第31回 理論懇シンポジウム「宇宙物理の標準理論：未来へ向けての再考」

その拡張

一般相対論と小林 努

一般相対論は学部4年生でも知っているので
今さら説明する必要はないですね。。。

宇宙物理理論の研究室の「標準カリキュラム」
・4年生 ‒ 場の古典論
・M1 ‒ Wald• 現状、一般相対論で特に問題はない• 困るのは、ロマンがないことぐらい• なぜ「標準理論」を拡張したいのか？
• この分野の近年の動向• どの非標準(拡張)理論が生き残っているのか・・・という話をします。

Plan
なぜ修正重力？
重力を拡張(修正)するとは？
“まとも”な重力理論に要請されること
近年の研究の動向 ‒‒ Horndeski & beyond
GW170817以降の修正重力
まとめ

c=~=1
<latexit sha1_base64="RR9N/6q5SyvJJ9R/soJ246BHxKk=">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</latexit>なぜ修正重力？
重力を拡張(修正)するとは？
“まとも”な重力理論に要請されること
近年の研究の動向 ‒‒ Horndeski & beyond
GW170817以降の修正重力まとめ動機1: 宇宙の加速膨張宇宙項で説明できるが、修正重力でも説明できる
起源がわからないので、あり得る可能性は検討したい

Planck動機2: 一般相対論は“低エネルギー有効理論”であると考えられる
高エネルギー/強重力、どこかで一般相対論による記述は悪くなるはず (プランクスケールかもしれないが)量子重力は万人の夢
初期宇宙/ブラックホールに痕跡があるかもしれない動機3: 一般相対論のテスト
太陽系実験、天文学的観測、宇宙論的観測、重力波、、、
一般相対論のテストをするためには比較相手が必要「理論/モデルの相対化」は科学として健全な態度

動機4: インフレーション
観測とよく合うモデルは“修正重力”インフレーション
インフラトンと重力場の相互作用は「標準的」でない可能性が高い0.20

Planck Collaboration: Constraints on Inflation

TT,TE,EE+lowE+lensing
TT,TE,EE+lowE+lensing+BK14
TT,TE,EE+lowE+lensing+BK14+BAO
Natural inflation
Hilltop quartic model
attractors
Power-law inflation
R2 inflation

Co
n

ca

ve

0.10

0.15

Co
n

V

2

V
V

4/3

2/3
V
Low scale SB SUSY
N =50
N =60

0.05
0.00

Tensor-to-scalar ratio (r0.002 )テンソル-スカラー比

ve
x

0.94

0.96
Primordial tilt (ns )

0.98

1.00

Planck 2018

Fig. 8. Marginalized joint 68 % and 95 % CL regions for ns and r at k = 0.002 Mpc 1 from Planck alone and in combination with
BK14 or BK14 plus BAO data, compared to the theoretical predictions of selected inflationary models. Note that the marginalized
joint 68 % and 95 % CL regions assume dns /d ln k = 0.

スカラー揺らぎのスペクトル指数

動機5: 重力を深く理解したい
純粋に理論的な興味であったとしても・・・
Massive spin-2理論の構築を試みることで、masslessspin-2理論がどう特別なのか理解できる
高次元時空を研究することで、4次元時空(の特殊性)をさらに深く理解できる
宇宙物理・天文とは直接の関係がなくても、物理(場の理論、幾何学、、、)としては面白
いこともあるなぜ修正重力？
重力を拡張(修正)するとは？
“まとも”な重力理論に要請されること
近年の研究の動向 ‒‒ Horndeski & beyond
GW170817以降の修正重力
まとめ

11/54

Lovelockの定理 (1971, 1972)
4次元
一般相対論 (+宇宙項)
ラグランジアン密度は計量だけに依存: L[gµ⌫ ]<latexit sha1_base64="o6C3otua+JrNKZUbsgi0yE/c4zU=">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</latexit>
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2階の運動方程式<latexit sha1_base64="IRhchKDeuCMGUANo4ISWcqUwSJk=">AAADVXicjVJNa9RAGH67W2u7fuxWLwUvwaVSC5bJHrQIQmkFPVRot25baMoyyc6uQyeTkGRXasgf6B/w4ElBUPwZXjx5k9qfIB4reFHxyWzqV2lxQjLvPO/7PDNP5nVDJeOEsYORUnn0zNjZ8YnKufMXLlZrk5fW46AfeaLlBSqINl0eCyW1aCUyUWIzjAT3XSU23J2lPL8xEFEsA/0w2Q3Fts97WnalxxNA7dqi44qe1ClXsqdns0rqeFxZy9kdpxtxL7Wz1L7phNK6l800bzScZSh3+PWKI3TniNOu1dkcM8M6HthFUKdirAS1H+RQhwLyqE8+CdKUIFbEKcazRTYxCoFtUwosQiRNXlBGFlXA7qNOoIYD38G3h9VWgWqsc9XY8D3so/BGhjvNPrDX7JC9Y2/YJ/btFLXUqOTn2cXsDtkibFf3pta+/gfPx5zQo988wzmZdVQfIdvFm51an3tNUDdvPEp4Dg2Su/eGew6ePD1cu92cTq+xF+wzfD9nB+wtnOvBF+/lqmg+K1xosB6b/+SbU2jczSxUHboLpdREETIW8OzXDZzM8nEWZZgPqP0He8WgebvY/zbH8WC9MWcjXm3UFxaLxhmnK3SVZtAdt2iB7kOvhf1f0Xvap4+l/dL38mh5bFhaGik4l+mvUa7+BKsDwxY=</latexit>
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Lovelockの定理 (1971, 1972)
4次元
ラグランジアン密度は計量だけに依存: L[gµ⌫ ]<latexit sha1_base64="o6C3otua+JrNKZUbsgi0yE/c4zU=">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</latexit>
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2階の運動方程式

計量以外の場(スカラー場、ベクトル場、テンソル場、、、)を導入する
スカラー・テンソル理論
ベクトル・テンソル理論
etc.

