matter bispectrum beyond horndeski
TRANSCRIPT
![Page 1: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/1.jpg)
MOGRA2018, August 8-10 ,2018
Shin’ichi Hirano Rikkyo U.
SH, T. Kobayashi, S. Yokoyama (Nagoya U.), T. Hiroyuki (Nagoya U.) 1801. 07885
SH, T. Kobayashi, D. Yamauchi (Kanagawa U.), S. Yokoyama, in preparation
based on
(進一 平野)
MATTER BISPECTRUM BEYOND HORNDESKI
![Page 2: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/2.jpg)
Introduction
1/18
■ Modified gravity: alternative to cosmological constant
・ Cosmological scale: late-time acceleration
・Most general scalar-tensor theory with 2nd-order EoMs
■ Horndeski theory Horndeski (1972), Kobayashi+ (2011), Deffaiyet+ (2011)
・Vainshtein screening thanks to 2nd-order derivative non-linear ints.
Lp�g
=f(�)
2R+G2(�, X)�G3(�, X)⇤�
・GW170817, GRB170817A: |cT � 1| < 10�15 Abbott+ (2017)
・ Small scale: recovering the result of gravitational test
⇒ screening mechanism
![Page 3: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/3.jpg)
■ Beyond Horndeski (higher-order EoMs, no Ostrogradski ghost)
GLPV theory Gleyzes+ (2014), Gleyzes+ (2015)
DHOST theory Langlois & Noui (2015), Achour+ (2016), Achour+ (2016)
✓ Models with and cT = 1(@@�)2
✓ Partial breaking of Vainshtein screening inside matter ( )Kobayashi+ (2015), Langlois+ (2017), …
Our aim
Matter bispectrum beyond Horndeski
non-linear int.
Recent progress & our work
How much is the effect of non-linear ints. at “cosmological scale” ?� ⌧ 1
2/18
� � 1<latexit sha1_base64="wKpvRKEKZEjv2fRd/KVFUjzD7CI=">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</latexit><latexit sha1_base64="wKpvRKEKZEjv2fRd/KVFUjzD7CI=">AAACmnicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQuYVseADYkuSk+KrTbQszAAAR0DPSNURm1MSmpOSaJCTHq6gmFtvIAyTKkCTAWCYagHpg2UGaAgIF9gOUMMQwpDPkMyQylDLkMqQx5DCZCdw5DIUAyE0QyGDAYMBUCxWIZqoFgRkJUJlk9lqGXgAuotBapKBapIBIpmA8l0IC8aKpoH5IPMLAbrTgbakgPERUCdCgyqBlcNVhp8NjhhsNrgpcEfnGZVg80AuaUSSCdB9KYWxPN3SQR/J6grF0iXMGQgdOF1cwlDGoMF2K2ZQLcXgEVAvkiG6C+rmv452CpItVrNYJHBa6D7FxrcNDgM9EFe2ZfkpYGpQbMZuIARAAtlBdyMMCM9QwM9w0ATZQcnaFRwMEgzKDFoAMPbnMGBwYMhgCEUaO9shoMMpxhOM8kwOTF5MnlDlDIxQvUIM6AAphAAPJCaAw==</latexit><latexit sha1_base64="wKpvRKEKZEjv2fRd/KVFUjzD7CI=">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</latexit><latexit sha1_base64="wKpvRKEKZEjv2fRd/KVFUjzD7CI=">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</latexit>
![Page 4: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/4.jpg)
Plan of talk
■ Our setup
■ Matter bispectrum beyond Horndeski
■Summary
3/18
■ Cosmological perturbations
![Page 5: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/5.jpg)
OUR SETUP
![Page 6: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/6.jpg)
quadratic DHOSTLqDp�g
= G2(�, X)�G3(�, X)⇤�+G4(�, X)R+ Cµ⌫⇢�(2) �µ⌫�⇢�
Langlois, Noui (2015,2016), Koyama+ (2016), de Rham, Matas (2016)
4/18
X = �1
2(r�)2,�µ = rµ�,⇤� = r2�,�µ⌫ = r⌫rµ�( ) non-linear ints.
Cµ⌫⇢�(2) �µ⌫�⇢� = a1(�, X)�2
µ⌫ + a2(�, X)(⇤�)2+a3(�, X)⇤�(�µ�µ⌫�⌫)
+a4(�, X)�µ�µ⌫�⌫⇢�⇢ + a5(�, X)(�µ�µ⌫�
⌫)2
■ includes Horndeski and GLPV at Lagrangian level
Horndeski: a1 = �a2 = �G4X , a3 = a4 = a5 = 0 , GLPV: …
(G4, C(2))■ The non-trivial relation between arbitrary func.
in order to evade Ostragradski ghost ← “degeneracy conditions”
![Page 7: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/7.jpg)
Viable DHOST after GW170817
5/18
Degenerate Scalar-Tensor theory
DHOST
stable cosmological sol.
Horndeski (2nd-order EoMs)
quintessence, f(R), KGB
Covariant Galileon, …
mimetic gravity, extended mimetic gravity
de Rham & Matas (2016)
class I class II, III
GLPV
(higher-order EoMs)
![Page 8: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/8.jpg)
5/18
Degenerate Scalar-Tensor theory
DHOST
stable cosmological sol.
