第50回 天文・天体物理若手夏の学校 8/24/2020

一般相対論の拡張と検証

小林 努
立教大学

今日の話

c = ~ = 1,

2
8⇡G = MPl

なぜ一般相対論の拡張理論を考えたいのか？
近年の“重力理論 beyond GR”界隈の動向
主にダークエネルギーへの応用を念頭に置きながら、理論的な枠組と実験的・観測的検証の話題
今後の展望
…など
今日の話を勉強したい人向けの良いレビュー論文
T. Clifton, et al., Modified Gravity and Cosmology,
Phys. Rept. 513 (2012) 1 [1106.2476]
A. Joyce, et al., Beyond the Cosmological Standard Model,
Phys. Rept. 568 (2015) 1 [1407.0059]
TK, Horndeski theory and beyond: a review,
Rept. Prog. Phys. 82 (2019) 8, 086901[1901.07183]

目次
1. Why: 一般相対論を拡張する動機
2. How: 一般相対論を拡張するとはどういうことか
3. Theory: 一般的な重力理論の枠組
4. Tests: 重力理論の検証 (重力波の話題を中心に)
5. まとめ1. Why: 一般相対論を拡張する動機
2. How: 一般相対論を拡張するとはどういうことか
3. Theory: 一般的な重力理論の枠組
4. Tests: 重力理論の検証 (重力波の話題を中心に)
5. 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

以降、適当に拾ってきた
クレジットのよくわからん絵は消しておきます。

Planck動機2: 一般相対論は低エネルギー
有効理論に過ぎない高エネルギー/強重力、どこかで一般相対論による記述は必ず破綻する。これは絶対にそう！
物理をやるからには量子重力に一度は興味持つでしょ？
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」でない可能性が十分にある
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私自身の動機は…

重力を深く理解したい が大部分
次点で 一般相対論がどこまで正しいかテストしたい

一般相対論が低エネルギー有効理論に過ぎないのは当たり前
ダークエネルギーが修正重力かも、などとは本心では全く考えていない1. Why: 一般相対論を拡張する動機
2. How: 一般相対論を拡張するとはどういうことか
3. Theory: 一般的な重力理論の枠組
4. Tests: 重力理論の検証 (重力波の話題を中心に)
5. まとめLovelockの定理 (1971, 1972)
4次元, diﬀ. invariant
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="IRhchKDeuCMGUANo4ISWcqUwSJk=">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</latexit>
sha1_base64="fWX03+2KFvXoieyiKe4QcJ615II=">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</latexit>

1
L=
(R
16⇡G

2階の運動方程式

ここにアインシュタイン方程式の絵があった

2⇤)

Lovelockの定理 (1971, 1972)
4次元, diﬀ. invariant
ラグランジアン密度は計量だけに依存: L[gµ⌫ ]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, diﬀ. invariant
ラグランジアン密度は計量だけに依存: L[gµ⌫ ]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, diﬀ. invariant
ラグランジアン密度は計量だけに依存: L[gµ⌫ ]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>
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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=">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</latexit>
sha1_base64="y7hufFiGGHdlWIV3qZ2N/TIGZjo=">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</latexit>f ( )R, · · · , Lmatter [⌦ ( )gµ⌫ , ]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=">AAADenicjVLLbtNAFL2NgZbwaGg3SGyiRkVQqug6Cq2LhFQBCxYg9UHaSnFk2e7UHXU8tvwIKpZ/gB9gwQokFojPYMOGJYt+AmIZJDYBcT1xeFWtGMueM+fec2au5zqh4HGCeDRR0c6cPTc5db564eKly9O1KzNbcZBGLuu4gQiiHceOmeCSdRKeCLYTRsz2HcG2nYP7RXy7z6KYB/JJchiynm97ku9x106Ismqe6TCPy8wW3JMLeTUzXVvUH+V3x6DrWZnpp6ZM80XiCsNu5Dm9DJtGe8loG4vYbOHSslEAXV9Zud3OzXCf572qyeTu2NiqNbCJatSPA70EDSjHWlD7ASbsQgAupOADAwkJYQE2xPR0QQeEkLgeZMRFhLiKM8ihDlVSp5THKMcm/oC+Hq26JStpXbjGSu/SPoLeSGnn8RO+xQF+wHf4GYenuGXKpTjPIc3OSM1Ca/r51c1v/6HzaU5g/7dOaU5WjfMjiu7Rm5+aX9SaUJ6hauRUc6iYonp3tGf/2YvB5p2N+ew6vsYvVPcrPML3VLnsf3XfrLONl2UVklRP1X/y1Skk3c0CuZrwgJwyhSKK1InPf93AySqfziKU8jFYf6jXFFu0i/5vcxwHWy3quCautxqr98rGmYJrMAc3qDuWYRUekl+H9v8IAxjC98pQm9NuardGqZWJUjMLfw2t/RM06s8M</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=">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</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=">AAADOXicjVK7TtxAFL04L7KEsEmaSGlWrEARBZqlAaVChIIGidcCEkaWbYbNaMdjy4+NwPIPRPQUqYiUAtGlhY4mP5CCMh2QciPRIJTjWZPwEChj2XPm3HvuzPFcJ5Aiihk76jIePHz0+En301LPs97nfeUXLxcjPwldXnd96YfLjh1xKRSvxyKWfDkIue05ki85zfd5fKnFw0j4aiHeCPiqZzeUWBeuHYOyyoOmwxtCpbYUDTWUlRas1PQSUyVZyeRq7ZK3ylU2zPSo3Aa1AlSpGDN++YJMWiOfXErII06KYmBJNkV4VqhGjAJwq5SCC4GEjnPKqEIlqBPkceTY4Jv4NrBaKViFdV410noX+0i8odYOsB9sl7XZd7bHTtj5PdVSXSU/zwZmp6PmgdX36fX82X/oPMwxffin05q7VZf5IaLreLN783OvMfLGtEcBz4FmcvduZ8/W5nZ7/t3cQDrIvrBf8L3DjtghnKvWb/frLJ/7XLhQUH3U/8nTp1C4myFUNWkSlVKNQkQq4LO/N3C3ysNZpFZOk3VFPaPZvF1qN5vjNlgcGa4Bz45UxyeKxummN9RPb9EdozROU6hXx/5b9I326cDYN34ax8ZpJ9XoKjSv6Now2n8ATrW8Ow==</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>
sha1_base64="83tsZg8gfo0wKLJ3rsUEMs+7ibM=">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</latexit>µ⇠ GN Tµも励起される<latexit sha1_base64="VmtuCxcJUQKtz93w6Cuemp2fbzk=">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</latexit>
sha1_base64="FkhNrPt4s/+Ghit+EGGGTJSe+J4=">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</latexit>

