Large Scale Structure in
Degenerate Higher-order
Scalar-Tensor Theories
立教大学 D3
共同研究者

平野進一

小林努(立教大), 田代寛之(名大),山内大介(神奈川大), 横山修一郎(名大)理論懇シンポジウム 2019年12月25~27日

修正重力理論■ 後期加速膨張の存在
・後期加速膨張 (Ia Supernova, LSS, CMB)・一般相対論では、後期加速膨張 = 宇宙項→ 宇宙項問題, coincidence, …

■ 修正重力理論 (宇宙項の代替として)・ 宇宙論的スケール: 後期加速膨張
・ 小スケール: スクリーニング機構 → ニュートンの逆二乗則を回復
・ 重力理論の拡張〜

新しい自由度の追加→ シンプルな拡張: スカラーテンソル理論 (gμν + ϕ)1

Horndeski theory

Horndeski (1972) , Kobayashi+ (2011), Deffyet+ (2011)

Lp ℒ = G2 ( , X)grµ = µ , r⌫ rµ = µ⌫ ,, ∇µμµ∇=⇤
G3 ( , X)⇤
μϕ = □ ϕ
1 µ⌫
⇥
⇤
X
=
g rµ r⌫2
2+ G4 ( , X)R + G4X (⇤ )2
µ⌫
⇤
G5X ⇥
µ⌫
3
2
3
+ G5 ( , X)G(⇤ )3(⇤ ) µ⌫ + 2 µ⌫µ⌫6<latexit sha1_base64="(null)">(null)</latexit>✓運動方程式が2階の最も一般的なスカラーテンソル理論 (2+1自由度).
✓ 任意関数の形を決めると具体的なモデルを再現✓ 加速膨張解が存在(例) f(R), DGP(例) De Felice-Tsujikawa (2009, 2010)✓ 小スケールでスカラー場の伝搬を抑える → Vainshtein screening
Kimura+ (2012), Narikawa+ (2013), Koyama+ (2013)( □ ϕを含む非線形相互作用)2

Whether go beyond Horndeski or not?
▪ Horndeski theory:
運動方程式が2階の最も一般的なスカラーテンソル理論▪ General higher-order EoMs: Ostrogradsky ghost

Ostrogradsky (1850)

縮退条件 (式の組み合わせで高階微分を消す)
2nd-order system
Degenerate Higher Order Scalar-Tensor (DHOST) theories
Langlois-Noui (2015,16), Crisostomi+ (2016)3

Quadratic DHOST

Langlois-Noui (2015,2016),
Crisostomi+ (2016)■ G4 の一般化 ( G2 , G3 + 6 funcs. )rµ =<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

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

G3 ( , X)⇤+f ( , X)R +<latexit sha1_base64="(null)">(null)</latexit>

5
X
i=1

µ , r⌫ r µ

=

µ⌫ ,

µ
µ =⇤

⇠ (rr )2<latexit sha1_base64="(null)">(null)</latexit>

Li

LL11 = A1 ( , X) µ⌫ µ⌫
LL22 = A2 ( , X)(⇤ )2
LL33 = A3 ( , X)⇤ µ µ⌫ ⌫LL44 = A4 ( , X) µ µ⇢ ⇢⌫ ⌫LL55 = A5 ( , X)( µ µ⌫ ⌫ )2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>■ healthy higher-order theory<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

縮退条件:<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

3 relations between f and Ai<latexit sha1_base64="(null)">(null)</latexit>

independent funcs.

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

G 2 , G3
<latexit sha1_base64="(null)">(null)</latexit>

andf, A1, A3(6 -3 =3)4

重力波からの制限■ GW170817, GRB 170817: NS-NS merger|c2GW<latexit sha1_base64="(null)">(null)</latexit>

1| < O(10 15 )Abbott+ (2017)

ほぼ光速

A1 = 0c2GW = 1<latexit sha1_base64="(null)">(null)</latexit>■ Catastrophic GWs decay into scalar field··
Hint ∼ A3 hij∂iπ∂jπ
hij

⇡

→

Γhππ
H0

Creminelli+ (2018, 19)

ωGW
∼ A3
( H0 )

2

≪1
要請

→

A3 = 0<latexit sha1_base64="(null)">(null)</latexit><latexit⇡<latexit sha1_base64="(null)">(null)</latexit><latexit

5

DHOST theory evading GW constraints
Creminelli+ (2018, 2019)cGW = 1 ( A1 = 0 ) + no GW decay ( A3 = 0 )<latexit sha1_base64="(null)">(null)</latexit>