Horndeski (2nd-order EoMs)
quintessence, f(R), KGB
Covariant Galileon, …
mimetic gravity, extended mimetic gravity
de Rham & Matas (2016)
class I class II, III
GLPV
(higher-order EoMs)
conformal &disformal trans.gµ⌫ = ⌦(�, X)gµ⌫ + �(�, X)�µ�⌫disformal �X 6= 0
<latexit sha1_base64="wslD7Z6uGROVXiT/Qme0KjwS0FQ=">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</latexit><latexit sha1_base64="wslD7Z6uGROVXiT/Qme0KjwS0FQ=">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</latexit><latexit sha1_base64="wslD7Z6uGROVXiT/Qme0KjwS0FQ=">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</latexit><latexit sha1_base64="LoN+so8WU7mEePEt6DB6+aSkNMs=">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</latexit>
Viable DHOST after GW170817
![Page 9: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/9.jpg)
5/18
Degenerate Scalar-Tensor theory
DHOST
stable cosmological sol.
Horndeski (2nd-order EoMs)
quintessence, f(R), KGB
Covariant Galileon, …
mimetic gravity, extended mimetic gravity
de Rham & Matas (2016)
class I class II, III
GLPV
(higher-order EoMs)
disformalGW180817, GRB 180817
cT = 1<latexit sha1_base64="jcS5STXAzGxLsmadWuBy6duFX1I=">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</latexit><latexit sha1_base64="jcS5STXAzGxLsmadWuBy6duFX1I=">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</latexit><latexit sha1_base64="jcS5STXAzGxLsmadWuBy6duFX1I=">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</latexit><latexit sha1_base64="o0yFrjiO+6o1/mt4fCjtLSM0oHQ=">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</latexit>
Target
Viable DHOST after GW170817
![Page 10: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/10.jpg)
ParametrizationSe↵ =
Zd4xp�M
2
2
"�K2 + c
2TR
(3) +H2↵K�N
2 + 4H↵B�K�N
+ (1 + ↵H)R(3)�N + (1 + ↵V )�N�K2 + 4�1�KV + �2V
2 + �3a2i
#.
K2 := K2ij �K2, V :=
1
N(N �N i@iN), ai := @iN/N depend on
through degeneracy cond.�1
■ alpha-parameters
↵K : kineticity … non-standard kinetic terms
↵M : time evolution of M
↵B : braiding … kinetic mixing between scalar and metric
: conformal & disformal coupling to matter → DHOST�1
: disformal coupling to matter → GLPV↵H
�↵H
6/18
Bellini & Sawicki (2014), Gleyzes et al. (2015)Langlois+ (2017), Dima & Vernizzi (2017)
![Page 11: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/11.jpg)
■ Early time
■ Late time (after MD)
M2pl/2 (GR)
M2⇡ 2G4 := O(M2
pl), (↵i,�1) ⌧ 1
Vainshtein screening around matter, its breaking inside matter
⇒ , ↵i = O(1), �1 = O(1)3M2H
2 ⇡ ⇢�, ⇢� � ⇢m<latexit sha1_base64="M9NZe37H3zrYKL3BX1SMHtD4sqM=">AAACnnicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQtYGvvGGXnEGSnEJBYUFOVXKMQUZeTHV8cUZGTW6tQhcWLS02FyRbkKubUK8QLKBnoGYKCAyTCEMpQZoCAgX2A5QwxDCkM+QzJDKUMuQypDHkMJkJ3DkMhQDITRDIYMBgwFQLFYhmqgWBGQlQmWT2WoZeAC6i0FqkoFqkgEimYDyXQgLxoqmgfkg8wsButOBtqSA8RFQJ0KDKoGVw1WGnw2OGGw2uClwR+cZlWDzQC5pRJIJ0H0phbE83dJBH8nqCsXSJcwZCB04XVzCUMagwXYrZlAtxeARUC+SIboL6ua/jnYKki1Ws1gkcFroPsXGtw0OAz0QV7Zl+SlgalBsxm4gBFgiB7cmIwwIz1DAz3DQBNlBydoVHAwSDMoMWgAw9ucwYHBgyGAIRRo73yGowznGM4zKTC5Mfky+UOUMjFC9QgzoACmCAA2oKAK</latexit><latexit sha1_base64="M9NZe37H3zrYKL3BX1SMHtD4sqM=">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</latexit><latexit sha1_base64="M9NZe37H3zrYKL3BX1SMHtD4sqM=">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</latexit><latexit sha1_base64="M9NZe37H3zrYKL3BX1SMHtD4sqM=">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</latexit>
� ⇠ Mpl, � ⇠ MplH0, � ⇠ MplH20 ,
G2 ⇠ M2plH
00 , G3 ⇠ Mpl, G4 ⇠ M
2pl, · · ·
⇒ 3M2H
2 ⇡ ⇢m, ⇢m � ⇢� ※ we do not consider quintessential inflation and early dark energy scenarios.
Kimura+ (2011) Kobayashi+ (2015)Viable conditions
7/18
![Page 12: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/12.jpg)
COSMOLOGICAL PERTURBATIONS
![Page 13: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/13.jpg)
Cosmological perturbations
ds2 = �(1 + 2�)dt2 + a2(t)(1� 2 )dx2.
�(t,x) = �(t) + ⇡(t,x), ⇢(t,x) = ⇢(t)[1 + �(t,x)].