Lovelockの定理 (1971, 1972)
4次元
ラグランジアン密度は計量だけに依存: L[gµ⌫ ]<latexit sha1_base64="o6C3otua+JrNKZUbsgi0yE/c4zU=">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</latexit>
sha1_base64="RGRO13WahVsN2qMj6NPHdn5hiig=">AAADYHicjVK7btRAFL1Z81gWwm5Cg6BZsQqKIrSaTUOSagUUFETKg00irVeW7UycUcZjy4+NguUf4AcoqECiQJR8Ag0/QLFtqBBlkGgQ4njWCY8oEWPNzJlz77l3ruc6oRRxwthoomJcuHjpcvVK7eq1yev1xtT0Rhykkct7biCDaMuxYy6F4r1EJJJvhRG3fUfyTWfvYWHfHPIoFoF6mhyEfODbnhI7wrUTUFZj2XS4J1RmS+GpubyWmTpmP/KcQcbai4v3TpYcNls2n+R9z8pMPzVVmg/ymsnV9rHcarRYm+nRPA06JWh1aTxWgsZPMmmbAnIpJZ84KUqAJdkU4+tThxiF4AaUgYuAhLZzyqlJNahT+HH42OD3sHo49UtW4VxEjbXeRR6JGWntDPvE3rIj9pG9Y1/Yj3OiZTpKcZ8D7M5YzUOr/vzm+vf/0PnYE9r9rdOas1XH/hGsO5j5uf5FrQn8FnSNAjWHmimqd8c5h89eHK0vrc1kd9lr9hV1v2Ij9gGVq+E3980qX3tZVqGg2tf/yde3UHibOUQ16REiZRpFsDTB5ycvcLbKx12kVi6T9Yd6RbNFu3T+bY7TYGO+3QFenW91H5SNU6XbdIdm0R33qUuPEa+H/O9pRIf0uXJoVI26MTV2rUyUmhv01zBu/QLqI8bm</latexit>

2階の運動方程式

高次元
高次元空間の“体積”は4次元から見たら新しい場Lovelockの定理 (1971, 1972)
4次元
ラグランジアン密度は計量だけに依存: L[gµ⌫ ]<latexit sha1_base64="o6C3otua+JrNKZUbsgi0yE/c4zU=">AAADYHicjVK7btRAFL1Z81iWx26gQdCsWAWhCK1m00CoIqCgIFIebBJpvbJsZ+KMMh5bY3tRsPwD/AAFFUgUiJJPoOEHKLYNFaIMEg1CHM864RElYqyZOXPuPffO9VwvliJJGRtP1axTp8+crZ9rnL9w8VKzNX15LYky7fO+H8lIb3huwqVQvJ+KVPKNWHM39CRf93YelPb1EdeJiNSTdDfmw9ANlNgSvpuCclqLtscDoXJXikDNFo3cNjEHOvCGOevOz98+XArYXNl+XAwCJ7fDzFZZMSwaNlebB3Kn1WFdZkb7KOhVoEPVWIpaP8mmTYrIp4xC4qQoBZbkUoJvQD1iFIMbUg5OAwlj51RQmxpQZ/Dj8HHB72ANcBpUrMK5jJoYvY88ElMb7Qz7xN6yffaRvWNf2I8TouUmSnmfXezeRM1jp/n86ur3/9CF2FPa/q0zmuNVB/4a1i3M4kT/stYUfndNjQI1x4Ypq/cnOUfPXuyv3luZyW+y1+wr6n7FxuwDKlejb/6bZb7ysqpCQfXU/KfQ3ELhbWYR1aaHiJQbpGFpgy8OX+B4VYi7SKNcJOcP9ZJhy3bp/dscR8HaXLcHvDzXWbhfNU6drtMNuoXuuEML9Ajx+sj/nsa0R59re1bdalrTE9faVKW5Qn8N69ovSKPGpg==</latexit>
sha1_base64="RGRO13WahVsN2qMj6NPHdn5hiig=">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</latexit>

2階の運動方程式

高階の運動方程式

新しい場を導入するのと同じ.... ...
x + x + ẍ + · · · = 0ẍ = y,<latexit sha1_base64="pBmuzDK1uRPjUWefkfRJsVshVRo=">AAADVXicjVJNaxRBEK3sxhg36q7xIngZXCISIdTmoghCiB5yEfLhJoFMWGZmO5smPT3DTO9qMuwf8A948BRBUPIzcvHkTWJ+gnhMwIuKb3onfoUEa5ip16/qVXdNlx8rmRrmw6FSefjCyMXRS5Wxy1euVmvXxpfTqJsEohlEKkpWfS8VSmrRNNIosRonwgt9JVb8rUd5fKUnklRG+qnZjsV66HW03JCBZ0C1arOuLzpSZ56SHT3Zr7htWGSc585dt0A5GPgAPnUeOlxxhW6faFq1Ok+xNec0aBSgToXNR7Uf5FKbIgqoSyEJ0mSAFXmU4lmjBjHF4NYpA5cASRsX1CeHKlB3kSeQ44HfwreD1VrBaqzzqqnVB9hH4U2sdoI/8js+4ve8x5/52znVMlslP882vD9Qi7hVfXFj6et/6EJ4Q5u/dVZztuokP0F0A2//3Py8V4O8+7ZHiZ5jy+TdB4M9ezsvj5YeLE5kt/k1f0Hfu3zI++hc946DNwti8VXRhYbqmf1PoT2Fxt1MoqpLj1EpsyhBxAHf/3UDZ6tCnEVZ5RNq/aGet2w+Lo1/h+M0WJ6eagAvTNdnZovBGaWbdIvuYDru0QzNoV4T+7+lD3RAn0oHpe/l4fLIILU0VGiu019Wrv4EZ+LC/g==</latexit>
sha1_base64="hgsGhFO8YPTzcmQweuTCV/NBDyw=">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</latexit><latexit sha1_base64="vLBpf5GrJbhSJOozX9Cd/KVlLtc=">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</latexit>
sha1_base64="DH0ZXl1jDZVy9u/UcIR2brcSHiw=">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</latexit>

例: f (R) 重力<latexit sha1_base64="TWhXgwC/2QF+dfmMMtaRQfpC3Ao=">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</latexit>
sha1_base64="O8uqqgVq2eR0mmtQfJW/dBplYjk=">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</latexit>ÿ + ẏ + · · · = 0

スカラー・テンソル理論

スカラー・テンソル理論
多くの修正重力理論は、少なくともeﬀectiveにはスカラー・テンソル理論で記述される:<latexit sha1_base64="1CijZDOpzsqzkcg9PFt8BWtFmLs=">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</latexit>
sha1_base64="m6QREmmHOSqP8uiq8e4UqkA2xIc=">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</latexit>

L = L[gµ⌫ , ]テンソル自由度2 (重力波の+とx) + スカラー自由度1
スカラー場が曲率ないしは物質と(重力と同じ O(GN ) で)非最小結合することで重力が修正される:<latexit sha1_base64="8d82vF8c9nwzSqZQ8vjmEXHBQ/c=">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</latexit>

