第3回 若手による重力・宇宙論研究会　2019年 2月27日 ~ 3月１日

SH, T. Kobayashi, D. Yamauchi (Kanagawa U.), S. Yokoyama (KMI, IPMU)

Shin’ichi Hirano　Rikkyo U., D2　

ONE-LOOP POWER SPECTRUM BEYOND HORNDESKI AND VAINSHTEIN SCREENING(work in progress)

Clue of modified gravityModified Gravity: accelerating universe → ~ΛCDM universe

Intermediate regime:

screening mechanism → ~ GR at a small scale

deviation from GR, no screening?

・BH falling into a galaxy, void profile e.g.) Sakstein+ (2018)

・Inner structure of compact star e.g.) Kobayashi, Hiramatsu (2018)

・GWs: scalar/vector polarization e.g.) Cardoso+ (2018)

・LSS: matter power spectrum/bispectrum e.g.) Hirano+ (2018)

LSS: non-linear property of gravity

cited from web page of M. Norman

— linear power

quasi non-linear regime : our target

Perturbative expansion:

Matter power spectrum:

solutions at each order

⇔ can trace non-linearproperty of gravity

< �(t,k)�(t, k̃) >

= (2⇡)3�(3)(k + k̃)P��(t, k)(null)(null)(null)(null)(null)

� = �1 + �2 + �3 + · · ·(null)(null)(null)(null)(null)

P�� =P�1�1 + P�2�2+ P�1�3 + P�3�1 + · · ·

(null)(null)(null)(null)(null)

Lp�g = G2(�, X)�G3(�, X)⇤�

+G4(�, X)R+G4X⇥(⇤�)2 � �2µ⌫

⇤

+G5(�, X)Gµ⌫�µ⌫ �

G5X6

⇥(⇤�)3 � 3(⇤�)�2µ⌫ + 2�3µ⌫

⇤

rµ� = �µ,r⌫rµ� = �µ⌫ ,�µµ = ⇤�( )

(GW170817, GRB 170817)|cT � 1| < 10�15

■ Most general ST theory with 2nd order EoM (no Ostrogradski ghost)

⇒ G5 Kimura+ (2011), Koyama+ (2013)■ Vainshtein screening via non-linear int.

■ Propagation speed of graviton changes from that of photon

Horndeski theory Horndeski (1972), Kobayashi+ (2011), Deffaiyet+ (2011)

⇒ G4X = 0(null)(null)(null)(null)(null)

■ Beyond Horndeski (higher-order EoM, no Ostrogradski ghost)

GLPV theory Gleyzes+ (2014), Gleyzes+ (2015)

DHOST theory Langlois & Noui (2015), Achour+ (2016), Achour+ (2016)

✓ Partially breaking of Vainshtein screening inside matter ( )Kobayashi+ (2015), Langlois+ (2017), …

� > 1

■ This work

One-loop matter power spectrum, beyond Horndeski

our work

How much is the effect of non-linear int. at “cosmological scale” ?� ⌧ 1

✓ non-linear ints. with and cT = 1@@�

Class I q. DHOSTLqDp�g = G2(�, X)�G3(�, X)⇤�+G4(�, X)R+ C

µ⌫⇢�(2) �µ⌫�⇢�

Langlois, Noui (2015,2016), Koyama+ (2016), de Rham, Matas (2016)

X = �12(r�)2,�µ = rµ�,⇤� = r2�,�µ⌫ = r⌫rµ�( )

Cµ⌫⇢�(2) �µ⌫�⇢� = a1(�, X)�2µ⌫ + a2(�, X)(⇤�)2+a3(�, X)⇤�(�µ�µ⌫�⌫)

+a4(�, X)�µ�µ⌫�

⌫⇢�⇢ + a5(�, X)(�µ�µ⌫�

⌫)2

■ healthy higher-order theory ⇒ degeneracy conditions

■ stable cosmological solution, partial breaking of Vainshtein screening,cT = 1

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

non-trivial non-linear ints. !!

■ includes Horndeski and GLPV at Lagrangian level

Horndeski: a1 = �a2 = �G4X , a3 = a4 = a5 = 0 , GLPV: …

ParametrizationSe↵ =

Zd4xp�M

2

2

"�K2 + c2TR(3) +H2↵K�N2 + 4H↵B�K�N

+ (1 + ↵H)R(3)

�N + (1 + ↵V )�N�K2 + 4�1�KṼ + �2Ṽ 2 + �3a2i

#.

K2 := K2ij �K2, Ṽ :=1

N(Ṅ �N i@iN), ai := @iN/N depend on

through degeneracy cond.�1

■ 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 (E. frame) → DHOST�1

: disformal coupling to matter (Einshtein frame) → GLPV↵H

�↵H

Bellini & Sawicki (2014), Gleyzes et al. (2015)Langlois+ (2017), Dima & Vernizzi (2017)

16/32

ΛCDM

Quintessence

KGB, cubic Galileon

Generalized Brans-Dicke

Horndeski after GW170817

GLPV

f(R)

DHOST

✓

✓✓

✓ ✓ ✓

✓ ✓ ✓

✓ ✓ ✓ ✓

✓✓

✓ ✓ ✓ ✓ ✓ ✓

↵K ↵B ↵M ↵H �1

Cosmological modelswith Vainshtein screening

Road to one-loop matter power

(usual forms)③ Fluid equations: @�(t,x)