■ perturbationsQ = H⇡/�
Sub-horizon ( )aH ⌧ k , late time (after MD)
■ Quasi-static approximation (QSA)
Note: 0 6= ↵i ⌧ 1 ⇒cf) f(R) Ge↵ = Ge↵(k, t)
H2✏2 ⇠ ↵ik
2✏2
⇒ |✏| ⇡ |H✏|, ✏ = ,�, Q
| |2, |�|2, |Q|2 ⌧ k2 2, k2�2, k2Q2
ksh :=aH
cs⌧ k
8/18
↵i ⇠ ↵j = O(1)
In this work
![Page 14: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/14.jpg)
Evolution of density fluctuations
⇒ One obtain the evolution equation of density contrast
1. Perturbative expansion: ✏ = ✏1 + ✏2, ✏ = ,�, Q, �
continuity/ Eular (usual forms)
3. Fluid equations: @�(t,x)
@t+
1
a@i[(1 + �)ui(t,x)] = 0,
@ui
@t+Hu
i +1
auj@ju
i = �1
a@i�(t,x)
2. EoMs: � , ��, �Q ⇒ �1,�2
9/18
↑ include the effect of modified gravity
![Page 15: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/15.jpg)
GT@2 + A2@
2Q�A6@
2�+A8@2Q
H� a
2
2⇢m� = � B2
2a2H2Q2 +
B5
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
(��)
FT@2 � GT@
2�� A1@2Q+A4
@2Q
H=
B1
2a2H2Q2 +
B4
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
(� )
A0@2Q�A1@
2 �A2@2��A4
@2
H+A8
@2�
H� A9
@2Q
H�A9
@2Q
H2
= � B0
a2H2Q2 +
B2
a2H2(@2�@2
Q� @i@j�@i@jQ)
� B4
a2H2
�@2 @2
Q+ @iQ@i@2
�+
B5
a2H2(@2�@2
Q+ @iQ@i@2�)
� B6
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
� B6
a2H2
1
H
⇣@2Q@
2Q+ 2@iQ@
i@2Q+ 2@i@jQ@
i@jQ+ @i@
2Q@
iQ
⌘
(�Q)
EoMs of gravitational fields
10/18
A,B � ↵i,�1<latexit sha1_base64="rWWsxF6KLrDmcemfsX1sI1NYG0I=">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</latexit><latexit sha1_base64="rWWsxF6KLrDmcemfsX1sI1NYG0I=">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</latexit><latexit sha1_base64="rWWsxF6KLrDmcemfsX1sI1NYG0I=">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</latexit><latexit sha1_base64="rWWsxF6KLrDmcemfsX1sI1NYG0I=">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</latexit>
Green: GLPV, Red: DHOST
![Page 16: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/16.jpg)
EoMs of gravitational fields
GT@2 + A2@
2Q�A6@
2�+A8@2Q
H� a
2
2⇢m� = � B2
2a2H2Q2 +
B5
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
(��)
FT@2 � GT@
2�� A1@2Q+A4
@2Q
H=
B1
2a2H2Q2 +
B4
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
(� )
A0@2Q�A1@
2 �A2@2��A4
@2
H+A8
@2�
H� A9
@2Q
H�A9
@2Q
H2
= � B0
a2H2Q2 +
B2
a2H2(@2�@2
Q� @i@j�@i@jQ)
� B4
a2H2
�@2 @2
Q+ @iQ@i@2
�+
B5
a2H2(@2�@2
Q+ @iQ@i@2�)
� B6
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
� B6
a2H2
1
H
⇣@2Q@
2Q+ 2@iQ@
i@2Q+ 2@i@jQ@
i@jQ+ @i@
2Q@
iQ
⌘
(�Q)
■ linear level, GR
0
0
0
GT = FT = M2pl
Poisson equation
trace component of Einstein tensor
�Q = 0
11/18
![Page 17: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/17.jpg)
GT@2 + A2@
2Q�A6@
2�+A8@2Q
H� a
2
2⇢m� = � B2
2a2H2Q2 +
B5
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
(��)
FT@2 � GT@
2�� A1@2Q+A4
@2Q
H=
B1
2a2H2Q2 +
B4
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
(� )
A0@2Q�A1@
2 �A2@2��A4
@2
H+A8
@2�
H� A9
@2Q
H�A9
@2Q
H2
= � B0
a2H2Q2 +
B2
a2H2(@2�@2
Q� @i@j�@i@jQ)
� B4
a2H2
�@2 @2
Q+ @iQ@i@2
�+
B5
a2H2(@2�@2
Q+ @iQ@i@2�)
� B6
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
� B6
a2H2
1
H
⇣@2Q@
2Q+ 2@iQ@
i@2Q+ 2@i@jQ@
i@jQ+ @i@
2Q@
iQ
⌘
(�Q)
■ linear level, Horndeski
0
0
0
contribution of scalar field
as anisotropic stress
⇒ increase of gravitational constant
EoMs of gravitational fields
11/18
![Page 18: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/18.jpg)
GT@2 + A2@
2Q�A6@
2�+A8@2Q
H� a
2
2⇢m� = � B2
2a2H2Q2 +
B5
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
(��)
FT@2 � GT@
2�� A1@2Q+A4
@2Q
H=
B1
2a2H2Q2 +
B4
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
(� )
A0@2Q�A1@
2 �A2@2��A4
@2
H+A8
@2�
H� A9
@2Q
H�A9
@2Q
H2
= � B0
a2H2Q2 +
B2
a2H2(@2�@2
Q� @i@j�@i@jQ)
� B4
a2H2
�@2 @2
Q+ @iQ@i@2
�+
B5
a2H2(@2�@2
Q+ @iQ@i@2�)
� B6
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
� B6
a2H2
1
H
⇣@2Q@
2Q+ 2@iQ@
i@2Q+ 2@i@jQ@
i@jQ+ @i@
2Q@
iQ
⌘
(�Q)
■ linear level, beyond Horndeski
0
0
0
O(�)
O(�)
⇒ increase of gravitational constant,
additional friction term11/18
EoMs of gravitational fieldsGreen: GLPV, Red: DHOST
![Page 19: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/19.jpg)
1st-order solution Kobayashi+ (2015), D’Amico+ (2017), Chrisostomi & Koyama (2017)
■ change in the growth of density fluctuation due to &
cf.) improvement of f�8 Tsujikawa (2015), D’Amico+ (2017)
■ growing mode: �1(p, t) = D+(t)�L(p)
D+(t) : growth factor, �L(p) : initial density fluc.