2<latexit sha1_base64="nFpFEop/BKzdl89D5r4KOi3oQnw=">AAADdXicjVLLbtNAFL2NeZTwaIANEkKyCK1KFZVJNiDYVMCCBYg2JW2lTLBsZ+KOOjO2xpOgYvkH+AEWrEBigfgMNv0BFv0EhMSmSGGBENcTl1fVirHsOXPuPXfmeG6QCJ4aQnanKs6x4ydOTp+qnj5z9txM7fyFtTQe6pB1wljEeiPwUya4Yh3DjWAbiWa+DARbD7buFfH1EdMpj9UTs52wnvQjxQc89A1SXi2gAYu4ynzBI7WQVwfzNNnk19sNGvZjkzbonYyGvnAf5l5GtXSlbwzTeZc+lizyn7Ym6REG5ZCqYd6gScp7VcpUf7+mV6uTRWKHexA0S1CHcizHtR9AoQ8xhDAECQwUGMQCfEjx6UITCCTI9SBDTiPiNs4gBxeqqB5iHsMcH/kt/Ea46paswnVRNbX6EPcR+GqrnSUfyTuyR3bIe/KJfD+iWmarFOfZxjmYqFnizby4tDr+D53E2cDmb53VHK7az9cYHeCbH5lfeDWYd8t65Og5sUzhPpzsOXr+cm/1dns2myNvyGf0/Zrskg/oXI2+hm9XWPtV6UKh6pn9T9KeQuHdLGBVCvexUmaRxoiLfP7rBg5XSTyLsMpH4P2hXrZs0S7Nf5vjIFhrLTYRr7TqS3fLxpmGy3AV5rE7bsISPMB6Hdx/B77AGL5Vxs4V55ozN0mtTJWai/DXcG78BKVC0DY=</latexit>
sha1_base64="y7hufFiGGHdlWIV3qZ2N/TIGZjo=">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</latexit>f ( )R, · · · , Lmatter [⌦ ( )gµ⌫ , ]Tµ⌫ がいると、gµ⌫ だけでなく<latexit sha1_base64="LBbrlsyamPilOZdlNjDncta98Qw=">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</latexit>
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sha1_base64="AuZZqbcbKu0BgcSmn4ndFkNfvJg=">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</latexit>Gµ⌫ ⇠ @ gµ⌫ ⇠ GN Tµ⌫ , ⇤2<latexit sha1_base64="LR+eg0gjDla5L1yCN73LX4YLG0M=">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</latexit>
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sha1_base64="c6va19IIXTQr/rHLaC76K43Vwrk=">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</latexit>

F~ ⇠ F~grav~
mrFifth forceµ⇠ GN Tµも励起される<latexit sha1_base64="VmtuCxcJUQKtz93w6Cuemp2fbzk=">AAADM3icjVK7TtxAFL04PJfXEhokmhUrEKJAszSJqFCSIg0SrwUkjFa2GZYR47FlexcRa9sU+YEUVCBRRKElP5CGH6BA6SIahKhAokGI41nzFoix7Dlz7j135niu7UsRRowdNRjvGpuaW1rbMu0dnV3d2Z7386FXCRxedDzpBYu2FXIpFC9GIpJ80Q+45dqSL9jrn5P4QpUHofDUXLTp82XXKiuxKhwrAlXK5kybl4WKLSnKaqSWMf01kTG5WrllStk8G2V65J6DQgrylI4pL3tNJq2QRw5VyCVOiiJgSRaFeJaoQIx8cMsUgwuAhI5zqlGOMlBXkMeRY4Ffx7eM1VLKKqyTqqHWO9hH4g20dpAdsl/snB2w3+yEXb1SLdZVkvNsYrbrau6Xun/0zV6+QedijmjtXqc1L6tu8wNEV/HWXs1PvEbI+6g9Cnj2NZO4d+p7Vr/9PJ8dnxmMh9gOO4XvbXbE/sK5ql44u9N8Zit1oaDa0P/J1adQuJsRVDXpCyrFGgWI5MDX7m7gZZWLs0itnKTSA/WUZpN2KTxtjudgfmy0ADw9lp/4lDZOK/XTAA2jOz7QBH1FvSL2/057tE9/jH3jn/HfOK6nGg2pppceDePsBkUVuXQ=</latexit>
sha1_base64="FkhNrPt4s/+Ghit+EGGGTJSe+J4=">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</latexit>なぜ修正重力？
重力を拡張(修正)するとは？
“まとも”な重力理論に要請されること
近年の研究の動向 ‒‒ Horndeski & beyond
GW170817以降の修正重力
まとめ

17/54

“まとも”な重力 Consistent, healthy, viable, …
(加速膨張を実現するとか、そういう目的にかなうことは当然として…)

然るべき極限で一般相対論/標準宇宙論を再現する
ニュートン重力、太陽系実験 (post-Newtonian)、重力波
CMB etc.
Ostrogradskyゴーストがない(安定性、後述)

宇宙論解(などこの世に安定に存在しているものを表す解)
が安定
etc.

“まとも”な重力 Consistent, healthy, viable, …
(加速膨張を実現するとか、そういう目的にかなうことは当然として…)

然るべき極限で一般相対論/標準宇宙論を再現する
ニュートン重力、太陽系実験 (post-Newtonian)、重力波

初期宇宙に適用するならどうでもよい要請
CMB etc.

Ostrogradskyゴーストがない(安定性、後述)

初期宇宙でも重要な要請
宇宙論解(などこの世に安定に存在しているものを表す解)
が安定
etc.

スクリーニング機構
スカラー自由度が励起されていると、光の曲がりなどに抵触
物体のまわりでスカラー自由度を“隠す”必要|F~ | ⌧ |F~grav |(⇤<latexit sha1_base64="EPhlZa71j6ONPfPaj6GQBSU+Fdg=">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</latexit>
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m2e↵ ) ⇠ GN Tµµ~m|eff⇠r |F~grav ||F⇠ (GN M/r)e<latexit sha1_base64="StG2AIyCY5Vp2kHOYPlvX1xKjjg=">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</latexit>
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スクリーニングの2つのアイデア: 物体のまわりで…
i. Eﬀectively massiveGµ⌫ ⇠ GN Tµ⌫ , ⇤ ⇠ ge↵ Tµµii. Eﬀectively weakly coupled<latexit sha1_base64="X9rJH8xUQuhVl/30LwyYy5EQ5BA=">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</latexit>
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スクリーニング機構 (i)カメレオン/シンメトロン機構 = “eﬀectively massive”
00
Lmatter [⌦2 ( )gµ⌫ , ] ! Ve↵ = Ve↵ ( , ⇢) ! m2e↵ = Ve↵( , ⇢)<latexit sha1_base64="wYvBs+K65IHgGUw2NMZJ46XsWQs=">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</latexit>

