SKAとVLBIによる パルサー研究の未来

高橋慶太郎 熊本大学 2018.7.23

目次 １、SKAによるパルサー観測 ２、重力波直接検出 ３、相対論検証 ４、まとめ

１、SKAによるパルサー観測

Pulsars and the SKA Michael Kramer

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Figure 1: Pulsar-related discoveries as a function of time. The time of the first SKA Science Book is markedand some important (selected) discoveries since are marked. The right panel puts the current numbers intoperspective with those expected for the SKA.

2. Science enabled by the discovery & study of pulsars and radio emitting neutronstars with the SKA

The pulsar key science described in the first SKA Science Book had a number of relatedcomponents, which were summarised under the theme of “Testing Gravity”. With pulsars beingstrongly self-gravitating bodies and precision clocks at the same time, timing observations of bi-nary and isolated millisecond pulsars allow unprecedented strong-field experiments. These includetesting general relativity and alternative theories of gravity using binary pulsars and (the yet to bediscovered) pulsar-black hole systems as well as the direct detection of gravitational waves usinga “Pulsar Timing Array” (PTA) experiment. Given the advances in recent years, prospects are nowdescribed in two separate chapters by Shao et al. (2015) and Janssen et al. (2015), respectively.In addition to those, we provide here a summary of the rich and varied science goals for the SKAdescribed in the appropriate chapters:

Chapter 37 — Gravitational wave astronomy with the SKA — Janssen et al. (2015) APulsar Timing Array (PTA) is used as a cosmic gravitational wave (GW) detector. As describedin the chapter by Janssen et al. (2015), Phase I essentially guarantees the direct detection of aGW signal. This may appear as a stochastic background from binary super-massive black holes inthe process of early galaxy evolution, or it may be bright individual source(s) of this kind. Exoticphenomena like cosmic strings may also be expected to produce measurable GW signals, shouldthey exist. The last ten years have seen a much better understanding of the source population, thedetection procedures and the use of a PTA for fundamental physics (such as graviton properties,e.g. Lee et al. 2010) or single source localisation capabilities (e.g. Lee et al. 2011), all of which isdescribed in the corresponding chapter.

Chapter 38 — Understanding pulsar magnetospheres with the SKA — Karastergiou etal. (2015) Considerable progress has been made with our understanding of the pulsar emissionmechanism in the last decade. However, the wide bandwidth and exceptional sensitivity of theSKA will revolutionise our understanding of radio emission from all types of radio emitting neu-

3

パルサー観測の今後

観測戦略 ・低周波の方が明るい ・低周波で遅延、散乱が大きい →銀河面は高周波、面外は低周波 SKAによるパルサー観測 　全天サーベイ 　　・SKA-mid銀河面サーベイ 　　・SKA-low銀河面外サーベイ 　ターゲット観測 　　・銀河系中心 　　・球状星団 　　・系外銀河 　　・タイミング観測（ミリ秒パルサー） 　　・Fermi未同定天体

SKAによるpulsar観測

SKAによるpulsar観測SKA1サーベイ ・9,000 normal pulsars ・1,400 millisecond pulsars SKA2サーベイ ・30,000 normal pulsars ・3,000 millisecond pulsars これだけたくさんあると・・・ ・統計　　　　　・珍しいもの 　‐光度関数　　　‐質量上限、下限 　‐質量関数　　　‐submsecパルサー 　‐空間分布　　　‐惑星系 　‐周期分布　　　‐珍しい連星

・パルサー国勢調査 ・基礎物理の探求 　‐重力波直接検出 　‐強重力での相対論検証 　‐核物質の状態方程式 ・パルサー磁気圏 ・パルサー風 ・中性子星の誕生、進化 ・銀河系の構造（ガス・磁場） ・銀河間ガス パルサーVLBIの寄与（イメージング観測） 　・重力波直接検出 　・強重力での相対論検証 　・新たな銀河系地図

SKA highlights

SKA pulsar science

２、重力波直接検出

pulsar timing array

多波長重力波天文学

CMB PTA 宇宙干渉計地上干渉計

10-17Hz ～1nHz 1mHz-0.1Hz 100Hz

原始重力波

巨大ＢＨ連星

コンパクト天体連星

超新星爆発

宇宙ひも

pulsar timing array 原理：重力波が通過すると地球とパルサーの時空の 　　　変化でパルスの到達するタイミングがずれる 重力波周波数：観測頻度と観測期間で決まる 　(１日)-1 ～(数年)-1 　→0.1µHz ～1nHz 　→超巨大BH連星、宇宙ひも 感度：10nsecの精度でタイミング観測、10年継続 　→ 10 ns / 10 yr ～3×10-16 タイミングモデル パルサーの周期・周期の変化・位置・DMの値 →　モデルからのズレをtiming residualという

pulsar timing array timing residual: 重力波がないと仮定した タイミングモデルからの ずれ 上：シングルソース 下：背景重力波 （パルサーと波源の 位置関係で変わる）