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=">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</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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F~ ⇠ F~grav~
mrFifth forceµ⇠ GN Tµも励起される<latexit sha1_base64="VmtuCxcJUQKtz93w6Cuemp2fbzk=">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</latexit>
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“まとも”な重力理論(加速膨張を実現するとか、使用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”な重力理論(加速膨張を実現するとか、使用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.

注1
例えば、インフレーションを記述する理論
Ostrogradskyゴーストがないに対し、太陽系実験の制限を満たすことを(安定性、後述)
要求するのはナンセンス
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”な重力理論(加速膨張を実現するとか、使用目的にかなうことは当然として…)

注2
然るべき極限で一般相対論/標準宇宙論を再現する
不安定モードの質量が非常に重かったり、不安定性の成長のタイムスケールが非常にニュートン重力、太陽系実験(post-Newtonian)、重力波
長かったりすれば気にする必要はない
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.

太陽系(など)実験
ds2 =[1 + 2 (~x)]dt2 + [1
重力ポテンシャル

一般相対論:

=

GN M
r

=

シャピロ時間遅れ・光の曲がり
PPNパラメータ

:=

/ に制限

重力定数の精密測定|ĠN /GN | < O(10 3 ) ⇥ H0
(Lunar laser ranging, Mars ephemeris)

弱重力場での重力理論検証

2 (~x)]d~x2
空間の曲がりtaken from

Ostrogradskyの定理
古典力学

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

変数変換:<latexit sha1_base64="liFxlWC+RgRDL4XABb/WhxMU0cs=">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</latexit>
sha1_base64="/V6GZtKz9wfUFLeP7V6lQlb7htE=">AAADf3icjVLLbtNAFL2JeRTzaCgbJDYRUVEoJRpnQ4uEFAELNkh9EFqpU1njySQddTx2xk5QsfwD/AALViCxQHwGG/aIRT8BsSwSQkKI64nLq0rFWPbcOfecM3M9N4iVTFJC9itV58TJU6dnzrhnz52/MFu7OPc4iUaGiy6PVGQ2A5YIJbXopjJVYjM2goWBEhvB7r0ivzEWJpGRfpTuxWI7ZAMt+5KzFCG/FtJADKTOmJIDvZC7wzu0bxjPvDyjydCkWTvPmzROpO/dsFP7+qJLhyPWcynfkdPZN0u2S4XuHdr7tQZpETvqRwOvDBodmIyVqPYDKPQgAg4jCEGAhhRjBQwSfLbAAwIxYtuQIWYwkjYvIIc6uKgeIU8ghyG+i98BrrZKVOO6cE2snuM+Cl9jtfPkI3lDDsh78pZ8It+PccusS3GePZyDiVrE/uyzy+tf/0MX4pzCzm+d1UxXHfINZvv45sfyi1pT5C3ZGiXWHFukqJ5P9hw/fX6wfnttPrtGXpHPWPdLsk/eYeV6/IW/XhVrL8oqNKqe2P8U2lNovJsFdKVwH50yGxnM1BHPf93AdFWIZ1FW+RD8P9QrFi3axfu3OY4G3XZrueWtthudu2XfzMAVuApNbI5b0IEHaNfF7T/At0q14jhVp+m0HDKhViul5hL8NZzlnyV/0WQ=</latexit>L=<latexit sha1_base64="idx/fFteyNh/DkgNyYmyA9depNI=">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</latexit>
sha1_base64="SJ+z1RxuxH6wyrr3T7SAaTk9n/E=">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</latexit>

1
q= p ( 1+2
1 ˙2 1 ˙2
1 +
2
2
2

2 ),

V ( 1,1=p ( 12
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次のラグランジアンから判定できる:<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
伝播速度が実数 , cs (t) := B/A > 0<latexit sha1_base64="(null)">(null)</latexit>