スカラーのeﬀective massがエネルギー密度に依存
高密度環境下ほど me↵ % となるポテンシャルならOK<latexit sha1_base64="V4l2fUMeErFsUA0jteJEHYN+yGs=">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</latexit>f (R) 重力はこのタイプ:<latexit sha1_base64="thtZiw7TD9nDrhIB0CXfNXG6eeU=">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</latexit>f (R) の関数の形 , Ve↵ ( , ⇢) の関数の形<latexit sha1_base64="eQZanlKMxpmpiJoeVR7KR136nPg=">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</latexit><latexit sha1_base64="thtZiw7TD9nDrhIB0CXfNXG6eeU=">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</latexit>Ve↵ ( , ⇢)<latexit sha1_base64="u3GGIcDZHi+rJ559M0eV6tj3NBg=">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</latexit>⇢小<latexit sha1_base64="kU1SKz8pPETFtkgizQAT7qEB4d0=">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</latexit>⇢大<latexit sha1_base64="kU1SKz8pPETFtkgizQAT7qEB4d0=">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</latexit>

スクリーニング機構 (ii)
Vainshtein機構 = “eﬀectively weakly coupled”
スカラーの非線形微分相互作用 (“ガリレオン”タイプの理論)⇤
1 ⇥(@ )2 + rc2 (@ )2 ⇤ + Tµµ
典型的な例: L =16⇡G<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>⇤ ⇠ G⇢

rc2 2<latexit sha1_base64="(null)">(null)</latexit>

12ge↵ ⇠ G/(rc ⇤)⌧G<latexit sha1_base64="k1ViBQI7jG/9eH79ZZzPZNXSQOU=">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</latexit>

Vainshtein半径<latexit sha1_base64="(null)">(null)</latexit>

rc2 ⇤ ⇠ rc2 · GM/r3

1<latexit sha1_base64="(null)">(null)</latexit>

Screened<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>rc ⇠ H 0 1 , M ⇠ M

rc2 ⇤ ⌧ 1

rV := (rc2 · GM )1/3

) rV ⇠ 100 pc

Ostrogradskyの定理
古典力学

L = L(q̇, q)<latexit sha1_base64="OUCq8WmllW06Iv2lHZbPPPbPHao=">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</latexit>
sha1_base64="ZMW25ccvFPfVjvTLq1cURGWiEis=">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</latexit>

運動方程式は2階 (2個の初期条件): 1自由度

L = L(q̈, q̇, q)<latexit sha1_base64="x3gn+zkieNxLith43Y6m+XHYRWo=">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</latexit>
sha1_base64="zNY3n9I8Kbb9amgx/mmHbr7sBDY=">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</latexit>

運動方程式は4階 (4個の初期条件): 1+1自由度「高階EOMを与えるラグランジアンにより記述される系には、一般にゴースト(不安定な自由度)が現れる」
Ostrogradsky (1850); Woodard 1506.02210

ラグランジアンの形を制限する指針

解析力学の例<latexit sha1_base64="Aq438gMTkqTelZ/IUI0CHNyzIoQ=">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</latexit>
sha1_base64="sp7MNTHg+0DanzFod/ZCr9WJO4M=">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</latexit>

1 2
L = q̈
2

U (q)

1
2

L = q̈
=

˙ q̇

2

U (q)

V (q, )

<latexit sha1_base64="tIzutGzcXmAgc5aZn/U1Xeyto48=">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</latexit>
sha1_base64="tRmZIL6/UpKUysPX6qwoMlgafGQ=">AAADxXiclVJLb9NAEB7XPIp5NMAFxCUiahSqJlpbJbhIFRFwCBJIfZC2UjZEtrNJV7XXju0EFcvizh/gwAkkDogjP4ELf4BDfwLiWCQuCDHeuLyqVrDWeme/me/bmd2xA5dHMSG7ypR67PiJk9OntNNnzp6bKZy/sB75o9BhLcd3/XDTtiLmcsFaMY9dthmEzPJsl23Y23cy/8aYhRH3xcN4J2AdzxoI3ueOFSPULbyjNhtwkVguH4i5VLtfXtISKnXb4cDuJKRmLtRNw5gnNYPU6+YCGrq+aF4nKXW2eEp7PT8uDqu0H1pOoqeJkf4P/5GRVluV4TWNUq28VKUolsF0IrpeGc5nW3Qz0dtPslsokRqRo3jQ0HOj1IDJWPYL34FCD3xwYAQeMBAQo+2CBRF+bdCBQIBYBxLEQrS49DNIoQgaskcYxzDGQnwb/wPctXNU4D5TjSTfwXNcnKHkzpKP5A3ZIx/IW/KJfDtCLZEqWT47uNoTNgu6M88urX39B56Hawxbv3iSczhrPz5Ebx9nemR8VmuMcaaskWPNgUSy6p3JmeMnz/fWbq7OJmXyinzGul+SXfIeKxfjL87rFbb6Iq9CIOuxvCdPZiHwbeZQlcJdVEqkFaKniHj68wUOZ3mYiyuZD6D7G3tZolm76H83x0GjZdQWa/qKUWrczvtmGq7AVahgc9yABjRRrgWOclm5pTSVe2pTFWqsjiehU0rOuQh/DPXpD3yJ4jM=</latexit>1
変数変換: q = p ( 1 +
2

2 ),

1
=p ( 1
2

V ( 1,

2)<latexit sha1_base64="liFxlWC+RgRDL4XABb/WhxMU0cs=">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</latexit>
sha1_base64="/V6GZtKz9wfUFLeP7V6lQlb7htE=">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</latexit>L=<latexit sha1_base64="idx/fFteyNh/DkgNyYmyA9depNI=">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</latexit>
sha1_base64="SJ+z1RxuxH6wyrr3T7SAaTk9n/E=">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</latexit>

1 ˙2 1 ˙2
1 +
2
2
2

2)

(2自由度)

運動エネルギーが負の自由度 (= ゴースト)

宇宙論解の安定性
Ostrogradsky不安定性がないように理論を作ったとしても、宇宙論解など安定であってほしい解が安定とは限らない
摂動の2次の作用から判定できる:<latexit sha1_base64="(null)">(null)</latexit>

gij = a2 (t) {[1 + 2⇣(t, ~x)] ij + hij (t, ~x)}
曲率揺らぎ(スカラー摂動)<latexit sha1_base64="(null)">(null)</latexit>hL(2) = a3 A(t)⇣˙ 2

a 2 B(t)(@⇣)2

重力波(テンソル摂動)i(重力波も同様)

理論が与えられば計算できる係数(時間依存)