�1 + (2 + &)H �1 � 4⇡Ge↵⇢m�1 = 0
Ge↵(t) G (GR) , (Horndeski, beyond Horndeski)G ! Ge↵■ :
: 0 (GR, Horndeski), (beyond Horndeski)&(t) / ↵H ,�1 0 ! &0<latexit sha1_base64="L3FlIFIkqiRDCrACQLcfDZvlaB4=">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</latexit><latexit sha1_base64="L3FlIFIkqiRDCrACQLcfDZvlaB4=">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</latexit><latexit sha1_base64="L3FlIFIkqiRDCrACQLcfDZvlaB4=">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</latexit><latexit sha1_base64="L3FlIFIkqiRDCrACQLcfDZvlaB4=">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</latexit>
12/18
![Page 20: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/20.jpg)
GT@2 + A2@
2Q�A6@
2�+A8@2Q
H� a
2
2⇢m� = � B2
2a2H2Q2 +
B5
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
(��)
FT@2 � GT@
2�� A1@2Q+A4
@2Q
H=
B1
2a2H2Q2 +
B4
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
(� )
A0@2Q�A1@
2 �A2@2��A4
@2
H+A8
@2�
H� A9
@2Q
H�A9
@2Q
H2
= � B0
a2H2Q2 +
B2
a2H2(@2�@2
Q� @i@j�@i@jQ)
� B4
a2H2
�@2 @2
Q+ @iQ@i@2
�+
B5
a2H2(@2�@2
Q+ @iQ@i@2�)
� B6
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
� B6
a2H2
1
H
⇣@2Q@
2Q+ 2@iQ@
i@2Q+ 2@i@jQ@
i@jQ+ @i@
2Q@
iQ
⌘
(�Q)
0
0
0
GT = FT = M2pl■ non-linear level, GR
�Q = 0
⇒ Non-linearity only derives from fluid equations
remains usual form
13/18
remains usual form
Poisson equation
trace component of Einstein tensor
EoMs of gravitational fields
![Page 21: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/21.jpg)
GT@2 + A2@
2Q�A6@
2�+A8@2Q
H� a
2
2⇢m� = � B2
2a2H2Q2 +
B5
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
(��)
FT@2 � GT@
2�� A1@2Q+A4
@2Q
H=
B1
2a2H2Q2 +
B4
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
(� )
A0@2Q�A1@
2 �A2@2��A4
@2
H+A8
@2�
H� A9
@2Q
H�A9
@2Q
H2
= � B0
a2H2Q2 +
B2
a2H2(@2�@2
Q� @i@j�@i@jQ)
� B4
a2H2
�@2 @2
Q+ @iQ@i@2
�+
B5
a2H2(@2�@2
Q+ @iQ@i@2�)
� B6
a2H2
⇥(@i@jQ)2 + @iQ@
i@2Q⇤
� B6
a2H2
1
H
⇣@2Q@
2Q+ 2@iQ@
i@2Q+ 2@i@jQ@
i@jQ+ @i@
2Q@
iQ
⌘
(�Q)
■ non-linear, Horndeski Q2 = (@2Q)2 � (@i@jQ)2
13/18
EoMs of gravitational fields
additional non-linearity from scalar field ⇒ novel probe of modified gravity !
(at a quasi non-linear regime)
![Page 22: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/22.jpg)
2nd-order solution
�21�2 + (2 + &)H �2 � 4⇡Ge↵⇢m�2 = S�
�2(p, t) = D2+(t)[(t)W↵(p) + �(t)W�(p)]⇒
:�(t) (GR)1 , (Horndeski, beyond Horndeski)1 ! �0 6= 1
1 (GR, Horndeski), (beyond Horndeski)1 ! 0 6= 1New! :(t) � ↵H ,�1
DHOST: Hirano+ in prep.
14/18
Primordial fluc. : Gaussian ⇒ inhomogeneous sol.