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Fig. 4.— Examples of the four classes of GW signals predicted in the pulsar timing band. Each graphicshows the induced timing residuals before parameter fitting (top panel) and after fitting for pulsar spin pe-riod and period derivative (bottom panel), for simulated data from pulsars PSRJ0437–4715 (red asterisks),J1012+5307 (blue dots), and J1713+0747 (black triangles) to demonstrate the expected quadrupolar signa-ture. All discrete GW sources (b–d) were injected in the same sky location. In all simulated pulsar residuals,�n = 1 ns of white noise and no red noise was injected. Panels are: (a) A GWB with hc = 10�15 and↵ = �2/3; (b) A continuous wave from an equal-mass 109 M� BSMBH at redshift z = 0.01. The distortionfrom a perfect sinusoid is caused by the lower-frequency pulsar term. (c) A memory event of h = 5 ⇥ 10�15,whose wavefront passes the Earth on day 1500. (d) A burst source with an arbitrary waveform.

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Fig. 4.— Examples of the four classes of GW signals predicted in the pulsar timing band. Each graphicshows the induced timing residuals before parameter fitting (top panel) and after fitting for pulsar spin pe-riod and period derivative (bottom panel), for simulated data from pulsars PSRJ0437–4715 (red asterisks),J1012+5307 (blue dots), and J1713+0747 (black triangles) to demonstrate the expected quadrupolar signa-ture. All discrete GW sources (b–d) were injected in the same sky location. In all simulated pulsar residuals,�n = 1 ns of white noise and no red noise was injected. Panels are: (a) A GWB with hc = 10�15 and↵ = �2/3; (b) A continuous wave from an equal-mass 109 M� BSMBH at redshift z = 0.01. The distortionfrom a perfect sinusoid is caused by the lower-frequency pulsar term. (c) A memory event of h = 5 ⇥ 10�15,whose wavefront passes the Earth on day 1500. (d) A burst source with an arbitrary waveform.

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systematics 重力波以外のタイミングのずれの要因 　・地球の運動、太陽系惑星の位置 　・パルサーの運動 　・phase jitter 　・グリッチ 　・パルス形状の不定性 　・ISMによる散乱 　・red noise ノイズの性質 　・予測可能性 　・時間スケール 　・観測波長への依存性 　・異なるパルサー間の相関

red noise 原因不明の長いタイムスケールのノイズ・パルサーの何らかの不安定性？・パルサーに小惑星帯が存在？・重力波と似た形になることがあるのでかなりやっかい

重力波を抜き出す

θ

・複数のパルサーを同時に 　fitする ・２つのパルサーの到着 　時刻残差の相関を見る 　→他のノイズを除去 ・できるだけ多く ・天球面上で一様に 　→　SKAを待つ

パルサー

パルサー重力波

θ

Hellings & Downs curve

感度予想

Kramer原理的な感度としては ・SKA以前に検出される 　可能性はある ・SKA1なら確実 現実にはred noiseが limitしている

重力波検出においてパルサー距離不定性は大きな誤差要因

パルサーVLBI観測

pulsarterm ⎟⎟⎠

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パルサーの距離の測定精度が重力波波長 （0.01pc – 1pc）よりよくないと、位相が決まらない。 →　距離の情報がないとノイズとして扱われる →　・シグナルと同程度のノイズ 　　・波源の位置決定精度が格段に悪くなる →　パルサーの距離測定で大幅改善

観測量　　　earth term　pulsar term

Zhu+ 2015 シミュレーション ・PPTA DR1のノイズ ・シグナルを入れる（□） ・P最大（○） ・パルサーの位置（☆） パルサーがたくさんある 方向は重力波の方向が よく決まる（数十平方度） 少ないところはあまり 決まらない（数千平方度）

方向決定精度

8 X.-J. Zhu et al.

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Figure 3. Empirical CDF (thick solid black) and its 2–� confidence region (thin solid blue) for the whitened A+,⇥(t) data obtained

from the PPTA DR1 data set, compared against the standard Gaussian distribution (red dash).