1. Why: 一般相対論を拡張する動機
2. How: 一般相対論を拡張するとはどういうことか
3. Theory: 一般的な重力理論の枠組
4. Tests: 重力理論の検証 (重力波の話題を中心に)
5. まとめ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？ (4次元以下で)<latexit sha1_base64="k47f8GLin8WoAHkmZOJCZa6BDJs=">AAADpnicjVJPa9RAFH9p1Nb4p6teBA8uLpW1lGWyF0tBKOrBg0r/uN3CZhuT2Wk6dDIJyWSlhnwBrx48eFLwIH4ML34BD/0I4rGCCCK+zGZra21xQjLv/d7v9+a9zPNjwVNFyK4xYZ46fWZy6qx17vyFi9O1S5fX0ihLKOvQSETJuu+lTHDJOoorwdbjhHmhL1jX375XxrtDlqQ8kk/UTsz6oRdIvsmppxByay8dnwVc5p7ggZwtrNyhnqg/LO6MjZ4VuLkTZo7Mijkn9hLFEf8HttE+CNJBpNI5y4m3+D7jkLPRHrma2LccJgfjItxag7SIXvWjhl0ZDajWUlT7BQ4MIAIKGYTAQIJCW4AHKT49sIFAjFgfcsQStLiOMyigDhaqM+Qx5HiIb+M3QK9XoRL9Mmuq9RTPEfgmWjtDPpP3ZI98Ih/IF/LzhGy5zlLWs4O7P1Kz2J1+cXX1+3/oQtwVbP3Rac3xqjE/wegmvsWJ/LJXhbx53SPHnmONlN3T0ZnD56/2VhdWZvKb5C35in2/IbvkI3Yuh9/ou2W28rrqQqLqmf5Poa5C4t3MYlYH7mOmXFsJRuqIF/s3cLwqxFqEVj4C94B6SaPluNh/D8dRY63dsu2WvdxuLN6tBmcKrsENaOJ03IZFeID5Onj+D+O60TRumU3zsdkxuyPqhFFprsChZT79DetW4Vk=</latexit>
sha1_base64="Gs1PnaMDQ6w0zAcjX4Y+qlrUjJI=">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</latexit>

L = L[gµ⌫ , @gµ⌫ , @ 2 gµ⌫ , · · · , , @ , @ 2 , · · · ]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？ (4次元以下で)<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
2011年に「一般化ガリレオン」(今日知られている形)として再発見
Horndeski, Int. J. Theor. Phys. 10, 363 (1974)

Horndeski理論
L = G2 ( , X)<latexit sha1_base64="xIAKrElTsTC1R2V6a7vPG/xcWR0=">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</latexit>
sha1_base64="91uvfA53Z/R5CJHVoOmawnlum8E=">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</latexit>

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=">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</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2階のスカラー・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1974

Horndeskiにより定式化
以降、37年もの間、完全に忘れ去られる

2011

Deﬀayet, et al. 1103.3260 「一般化ガリレオン」

2011 Charmousis, et al. 1106.2000 Horndeski発掘(「ググった」と言っていた)2011 TK, Yamaguchi, Yokoyama 1105.5723
Horndeski理論と「一般化ガリレオン」の等価性の証明
2020年現在、未達成
高次元への拡張
複数スカラー場への拡張-

誰か考えてください！

(2スカラーのEOM: Ohashi, Tanahashi, TK, Yamaguchi 1505.06029)

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

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
解析力学の例:<latexit sha1_base64="HCH+rAMnjKh/AhXRiYfVlyEraqI=">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</latexit>
sha1_base64="i55Vk0EJYeyngxVwzlhWswjdltE=">AAAD5XiclVLNa9RAFH9p/Kjxo6teBC+LS0W6ukyWtU0FYVEPHhT64baFzXZJsrPp0GQSJ9m1a8g/IHgTPXgQBQ/in+HFs9BD/4TisYIXEV8mqdqWFp2QvDe/936/eS/z7NBjUUzIljKmHjt+4uT4Ke30mbPnJkrnLyxFwUA4tOUEXiBWbCuiHuO0FbPYoyuhoJZve3TZXr+bxZeHVEQs4I/iUUg7vuVy1meOFSPULX0xbeoynlgec/lUqj24bfaF5SSJKbXbwrU7CakZjWmjYVwntTqZnjEyR9dnZ282UrPXC+Lyxmo9TZN6WjWjxyJG5z/paNGMqmYhplXzKnIYxVHyRg5t7NkVMc2kvLfbQ7dUITUiV/mgoxdOpQn5mgtKP8GEHgTgwAB8oMAhRt8DCyJ82qADgRCxDiSICfSYjFNIoQwasgeYRzHHQnwdvy7u2gXKcZ+pRpLv4DkevkJyJ8km+UB2yGfykWyTH0eoJVIlq2eE1s7ZNOxOPLu0+P0feD7aGNb+8CTncNZuvsBoH9/0yPys1xjzDNkjw55DiWTdO/mZw6evdhZvLUwmV8k78hX7fku2yCfsnA+/Oe/n6cLroguOrCfyP/myCo53M4WqJtxDpUR6AiNlxNPfN3A4y8daPMl8CN2/2HMSzcZF3z8cB52lOo5sTZ+vV5p3isEZh8twBa7hdMxAE+6jXgscxVBWFVdZU131ufpCfZmnjikF5yLsWeqbX7lq8hE=</latexit>

2
ẍ2 p
ẏ
L=
+ 2ẍẏ + ẋ2 +22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)

x2
2

y2
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>
sha1_base64="WoQn1DjeB3vPy+se/OY0+wqB8EM=">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</latexit><latexit sha1_base64="OjbTGtF92Ii9mxgvsT/08EeBivA=">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</latexit>
sha1_base64="ycXVLFW+FMm4duucdeeqG+w92Zw=">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</latexit>

