanical-loss materials in the test masses and their

suspensions: the test masses are 40-kg fused silica substrateswith low-loss dielectric optical coatings [58,59], and aresuspended with fused silica fibers from the stage above [60].To minimize additional noise sources, all components

other than the laser source are mounted on vibrationisolation stages in ultrahigh vacuum. To reduce opticalphase fluctuations caused by Rayleigh scattering, thepressure in the 1.2-m diameter tubes containing the arm-cavity beams is maintained below 1 μPa.Servo controls are used to hold the arm cavities on

resonance [61] and maintain proper alignment of the opticalcomponents [62]. The detector output is calibrated in strainby measuring its response to test mass motion induced byphoton pressure from a modulated calibration laser beam[63]. The calibration is established to an uncertainty (1σ) ofless than 10% in amplitude and 10 degrees in phase, and iscontinuously monitored with calibration laser excitations atselected frequencies. Two alternative methods are used tovalidate the absolute calibration, one referenced to the mainlaser wavelength and the other to a radio-frequency oscillator

(a)

(b)

FIG. 3. Simplified diagram of an Advanced LIGO detector (not to scale). A gravitational wave propagating orthogonally to thedetector plane and linearly polarized parallel to the 4-km optical cavities will have the effect of lengthening one 4-km arm and shorteningthe other during one half-cycle of the wave; these length changes are reversed during the other half-cycle. The output photodetectorrecords these differential cavity length variations. While a detector’s directional response is maximal for this case, it is still significant formost other angles of incidence or polarizations (gravitational waves propagate freely through the Earth). Inset (a): Location andorientation of the LIGO detectors at Hanford, WA (H1) and Livingston, LA (L1). Inset (b): The instrument noise for each detector nearthe time of the signal detection; this is an amplitude spectral density, expressed in terms of equivalent gravitational-wave strainamplitude. The sensitivity is limited by photon shot noise at frequencies above 150 Hz, and by a superposition of other noise sources atlower frequencies [47]. Narrow-band features include calibration lines (33–38, 330, and 1080 Hz), vibrational modes of suspensionfibers (500 Hz and harmonics), and 60 Hz electric power grid harmonics.

PRL 116, 061102 (2016) P HY S I CA L R EV I EW LE T T ER S week ending12 FEBRUARY 2016

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properties of space-time in the strong-field, high-velocityregime and confirm predictions of general relativity for thenonlinear dynamics of highly disturbed black holes.

II. OBSERVATION

On September 14, 2015 at 09:50:45 UTC, the LIGOHanford, WA, and Livingston, LA, observatories detected

the coincident signal GW150914 shown in Fig. 1. The initialdetection was made by low-latency searches for genericgravitational-wave transients [41] and was reported withinthree minutes of data acquisition [43]. Subsequently,matched-filter analyses that use relativistic models of com-pact binary waveforms [44] recovered GW150914 as themost significant event from each detector for the observa-tions reported here. Occurring within the 10-ms intersite

FIG. 1. The gravitational-wave event GW150914 observed by the LIGO Hanford (H1, left column panels) and Livingston (L1, rightcolumn panels) detectors. Times are shown relative to September 14, 2015 at 09:50:45 UTC. For visualization, all time series are filteredwith a 35–350 Hz bandpass filter to suppress large fluctuations outside the detectors’ most sensitive frequency band, and band-rejectfilters to remove the strong instrumental spectral lines seen in the Fig. 3 spectra. Top row, left: H1 strain. Top row, right: L1 strain.GW150914 arrived first at L1 and 6.9þ0.5

−0.4 ms later at H1; for a visual comparison, the H1 data are also shown, shifted in time by thisamount and inverted (to account for the detectors’ relative orientations). Second row: Gravitational-wave strain projected onto eachdetector in the 35–350 Hz band. Solid lines show a numerical relativity waveform for a system with parameters consistent with thoserecovered from GW150914 [37,38] confirmed to 99.9% by an independent calculation based on [15]. Shaded areas show 90% credibleregions for two independent waveform reconstructions. One (dark gray) models the signal using binary black hole template waveforms[39]. The other (light gray) does not use an astrophysical model, but instead calculates the strain signal as a linear combination ofsine-Gaussian wavelets [40,41]. These reconstructions have a 94% overlap, as shown in [39]. Third row: Residuals after subtracting thefiltered numerical relativity waveform from the filtered detector time series. Bottom row:A time-frequency representation [42] of thestrain data, showing the signal frequency increasing over time.