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Figure 5. Sky map of signal-to-noise ratios (⇢) for simulated data set that includes a strong signal injection made in the least (a) ormost (b) sensitive sky region. The signal is injected at the location indicated by a “⇤” and the maximum ⇢ is found at “�”. Sky locationsof the 20 PPTA pulsars are marked with “?”.

c� 2014 RAS, MNRAS 000, 1–13

・1kpc、10mJyのパルサーの距離測定精度は？ ・パルサーの距離がわかるとどの程度改善するか？ まだできてません・・・。

方向決定精度

３、一般相対論検証

銀河系中心のパルサー

PSR J1745−2900 　・マグネター（1014G） 　・SgrA*から0.1pc (2.4asec) 　・周期：3.76秒 　・DM: 1778 pc/cm3 　・RM: -66960 rad/m2 銀河系中心パルサーはいくつある？ 　・電波マグネターはパルサーの0.2% 　　→　SgrA*周辺にはたくさんパルサー？ 　・Wharton+ 2012 ~ 1000 in (1pc)3 　・Zhang+ 2014 ~ 200 in (0.01pc)3 　・星密度が高いのでミリ秒パルサーが多いだろう

Galactic Centre Pulsars with the SKA R. P. Eatough

Figure 2: The positions of detected radio pulsars in the central 0.5◦ toward the GC overlaid on a 10.55 GHzcontinuum map made with the Effelsberg telescope (Seiradakis et al. 1989). Assuming a GC distance of8.3kpc, 0.5◦corresponds to a projected distance of ∼ 70pc. Even PSR J1745−2900, which overlaps theposition of Sgr A* on this scale (∼ 3′′≡ 0.1 pc offset), is too distant for gravity tests. Figure courtesy ofB. Klein.

reactions should occur as they do in globular clusters. Voss & Gilfanov (2007) show that tidalcapture of NSs by low-mass main-sequence stars is the dominant mechanism for forming binaryNS in approximately the central 1′ of M31 and these can produce millisecond pulsars (MSPs)via long-term accretion. Along with similarly captured stellar-mass BHs, accreting objects in thecentral part of M31 can account for the excess number of point X-ray sources in the inner bulge ofM31. The same processes should occur in the GC over a similar-sized region (0.1–0.2 kpc) (Munoet al. 2005).

Arguably the strongest evidence for a significant NS population in the central parsec nowcomes from the recent detection of a radio loud magnetar - PSR J1745−2900, an inherently rareclass of pulsar, just ∼3′′(0.1 pc) from Sgr A* (Kennea et al. 2013; Mori et al. 2013; Eatoughet al. 2013a; Shannon & Johnston 2013). Along with the five pulsars previously discovered inproximity to Sgr A* (at 10′−15′- see Figure 2 (Johnston et al. 2006; Deneva et al. 2009)) theseobjects cannot be explained as part of a Galactic disk population because the number of foregroundpulsars expected in the pulsar surveys to date is much less than one. Consequently, the question ofthe GC pulsar population has been recently revisited by a number of authors (Chennamangalam &Lorimer 2014; Zhang et al. 2014; Dexter & O’Leary 2014). As will be described in Section 3, it ispulsars in compact orbits (Porb < 1 yr - corresponding to a semi-major axis of ∼ 160 au) aroundthe BH that will enable the most precise tests of GR. Recent simulations suggest that up ∼ 200pulsars could orbit within 4000 au of the BH, the closest of which could have a semi-major axis of∼ 120 au (Zhang et al. 2014). It is only through observations with the SKA that the full GC pulsar

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相対論の検証・新たな重力理論の発見 ブラックホールの基本的な定理 ・no-hair theorem (Wheeler) 　ブラックホールの性質は質量、 　スピン、電荷だけで決まる ・cosmic censorship conjecture (Penrose) 　‐ブラックホールの回転速度が 　　大きすぎると裸の特異点が出て 　　しまい理論が破綻 　‐一般相対論の解としては存在する 　‐現実の宇宙ではそのような解は 　　実現されないであろう 　　→ブラックホールの回転速度に上限

最も相対論らしいもので相対論を検証

ブラックホールの形を測る

銀河中心の巨大ブラックホール近傍のパルサー（1mpc） →パルスのタイミング →パルサーの軌道要素（Kepler, post-Kepler） →巨大ブラックホールの質量、スピン、形（四重極）を 　精密に測る

+ +

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Probing Sgr A* with a pulsars, stars, and the EHT 15

Fig. 9.— Signature of the black hole quadrupole moment, foran extreme Kerr black hole (� = 1). Simulations have been donefor two periapsis passages, with the above figure zooming into thefirst one. 10µs TOAs have been created only within ±15 daysaround the periapsis passages for a pulsar in an eccentric (e = 0.8)orbit with Pb = 0.5 yr. The orientation of the spin is taken asin Figure 5. The residuals are a result of a fit for the orbitaland frame-dragging contributions. For demonstration purposes wehave used a high timing cadence, to densely map the quadrupolesignature. In practice, such a coverage would be the result of manyperiapsis-passage observations over a few years.