係数がそろっている<latexit sha1_base64="hpHQI17tdLp7kFA9N2h1KP8Zou0=">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</latexit>
sha1_base64="hgZrqHD+VGxfZeRrHf1iG7v4ZpM=">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</latexit>

d
(2)
dt

(1), · · ·縮退理論: beyond Horndeski
Degenerate Higher-Order Scalar-Tensor Theory(DHOST) Langlois & Noui (2015); Crisostomi, et al. (2016); Ben Achour, et al. (2016)<latexit sha1_base64="wFwPDvN1hA1WFUnS0EYn3diDYlw=">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</latexit>
sha1_base64="ADCyRcd1ISWtYC8mjyf3LFE6kQM=">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</latexit><latexit sha1_base64="0h+GJMAuWVDOb7Dvc9FdL4GR1Lk=">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</latexit>
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>⇠ det(. . . ) = 0
3次の多項式まで分類済み、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=">AAADOXicjVI9TxVBFL2sgvj4ekhjYvPCC4RQvMzSYKiIWtCY8OEDEpZsdpdhnTA7u9mPZ3Czf8DYW1hpYmHsbLGz8Q9YUNopls+ExhjOzltAIRDvZnbunHvPnTkz142kSFLGDvuMGzf7B24N3q4NDY+MjtXH76wnYRZ7vO2FMow3XSfhUijeTkUq+WYUcydwJd9w9x6W8Y0OjxMRqifpfsS3A8dXYld4TgrIrk9bLveFyh0pfDVb1Hw7t4LMUllRs7jaOcXtepO1mLbGZcesnCZVthzW/5BFOxSSRxkFxElRCl+SQwm+LTKJUQRsm3JgMTyh45wKalAN7Ax5HDkO8D38fay2KlRhXVZNNN/DPhIj1twp9pW9Z132hX1gP9jva6rlukp5nn3Mbo/NI3vsxd214//gBZhTenrO05yrWaf5MaK7GMW1+aXWFHn3tUYBzZFGSvVeb8/O81fdtYXVqXyavWU/ofsNO2SfoVx1fnnvVvjq60qFAuuZvqdAn0LhbWZR1aJHqJRrL0akAbw4e4GrWQHOIjXzMdl/sZc1WraLebE5Ljvrcy3TbJkrc83FB1XjDNI9mqQZdMc8LdIS6rWx/0v6SAf0yTgwvhnfjaNeqtFXcSboHzO6J4OTvE8=</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); Langlois, et al. (2017)

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)

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

DHOST

–

の中の理論だけ調べれば十分(?)

最近の話題: 4D Gauss-Bonnet?
Gauss-Bonnet項:

LGB = Rµ⌫⇢ Rµ⌫⇢

4Rµ⌫ Rµ⌫+R

2

D≥5ではEOMに寄与するが、D=4では全微分
4D Gauss-Bonnet from coupling rescaling?

Gµ⌫ + ↵Hµ⌫ in D ≥ 5
0

↵
↵=
D 4
0

and

Gµ⌫ + ↵ (???)µ⌫

(Hµ⌫

2

@ gµ⌫ )

D!4
in D = 4

Glavan & Lin 1905.03601

最近の話題: 4D Gauss-Bonnet?
今年の3月から4月にかけて大量の論文が出たが…

Gµ⌫ + ↵0 (???)µ⌫
Lovelockの定理によると、4次元で、 gµ⌫ だけから作られ、⌫r(???)µ⌫ = 0 を満たすテンソル
2階微分について準線形、は (アインシュタインテンソルと計量以外に) 存在しない
つまり、Glavan & Linの理論はwell-deﬁnedでない
スカラー・テンソル理論(Horndeski理論の一例)として書くことができ、well-deﬁnedにできる
Lu & Pang 2003.11552; TK 2003.12771; Fernandes, et al. 2004.08362; Hennigar et al.,2004.09472; Bonifacio, et al. 2004.10716; See also Aoki, et al. 2005.03859

ここまでは理論の一般的な枠組作りの話「計量+スカラー場」の系は宇宙論・宇宙物理で頻出
例えば、インフレーションなど多方面への応用が可能

1. Why: 一般相対論を拡張する動機
2. How: 一般相対論を拡張するとはどういうことか
3. Theory: 一般的な重力理論の枠組
4. Tests: 重力理論の検証 (重力波の話題を中心に)
5. まとめGW170817

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

1| < O(1015)(ダークエネルギーへの応用を念頭に置くと…)

任意関数のいくつかに非常に強い制限

GW170817以降

Horndeski
GR (2 dofs)

DHOST<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=">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</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>
sha1_base64="CmmaAe2w/FM/6TvZ1b9+BuV3dmw=">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</latexit><latexit sha1_base64="OAC4+pQ+7gKNtehtaFjkGoyNbqE=">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</latexit>
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>
sha1_base64="lFNVPrMdjB3pjEniP2NOha+lysQ=">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</latexit>µ ⌫G3 ( , X)⇤µ⌫ + · · ·( , X) の任意関数4つまで追い詰めた<latexit sha1_base64="lfe1OYv862XDnK33kwrYhSMvonQ=">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</latexit>
sha1_base64="cAcrB92YRd1KYQzTIV0tcpao3e8=">AAADN3icjVI9T9xAEB2ckMDlg4M0kdKcuIAIik5rGiKqU0hBE4mvg5MwOtlmOVas15btu4hY9wf4AxRUIFFEadKSNg1/gIKKDoFSIZDSIJTnPQNJEChjrXf2zbzZfbvjBFJEMWMHHcaDh52PHnd15548ffa8J9/bNxf5jdDlFdeXflh17IhLoXglFrHk1SDktudIPu+sjqfx+SYPI+Gr2Xgt4IueXVdiWbh2DKiWf205vC5UYktRV8Ot3JAVrIi31Tc5i6ulK7SWL7IS01a47ZiZUyxT2yb9/CVZtEQ+udQgjzgpiuFLsinCt0AmMQqALVICLIQndJxTiwqUA7uBPI4cG/gq/nWsFjJUYZ1WjTTfxT4SI9TcAbbPvrAztse+shN2cU+1RFdJz7OG2WmzeVDrWX858+s/eB7mmFZueJpzN+sqP0R0GaN1b36qNUbeO61RQHOgkVS9296z+XnjbGZseiAZZNvsJ3RvsQP2A8pV89zdmeLTm5kKBdYnfU+ePoXC2wyjqkUfUCnRXohIAXjr+gXuZnk4i9TMj1T7gz2p0bRdzH+b47YzN1IyzZI5NVIsv88ap4teUT8NoTtGqUwTqFfB/uv0jXbpu7FrHBpHxnE71ejIOC/oLzNOfwPjYrqy</latexit>