ゴーストがない , A(t) > 0<latexit sha1_base64="(null)">(null)</latexit>2
伝搬速度が実数 , cs (t) := B/A > 0<latexit sha1_base64="(null)">(null)</latexit>なぜ修正重力？
重力を拡張(修正)するとは？
“まとも”な重力理論に要請されること
近年の研究の動向 ‒‒ Horndeski & beyond
GW170817以降の修正重力
まとめ

26/54

スカラー・テンソル理論の統一的な枠組
計量とスカラー場からなり、運動方程式が2階になる
最も一般的なラグランジアンは？
<latexit sha1_base64="k47f8GLin8WoAHkmZOJCZa6BDJs=">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</latexit>
sha1_base64="Gs1PnaMDQ6w0zAcjX4Y+qlrUjJI=">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</latexit>

L = L[gµ⌫ , @gµ⌫ , @ 2 gµ⌫ , · · · , , @ , @ 2 , · · · ]実は、1974年にHorndeskiにより作られていたHorndeskiは、その後ゴッホの絵に感銘を受けて物理をやめ、画家に転身したため、論文は忘れられていた検索技術(Google)の進展により、2011年に発掘された
Horndeski, Int. J. Theor. Phys. 10, 363 (1974)

Horndeski理論
L = G2 ( , X)<latexit sha1_base64="xIAKrElTsTC1R2V6a7vPG/xcWR0=">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</latexit>
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G3 ( , X)⇤⇥⇤2
2+ G4 ( , X)R + G4X (⇤ )(rµ r⌫ )
⇥
⇤
1
µ⌫
3
2
3
+ G5 ( , X)G rµ r⌫
G5X (⇤ )3⇤ (rµ r⌫ ) + 2(rµ r⌫ )6<latexit sha1_base64="fqb1Jf3JmXAyc1EJ3S9FQlC7EYM=">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</latexit>
sha1_base64="TH8WYRJ8HRMUJ0Er+suTEygjzzo=">AAADM3icjVIxSxxBFH5uYtSLiWdshDRHDkOwOGZtFKvDWNgE7jSngivH7jqeg7Ozy+7eiS7XpvAPWFglkEJMq3/Axj9gIXaSRiSVgTQh5Nu5NZqIkrfMzpvvve/NfDPPCaSIYsZOu4xHj7uf9PT25Z72P3s+kB98MR/5zdDlNdeXfrjo2BGXQvFaLGLJF4OQ254j+YKz/jaNL7R4GAlfvY83A77s2Q0lVoVrx4Dq+YLl8IZQiS1FQ422c1awJnIWVyvXSD1fZCWmrXDXMTOnWKaOVfz8L7JohXxyqUkecVIUw5dkU4RviUxiFABbpgRYCE/oOKc2FSgHdhN5HDk28HX8G1gtZajCOq0aab6LfSRGqLkj7ITtsSt2zPbZBfv5QLVEV0nPs4nZ6bB5UB/YHp778R88D3NMazc8zbmfdZ0fIrqK0X4wP9UaI29CaxTQHGgkVe929mxt7VzNTc6OJK/ZJ3YJ3R/ZKTuCctX67n6u8tndTIUCa0Pfk6dPofA2o6hq0TQqJdoLESkAb/95gftZHs4iNfMd1W+xKxpN28X8tznuOvNjJdMsmdWxYnkqa5xeekmv6A26Y5zKNIN6Nez/gb7QAR0aB8aZcW587aQaXRlniP4y49tv5ue5tQ==</latexit>と X :=g µ⌫ @µ @⌫ /2 の4つの任意関数<latexit sha1_base64="QfVe7qXabVVr68XZQNZY1VwpIYo=">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</latexit>
sha1_base64="jayjperGfHbdhc3EdFrliUDbUyQ=">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</latexit>

すべてのスカラー・テンソル理論を統一的に記述することができる
それまではモデルごとにやっていた計算(例: 密度揺らぎの発展方程式の導出など)を効率化
モデルに依らない一般的性質を見つけ出すのに有用

Rij = 0

Everything that can beconserved
is<latexit sha1_base64="fqnqzQQiyssdnYZk77pUeoYoPdM=">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</latexit>
sha1_base64="PJdD8LXBelHlkj4RvNRse0gqt7U=">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</latexit><latexit sha1_base64="C7cNO6oxiSoAe3LGKUbI6J4AAIk=">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</latexit>
sha1_base64="UKtfSfWvluu3W4x/OT8YMyVpuFg=">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</latexit>p~i =スカラー・テンソル理論の空間

2011~

Horndeski(2階の運動方程式)
一般相対論 (2自由度)

高階の運動方程式

Ostrogradskyゴースト

TK, M. Yamaguchi, J. Yokoyama (2011)

応用1: 宇宙論的摂動と安定性
作用を摂動について2次まで展開
例えば重力波(テンソル摂動)の場合…

3
a
B(t)
2
2
L=
A(t)ḣij
(@
h
)
k ij
2
4
a
<latexit sha1_base64="(null)">(null)</latexit>

A := G4
B := G4<latexit sha1_base64="(null)">(null)</latexit>⇣

2XG4X X H ˙ G5X⇣⌘
X ¨G5X + G5

G5⌘

重力波の伝搬速度

安定性の判定B
2
cGW =A > 0, c2GW > 0

A<latexit sha1_base64="(null)">(null)</latexit>

2c
どういう理論で GW 6= 1 なのか<latexit sha1_base64="(null)">(null)</latexit>

系統的に理解できる<latexit sha1_base64="(null)">(null)</latexit>

便利な公式集みたいなもの

応用2: インフレーション(single-ﬁeld)インフレーション = 計量とスカラー場で記述される系 = Horndeski理論 TK, M. Yamaguchi, J. Yokoyama (2011)“修正重力”インフレーションは今や当たり前
0.20

Planck Collaboration: Constraints on Inflation

ConTT,TE,EE+lowE+lensing
TT,TE,EE+lowE+lensing+BK14
TT,TE,EE+lowE+lensing+BK14+BAO
Natural inflation
Hilltop quartic model
attractors
Power-law inflation
R2 inflation

ve

ve

0.10

0.15

ca

V

2

V
V

4/3

2/3
V
Low scale SB SUSY
N =50
N =60

0.05
0.00

Tensor-to-scalar ratio (r0.002 )

Co
n

x

0.94

0.96
Primordial tilt (ns )

0.98

1.00

Planck 2018

Fig. 8. Marginalized joint 68 % and 95 % CL regions for ns and r at k = 0.002 Mpc 1 from Planck alone and in combination with
BK14 or BK14 plus BAO data, compared to the theoretical predictions of selected inflationary models. Note that the marginalized
joint 68 % and 95 % CL regions assume dn /d ln k = 0.