Wi(p) =1
(2⇡)3
Zd3k1d
3k2 �(3)(k1 + k2 � p)E(k1 · k2)�L(k1)�L(k2)
↵(k1,k2) = 1 +(k1 · k2)(k21 + k22)
2k21k22
, , �(k1,k2) = 1� (k1 · k2)2
k21k22
E = ↵, �<latexit sha1_base64="eRyvF8gPNCcC5Huj2IbNwzUMvQk=">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</latexit><latexit sha1_base64="eRyvF8gPNCcC5Huj2IbNwzUMvQk=">AAACfnicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQtIVsckJ+YouNYq2CrEJOYUZCTqKMSkJ+bmJsYLKBvoGYCBAibDEMpQZoCCgHyB5QwxDCkM+QzJDKUMuQypDHkMJUB2DkMiQzEQRjMYMhgwFADFYhmqgWJFQFYmWD6VoZaBC6i3FKgqFagiESiaDSTTgbxoqGgekA8ysxisOxloSw4QFwF1KjCoGlw1WGnw2eCEwWqDlwZ/cJpVDTYD5JZKIJ0E0ZtaEM/fJRH8naCuXCBdwpCB0IXXzSUMaQwWYLdmAt1eABYB+SIZor+savrnYKsg1Wo1g0UGr4HuX2hw0+Aw0Ad5ZV+SlwamBs1m4AJGgCF6cGMywoz0DA30DANNlB2coFHBwSDNoMSgAQxvcwYHBg+GAIZQoL31DEsZ1jGsZ2JgUmPSZdKHKGVihOoRZkABTBYANgWSFQ==</latexit><latexit sha1_base64="eRyvF8gPNCcC5Huj2IbNwzUMvQk=">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</latexit><latexit sha1_base64="eRyvF8gPNCcC5Huj2IbNwzUMvQk=">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</latexit>
Horndeski: Takushima+ (2013) GLPV: Hirano+ (2018)
![Page 23: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/23.jpg)
MATTER BISPECTRUM BEYOND HORNDESKI
Hirano+ (2018), Hirano+ in prep.
![Page 24: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/24.jpg)
■ Correlation function
h�(t,k1)�(t,k2)�(t,k3)i := (2⇡)3�(k1 + k2 + k3)B(t,k1,k2,k3)
■ Leading order (tree-level)
Kernel F2(t,k1,k2) = (t)↵(k1,k2)�2
7�(t)�(k1,k2)
W↵, W�P11(k) : initial power spectrum
B(t, k1, k2, k3) = 2D4+F2(t,k1,k2)P11(k1)P11(k2) + 2 cyclic terms
Matter bispectrum cf) Scoccimarro+ (1998) Barnardeau+ (2000)
15/18
![Page 25: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/25.jpg)
■ Reduced bispectrum
is sensitive to time evolution of and <latexit sha1_base64="43dSTpPQ2Z9ev3CY2uuBF+6ooHo=">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</latexit><latexit sha1_base64="43dSTpPQ2Z9ev3CY2uuBF+6ooHo=">AAAClHichVFNS8NAEH3G7/pVFUTwUiyKB4lTERTxUBTBk/hVFdpSkritoWkSkrRQQ/+Ad/HgSUFE/AkeVfAPePAniEcFLx6cpC3iQZ1ld96+mTc7u6vahu56RM8tUmtbe0dnV3ekp7evfyA6OLTrWmVHEynNMixnX1VcYeimSHm6Z4h92xFKSTXEnlpcCeJ7FeG4umXueFVbZEtKwdTzuqZ4TOWiM34mLJJ2CmrWJ3mBApsmefYnqGWKim0rtVw03kyKNWPfICGHnuJo2IYVvUIGB7CgoYwSBEx4jA0ocHmkkQDBZi4LnzmHkR7GBWqIsLbMWYIzFGaLvBZ4l26wJu+Dmm6o1vgUg6fDyhgm6Imu6Y0e6YZe6PPXWn5YI+ilyl6ta4WdGzge3f74V1Vi7+HwW/Vnzx7yWAh71bl3O2SCW2h1feXo9G17cWvCn6QLeuX+z+mZ7vgGZuVdu9wUW2eI8Ac0Xzn2O9idlRMkJzbn4snlxld0YQzjmOL3nkcSa9hAis89wS3u8SCNSEvSirRaT5VaGpph/DBp/QvUcJgv</latexit><latexit sha1_base64="43dSTpPQ2Z9ev3CY2uuBF+6ooHo=">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</latexit><latexit sha1_base64="43dSTpPQ2Z9ev3CY2uuBF+6ooHo=">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</latexit> �
<latexit sha1_base64="P3ukyJeHeK7yXQcajIXWx9gAl/Y=">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</latexit><latexit sha1_base64="P3ukyJeHeK7yXQcajIXWx9gAl/Y=">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</latexit><latexit