termine the spin of Sgr A*.4 In this section we relaxthis assumption, like we have done above for the spinmeasurement. For our simulations we added the im-plementation for the quadrupole moment of Liu et al.(2012) to our aforementioned extension of TEMPO. Fig-ure 9 is the result of a simulation, where we use timingdata only near periapsis, and allow for an undeterminedoverall precession of the orbit due to some unknown ex-ternal perturbations. Figure 9 clearly shows that, afterfitting for the pulsar spin parameters, orbital parameters,and frame dragging, there is still a distinctive signal inthe residuals as a result of the quadrupole moment ofthe black hole. As a general result, depending on thetiming-precision and the periapsis distance, we will stillbe able to extract the quadrupole moment of Sgr A*. Ofcourse, this also depends on the actual value of the spinof Sgr A*, which is poorly constraint to date. In fact,the strength of the quadrupole e↵ect scales with �

2, andis therefore clearly less prominent for a slowly rotatingblack hole (see Figure 10). Depending on the timingprecision, however, the quadrupole moment can still bedetermined with high precision.

We have conducted extensive mock data simulations tostudy the joint measurability of spin and quadrupole mo-ment. Like in the simulations for the spin measurement,we have assumed three TOAs per day. Figure 11 showsthe timing coverage of the spin and quadrupole signatureduring one periapsis passage. Some of the results are il-lustrated in the contour plots of Figure 12. We concludethat, even for the conservative situation of a comparablylow timing precision (�TOA = 100µs) and the presenceof external perturbations, a quantitative test of the Kerrhypothesis is possible after only a few periapsis passages.If we have a better timing precision or can make use oftiming measurements along the whole orbit, the spin and

4 Liu et al. 2012 have demonstrated a way to identify the pres-ence of external perturbations in the secular changes of the pulsarorbit.

Fig. 10.— Same as in Figure 9, but this time with � = 0.2.In this case, the quadrupole moment of the black hole leads to aconsiderably less prominent signal in the residuals, but can still bemeasured accurately given the assumed TOA error of 10µs.

Fig. 11.— Mock-data TOA (�TOA = 100µs) coverage of the spin(top) and quadrupole (bottom) signal during a periapsis passage.The simulated data cover three orbits. We used � = 0.6, P =0.5 yr, e = 0.8, ⇥ = 60�, ⌥0 = !0 = 45�, and � = 55�.

quadrupole moment can be determined with high preci-sion after a few orbits. The latter agrees with the findingsin Liu et al. (2012).

4.4. Distance Measurement with Pulsar Timing

Given the large size of the pulsar orbit (⇠ 102 au),the orbital parallax (Kopeikin 1995), which is of order⇠ a

2/2cD, will lead to a significant contribution to the

timing observations, even for a moderate timing preci-sion. This timing e↵ect depends only on well determinedorbital parameters and the distance to Sgr A*, D, andconsequently can give independent access to D (cf. dis-cussion in Subsection 4.1). The orbital parallax is a pe-riodic signal in the timing residuals, and therefore, ifwe have N equally distributed TOAs with uncertainty�TOA, its measurement scales proportional to �TOA andp

N . Consequently we find

�D⇠ 2c�TOAp

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四重極を考慮に入れないときのresidual

Psaltis+ (2016)

ブラックホールの形を測る

ブラックホール質量 = 4*106 Msun a/GM = 5000 (a/1mpc) 軌道長半径 = 1mpc 近点 = 0.2mpc (e = 0.8) 軌道周期 = 1.5yr (a/1mpc)3/2 ブラックホールパラメータ 決定精度の予測 ・ToA:100µsec、3/day 　→ normal pulsarを想定 ・パルサー以外の天体の 　影響で近点以外では 　ブラックホールが測れ 　ないかもしれない ・曲線は相対論の予言

16 Psaltis et al.

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Fig. 12.— The posterior likelihood of measuring the spin andquadrupole moment of SgrA* using pulsar timing. In the top panelthe dashed curves show the 68% and 95% confidence limits while,in the bottom panel, the solid curves show the 95% confidence lim-its. The solid curve shows the expected relation between these twoquantities in the Kerr metric. The filled circle marks the assumedspin and quadrupole moment (�= 0.6, |q| = 0.36). The pulsar isassumed to have an orbital period of 0.5 yr (orbital separation of2400GM/c2) and an eccentricity of 0.8, while three TOAs per daywith equal timing uncertainty of 100µs have been simulated. Thetop panel compares the uncertainties in the measurement whenonly three periastron passages have been considered in the timingsolution to those when the three full orbits are taken into account.The bottom panel shows the increase in the precision of the mea-surement when the number of periastron passages is increased fromthree to five.