このあたりのモデル<latexit sha1_base64="(null)">(null)</latexit>

L = R + ↵R<latexit sha1_base64="(null)">(null)</latexit>

L

2

⇠ 2R

こういうモデルを統一的に議論できるTK (2016); S. Akama, TK (2017,2018) S. Akama = 赤間進吾 (立教大D1)

応用3: 特異点のない宇宙は不安定
インフレーションの代替として初期特異点のないシナリオ(バウンス、サイクリック、etc.)が提案されているが…
Horndeski: 計量 + スカラー場の系、運動方程式が2階

特異点のない宇宙論解は必ず不安定[証明] 「摂動の安定性の条件:
A(t) > 0, c2s (t) > 0, · · ·<latexit sha1_base64="(null)">(null)</latexit>

は、特異点のない宇宙では 1 < t < 1 のどこかで必ず破れてしまう」ということを示せる<latexit sha1_base64="(null)">(null)</latexit>

その他、応用いろいろ
線形密度揺らぎの進化

計量摂動の空間成分¨ + 2H ˙ = 4⇡Ge↵ (t; k) ⇢,時間成分= (t; k)<latexit sha1_base64="(null)">(null)</latexit>

どういう変更が現れ得るのか/便利な公式集みたいなもの
A. De Felice, TK, S. Tsujikawa (2011)

ブラックホール摂動
重力波とスカラー波
球対称時空の安定性

etc.

TK, H. Motohashi, T. Suyama (2012, 2014)

Zumalacárregui et al. arXiv:1605.06102

J

Horndeski対応ボルツマンコード
ournal of Cosmology and Astroparticle Physics

An IOP and SISSA journal

hi class: Horndeski in the Cosmic
Linear Anisotropy Solving Systema Nordita, KHT Royal Institute of Technology and Stockholm University,

JCAP08(Roslagstullsbacken 23, SE-106 91 Stockholm, Swedenb Berkeley Center for Cosmological Physics, LBL and University of California at Berkeley,
Berkeley, CA 94720, U.S.A.
c Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg,
Philosophenweg 16, 69120 Heidelberg, Germanyd Institut de Ciènces del Cosmos, Universitat de Barcelona, IEEC-UB,
Martı̀ i Franquè 1, E-08028 Barcelona, Spaine Astrophysics, University of Oxford,
Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, U.K.
f Central European Institute for Cosmology and Fundamental Physics,
Fyzikálnı́ ustáv Akademie věd ČR,
Na Slovance 2, 182 21 Praha 8, Czech Republicg Institut für Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University,
D-52056 Aachen, Germany
E-mail: miguelzuma@berkeley.edu, emilio.bellini@physics.ox.ac.uk,
Figure 2. E↵ect of the braiding on CMB temperature power spectra (left panel) and the matter powerignacy.sawicki@fzu.cz,
lesgourg@physik.rwth-aachen.de,
spectrum at z = 0 (right panel). In the bottom panels we show the relative di↵erence between the

JCAP08(2017)019

Miguel Zumalacárregui,a,b,c Emilio Bellini,d,e Ignacy Sawicki,fJulien Lesgourguesg and Pedro G. Ferreiraeスカラー・テンソル理論の空間

Horndeski(2階の運動方程式)
一般相対論 (2自由度)

高階の運動方程式

Ostrogradskyゴースト

当
本

?
…

縮退理論: beyond Horndeski「運動方程式が2階」は「Ostrogradskyゴーストがない」
ための十分条件
ループホール: 縮退理論
= 運動方程式は高階だが、組み合わせると2階に落ちる
2
2
2
ẍ2 p
ẏ
x
y
2
L
=
+
2ẍ
ẏ
+
ẋ
+
解析力学の例:2222<latexit sha1_base64="HCH+rAMnjKh/AhXRiYfVlyEraqI=">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</latexit>
sha1_base64="i55Vk0EJYeyngxVwzlhWswjdltE=">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</latexit>

運動方程式は高階だが…
.... p ...
x + 2 y ẍ x = 0 · · · (1)p... p
x + 2ÿ + y/ 2 = 0 · · · (2)2階の系に落ちるpẍ + ẏ/ 2 + x = 0pÿ
2ẋ + y = 0<latexit sha1_base64="Fr6NXsk9umjWk2GFbtUGyIMKJmE=">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</latexit>
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係数がそろっている<latexit sha1_base64="hpHQI17tdLp7kFA9N2h1KP8Zou0=">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</latexit>
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d
(2)
dt

(1), · · ·縮退理論: beyond Horndeski
Degenerate Higher-Order Scalar-Tensor Theory
Langlois & Noui (2015); Crisostomi, et al. (2016)(DHOST)<latexit sha1_base64="wFwPDvN1hA1WFUnS0EYn3diDYlw=">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</latexit>
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sha1_base64="CyLxPWOvUQRj+jrcxWFj4+IaJOs=">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</latexit>の多項式であることを仮定

rµ r ⌫<latexit sha1_base64="w3MGZ470sYRgHTlDRbeG2qat2yQ=">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</latexit>
sha1_base64="IdbJxlPaAU4qRR4q+6zZXbmetCY=">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</latexit>µ := rµL = · · · + f ( , X)R + A1 ( , X) µ⌫ µ⌫ + · · · + A5 ( , X)( µ ⌫ µ⌫ )2 + · · ·縮退 , 係数の関数の間にいくつかの関係式<latexit sha1_base64="wDcbiPIc8ftTy5z8XY5WYWox42c=">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</latexit><latexit
sha1_base64="lY3gEXmjEYXkd5kgAXP4UOrOnu4=">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</latexit>

完全な一般形はわかっていない2015~
2+1自由度の縮退理論⇠ (rr )4 ?, (rr )5 ?, · · ·Horndeski(2階の運動方程式)
一般相対論(2自由度)

Quadratic & cubic DHOST(高階の運動方程式)

Ostrogradskyゴーストのある理論

= Horndeskiとdisformal変換でつながる理論
安定な宇宙論解はの内部にのみ存在

Horndeski＊disformal変換(2階の運動方程式)

gµ⌫<latexit sha1_base64="mPwJPGr914YQdyZJXW15J9b4L04=">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</latexit>
sha1_base64="6neg+Fi9JUUW+A4ytQf37i8axY8=">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</latexit>

場の再定義

一般相対論(2自由度)

Quadratic & cubic DHOST

A( , X)gµ⌫ + B( , X)@µ @⌫<latexit sha1_base64="nx17kFMgYLywcJjxmw59UyAl2jk=">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</latexit>
sha1_base64="U66sNXAWiq841YJ2kccjOmFC08o=">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</latexit>(高階の運動方程式)= 宇宙論解が必ず不安定
de Rham & Matas (2016)

さらに一般的な理論を
systematicに作って調べたが、かなり普遍的に不安定なようだ…

K. Takahashi & TK (2017)
K. Takahashi = 高橋一史 (学振PD@立教)

Extended mimetic gravity(高階の運動方程式)

Horndeski(2階の運動方程式)

Quadratic & cubic DHOST(高階の運動方程式)= 宇宙論解が必ず不安定Horndeski理論、およびそれと(可逆)disformal変換でつながっている理論以外、宇宙論解が不安定 (状況証拠)

Horndeski(2nd-order EOMs)–の中の理論だけ調べれば十分(?)