sha1_base64="P3ukyJeHeK7yXQcajIXWx9gAl/Y=">AAAClXichVHNSsNAEP6M//Wv6kHBS7EoHiRMRVAEQVTEmz+1KrSlJHGtwTQJSVrQ0BfwAfTgSUFEfAVvovgCHvoI4lHBiwcnaYv0UJ1ld2a/me/b2V3VNnTXI6q0SK1t7R2dXd2Rnt6+/oHo4NCuaxUdTaQ0y7CcfVVxhaGbIuXpniH2bUcoBdUQe+rxSpDfKwnH1S1zxzuxRbag5E39UNcUj6FclPxMKJJ28mrWJ3meApsmeaYxKGcMFj1QyrlovF4Vqyd/g4QceoqjZptW9BYZHMCChiIKEDDhcWxAgcsjjQQINmNZ+Iw5HOlhXqCMCHOLXCW4QmH0mNc879I11OR9oOmGbI1PMXg6zIxhgl7pjj7ohe7pjb6bavmhRtDLCXu1yhV2buBsNPn1L6vA3sPRL+vPnj0cYj7sVefe7RAJbqFV+aXTi4/kwvaEP0nX9M79X1GFHvkGZulTu9kS25eI8AfUXznWPNidkRMkJ7Zm40vLta/owhjGMcXvPYclrGMTKT73HA94wrM0Ii1Kq9JatVRqqXGG0WDSxg+vuZiN</latexit><latexit sha1_base64="bs8QlLljyYde7WbNFuyzJJm+ZMk=">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</latexit><latexit sha1_base64="GPvqPA+Kf8fPfy6TNAsL3s+TY2g=">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</latexit><latexit sha1_base64="Gce453J0f4ziIKX5FNDUbowEPOk=">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</latexit><latexit sha1_base64="8nFmSOHyAJ4w2IYRk4zqgHEFO+4=">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</latexit><latexit sha1_base64="P3ukyJeHeK7yXQcajIXWx9gAl/Y=">AAAClXichVHNSsNAEP6M//Wv6kHBS7EoHiRMRVAEQVTEmz+1KrSlJHGtwTQJSVrQ0BfwAfTgSUFEfAVvovgCHvoI4lHBiwcnaYv0UJ1ld2a/me/b2V3VNnTXI6q0SK1t7R2dXd2Rnt6+/oHo4NCuaxUdTaQ0y7CcfVVxhaGbIuXpniH2bUcoBdUQe+rxSpDfKwnH1S1zxzuxRbag5E39UNcUj6FclPxMKJJ28mrWJ3meApsmeaYxKGcMFj1QyrlovF4Vqyd/g4QceoqjZptW9BYZHMCChiIKEDDhcWxAgcsjjQQINmNZ+Iw5HOlhXqCMCHOLXCW4QmH0mNc879I11OR9oOmGbI1PMXg6zIxhgl7pjj7ohe7pjb6bavmhRtDLCXu1yhV2buBsNPn1L6vA3sPRL+vPnj0cYj7sVefe7RAJbqFV+aXTi4/kwvaEP0nX9M79X1GFHvkGZulTu9kS25eI8AfUXznWPNidkRMkJ7Zm40vLta/owhjGMcXvPYclrGMTKT73HA94wrM0Ii1Kq9JatVRqqXGG0WDSxg+vuZiN</latexit><latexit sha1_base64="P3ukyJeHeK7yXQcajIXWx9gAl/Y=">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</latexit><latexit sha1_base64="P3ukyJeHeK7yXQcajIXWx9gAl/Y=">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</latexit><latexit sha1_base64="P3ukyJeHeK7yXQcajIXWx9gAl/Y=">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</latexit><latexit sha1_base64="P3ukyJeHeK7yXQcajIXWx9gAl/Y=">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</latexit><latexit sha1_base64="P3ukyJeHeK7yXQcajIXWx9gAl/Y=">AAAClXichVHNSsNAEP6M//Wv6kHBS7EoHiRMRVAEQVTEmz+1KrSlJHGtwTQJSVrQ0BfwAfTgSUFEfAVvovgCHvoI4lHBiwcnaYv0UJ1ld2a/me/b2V3VNnTXI6q0SK1t7R2dXd2Rnt6+/oHo4NCuaxUdTaQ0y7CcfVVxhaGbIuXpniH2bUcoBdUQe+rxSpDfKwnH1S1zxzuxRbag5E39UNcUj6FclPxMKJJ28mrWJ3meApsmeaYxKGcMFj1QyrlovF4Vqyd/g4QceoqjZptW9BYZHMCChiIKEDDhcWxAgcsjjQQINmNZ+Iw5HOlhXqCMCHOLXCW4QmH0mNc879I11OR9oOmGbI1PMXg6zIxhgl7pjj7ohe7pjb6bavmhRtDLCXu1yhV2buBsNPn1L6vA3sPRL+vPnj0cYj7sVefe7RAJbqFV+aXTi4/kwvaEP0nX9M79X1GFHvkGZulTu9kS25eI8AfUXznWPNidkRMkJ7Zm40vLta/owhjGMcXvPYclrGMTKT73HA94wrM0Ii1Kq9JatVRqqXGG0WDSxg+vuZiN</latexit>
Q123(t, k1, k2, k3) =B(t, k1, k2, k3)
D4+(t)[P11(k1)P11(k2) + 2 cyclic terms]
⇒ k1 = (0, 0, k1) k2 = (0, k2 sin ✓12, k2 cos ✓12)k3 = (0,�k2 sin ✓12,�k1 � k2 cos ✓12)
, ,✓
16/18
✓ We estimate matter bispectrum on the given (z=0) at and . k1 = k2 = 0.01h/Mpc k1 = 5k2 = 0.05h/Mpc
, �
(cosmological parameters: Planck 2015)
Matter bispectrum cf) Scoccimarro+ (1998) Barnardeau+ (2000)