⇠ 20 pc

✓�TOA

102 µs

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◆�1/2 ⇣a

102 au

⌘�2

, (57)

where we have used D = 8.3 kpc.External perturbations can also lead to changes of the

orbit, which could in principle partly mimic the abovee↵ects. This, however, depends highly on the specifics ofthe perturbation, and we will not discuss this in furtherdetail in this paper. On the other hand, as argued byLiu et al. (2012), a precise measurement of the Sgr A*

mass from pulsar timing can be converted into a precisedetermination of the distance to Sgr A*, when combinedwith high-precision astrometric observations in the in-frared. For instance, a high precision measurement ofM• in combination with the (angular) size of the S2-starorbit in Gillessen et al. (2009b) can be converted into adirect measurement of the Galactic center distance withan error of ⇠100 pc. Future 10 µas astrometry promisesa precision of order one parsec or even better.

5. PROBING THE SPACETIME OF SgrA* WITH THE EHT

The EHT will image the millimeter emission fromSgr A* with horizon-scale resolution. There have beenat least three proposals for using EHT observations tomap the spacetime of this black hole and, in particu-lar, to measure di↵erent combinations of its spin andquadrupole moment.

The first approach utilizes the detailed shape of theshadow cast by the black hole on the surrounding emis-sion (Johannsen & Psaltis 2010b). Because of the com-bined e↵ects of frame dragging and of the quadrupoledeformation of the spacetime, the shadows of Kerr blackholes are nearly circular, independent of the black-holespin and the orientation of the observer (Bardeen 1973).The shadows of spacetimes that violate the no-hair theo-rem, however, can be significantly asymmetric, with themagnitude of asymmetry providing a measure of the de-gree of violation of relation (2); see Johannsen & Psaltis2010b5. The shape of the shadow can be measured us-ing the interferometric data either via an edge detectionscheme (Psaltis et al. 2015a) or via fitting phenomeno-logical geometric models (Ricarte & Dexter 2015).

In a second approach, simulated images of the accre-tion flow are fitted against the measured complex in-terferometric visibilities. The characteristic scale of thebrightness in the accretion flow is set by the radius ofthe innermost stable circular orbit (see, e.g., Brodericket al. 2009; Dexter et al. 2010). For a general spacetime,this radius is determined, in turn, by a particular com-bination of the black-hole spin and quadrupole moment(see, e.g., Johannsen & Psaltis 2010a). Even the current,limited imaging data at 1.3 mm provide a glimpse of howthis method can be used to constrain the properties ofthe black-hole spacetime by measuring the location ofits innermost stable circular orbit (Broderick et al. 2014)and this technique will flourish as the full EHT arraybecomes operational.

Finally, if either GRAVITY or the EHT finds evidencefor short-lived, compact emission regions (“hot spots”)that are advected with the accretion flow, tracing theirorbits will lead to a measurement of the spacetime prop-erties in a way that is similar to those discussed in §3and §4 for orbits of stars and pulsars (Broderick & Loeb2006; Vincent et al. 2011). The dynamical timescale inthe vicinity of the horizon of Sgr A* is equal to a few tensof minutes, i.e., much smaller than the time it will take

5 A number of studies have explored the shapes and sizes ofblack-hole shadows in modified gravity theories as well as in para-metrically modified Kerr-like metrics (see, e.g., Bambi & Freese2009; Bambi & Yoshida 2010; Johannsen 2012; Abdujabbarov etal. 2013; Amarilla & Eiroa 2013; Ghasemi-Nodehi et al. 2015). Inthis article, we focus on work that aims specifically to measure thequadrupole moment of the black-hole spacetime using its shadowproperties.

Psaltis+ 2016

VLBIでパルサー観測

軌道の見かけの大きさ = 50mas (a/1mpc) →　パルサーのイメージ観測で軌道要素を決める 位置決定精度（10GHz, 6000km, S/N=100） → 5µas 　・軌道長半径の決定精度〜0.5µas（？） 　　→ BH質量決定精度〜10-5（？） 　　→ pulsar timingと同程度！ 　・近点移動 　　→ BHスピン決定精度〜O(1)%（？） 　　→ pulsar timingより良い！

バイアス

PTAの経験によると、性質の 良いミリ秒パルサーでも O(year)のタイムスケールの red noiseが存在する。

16 Psaltis et al.

1.0

0.8

0.6

0.4

0.2

0.0

Blac

k H

ole

Qua

drup

ole

Mom

ent

0.80.70.60.50.4

Black Hole Spin

Kerr

Periapsisonly

Fullorbit

1.0

0.8

0.6

0.4

0.2

0.0

Blac

k H

ole

Qua

drup

ole

Mom

ent

0.80.70.60.50.4

Black Hole Spin

3 Periapsispassages

5 Periapsispassages

Fig. 12.— The posterior likelihood of measuring the spin andquadrupole moment of SgrA* using pulsar timing. In the top panelthe dashed curves show the 68% and 95% confidence limits while,in the bottom panel, the solid curves show the 95% confidence lim-its. The solid curve shows the expected relation between these twoquantities in the Kerr metric. The filled circle marks the assumedspin and quadrupole moment (�= 0.6, |q| = 0.36). The pulsar isassumed to have an orbital period of 0.5 yr (orbital separation of2400GM/c2) and an eccentricity of 0.8, while three TOAs per daywith equal timing uncertainty of 100µs have been simulated. Thetop panel compares the uncertainties in the measurement whenonly three periastron passages have been considered in the timingsolution to those when the three full orbits are taken into account.The bottom panel shows the increase in the precision of the mea-surement when the number of periastron passages is increased fromthree to five.