3f X2 μℒ = G2(ϕ, X) − G3(ϕ, X) □ ϕ + f(ϕ, X)R + ϕ ϕμρϕ νρϕν
2f

Our work
Toward tests of DHOST theory evading GW constraints
・仕事1: 線形密度揺らぎからの制限
・仕事2: 非線形密度揺らぎへの影響 (matter bispectrum)6仕事1: 線形密度揺らぎ
からの制限
SH, T. Kobayashi, D. Yamauchi, S. Yokoyama,
Phys. Rev. D 99 (2019) no.10, 104051

Setup■ Quadratic DHOST, sub-horizon ( aH ⌧ k) , 物質優勢以降■ perturbations (一様等方時空＋摂動)

ds2 =(1 + 2 )dt2 + a2 (t)(12 )dx2 .

(t, x) = (t) + ⇡(t, x), ⇢(t, x) = ⇢(t)[1 + (t, x)].
■ Quasi-static approximation (QSA) ← 仮定|✏|
˙ ⇡ |H✏|, ✏ =π, ,Qπ· |22 ⌧ k2
| ˙ |2 , | ˙ |2 , ||Q̇|

2

, k2

2

, k2 Q
π 22
7

Evolution of density ﬂuctuations■ Perturbative expansion: ϵ = ϵ1, ϵ = Φ, Ψ, π, δ
■ EoMs: δΦ, δΨ, δπ

Φ1↑ include the effect of modified gravity(modified Poisson eq., anisotropic stress)■ Fluid equations:continuity/ Eular(standard forms)

u i : velocity field@ (t, x) 1
+ @i [(1 + )ui (t, x)] = 0,
@t
a

@ui
1 j
i
+ Hu + u @j ui =
@t
a

1 i
@ (t, x)aevolution equation of density flues.
8

1st-order solution¨1 + (2 + &)H ˙1

Kobayashi+ (2015), D’Amico+ (2017),
Crisostomi-Koyama (2017), Hirano+ (2019)4⇡Ge↵ ⇢m 1 = 0■ Ge↵ (t): G (GR) , Geﬀ ≠ G (Horndeski, DHOST): ↵0H (GR,&(t) /, 1 : Horndeski), ς ≠ 0 (DHOST)■

growing mode:

1 (p, t) = D+ (t) L (p)

D+ (t) : growth factor,

L (p) : initial density fluc.

9Linear growth of density ﬂucs.
Ḋ+■ Linear growth rate f :=(物質優勢期でf = 1)
D+ H
<latexit sha1_base64="(null)">(null)</latexit>■ Growth index<latexit sha1_base64="(null)">(null)</latexit>

ρmd ln fΩm :=:=(3Mpl2 H 2 , 物質優勢期で1)
d ln ⌦m Ωm=1

γGR = 6/11 = 0.545■ Observational data
BOSS DR12

Sanchez+ (2016)

BOSS DR14

Zhao+ (2018)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>= 0.609 ± 0.079= 0.580 ± 0.082

10やったことHirano+ (2019)■ Typical DHOST models✔ GW170817, GRB 170817

c2GW = 1<latexit sha1_base64="(null)">(null)</latexit>✔ G2 ( , X) = X p , p = const.
<latexit sha1_base64="(null)">(null)</latexit>✔ tracker ansatz: H ˙ q = const.

e.g.) De Felice-Tsujikawa (2010, 2011)<latexit sha1_base64="(null)">(null)</latexit>

2X fX
X= cH(1 − Ωm), β1 := ( fX + XA3) = β(1 − Ωm)✔ models with αH := −ff

↑ DHOST固有の11やったことHirano+ (2019)■ Typical DHOST models✔ GW170817, GRB 170817

c2GW = 1<latexit sha1_base64="(null)">(null)</latexit>✔ G2 ( , X) = X p , p = const.
<latexit sha1_base64="(null)">(null)</latexit>✔ tracker ansatz: H ˙ q = const.

e.g.) De Felice-Tsujikawa (2010, 2011)<latexit sha1_base64="(null)">(null)</latexit>

2X fX
X= cH(1 − Ωm), β1 := ( fX + XA3) = β(1 − Ωm)✔ models with αH := −ff

↑ DHOST固有のγ = γ(p, q, cH, β)を計算し、given p, qの下、c_H-β 平面に| γ − 6/11 | < 0.1 の領域をプロット.