k1<latexit sha1_base64="PKUMMEWGB1mFWG2H8WK+mEgnCPg=">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</latexit><latexit sha1_base64="PKUMMEWGB1mFWG2H8WK+mEgnCPg=">AAACdXichVHLSsNQED2Nr1ofrboRRCjWFld1IoLiqujGpa+2gkpJ4m29NE1CkhZq8Qf8ARe6UVARP8ONP+DCTxCXCm5cOE0DoqJOyL1zz50zc+6M7pjS84keI0pXd09vX7Q/NjA4NBxPjIwWPLvuGiJv2KbtbuuaJ0xpibwvfVNsO67Qaropinp1pX1fbAjXk7a15TcdsVfTKpYsS0PzGZJo4RI6ykiiiiOUoJYSKcpSYMmfjho6KYS2ZieusYt92DBQRw0CFnz2TWjw+NuBCoLD2B4X0+CyJ4N7wQVjzK1zlOAIjdEqrxU+7YSoxed2Ti9gG1zF5N9lZhJpeqAbeqF7uqUnev81VyvI0dbS5F3vcIVTih+Pb779y6rx7uPgk/WnZp9buRholazdCZD2K4wOv3F48rK5tJFuZeiCnln/OT3SHb/Aarwal+ti4xQxHoD6vd0/ncJcVqWsuj6fyi2Ho4hiAlOY4X4vIIdVrCHPdRs44xFfRd6USWVayXRClUjIGcMXU2Y/ALE/jbE=</latexit><latexit sha1_base64="PKUMMEWGB1mFWG2H8WK+mEgnCPg=">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</latexit><latexit sha1_base64="11wyiyCS1UQWWSMSAatnHauFVdE=">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</latexit>
k2<latexit sha1_base64="L7ffxPkCpDe5og7b7lywIuyjs6Q=">AAACd3ichVFLSgNBEH0Z//GTqBvBhcGguAo1QVBcBd241MQkgkqYGVsdMz9mJkENXsALuBAEhRDFY7jxAi5yBHEZQQQXViYDoqJW093Vr+tVva5WHUP3fKJmROrq7unt6x+IDg4Nj8Tio2MFz664mshrtmG7m6riCUO3RN7XfUNsOq5QTNUQRbW80r4vVoXr6ba14R87YsdU9i19T9cUn6FD1FCHij0kUMYpSkgjWoonKUWBJX46cugkEdqaHW9gG7uwoaECEwIWfPYNKPB4bEEGwWFsh8spcNnTg3vBJaPMrXCU4AiF0TKv+3zaClGLz+2cXsDWuIrB02VmAjP0SLfUoge6oyd6/zVXLcjR1nLMu9rhCqcUO5vIvf7LMnn3cfDJ+lOzz81cDLTqrN0JkPYrtA6/enLeyi1lZ2qzdE3PrP+KmnTPL7CqL1p9XWQvgg+Qv7f7p1NIp2RKyevzycxy+BX9mMQ05rjfC8hgFWvIc90jXKKBm8ibNCXNSnOdUCkScsbxxST5AxdQjc8=</latexit><latexit sha1_base64="L7ffxPkCpDe5og7b7lywIuyjs6Q=">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</latexit><latexit sha1_base64="L7ffxPkCpDe5og7b7lywIuyjs6Q=">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</latexit><latexit sha1_base64="3RztRoVDNQxPGctgB2j5w888nkI=">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</latexit>
k3<latexit sha1_base64="wHc+if2M1adkz6en9X6t7c2dv5g=">AAACd3ichVHNSsNAEP4a/2r9q3oRPFgslZ7KRAXFU9GLx/rTWqilJHGrsWkSkrSoxRfwBTwIgkJR8TG8+AIe+gjiUUEED07TgKios+zu7LfzzXw7q9qG7npErZDU1d3T2xfujwwMDg2PREfHcq5VczSR1SzDcvKq4gpDN0XW0z1D5G1HKFXVEFtqZaV9v1UXjqtb5qZ3aItiVdk19bKuKR5D+2igCRVlxFDBMUqYQ6QUjVOKfIv9dOTAiSOwjBW9wjZ2YEFDDVUImPDYN6DA5VGADILNWJHLKXDY0/17wSUjzK1xlOAIhdEKr7t8KgSoyed2Ttdna1zF4OkwM4YEPdANPdM93dIjvf+aq+HnaGs55F3tcIVdGjmZ2Hj9l1Xl3cPeJ+tPzR43c9HXqrN220far9A6/PrR6fPG0nqiMUOX9MT6L6hFd/wCs/6iNdfE+pn/AfL3dv90crMpmVLy2nw8vRx8RRiTmEaS+72ANFaRQZbrHuAcV7gOvUlT0oyU7IRKoYAzji8myR8ZUo3Q</latexit><latexit sha1_base64="wHc+if2M1adkz6en9X6t7c2dv5g=">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</latexit><latexit sha1_base64="wHc+if2M1adkz6en9X6t7c2dv5g=">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</latexit><latexit sha1_base64="RjgbdfDRenWaH2S3k0CW16ssPx8=">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</latexit>
✓12<latexit sha1_base64="Hx1JBFiaO/OEvspmvjaKm5H5b3o=">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</latexit><latexit sha1_base64="Hx1JBFiaO/OEvspmvjaKm5H5b3o=">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</latexit><latexit sha1_base64="Hx1JBFiaO/OEvspmvjaKm5H5b3o=">AAACb3ichVHLSsNAFD2Nr1pfVRcKghSLoptyI4LiSnTjslWrgpWSxFGDaRKSaaEWf8AP0IULHyAifoYbf8BFP0FcSQU3LrxJA6Ki3mFmzpy5586ZGd21TF8S1WNKS2tbe0e8M9HV3dPbl+wfWPedsmeIvOFYjrepa76wTFvkpSktsel6QivpltjQD5aC/Y2K8HzTsddk1RXbJW3PNndNQ5NMFQpyX0itWFOnjxLFZJoyFEbqJ1AjkEYUWSd5gwJ24MBAGSUI2JCMLWjwuW1BBcFlbhs15jxGZrgvcIQEa8ucJThDY/aAxz1ebUWszeugph+qDT7F4u6xMoVxeqRbatAD3dETvf9aqxbWCLxUedabWuEW+46HV9/+VZV4ltj/VP3pWWIXc6FXk727IRPcwmjqK4enjdX5lfHaBF3RM/u/pDrd8w3syqtxnRMrZwg+QP3+3D/B+nRGpYyam0kvLEZfEccIxjDJ7z2LBSwjizyf6+IE57iIvShDyqiSaqYqsUgziC+hTH0AKGmOJA==</latexit><latexit sha1_base64="Hx1JBFiaO/OEvspmvjaKm5H5b3o=">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</latexit>