⇠ 20 pc

✓�TOA

102 µs

◆✓N

103

◆�1/2 ⇣a

102 au

⌘�2

, (57)

where we have used D = 8.3 kpc.External perturbations can also lead to changes of the

orbit, which could in principle partly mimic the abovee↵ects. This, however, depends highly on the specifics ofthe perturbation, and we will not discuss this in furtherdetail in this paper. On the other hand, as argued byLiu et al. (2012), a precise measurement of the Sgr A*

mass from pulsar timing can be converted into a precisedetermination of the distance to Sgr A*, when combinedwith high-precision astrometric observations in the in-frared. For instance, a high precision measurement ofM• in combination with the (angular) size of the S2-starorbit in Gillessen et al. (2009b) can be converted into adirect measurement of the Galactic center distance withan error of ⇠100 pc. Future 10 µas astrometry promisesa precision of order one parsec or even better.

5. PROBING THE SPACETIME OF SgrA* WITH THE EHT

The EHT will image the millimeter emission fromSgr A* with horizon-scale resolution. There have beenat least three proposals for using EHT observations tomap the spacetime of this black hole and, in particu-lar, to measure di↵erent combinations of its spin andquadrupole moment.

The first approach utilizes the detailed shape of theshadow cast by the black hole on the surrounding emis-sion (Johannsen & Psaltis 2010b). Because of the com-bined e↵ects of frame dragging and of the quadrupoledeformation of the spacetime, the shadows of Kerr blackholes are nearly circular, independent of the black-holespin and the orientation of the observer (Bardeen 1973).The shadows of spacetimes that violate the no-hair theo-rem, however, can be significantly asymmetric, with themagnitude of asymmetry providing a measure of the de-gree of violation of relation (2); see Johannsen & Psaltis2010b5. The shape of the shadow can be measured us-ing the interferometric data either via an edge detectionscheme (Psaltis et al. 2015a) or via fitting phenomeno-logical geometric models (Ricarte & Dexter 2015).

In a second approach, simulated images of the accre-tion flow are fitted against the measured complex in-terferometric visibilities. The characteristic scale of thebrightness in the accretion flow is set by the radius ofthe innermost stable circular orbit (see, e.g., Brodericket al. 2009; Dexter et al. 2010). For a general spacetime,this radius is determined, in turn, by a particular com-bination of the black-hole spin and quadrupole moment(see, e.g., Johannsen & Psaltis 2010a). Even the current,limited imaging data at 1.3 mm provide a glimpse of howthis method can be used to constrain the properties ofthe black-hole spacetime by measuring the location ofits innermost stable circular orbit (Broderick et al. 2014)and this technique will flourish as the full EHT arraybecomes operational.

Finally, if either GRAVITY or the EHT finds evidencefor short-lived, compact emission regions (“hot spots”)that are advected with the accretion flow, tracing theirorbits will lead to a measurement of the spacetime prop-erties in a way that is similar to those discussed in §3and §4 for orbits of stars and pulsars (Broderick & Loeb2006; Vincent et al. 2011). The dynamical timescale inthe vicinity of the horizon of Sgr A* is equal to a few tensof minutes, i.e., much smaller than the time it will take

5 A number of studies have explored the shapes and sizes ofblack-hole shadows in modified gravity theories as well as in para-metrically modified Kerr-like metrics (see, e.g., Bambi & Freese2009; Bambi & Yoshida 2010; Johannsen 2012; Abdujabbarov etal. 2013; Amarilla & Eiroa 2013; Ghasemi-Nodehi et al. 2015). Inthis article, we focus on work that aims specifically to measure thequadrupole moment of the black-hole spacetime using its shadowproperties.

red noiseはブラックホール パラメータへのバイアスを生む。 →　これはcritical！

相対論からのズレ 相対論が否定された！

このようなことが起こらない ようにVLBIで外部データを 与え、バイアスをなくす！

４、まとめ

まとめ

パルサーVLBI ・重力波「天文学」 　パルサーの距離決定でPTAの感度上昇、 　重力波源の方向決定 ・相対論検証 　SMBHまわりのパルサーの軌道要素決定 　感度上昇、pulsar timingのバイアス減少 ・新たな銀河系地図 　パルサーという異なるpopulation 国際的にほとんど検討されていない。チャンス！