11

Comparison to that of previous works
Hirano+ (2019)

Ours(p = 1, q = 1/2)
<latexit sha1_base64="(null)">(null)</latexit>

Babichev+ (2016),
Sakstein+ (2016)(from astrophysical phenomena
at a small scale)12Comparison to that of previous works
Hirano+ (2019)

Ours(p = 1, q = 1/2)
<latexit sha1_base64="(null)">(null)</latexit>

We obtain the constraint on DHOST theories as well as those ofprevious works and this bound at high-z is independent of those!
Babichev+ (2016),
Sakstein+ (2016)(from astrophysical phenomena
at a small scale)12仕事2: 非線形密度揺らぎ
への影響
SH, T. Kobayashi, H. Tashiro, S. Yokoyama,
Phys. Rev. D 97 (2018) no.10, 103517

Evolution of density ﬂuctuations■ Perturbative expansion: ϵ = ϵ1 + ϵ2, ϵ = Φ, Ψ, π, δ
■ EoMs: δΦ, δΨ, δπ

Φ1, Φ2↑ include the effect of modified gravity(modified Poisson eq., anisotropic stress)■ Fluid equations:continuity/ Eular(standard forms)

u i : velocity field@ (t, x) 1
+ @i [(1 + )ui (t, x)] = 0,
@t
a

@ui
1 j
i
+ Hu + u @j ui =
@t
a

1 i
@ (t, x)aevolution equation of density flues.
13

2nd-order solution Hirano+ (2018), Crisostomi+ (2019)¨2 + (2 + &)H ˙2214⇡Ge↵ ⇢m 2 = S

Primordial fluc. : Gaussian
2
(p,
t)
=
D
2
+ (t)
1
Wi (p) =(2⇡)3

i = ↵,,

Z

inhomogeneous sol.

(t)W↵ (p)d3 k1 d3 k2 (3) (k1 + k2

(k1 · k2 )(k12 + k22 )↵(k1 , k2 ) = 1 +2k12 k222(t)W (p)7p)E(ki(k11 ,· kk22) L (k1 ) L (k2 ), (k1 , k2 ) = 1(k1 · k2 )2
k12 k22(t) ： 1 (GR) , λ ≠ 1 (Horndeski, DHOST), 1 Horndeski), κ ≠ 1 (DHOST)
New! (t)：: ↵1H(GR,14

Matter bispectrumcf) Scoccimarro+ (1998)
Barnardeau+ (2000)■ 3pt correlation function: bispectrumh (t, k1 ) (t, k2 ) (t, k3 )i := (2⇡)3 (k1 + k2 + k3 )B(t, k1 , k2 , k3 )⟨δ2δ1δ1⟩ : leading order (tree-level)

15

Matter bispectrumcf) Scoccimarro+ (1998)
Barnardeau+ (2000)■ Reduced bispectrum

B(t, k1 , k2 , k3 )Q123 (t, k1 , k2 , k3 ) = 4
D+ (t)[P11 (k1 )P11 (k2 ) + 2 cyclic terms]
only depends on the time evolution of  and<latexit sha1_base64="43dSTpPQ2Z9ev3CY2uuBF+6ooHo=">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</latexit><latexit✓ k1 + k2 + k3 = 0<latexit sha1_base64="R5DMhSnXDVW5v4Qj/+UdgUCUQGY=">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</latexit>
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k1 = (0, 0, k1 ), k2 = (0, k2 sin ✓12 , k2 cos ✓12 ) ,k3 = (0, k2 sin ✓12 , k1
✓12

k1

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k2 cos ✓12 )k2<latexit sha1_base64="3RztRoVDNQxPGctgB2j5w888nkI=">AAACd3ichVHNSsNAEP4a/+tPq14EDxZLpacy6UXxVPTi0b9qQaUkcVvXpklI0mItvoAv4EEQFETFx/DiC3jwEcRjBRE8OE0DoqLOsruz38438+2s7pjS84keI0pXd09vX/9AdHBoeCQWHx3b8Oyaa4i8YZu2W9A1T5jSEnlf+qYoOK7QqropNvXKYvt+sy5cT9rWut9wxE5VK1uyJA3NZ2gfTVxCRwkJVHCEIrKIFuNJylBgiZ+OGjpJhLZsx6+wjV3YMFBDFQIWfPZNaPB4bEEFwWFsh8tpcNmTwb3gklHm1jhKcITGaIXXMp+2QtTiczunF7ANrmLydJmZQIoe6IZadE+39ETvv+ZqBjnaWhq86x2ucIqx44m1139ZVd597H2y/tTsczPnAq2StTsB0n6F0eHXD09aa/OrqeYMXdAz6z+nR7rjF1j1F+NyRayeBh+gfm/3T2cjm1Epo65kk7mF8Cv6MYlppLnfs8hhCcvIc90DnOEK15E3ZUqZUdKdUCUScsbxxRT1Axawjc0=</latexit>
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k3<latexit sha1_base64="RjgbdfDRenWaH2S3k0CW16ssPx8=">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</latexit>
sha1_base64="wHc+if2M1adkz6en9X6t7c2dv5g=">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</latexit>✓ We estimate matter bispectrum on the given ,(z=0)
at k1 = k2 = 0.01h/Mpc and k1 = 5k2 = 0.05h/Mpc .
(cosmological parameters: Planck 2015)15