k1 + k2 + k3 = 0<latexit sha1_base64="kf2JJ4vkKOllLaQ2NfiWcUCMZl4=">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</latexit><latexit sha1_base64="kf2JJ4vkKOllLaQ2NfiWcUCMZl4=">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</latexit><latexit sha1_base64="kf2JJ4vkKOllLaQ2NfiWcUCMZl4=">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</latexit><latexit sha1_base64="R5DMhSnXDVW5v4Qj/+UdgUCUQGY=">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</latexit>
![Page 26: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/26.jpg)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
θ12 /π
Q
ー: LCDM
� = 1.1- - :
� = 0.9- - :
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
θ12 /π
Q
= 0.9- - :
= 1.1- - :
Effect of κ
k1 = k2 = 0.01h/Mpc
� = 1
(beyond H.)
= 1(Horndeski)
ー: LCDM
� = 1.1- - :
� = 0.9- - :
0.0 0.2 0.4 0.6 0.8 1.0
0.8
1.0
1.2
1.4
1.6
θ12 /π
Q
= 0.9- - :
= 1.1- - :
0.0 0.2 0.4 0.6 0.8 1.0
0.6
0.8
1.0
1.2
1.4
1.6
1.8
θ12 /π
Q
k1 = 5k2 = 0.05h/Mpc
Effect of κ
17/18
k-dependence Horndeski: Takushima+ (2013) GLPV: Hirano+ (2018)DHOST: Hirano+ in prep.
![Page 27: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/27.jpg)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
θ12 /π
Q
ー: LCDM
� = 1.1- - :
� = 0.9- - :
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
θ12 /π
Q
= 0.9- - :
= 1.1- - :
Effect of κ
k1 = k2 = 0.01h/Mpc
= 1(Horndeski)
� = 1
(beyond H.)
ー: LCDM
� = 1.1- - :
� = 0.9- - :
0.0 0.2 0.4 0.6 0.8 1.0
0.8
1.0
1.2
1.4
1.6
θ12 /π
Q
= 0.9- - :
= 1.1- - :
0.0 0.2 0.4 0.6 0.8 1.0
0.6
0.8
1.0
1.2
1.4
1.6
1.8
θ12 /π
Q
k1 = 5k2 = 0.05h/Mpc
Effect of κ
17/18
k-dependence Horndeski: Takushima+ (2013) GLPV: Hirano+ (2018)DHOST: Hirano+ in prep.
Equilateral shape
![Page 28: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/28.jpg)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
θ12 /π
Q
ー: LCDM
� = 1.1- - :
� = 0.9- - :
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
θ12 /π
Q
= 0.9- - :
= 1.1- - :
Effect of κ
k1 = k2 = 0.01h/Mpc
� = 1� = 1
(beyond H.)
= 1(Horndeski)
ー: LCDM
� = 1.1- - :
� = 0.9- - :
0.0 0.2 0.4 0.6 0.8 1.0
0.8
1.0
1.2
1.4
1.6
θ12 /π
Q
= 0.9- - :
= 1.1- - :
0.0 0.2 0.4 0.6 0.8 1.0
0.6
0.8
1.0
1.2
1.4
1.6
1.8
θ12 /π
Q
k1 = 5k2 = 0.05h/Mpc
Effect of κ
Equilateral shape
Folded shape
17/18
k-dependence Horndeski: Takushima+ (2013) GLPV: Hirano+ (2018)DHOST: Hirano+ in prep.
![Page 29: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/29.jpg)
SUMMARY
![Page 30: MATTER BISPECTRUM BEYOND HORNDESKI](https://reader031.vdocuments.net/reader031/viewer/2022020700/61f4c77352b0626eb2659e66/html5/thumbnails/30.jpg)
■ We discuss beyond Horndeski on density fluctuations
at cosmological scale under some assumptions (QSA, )
Summary
■ Non-linear int. … (small scale, early universe) Vainshtein screening
(cosmological scale) Matter bispectrum, …
■ Cosmological perturbations
non-linear level: new time-evolution on matter bispectrum(t)
k-dependence … folded shape (beyond Horndeski)
linear level: friction term &(t)
18/18
↵i ⇠ �1 = O(1)
<Future direction>
・Typical value of κ and λ beyond Horndeski? Hirano+ in prep.
・The effect of partial breaking on the density perturbations?