イメージングによるパルサー探査 電波天体 ・パルサー ・QSO（圧倒的多数） ・電波銀河 指標 ・spectral index ・compactness ・偏光率 これらの指標から候補選定 現在、criterionを決めてO(10万)の電波天体から 数十個の候補を絞り、検証のためParkesの観測準備中

日本の研究 Kumamoto, KT+, in prep

Artificial Neural Network

銀経

銀緯

spectral index

compactness

PSRである確率

QSOである確率

出力層中間層入力層xi

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ニューラルネットワークによるパルサー自動判定

Yonemaru, KT+, in prep 日本の研究

DR2から得られるdetection statisticsEpoch = 54000

Epoch = 57000Epoch = 56000

Epoch = 55000

Width = 3,000 daysで固定

重力波による宇宙ひも探査 ・PPTA Data Release 2 (2004-2016) ・銀河系内にGµ = 10-12の宇宙ひもが 　１本もない

日本の研究 Yonemaru, Kuroyanagi, KT+, in prep

これまでたくさんのパルサーサーベイがあった →　10-20GHz、100m口径、10時間積分 　　・@15 GHz by Macquart et al. (2010) 　　・@12−18 GHz by Siemion et al. (2013) 　　・@19 GHz by Eatough et al. (2013) しかし１つのマグネターを除いて見つかっていない →　missing pulsar問題は長年の謎

これまでの銀河系中心パルサー探査

周期-光度@1.4GHz ・発見されている 　パルサーを銀河中心 　に置いた時の散布図 ・これまでの探査 　(5, 15GHz)の感度 　(weak, strong)

2

tion, or are significantly under-luminous relative to thepopulation of slow pulsars that have been detected else-where in the Galaxy.In passing, we note that Chennamangalam & Lorimer

(2014) used a Bayesian analysis combined with an as-sumed log-normal pulsar luminosity function to find thatexisting pulsar surveys at the GC are not sufficientlydeep to eliminate the possibility that a substantial popu-lation of non-recycled low-luminosity pulsars might existat the GC. While this is broadly correct, the shape of thepulsar luminosity function is not at all well-constrainedat the low-luminosity end. As such, there is consider-able error in the extrapolation of the distribution to lowluminosities, and hence on such estimates of the totalsize of the normal pulsar population in the central par-sec. For example, an uncertainly of only 20% in the twoparameters of the log-normal distribution (Bagchi et al.2011; Chennamangalam & Lorimer 2014) yields an un-certainty of two orders of magnitude in the survey com-pleteness. Similar estimates of the number of potentialMSPs at the GC are plagued with an even greater degreeof uncertainty because the luminosity function of recy-cled pulsars in globular clusters is even less well known.In the present analysis, we will adopt a pragmatic ap-proach to the pulsar luminosity distribution of pulsars:our detection arguments are made only with reference tothe known luminosities of detected Galactic pulsars.In this paper, we explore the reasons for the paucity

of pulsar detections at the Galactic Center. In § 2, wediscuss the nature of the pulsar population at the GC re-gion, and argue that it is likely to be dominated by recy-cled millisecond pulsars (MSPs); present pulsar surveysof the GC have been largely insensitive to such a pop-ulation. Next, we present in § 3 detailed calculations ofthe expected signal-to-noise ratio for a wide-bandwidthsearch for MSPs at the Galactic Center with present andfuture telescopes, incorporating the frequency depen-dence of temporal smearing and sky temperature acrossthe different observing bands. Finally, the results of thiswork are summarized in § 4.

2. A MILLISECOND PULSAR POPULATION AT THEGALACTIC CENTER

Over the last two decades, several searches have beencarried out for pulsars at the GC, mostly at frequenciesbetween 1.4 GHz and 8 GHz (e.g. Johnston et al. 1995,2006; Kramer et al. 2000; Deneva et al. 2009; Deneva2010; Bates et al. 2011). Based on the high expectedtemporal smearing, Macquart et al. (2010) argued thathigher observing frequencies are more amenable to thediscovery of pulsars, with the optimal frequency rangefor searches for “normal” pulsars – those with periodsof ≈ 0.5 seconds – being 10 − 16 GHz. Following this,there have been a number of deep searches at frequenciesabove 10 GHz, at 15 GHz by Macquart et al. (2010), at12− 18 GHz by Siemion et al. (2013), and at 19 GHz byEatough et al. (2013). Remarkably, despite integrationtimes exceeding 10 hours with 100-m single-dish tele-scopes, none of these searches has discovered a singlepulsar in the Galactic Centre region!Using the temporal smearing estimates of

Lazio & Cordes (1998), Macquart et al. (2010) esti-mated that their 15 GHz search would have beensensitive to ≈ 15% of the Galactic center pulsar