Graviton decay

Creminelli et al. (2018, 2019)「重力波がスカラーに崩壊しない」という制限<latexit sha1_base64="GG3/RgUBpMyGlWstDDRfmoVUcBU=">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</latexit><latexit sha1_base64="WCtqxFc8PjHzM30oYJzF4a1a07I=">AAADbXicjVLNahRBEK7s+BPXn6yKICgyuERjDktvgiieFvXgQSE/bhJMx6FntrPbSU9P09O7Eod5AV/AgycFD+JLCF58AQ95AUE8RIjgRcSa3kn8CQn20NNVX9X3dVd3hVqK1BKyOVLxDh0+cnT0WPX4iZOnxmqnzyykSd9EvB0lMjFLIUu5FIq3rbCSL2nDWRxKvhiu3yniiwNuUpGoh3ZD85WYdZVYFRGzCAW1RzTkXaEyJkVXTebVjEZM+vfzx9nE9LWcpn2dcuvTTiexfi/IxFpONTNWMBkIqnti11srvCrlqrOjFdTqpEHc8PcazdKot2A4ZpLaT6DQgQQi6EMMHBRYtCUwSPFbhiYQ0IitQIaYQUu4OIccfKgiu495HHMY4uv476K3XKIK/UI1dfwI95E4jeOOk4/kDdkmH8hb8pn8OEAtcyrFeTZwDYdsroOxZ+fnv/8HL8bVQu83z3H2Z+3kG4yu4swPzC9qtZh309UosGbtkKL6aLjn4Onz7flbc+PZFfKKfMG6X5JN8h4rV4Nv0etZPveirEIh64m7p9idQuHbTKIqhbuolDnLYMRHPN99gf1ZMZ5FOuYDCP5gzzi0aJfmv82x11iYajSnG9dnp+qt22XjjMIFuAwT2B03oAX3UK+N+7+DT7AFXytb3jnvondpmFoZKTln4a/hXf0FbarOZg==</latexit>L(3)ḧij @i @jhij<latexit sha1_base64="GLY0NaefP6iV5HHR8XrdyESi7ws=">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</latexit><latexit sha1_base64="GG3/RgUBpMyGlWstDDRfmoVUcBU=">AAADM3icjVK/SxxBFH6u+WEuMZ6mCdgcOQxiccydRcTqUIs0AX/kVHDl2F3Hu8HZ2WV370SXay38B1KkUkgRYqv/gE3+gRRiJ2lEUinYiOTbudVERfEts/Pme+97M9/Ms30pwoixgw6j88nTZ8+7XmRevup+3ZPt7ZsNvUbg8IrjSS+Yt62QS6F4JRKR5PN+wC3XlnzOXhlP4nNNHoTCU5+jNZ8vulZNiWXhWBGgajZn2rwmVGxJUVNDrYzp10XG5GrpCqlm86zAtOXuOsXUyZepbZNe9pJMWiKPHGqQS5wURfAlWRTiW6AiMfKBLVIMLIAndJxTi3KUAbuBPI4cC/gK/jWsFlJUYZ1UDTXfwT4SI9DcAfaLfWen7Cf7wY7ZxQPVYl0lOc8aZrvN5n61Z/PtzPkjeC7miOr/eJpzP+sqP0B0GaP1YH6iNULeiNYooNnXSKLeae/ZXP9yOjM6PRC/Z9vsBLq32AHbh3LVPHO+TfHpr6kKBdaqvidXn0LhbYZQ1aQJVIq1FyCSA966foH7WS7OIjXzE1X/Y09qNGmX4u3muOvMlgrF4UJpqpQvj6WN00X99I4G0R0fqEwfUa+C/Tdoh3Zpz9g1Do0j43c71ehIOW/ohhl//gLn4bm4</latexit>(dimensionless function) . 10 10<latexit sha1_base64="pBt9NxMNTQToSG5fn1dNEHk2GH0=">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</latexit>

L = G2 ( , X)<latexit sha1_base64="N3MrUvxbuXhm4LqAyi2oo4QIcBc=">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</latexit>

A3 = 02
3fXµ
G3 ( , X)⇤ + f ( , X)R +rµ Xr X
2f

<latexit sha1_base64="FCy6RawDYCze0qBySUUYI0eCigU=">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</latexit>( , X) の任意関数3つまで追い詰めた<latexit sha1_base64="lfe1OYv862XDnK33kwrYhSMvonQ=">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</latexit>
sha1_base64="cAcrB92YRd1KYQzTIV0tcpao3e8=">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</latexit>

GW170817以降
Horndeski<latexit sha1_base64="kKRBujegymKBocimzrJgFI0mm/8=">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</latexit>
sha1_base64="UQxzZcC8HWTNjWPAnSdJKk5OYhc=">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</latexit>