なぜ修正重力？
重力を拡張(修正)するとは？
“まとも”な重力理論に要請されること
近年の研究の動向 ‒‒ Horndeski & beyond
GW170817以降の修正重力
まとめ

43/54

GW170817

2|cGW<latexit sha1_base64="YlSQXyfzUOz7k6iEqzmtnluha9I=">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</latexit>
sha1_base64="fIIeLmbda7hFoFgABH7E7OXP/ok=">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</latexit>

1| < O(1015)

任意関数のいくつかに非常に強い制限

GW170817以降

Horndeski(2nd-order EOMs)
GR (2 dofs)<latexit sha1_base64="kKRBujegymKBocimzrJgFI0mm/8=">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</latexit>
sha1_base64="UQxzZcC8HWTNjWPAnSdJKk5OYhc=">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</latexit>

cGW = c in

GW170817とHorndeski
L = G2 ( , X)

G3 ( , X)⇤⇥⇤2
2+ G4 ( , X)R + G4X (⇤ )(rµ r⌫ )
⇥
⇤
1
µ⌫
3
2
3
+ G5 ( , X)G rµ r⌫
G5X (⇤ )3⇤ (rµ r⌫ ) + 2(rµ r⌫ )6<latexit sha1_base64="xIAKrElTsTC1R2V6a7vPG/xcWR0=">AAAEbnicjVLdatRAFD5pV63xp62CCCIGS8u2dcvsbqsiCKUK64VCf2y72GnDJJ1NQyeTmE2qNeQFfAEvvFLwQnwK8cYX8KJPIOJNpYI3Ip5Ms9tq3cUJkzl/3zffmRkrEG4zImRH6+ktHDt+ou+kfur0mbP9A4Pnlpp+HNp80faFH9Yt1uTClXwxciPB60HImWcJvmxt3snyy1s8bLq+fBhtB3zVY450G67NIgyZg9ojanHHlQkTriPHUj2hNhPG/XTkds2sFGmw4V4z6qN6qWZW2x6d8Z9mtk6lHzHHoNQYoY9jtm7o4zVzsl03n7nJZD2lgjeilWILN7pWKRWpZJZgJvXiliXjPElD19mIVv9NP9Wmr60lCh2nnbj0Em2EzE7KaXI9RSlTKMVyHXFYSlUvVVteN1H6eKVLuprxhiiZy/XWWZoDQ2SCqGEcNcq5MQT5mPUHfgGFdfDBhhg84CAhQlsAgyZ+K1AGAgHGViHBWIiWq/IcUjBAR3SMdRxrGMY38e+gt5JHJfoZa1PhbdxH4AwVdph8Im/JHvlI3pEv5GcXtkSxZHq2cbX20Tww+59fXPjxHzgP1wg2DnAK0xnVqg8x28CZdq3Peo2w7qbq0cWeAxXJurf399x69mJv4db8cDJCXpOv2PcrskM+YOdy67v9Zo7Pv8y7kIh6os7JUyok3s0YslK4i0yJskLMGBhP2zfQGeWhFqGQD8A8hJ5V0ey5lP9+HEeNpcpEmUyU5ypD0zP5w+mDS3AVivg6bsA03EO+RbC199pnbVf71rtbuFC4XLiyX9qj5Zjz8McoFH8DpKoi8A==</latexit>
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0c2GW =G4

G4 X( ¨G5X + G5 )2XG4X X(H ˙ G5X G5 )Creminelli & Vernizzi;
Sakstein & Jain;
Ezquiaga & Zumalacarregui;
Baker, et al. (2017)<latexit sha1_base64="JX3r49UJtA0CsVzskM5SLkXThCI=">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</latexit>
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sha1_base64="4iVWmG26r0/HqyNqndLyGg9ZdDA=">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</latexit>

L = G4 ( )R + G2 ( , X)

G3 ( , X)⇤

注1: 「スカラー・テンソル理論の虐殺が起こった」などとTwitterに書かれていたが、殺されたのは目新しい理論ばかりで、昔からあるような理論は無傷だった
注2: z~0.01の話なので、初期宇宙における重力への制限にはならない

GW170817とDHOST
L = f ( , X)R + G2 ( , X)
+ A3 ( , X)⇤<latexit sha1_base64="2+vHkUrouoaBGlZkHD1QE8xfuBc=">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</latexit>
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sha1_base64="cAcrB92YRd1KYQzTIV0tcpao3e8=">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</latexit>

de Rham & Melville, arXiv:1806.09417

しかし、微妙な点もある
この手の理論の典型的なカットオフスケール<latexit sha1_base64="Lsw/PMqTbfWe8TSULYAD+VDCOwE=">AAADX3icjZK9btRAEMcnZyDhgMRAA6KxOAWFFMf4KBJBEwHFFUHKB5dEihPL9m2OVXbX1tp3KLH8ArwABRVIFIiOV6DhBZBIGTpEGSQahBjvOXxFiVjL3tn/zG92xzthIniaIe6N1KxTp8+Mjp2tnzt/YXzCvnhpJY37OmKdKBaxXguDlAmuWCfjmWBriWaBDAVbDbfvl/7VAdMpj9WjbCdhGzLoKb7FoyAjybfnvZD1uMoDwXtquqh788R2A8dLuZzyZCKcto+brZubuXvrdlGqjovo3c09LZ32LgFMdQ9p325gE81wjhpuZTSgGgux/QM86EIMEfRBAgMFGdkCAkjpWQcXEBLSNiAnTZPFjZ9BAQ7Uie5THKOYgPRt+vZotV6pitZl1tTwEe0j6NWGncQP+BoP8D2+wc/4/YRsuclSnmeH5nBIs8SfeHpl+dt/cJLmDB7/5gxzPHUYr8m7RW9xYnxZa0Zxs6ZGTjUnRimrj4Z7DnafHSzfWZrMb+BL/EJ1v8A9fEeVq8HX6NUiW3peVaGIemL+kzSnUHQ305TVgweUKTeWJo9DevHrBo6nJJ1FGPIh+H/QC0Yt28X9tzmOGiutpus23cVWY+5e1ThjcA2uwxR1xwzMQZvydWj/t/AR9uFTbd8atcYtexhaG6mYy/DXsK7+BJ9FxV4=</latexit>
sha1_base64="tdQoUGaglIbOr1I9WRXfLaBMdDM=">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</latexit>