@t+

1

a@i[(1 + �)u

i(t,x)] = 0,

@ui

@t+Hui +

1

auj@ju

i = �1a@i�(t,x)

② 3 EoMs: � , ��, �Q ⇒ solns.↑ include the effect of modified gravity

�1,�2,�3(null)(null)(null)(null)(null)

■ Assumptions: quasi-static approximation, O(↵i) = O(↵j) (i 6= j)(null)(null)(null)(null)(null)

■ Setup: sub-horizon ( )aH ⌧ k , period from MD, Newtonian gauge

① Perturbative expansion: ✏ = ✏1 + ✏2 + ✏3 (✏ = �, , Q, �)(null)(null)(null)(null)(null)

④ Take 2-pt correlation up to 3rd-order solutions:P one�loop�� = P�1�1 + P�2�2 + P�1�3 + P�3�1

(null)(null)(null)(null)(null)

GT@2 + Ã2@2Q�A6@2�+A8@2Q̇

H� a

2

2⇢m� = �

B2

2a2H2Q2 +

B5

a2H2

⇥(@i@jQ)

2 + @iQ@i@2Q⇤

(��)

FT@2 � GT@2�� Ã1@2Q+A4@2Q̇

H=

B1

2a2H2Q2 +

B4

a2H2

⇥(@i@jQ)

2 + @iQ@i@2Q⇤

(� )

A0@2Q�A1@2 �A2@2��A4

@2 ̇

H+A8

@2�̇

H� Ã9

@2Q̇

H�A9

@2Q̈

H2

= � B0a2H2

Q2 +B2

a2H2(@2�@2Q� @i@j�@i@jQ)

� B4a2H2

�@2 @2Q+ @iQ@

i@2

�+

B5

a2H2(@2�@2Q+ @iQ@

i@2�)

� B̃6a2H2

⇥(@i@jQ)

2 + @iQ@i@2Q⇤

� B6a2H2

1

H

⇣@2Q@

2Q̇+ 2@iQ@

i@2Q̇+ 2@i@jQ@

i@jQ̇+ @i@

2Q@

iQ̇

⌘

(�Q)

EoMs of gravitational fieldsA,B � ↵i,�1

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

Green: GLPV, Red: DHOST

+ · · ·(null)(null)(null)(null)(null)

1st-order solution previous works: Gleyzes+ (2015), D’Amico+ (2017), Chrisostomi & Koyama (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 H.)G ! Ge↵■ :: 0 (GR, Horndeski), (beyond H.)&(t) / ↵H ,�1 0 ! &

■　studying the linear evolution on typical BG solutions. in DHOSTHirano+ 1902. 02946

→ Gaussian

2nd-order solution

：�(t) (GR)1 , (Horndeski, beyond H.)1 ! �0 6= 1

�21�̈2 + (2 + &)H �̇2 � 4⇡Ge↵⇢m�2 = S�

⇒ �2(p, t) = D2+(t)(t)W↵(p)�

2

7�(t)W�(p)

�

1 (GR, Horndeski), (beyond H.)1 ! 0 6= 1New! ：(t) � ↵H ,�1

↵(k1,k2) = 1 +(k1 · k2)(k21 + k22)

2k21k22

, , �(k1,k2) = 1�(k1 · k2)2

k21k22

i = ↵, �

Horndeski: Takushima+ (2013) beyond H.: Hirano+ (2018, 2019?)

Wi(p) =1

(2⇡)3

Zd3k1d

3k2 �(3)(k1 + k2 � p)E(k1 · k2)�L(k1)�L(k2)i(k1,k2)

※ 3rd-order solution can be obtained with same way, but it is very complicatedSo, we skip the 3rd order level.

One-loop matter power spectrum

Takushima+ (2015)KGB modelScreening effect ?

n=5n=2n=1

■

(n→∞: Λ-CDM at BG level)

G2 = �X, G3 = MPl✓

r2cM2Pl

X

◆n,

G4 = M2Pl/2, a3 = 0

P one�loop�� = P�1�1 + P�2�2 + P�1�3 + P�3�1(null)(null)(null)(null)(null)

2nd/3rd-order solutions beyond Horndeski

⇒ Whether does screening effects appear on a power spectrum or not could depend on cosmological models.

�2(t,p) = D2+(t)

(t)W↵(p)�

2

7�(t)W�(p)

�Hirano+ (2019?)

> 1 ⇒ the effect of partial breaking appears on a power spectrum?⇒ power spectrum is suppressed/enhanced■

are not directly related with Vainshtein screening.(of course, related to non-linear interactions)

, �(null)(null)(null)(null)(null)

■

�3(t,k) = D3+(t)

3(t)W↵↵ �

2

7�3R(t)W↵�(p)�

2

7�3L(t)W↵�(p) + · · ·

�

(null)(null)(null)(null)(null)

� > 1 / � < 1(null)(null)(null)(null)(null)

■ We discuss beyond Horndeski on matter density fluctuationsat cosmological scale under some assumptions (QSA)

Summary

■ Non-linear int. … (small scale, early universe) Vainshtein screening(cosmological scale) Matter bispectrum, one-loop power

■ One-loop matter power spectrum

・screening effect appears on power spectrum? ⇒ model dependent?

・Partial breaking appears on power spectrum? ⇒ model dependent?

・Deviations from GR appear at a large or small scale in modified gravity