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

DHOST

No graviton decay

de Rham & Melville 1806.09417

有効理論の適用範囲に注意！
ダークエネルギーの代替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=">AAADX3icjZK9btRAEMcnZyDhgMRAA6KxOAWFFMf4KEDQnIDiiiDlg0sixYll+zbHKrtra+07lFh+AV6AggokCkTHK9DwAkikDB2iDBINQoz3HL6iRKxl7+x/5je7450wETzNEHfHataJk6fGJ07Xz5w9Nzlln7+wnMYDHbFuFItYr4ZBygRXrJvxTLDVRLNAhoKthFv3S//KkOmUx+pRtp2wdRn0Fd/kUZCR5NtzXsj6XOWB4H01W9S9OWJ7geOlXM54MhFOx8eN1vWN3L1xsyhVx0X07uaelk5nhwCmege0bzewiWY4hw23MhptGI352P4BHvQghggGIIGBgoxsAQGk9KyBCwgJaeuQk6bJ4sbPoAAH6kQPKI5RTED6Fn37tFqrVEXrMmtq+Ij2EfRqw07jB3yN+/ge3+Bn/H5MttxkKc+zTXM4olniTz29tPTtPzhJcwaPf3OGOZo6iNfk3aS3ODa+rDWjuNumRk41J0Ypq49Gew53nu0v3Vmczq/hS/xCdb/AXXxHlavh1+jVAlt8XlWhiHpi/pM0p1B0N7OU1YMHlCk3liaPQ3rx6waOpiSdRRjyIfh/0PNGLdvF/bc5DhvLrabrNt2FVqN9r2qcCbgCV2GGuuMWtKFD+bq0/1v4CHvwqbZnjVuTlj0KrY1VzEX4a1iXfwJA1MWe</latexit>

2 1/3⇤ ⇠ (MPl H0 )⇠ 100 Hz

LIGOの観測は、ダークエネルギー/修正重力の有効理論としての妥当性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 background. LI should be restored[28] and hence high-frequency δϕ waves should be exactly
luminal. The reason this is not manifest in Eq. (3) isbecause we are working within the EFT (2), which isonly consistent at frequencies much smaller than the
cutoff, M. Interestingly, in the context of the GW170817
detection, the frequency of the GWs span from 24 Hz to afew hundred Hz, which is perilously close to the strong
coupling scale associated with many Horndeski darkenergy models [29],→ さらに低周波での観測が重要パリティ対称性の破れ
重力セクターのパリティ対称性の破れは一般に重力波の伝播速度を変えてしまう
L

(2)

↵ ijk
✏ ḣil @j ḣkl + 3 ✏ijk @ 2 hil @j hkla⇤a ⇤微分展開のleading orderはこの2項で尽きている
2
|cGW

k
1| ⇠
|↵
a⇤

Chern-Simons重力は特別な場合( ↵ =Creminelli, et al. (2014)| . 1015

A. Nishizawa & TK 1809.00815)で、制限がつかない⇠ "µ⌫⇢ Rµ⌫↵ R↵ ⇢波形の情報を使うとさらに強い制限

Yamada & Tanaka 2006.11086

弱重力場のふるまい: overview
Vainshtein機構 = “eﬀectively weakly coupled”
スカラーの非線形微分相互作用が(弱場でも)重要⇤
1 ⇥典型的な例: L =(@ )2 + rc2 (@ )2 ⇤ + Tµµ
16⇡G<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>⇤ ⇠ G⇢

rc2 2⇤ ⇠ ge↵ ⇢1<latexit sha1_base64="k1ViBQI7jG/9eH79ZZzPZNXSQOU=">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</latexit><latexit sha1_base64="(null)">(null)</latexit>

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

rc2 ⇤ ⇠ rc2 · GM/r3

ge↵ ⇠ G/(rc2 ⇤ ) ⌧ G

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

Screened fifth force<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

Vainshtein機構のはたらき方
Horndeski<latexit sha1_base64="kKRBujegymKBocimzrJgFI0mm/8=">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</latexit>
sha1_base64="UQxzZcC8HWTNjWPAnSdJKk5OYhc=">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</latexit>

cGW = c in

GR (2 dofs)

DHOST

No graviton decay

基本2パターン: Horndeski か beyond Horndeski (DHOST) か2
例外1パターン: DHOST w/ cGW = 1 + no graviton decay

Vainshtein機構のはたらき方
Horndeski‒ Vainshtein機構が完璧にはたらく
Z r
GN M
0
0
02
0
=
=
,
M
=
4⇡
⇢
r
dr
r2
0

Kimura et al. (2012);
Narikawa et al. (2013);
Koyama et al. (2013); ……

<latexit sha1_base64="gceYr9Y1zAZXO2qOnYdBWnS5m1c=">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</latexit>

Generic DHOST ~ Horndeski + disformally coupled matter‒ 物体内部でのみVainshtein機構が完璧にははたらかない<latexit sha1_base64="h4kTGsz4/q9RnVNtHFkIqz9CZs4=">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</latexit>

00
G
M
G
M
N
N
0
=
+⌥1
2
r
4
0
G
M
5⌥
G
M
N
2
N
0
00
=
+ ⌥ 3 GN M
2r
4r

TK, Watanabe, Yamauchi (2015);
Crisostomi & Koyama (2017);
Langlois et al. (2017);
Dima & Vernizzi (2017)→ 太陽系実験では難しいが、アストロ的な検証は可能 (天体、銀河団 etc.)