2 1/3⇤ ⇠ (MPl H0 )⇠ 100 Hz

LIGOの観測は、ダークエネルギー/修正重力の低エ
ネルギー有効理論の記述が微妙になってくるあたりを見ているのかもしれない (= 制限にならない)

PHYSICAL REVIEW LETTERS 121, 221101 (2018)

insensitive to the sLB backg[28] and hence high-frequenc
luminal. The reason this isbecause we are working w
only consistent at frequenc
cutoff, M. Interestingly, in th
detection, the frequency of th
few hundred Hz, which is p
coupling scale associatedenergy models [29],GW170817以降
Horndeski<latexit sha1_base64="kKRBujegymKBocimzrJgFI0mm/8=">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</latexit>
sha1_base64="UQxzZcC8HWTNjWPAnSdJKk5OYhc=">AAADOnicjVK/axRBFH5ZfyWXaC7aCDaHR0JIccyl0QQChwraCPnheYFsWHYnk3PI7Owyu3cSl/0HLGwtrCJYiK1tbGz8ByxSW4WUCaQR8du5jb9Cgm+ZnTffe9+b+WZeECuZpIztDTkXLl66fGV4pDI6dvXaeHXi+tMk6hku2jxSkVkN/EQoqUU7lakSq7ERfhgo0Qm27hfxTl+YREb6Sbodi/XQ72q5KbmfAvKq024gulJnvpJdPZNXuJe5Jqw97OQLvOIKvXES8ap11mDWaqedZunUWzSwxaj6g1zaoIg49SgkQZpS+Ip8SvCtUZMYxcDWKQNm4EkbF5RTjSpg95AnkOMD38K/i9VaiWqsi6qJ5XPsozCM5U6yr+w9O2Rf2Ae2z76fUy2zVYrzbGMOBmwRe+Mvb64c/wcvxJzSs988yzmbdZJvEN3EyM/NL7SmyLtrNUpoji1SqOeDPfsvXh+uzC9PZlPsLTuA7h22xz5Due4f8XdLYvlNqUKD9dzeU2hPofE2M6jq0gNUyqxnEKkBz3+9wNmsEGdRlvmYvD/YixYt2qX5b3OcdtqzjblGc2m23rpX9s0w3aLbNI3muEMteoRybWz/ij7SLn1ydp1vzr5zMEh1hkrODfrLnKOfBby8jg==</latexit>

cGW = c in(2nd-order EOMs)
GR (2 dofs)

DHOST

このクラスの理論をテストするのは面白い

重力が物質内部でのみ変更される(星の内部などでスクリーニングが不完全)
TK, Y. Watanabe, D. Yamauchi (2015)

DHOSTのテスト
Newtonian極限 ‒ 内部でのみ重力変更
2<latexit sha1_base64="PpQtmPDMFMFef/O8AutNH9xuSWs=">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</latexit>
sha1_base64="DMgkq0DGH9cYN6px0A1QEHyEoPc=">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</latexit>

d
GM
Gd M
= 2 +⌥drr
4 dr2

外はスクリーンされていて変更なし(光の曲がりなどではテスト不可能)星の構造からの制限

2/3 < ⌥ < 1.6<latexit sha1_base64="Hj+cHj6rmbHgai4mZe9lx0+Ldgc=">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</latexit>
sha1_base64="/7+Y0nVR5K/xmMK/y8p8BIPIxf0=">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</latexit>星の中心で重力が引力であるべし
Saito et al. (2015)

水素燃焼が起こる最小質量< 観測されているred dwarfの最小質量
Sakstein (2015)

TK & T. Hiramatsu (2018)

FIG. 3: The mass (µ) versus central density diagram for
FIG. 5α = 0. The parameters are given by β = 2 × 10−3 , 10−3 , 4 ×
The pa10−4 , 2 × 10−4 , 10−4 , 0 (GR) (solid lines, from top to botlines)
tom), and β = −10−4 , −2×10−4 , −4×10−4 , −10−3 , −2×10−3
represeNewtonianではほとんど違いが出ないようなパラメータ(dashed lines, from top to bottom).

DHOSTと相対論的天体

でも大きく変わる (ただし、モデル依存性はある)
4<latexit sha1_base64="b1pBWSFeshpkmF60U+vRf70lDSY=">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</latexit>
sha1_base64="v4l5VdwV7/TOwihDnwKJ5Qc3044=">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</latexit>

↵=0

3
2.5
<latexit sha1_base64="yoNjNXoYrUEIw73iX3NQYqo+s5Q=">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</latexit>
sha1_base64="IWx7YW5/MDkSlzaOdYK3Tx7UBmA=">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</latexit>µ/Msol

質量 [M̲solar]

3.5= 2 ⇥ 10 3

2
1.5
1

=

GR

<latexit sha1_base64="pc3gCG/dwv9GtO6PoAseJgoPC+k=">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</latexit>
sha1_base64="0KgIMV+Vad6L0/zQH4Uz0hkfBXE=">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</latexit>

2 ⇥ 10 3

0.5
0

8

10

12

14
R [km]16

18

20

半径 [km]DHOSTの宇宙論的テスト
DHOST対応ボルツマンコードを平松君が開発中
T. Hiramatsu, in preparation

大規模構造
k1
‒ folded (<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

k2)でバイスペクトルを変更

k3

S. Hirano, TK, H. Tashiro, S. Yokoyama (2018)
S. Hirano, TK, D. Yamauchi, S. Yokoyama, in preparation
S. Hirano = 平野進一 (立教大D2)

重力波がスカラーに崩壊 (したら届かないということから制限)
P. Creminelli et al. 1809.03484<latexit sha1_base64="(null)">(null)</latexit>

hij

生き残っていた4つの関数のうちの1つ = 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>なぜ修正重力？
重力を拡張(修正)するとは？
“まとも”な重力理論に要請されること
近年の研究の動向 ‒‒ Horndeski & beyond
GW170817以降の修正重力
まとめ

まとめ
現状、一般相対論でよいが、拡張する動機は十分にある

一般的・統一的な理論の枠組(Horndeski/beyond Horndeski)* 有効理論の適用範囲なのかは要注意[現在の宇宙] GW170817以降、viableな理論はある程度絞られたが、まだ面白い理論が生き残っている[初期宇宙] “修正重力”インフレーションは今や当たり前