0.001 0.01 0.1 10.01

0.1

1

10

100

1000

Period (s)

1.4

GH

z Lu

min

osi

ty (m

Jy k

pc2

)

5 GHz

15 GHz

5 GHz15 GHz

Fig. 1.— The 1.4GHz luminosity (in mJy kpc2) of the known pul-sar population (Manchester et al. 2005) is plotted versus pulsar pe-riod. The 10σ sensitivities of previous 5GHz (Johnston et al. 2006)and 15GHz GBT (Macquart et al. 2010) searches of the GalacticCenter are shown by the green and red curves, respectively, withthe dashed and solid curves representing the “strong” and “weak”temporal smearing scenarios, respectively.

population, assuming the luminosity distribution ofknown Galactic pulsars. However, if the pulse smearingis benign, as suggested by the detection of the GCmagnetar, then even the earlier lower-frequency searcheswould have been sensitive to normal pulsars located nearthe GC. Indeed, Dexter & O’Leary (2014) estimate thatboth the 5 GHz search of Johnston et al. (2006) and the15 GHz search of Macquart et al. (2010) would bothhave been sensitive to ≈ 20% of the GC pulsar popula-tion, if its characteristics resemble those of the knownpulsar population (see Fig. 2 of Dexter & O’Leary2014). This “missing pulsar” problem was used byDexter & O’Leary (2014) to argue that the GC pulsarpopulation may be dominated by magnetars, i.e. thatthe population is very different from that in the restof the Galaxy (where, as noted earlier, only ≈ 0.2% ofknown radio pulsars are magnetars, although the mag-netar birth rate may be 10 − 50% of the total neutronstar birth rate in the Galaxy; e.g. Keane & Kramer2008).However, a number of studies have raised another pos-

sibility, namely that the GC pulsar population is domi-nated by recycledmillisecond pulsars (MSPs). The densestellar environment at the GC is likely to result in spin-ning pulsars up to millisecond periods by frequent closeinteractions with neighbouring stars, by analogy withthe population of MSPs detected in the dense stellar en-vironments of globular clusters (e.g. Alpar et al. 1982;Verbunt 1987; Camilo et al. 2000; Ransom et al. 2005).Note that the stellar density in the central parsec ofthe GC is ≈ 106 per cubic parsec (e.g. Genzel et al.1996; Schodel et al. 2007), a couple of orders of mag-nitude larger than the stellar density in globular clustercores (! 104 per cubic parsec), implying that close in-teractions are far more likely in the vicinity of the GC.The formation rate of low-mass X-ray binaries (LMXBs)in dense environments is also proportional to both thenumber density of neutron stars and the stellar den-sity (Verbunt & Hut 1987). Since the neutron star den-sity itself scales with the stellar density, this impliesthat the formation rate of LMXBs (and hence, that of

これまでなぜ見つからなかった？

normal pulsarがあれば発見できたが MSPは見つからなくても不自然でない （銀河系中心は星密度が高いので、 多くのパルサーがMSPであろうと予測される）

予測感度8

0.001 0.01 0.1 10.01

0.1

1

10

100

1000

Period (s)

1.4

GH

z Lu

min

osi

ty (m

Jy k

pc2

)

GBT X-band

JVLA X-band

SKA1-MID X-band

full SKA X-band

Fig. 4.— The 10σ sensitivities of GBT, VLA, SKA1-MID andSKA 30-hour X-band integrations to the known Galactic pul-sar population, if placed at the distance of the Galactic Cen-ter, assuming the weak-scattering case. As in Fig. 1, the dotsshow the 1.4GHz luminosity of the known pulsar population(Manchester et al. 2005) plotted versus pulsar period, while thesolid, dashed, dotted and dash-dotted curves show the 10σ sensi-tivities for the GBT, VLA, SKA1-MID and full SKA, respectively.It is clear that deep X-band observations with existing telescopes(the GBT and the VLA) would be sensitive to a significant fraction(! 30%) of the known MSP population (as well as to ! 65% of theentire known pulsar population), if located at the GC distance.

ever, the strong-scattering scenario would require the fullSKA to detect and time MSPs at the distance of theGalactic Center.

Parts of this research were conducted by the Aus-tralian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), through project numberCE110001020. NK acknowledges support from the De-partment of Science and Technology via a SwarnajayantiFellowship, and also thanks ICRAR for support duringa visit during which part of this work was carried out.JPM thanks Yuri Levin for engaging discussions relatingto the topic of this work. We also thank an anonymousreferee for suggestions that improved the clarity of thispaper.

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GBT, JVLA, SKA1, 2：30時間積分探査の感度（弱散乱）

GBTやJVLAでも30時間で明るい ミリ秒パルサーを見つけられそう。 SKA2なら暗いものも見つかる。