Hirano, TK, Yamauchi (2019); See also Crisostomi, Lewandowski, Vernizzi (2019)

Vainshtein機構のはたらき方
2c
DHOST w/ GW = 1 + no graviton decay‒ 物体外部でもパラメータのﬁne-tuningが必要‒ 物体内部と外部で重力定数が異なる
内部:
0

GN,in M
=
,
2
r

= (1

2 ) ,

1
GN,in :=
16⇡f (1)2<latexit sha1_base64="3vQDKdl+/Kp0zkqxQ0x+UjefSBM=">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</latexit>

外部:
0

=

0

GN,out M=,
2
r

1
GN,out :=16⇡f (1)= (1)GN,in<latexit sha1_base64="gzt+ONIlZ6hDBDdNFAvQ/AOaPtY=">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</latexit>:= XfX /f … “beyond Horndeski”具合を表すパラメータHirano, TK, Yamauchi (2019); See also Crisostomi, Lewandowski, Vernizzi (2019)

Vainshtein機構のはたらき方
誰かアストロ的な検証方法のアイデア
があれば一緒に論文を書きましょう！
c2 = 1

DHOST w/

GW

+ no graviton decay‒ 物体外部でもパラメータのﬁne-tuningが必要‒ 物体内部と外部で重力定数が異なる
内部:
0

GN,in M
=
,
2
r

= (1

2 ) ,

1
GN,in :=
16⇡f (1)2<latexit sha1_base64="3vQDKdl+/Kp0zkqxQ0x+UjefSBM=">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</latexit>

外部:
0

=

0

GN,out M=,
2
r

1
GN,out :=16⇡f (1)= (1)GN,in<latexit sha1_base64="gzt+ONIlZ6hDBDdNFAvQ/AOaPtY=">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</latexit>:= XfX /f … “beyond Horndeski”具合を表すパラメータHirano, TK, Yamauchi (2019); See also Crisostomi, Lewandowski, Vernizzi (2019)

パルサー観測からの制限
重力波にとっての “G” がニュートンの “G” と異なる
h
1
LGW =
ḣ2ij
64⇡GGW(@k hij )<latexit sha1_base64="iNUldV6pWd7c1g0JXAwofGRW1n4=">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</latexit>2i1+ hij T ij ,
2

1
GGW =
= (1
16⇡f

)GN,out

Weisberg et al. (2010);
Beltran Jimenez et al. (2016);
Dima & Vernizzi (2018)

四重極公式 / GGW<latexit sha1_base64="7KClxrnpzIRHNf28JNAs7vgit8c=">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</latexit>

Hulse-Taylorパルサーの公転周期減少から<latexit sha1_base64="rEFhtNFx1XtqIdbGz/keggM+8rI=">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</latexit>3

7.5 ⇥ 10GGW<
GN

1 < 2.5 ⇥ 10 3

| | < O(10 3 )<latexit sha1_base64="Agqtn4q2d9TuiE7sovEabvsc2aY=">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</latexit>

Horndeski/DHOSTへの宇宙論的観測からの制限
45
LCDM

40

10

LCDM

ournalボルツマンコードはあるので、いろいろ計算できる
of Cosmology and Astroparticle Physics`(` + 1)C` /(2⇡)

J

An IOP and SISSA journal

1,0 = 0.012

0.024
0.048
0.096

4000
3000
2000

20
15

10

100

0

1000

10

100

1000

0.01

40
35

3000
2000
1000

30
25

100

20
15

0

1000

10

100

`

0.1

1000

0.01

`

6000

45

10

LCDM

LCDM
40
↵
= 0.06↵= 0.06
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
5000
0.12
0.12
0.24
0.24
35
b Berkeley Center for Cosmological Physics, LBL and University of California at Berkeley,
0.48
0.48
4000
Figure2. E↵ect of the braiding on CMB temperature30power spectra (left panel) and the matter power 1
spectrum at z = 0 (right panel). In the bottom panels
we show the relative di↵erence between the25
Berkeley, CA 94720, U.S.A.
3000
DE/MG
models considered and a fiducial ⇤CDM. In 20
each model, we fix all the standard cosmologicalc Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg,
parameters
to some fiducial value. We use the lcdm 15
parameterization for the background expansion0.1
2000
history (i.e. wDE = 1) and propto omega for the evolution of the alphas. We fix ↵ˆ K = 1, ↵ˆM =10Philosophenweg 16, 69120 Heidelberg, Germany↵
ˆ T = 01000
and we vary ↵ˆ B in the stable region.
5d Institut de Ciènces del Cosmos, Universitat de Barcelona, IEEC-UB,
0
0
0.01
10
100
1000
10
100
1000
Martı̀ i Franquè 1, E-08028 Barcelona, Spain
`
`
6000
45
10
RUP-20-14
e Astrophysics, University of Oxford,
LCDM
LCDM
40
↵
= 0.06↵= 0.06
5000
0.12
0.12
0.24
Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, U.K.
0.24
35
0.48
0.48
4000
30
1
f Central European Institute for Cosmology and Fundamental Physics,
Testing gravity theories with cosmic microwave background in 3000the degenerate 25
Fyzikálnı́ ustáv Akademie věd ČR,
20
higher-orderscalar-tensor theory
2000
15
0.1
Na Slovance 2, 182 21 Praha 8, Czech
Republic
10
g Institut für Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University,
1000
5
1⇤
2†
0
0
0.01
D-52056 Aachen, Germany
Takashi Hiramatsu and Daisuke Yamauchi
10
100
1000
10
100
1000
`
`
E-mail: miguelzuma@berkeley.edu, emilio.bellini@physics.ox.ac.uk,
6000
45
10
LCDM
LCDM
1
40 ↵
= 0.06↵= 0.06ignacy.sawicki@fzu.cz,
lesgourg@physik.rwth-aachen.de,
Departmentof Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan
5000
0.12
0.12
0.24
0.24
35
0.48
2
0.48
pedro.ferreira@physics.ox.ac.uk
Faculty of Engineering, Kanagawa University, Kanagawa, 221-8686, Japan
4000
30
1