PRL 116, 061102 (2016) P HY S I CA L R EV I EW LE T T ER S week ending12 FEBRUARY 2016

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観測された重力波の波形

振幅

ブラックホールの合体propagation time, the events have a combined signal-to-noise ratio (SNR) of 24 [45].Only the LIGO detectors were observing at the time of

GW150914. The Virgo detector was being upgraded,and GEO 600, though not sufficiently sensitive to detectthis event, was operating but not in observationalmode. With only two detectors the source position isprimarily determined by the relative arrival time andlocalized to an area of approximately 600 deg2 (90%credible region) [39,46].The basic features of GW150914 point to it being

produced by the coalescence of two black holes—i.e.,their orbital inspiral and merger, and subsequent final blackhole ringdown. Over 0.2 s, the signal increases in frequencyand amplitude in about 8 cycles from 35 to 150 Hz, wherethe amplitude reaches a maximum. The most plausibleexplanation for this evolution is the inspiral of two orbitingmasses, m1 and m2, due to gravitational-wave emission. Atthe lower frequencies, such evolution is characterized bythe chirp mass [11]

M ¼ ðm1m2Þ3=5

ðm1 þm2Þ1=5¼ c3

G

!5

96π−8=3f−11=3 _f

"3=5

;

where f and _f are the observed frequency and its timederivative and G and c are the gravitational constant andspeed of light. Estimating f and _f from the data in Fig. 1,we obtain a chirp mass of M≃ 30M⊙, implying that thetotal mass M ¼ m1 þm2 is ≳70M⊙ in the detector frame.This bounds the sum of the Schwarzschild radii of thebinary components to 2GM=c2 ≳ 210 km. To reach anorbital frequency of 75 Hz (half the gravitational-wavefrequency) the objects must have been very close and verycompact; equal Newtonian point masses orbiting at thisfrequency would be only ≃350 km apart. A pair ofneutron stars, while compact, would not have the requiredmass, while a black hole neutron star binary with thededuced chirp mass would have a very large total mass,and would thus merge at much lower frequency. Thisleaves black holes as the only known objects compactenough to reach an orbital frequency of 75 Hz withoutcontact. Furthermore, the decay of the waveform after itpeaks is consistent with the damped oscillations of a blackhole relaxing to a final stationary Kerr configuration.Below, we present a general-relativistic analysis ofGW150914; Fig. 2 shows the calculated waveform usingthe resulting source parameters.

III. DETECTORS

Gravitational-wave astronomy exploits multiple, widelyseparated detectors to distinguish gravitational waves fromlocal instrumental and environmental noise, to providesource sky localization, and to measure wave polarizations.The LIGO sites each operate a single Advanced LIGO

detector [33], a modified Michelson interferometer (seeFig. 3) that measures gravitational-wave strain as a differ-ence in length of its orthogonal arms. Each arm is formedby two mirrors, acting as test masses, separated byLx ¼ Ly ¼ L ¼ 4 km. A passing gravitational wave effec-tively alters the arm lengths such that the measureddifference is ΔLðtÞ ¼ δLx − δLy ¼ hðtÞL, where h is thegravitational-wave strain amplitude projected onto thedetector. This differential length variation alters the phasedifference between the two light fields returning to thebeam splitter, transmitting an optical signal proportional tothe gravitational-wave strain to the output photodetector.To achieve sufficient sensitivity to measure gravitational

waves, the detectors include several enhancements to thebasic Michelson interferometer. First, each arm contains aresonant optical cavity, formed by its two test mass mirrors,that multiplies the effect of a gravitational wave on the lightphase by a factor of 300 [48]. Second, a partially trans-missive power-recycling mirror at the input provides addi-tional resonant buildup of the laser light in the interferometeras a whole [49,50]: 20Wof laser input is increased to 700Wincident on the beam splitter, which is further increased to100 kW circulating in each arm cavity. Third, a partiallytransmissive signal-recycling mirror at the output optimizes

FIG. 2. Top: Estimated gravitational-wave strain amplitudefrom GW150914 projected onto H1. This shows the fullbandwidth of the waveforms, without the filtering used for Fig. 1.The inset images show numerical relativity models of the blackhole horizons as the black holes coalesce. Bottom: The Keplerianeffective black hole separation in units of Schwarzschild radii(RS ¼ 2GM=c2) and the effective relative velocity given by thepost-Newtonian parameter v=c ¼ ðGMπf=c3Þ1=3, where f is thegravitational-wave frequency calculated with numerical relativityand M is the total mass (value from Table I).

PRL 116, 061102 (2016) P HY S I CA L R EV I EW LE T T ER S week ending12 FEBRUARY 2016

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周期:

115M_sun for τ=0.01sec

� = 2�r/v

r � 3rg

rg � 2GM/c2

v � c

重力波の伝搬

�gµ� � 10�21

重力波の振幅

rg ⇠ 300km ⇠ 3⇥ 107cmM = 100M�ブラックホールの質量

�gµ⌫ ⇠ rg/3r

光年r ⇠ 0.3⇥ 1021rg ⇠ 1028cm ⇠ 3⇥ 109

観測結果For robustness and validation, we also use other generic

transient search algorithms [41]. A different search [73] anda parameter estimation follow-up [74] detected GW150914with consistent significance and signal parameters.