k-dependence

Hirano+ (2018)

k1 = 5k2 = 0.05h/M pc

k1 = k2 = 0.01h/M pc
1.0

ー: GR

0.8

1.6

1.4

Q

(Horndeski)

Q

=1

0.6

0.4

0.2

0.0

--:

= 0.9

--:

= 1.1

0.0

0.2

0.4

--:

= 0.9

--:

= 1.1

1.2

1.0

ー: GR

0.8
0.6

0.8

1.0

0.0

0.2

0.4

θ12 /π

1.6

Q

Q(DHOST)- - :  = 1.1

1.4

0.6

- - :  = 0.9

0.4
0.2

1.2

Effect of κ

1.0

- - :  = 1.1

0.0

1.0

- - :  = 0.9

1.8

0.8

=1

0.8

θ12 /πEffect of κ

1.0

0.6

0.8
0.6

0.0

0.2

0.4

0.6
θ12 /π

0.8

1.0

0.0

0.2

0.4

0.6
θ12 /π

0.8

1.0

16

k-dependence

Hirano+ (2018)

k1 = 5k2 = 0.05h/M pc

k1 = k2 = 0.01h/M pc
1.0

ー: GR

0.8

shape
- Equilateral- : = 0.9

0.4

0.2

--:

0.0

0.0

Q

(Horndeski)

1.4

0.6

Q

=1

1.6

0.4

0.8

--:

= 1.1

1.2

ー: LCDM

0.8
0.6

= 0.9

1.0

= 1.1
0.2

--:

1.0

0.0

θ12 /π

0.2

0.4

0.6

0.8

1.0

θ12 /πEffect of κ

1.0

1.6

0.8

Q

(DHOST)- - :  = 1.1

1.4

0.6

Q

=1

- - :  = 0.9

1.8- - :  = 0.9

0.4
0.2

Effect of κ

1.0

- - :  = 1.1

0.0

1.2

0.8
0.6

0.0

0.2

0.4

0.6
θ12 /π

0.8

1.0

0.0

0.2

0.4

0.6
θ12 /π

0.8

1.0

16

k-dependence

Hirano+ (2018)

k1 = 5k2 = 0.05h/M pc

k1 = k2 = 0.01h/M pc
1.0

ー: GR

0.8

shape
- Equilateral- : = 0.9

0.4

0.2

--:

0.0

0.0

Q

(Horndeski)

1.4

0.6

Q

=1

1.6

0.4

0.8

--:

= 1.1

1.2

ー: LCDM

0.8
0.6

= 0.9

1.0

= 1.1
0.2

--:

1.0

0.0

θ12 /π

0.2

0.4

0.6

0.8

1.0

θ12 /πEffect of κ

1.0

1.6

0.8

Q

(DHOST)- - :  = 0.9

0.4
0.2

1.2

Effect of κ

1.0

- - :  = 1.1

0.0- - :  = 1.1

1.4

Folded shape

0.6

Q

=1

- - :  = 0.9

1.8

0.8
0.6

0.0

0.2

0.4

0.6
θ12 /π

0.8

1.0

0.0

0.2

0.4

0.6
θ12 /π

0.8

1.0

16

Summary■ 修正重力理論 大スケール→後期加速膨張
小スケール→修正の影響を遮蔽するスクリーニング■ DHOST理論

現在最も広いスカラーテンソル理論
重力波の制限も突破■ 仕事1: 線形密度揺らぎを用いたDHOSTの制限→growth index γを用いることで典型的なモデルに制限■ 仕事2: バイスペクトルにおけるDHOSTの影響→波数空間でfolded typeのピークを持つ