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

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

■ 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=">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</latexit><latexit sha1_base64="wKpvRKEKZEjv2fRd/KVFUjzD7CI=">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</latexit><latexit sha1_base64="wKpvRKEKZEjv2fRd/KVFUjzD7CI=">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</latexit>

Plan of talk

■ Our setup

■ Matter bispectrum beyond Horndeski

■Summary

3/18

■ Cosmological perturbations

OUR SETUP

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”

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)

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

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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)

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

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)

■ 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=">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</latexit><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>

� ⇠ 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

COSMOLOGICAL PERTURBATIONS

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

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

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

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

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

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

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

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

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)

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=">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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>

Horndeski: Takushima+ (2013) GLPV: Hirano+ (2018)

MATTER BISPECTRUM BEYOND HORNDESKI

Hirano+ (2018), Hirano+ in prep.

■ 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

■ Reduced bispectrum

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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)

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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>

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.

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

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.

SUMMARY

■ 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?