107 ⇥ [`(` + 1)]2 C` /(2⇡)`(` + 1)C` /(2⇡)`(` + 1)C` /(2⇡)

10

100

1000

`
LCDM
↵M,0 = 0.06
0.12
0.24
0.48

重力波と独立なhigh-zでの検証は重要`(` + 1)C` /(2⇡)

10

100

LCDM
↵T,0 = 0.06
0.12
0.24
0.48

10

100

`(` + 1)C` /(2⇡)

25

1000

`
LCDM
↵B,0 = 0.06
0.12
0.24
0.48

B,0

B,0

1000

`

107 ⇥ [`(` + 1)]2 C` /(2⇡)`(` + 1)C` /(2⇡)

107 ⇥ [`(` + 1)]2 C` /(2⇡)

T,0

T,0

`(` + 1)C` /(2⇡)

Zumalacárregui et al. 1605.06102
for Horndeski

M,0

M,0

1000

LCDM
↵H,0 = 0.06
0.12
0.24
0.48

1

5
10

100

10

LCDM
↵H,0 = 0.06
0.12
0.24
0.48

10

0

10

`

107 ⇥ [`(` + 1)]2 C` /(2⇡)

45
LCDM↵H,0 = 0.06
0.12
0.24
0.48

`(` + 1)C` /(2⇡)`(` + 1)C` /(2⇡)

4000

0.1

`

6000
5000

0.024
0.048
0.096

1

JCAP08(2017)019

a Nordita, KHT Royal Institute of Technology and Stockholm University,25`

JCAP08(2017)019

Miguel Zumalacárregui,a,b,c Emilio Bellini,d,e Ignacy Sawicki,fJulien Lesgourguesg and Pedro G. Ferreirae30

1,0 = 0.012

50hi class: Horndeski in the Cosmic
Linear Anisotropy Solving System

0.024
0.048
0.096

10

1000

LCDM

1,0 = 0.012

35`(` + 1)C` /(2⇡)

5000

107 ⇥ [`(` + 1)]2 C` /(2⇡)

6000

30003. Same as figure 2 but for the running of the Planck mass. We fix ↵Figureˆ K = 1, ↵
ˆB = ↵
ˆT = 0Received April 28, 201720
2
and we vary ↵ˆ M in the stable region, with M⇤,0
= 1 initially.
We
study
the
cosmicmicrowave
background
(CMB)
in
the
frameworkofthedegeneratehigher2000
15
0.1
Accepted July 23, 201710
orderscalar-tensor(DHOST)
theorytotestgravitytheories.
This
theoreticalframework
includes1000
Published August 16, 2017
We should also note that we have generated 5the plots for w = 1. In this limit, many

Hiramatsu & Yamauchi 2004.09520
the wide class of dark energy models such as the Horndeski theory and its extensions as certain
Abstract.
We present
the generalpublic versionof hi class(www.hiclass-code.net),
an extenforlimits,
DHOST
and therelativity
can bealso recovered. In thisstudy, to test gravity theories

DE

0.01
terms in0 the perturbation
equations 1000
cancel and0 therefore10 the modifications
to 1000
observables
10
100
100
` ⇤CDM can produce aare suppressed. Adding in` a background evolution that departs frommuch richer phenomenology. Similarly, considering simultaneously non-zero values of several

10

100

1000

`

sion of thecode
CLASS tothea broadof modifications
to generalrela- in the theory and their
withBoltzmann
CMB, weformulatelinearensemble
perturbationsof gravity
and matter
FIG. 1: Angular power spectra with varying 1,0 , ↵H,0 , ↵M,0 , ↵T,0 and ↵B,0 from top to bottom. From left to right, we stivity. In
particular,
hi class parameterised
can calculate predictions
for models basedon Horndeski’sTTEEe↵ective
description
by time-dependente↵ective
field theory (EFT) parameters, ↵ithe angular power spectra of temperature(C` ), E-mode (C`) and lensing potential (C` ).

ブラックホール

強重力場での重力理論検証

No hair 定理: (諸々の仮定のもとで) 自明な解しかない(GR)
gµ⌫ = gµ⌫,

= const.

定理の前提条件を外せば、多様な解の可能性
BH摂動の一般論 (安定性、準固有振動、…)- Horndeski/DHOST
TK, Motohashi, Suyama (2012, 2014); Ogawa, et al. (2016); Takahashi &Suyama (2017); Takahashi, Motohashi, Minamitsuji (2019)- EFT of BH perturbations in scalar-tensor theories
Franciolini, et al. (2019); Kuntz, et al. (2020)

これまで球対称解の摂動が中心
回転解とその摂動が重要だろう1. Why: 一般相対論を拡張する動機
2. How: 一般相対論を拡張するとはどういうことか
3. Theory: 一般的な重力理論の枠組
4. Tests: 重力理論の検証 (重力波の話題を中心に)
5. まとめまとめ
一般相対論の拡張理論を研究する豊富な動機がある。
一般的な重力理論の枠組作りはかなり成熟してきた。
そのおかげで、例えば、重力波の伝播速度が1の理論/1からズレる理論について系統的に理解することができた。
強重力場で重力理論の検証ができる時代に突入した。