B. Binary coalescence search

This search targets gravitational-wave emission frombinary systems with individual masses from 1 to 99M⊙,total mass less than 100M⊙, and dimensionless spins up to0.99 [44]. To model systems with total mass larger than4M⊙, we use the effective-one-body formalism [75], whichcombines results from the post-Newtonian approach[11,76] with results from black hole perturbation theoryand numerical relativity. The waveform model [77,78]assumes that the spins of the merging objects are alignedwith the orbital angular momentum, but the resultingtemplates can, nonetheless, effectively recover systemswith misaligned spins in the parameter region ofGW150914 [44]. Approximately 250 000 template wave-forms are used to cover this parameter space.The search calculates the matched-filter signal-to-noise

ratio ρðtÞ for each template in each detector and identifiesmaxima of ρðtÞwith respect to the time of arrival of the signal[79–81]. For each maximum we calculate a chi-squaredstatistic χ2r to test whether the data in several differentfrequency bands are consistent with the matching template[82]. Values of χ2r near unity indicate that the signal isconsistent with a coalescence. If χ2r is greater than unity, ρðtÞis reweighted as ρ̂ ¼ ρ=f½1þ ðχ2rÞ3&=2g1=6 [83,84]. The finalstep enforces coincidence between detectors by selectingevent pairs that occur within a 15-ms window and come fromthe same template. The 15-ms window is determined by the10-ms intersite propagation time plus 5 ms for uncertainty inarrival time of weak signals. We rank coincident events basedon the quadrature sum ρ̂c of the ρ̂ from both detectors [45].To produce background data for this search the SNR

maxima of one detector are time shifted and a new set ofcoincident events is computed. Repeating this procedure∼107 times produces a noise background analysis timeequivalent to 608 000 years.To account for the search background noise varying across

the target signal space, candidate and background events aredivided into three search classes based on template length.The right panel of Fig. 4 shows the background for thesearch class of GW150914. The GW150914 detection-statistic value of ρ̂c ¼ 23.6 is larger than any backgroundevent, so only an upper bound can be placed on its falsealarm rate. Across the three search classes this bound is 1 in203 000 years. This translates to a false alarm probability< 2 × 10−7, corresponding to 5.1σ.A second, independent matched-filter analysis that uses a

different method for estimating the significance of itsevents [85,86], also detected GW150914 with identicalsignal parameters and consistent significance.

When an event is confidently identified as a realgravitational-wave signal, as for GW150914, the back-ground used to determine the significance of other events isreestimated without the contribution of this event. This isthe background distribution shown as a purple line in theright panel of Fig. 4. Based on this, the second mostsignificant event has a false alarm rate of 1 per 2.3 years andcorresponding Poissonian false alarm probability of 0.02.Waveform analysis of this event indicates that if it isastrophysical in origin it is also a binary black holemerger [44].

VI. SOURCE DISCUSSION

The matched-filter search is optimized for detectingsignals, but it provides only approximate estimates ofthe source parameters. To refine them we use generalrelativity-based models [77,78,87,88], some of whichinclude spin precession, and for each model perform acoherent Bayesian analysis to derive posterior distributionsof the source parameters [89]. The initial and final masses,final spin, distance, and redshift of the source are shown inTable I. The spin of the primary black hole is constrainedto be < 0.7 (90% credible interval) indicating it is notmaximally spinning, while the spin of the secondary is onlyweakly constrained. These source parameters are discussedin detail in [39]. The parameter uncertainties includestatistical errors and systematic errors from averaging theresults of different waveform models.Using the fits to numerical simulations of binary black

hole mergers in [92,93], we provide estimates of the massand spin of the final black hole, the total energy radiatedin gravitational waves, and the peak gravitational-waveluminosity [39]. The estimated total energy radiated ingravitational waves is 3.0þ0.5

−0.5M⊙c2. The system reached apeak gravitational-wave luminosity of 3.6þ0.5

−0.4 × 1056 erg=s,equivalent to 200þ30

−20M⊙c2=s.Several analyses have been performed to determine

whether or not GW150914 is consistent with a binaryblack hole system in general relativity [94]. A first

TABLE I. Source parameters for GW150914. We reportmedian values with 90% credible intervals that include statisticalerrors, and systematic errors from averaging the results ofdifferent waveform models. Masses are given in the sourceframe; to convert to the detector frame multiply by (1þ z)[90]. The source redshift assumes standard cosmology [91].

Primary black hole mass 36þ5−4M⊙

Secondary black hole mass 29þ4−4M⊙

Final black hole mass 62þ4−4M⊙

Final black hole spin 0.67þ0.05−0.07

Luminosity distance 410þ160−180 Mpc

Source redshift z 0.09þ0.03−0.04

PRL 116, 061102 (2016) P HY S I CA L R EV I EW LE T T ER S week ending12 FEBRUARY 2016

061102-7

次世代ネットワーク KAGRA

日本

まとめ：重力波検出の意義• 一般相対論が予言する重力波を検出できた意義は大きい

• 一般相対論の正しさを示した

• ブラックホールの合体を示した

• 重力波を使ってブラックホールや中性子星の合体を研究できる

• レポート

• 　前半の講義の中からテーマをひとつ選び，何故興味をもったのか，また，これに関する研究について調べ理解できたことと出来なかったことをA4一枚程度にまとめよ。（締め切り12/21必着。メイルで提出）

• 12月4日から、徂徠さんが後半の講義をします