Introduction So far, GR seems compatible with all observations. - - PowerPoint PPT Presentation

Dark energy & Modified gravity

in scalar-tensor theories

David Langlois (APC, Paris)

Introduction

• So far, GR seems compatible with all observations.
• Several motivations for exploring modified gravity

– Quantum gravity effects – Explain cosmological acceleration (or possibly dark matter) – Explore alternative gravitational theories – Testing gravity

• Many models of dark energy & modified gravity:

quintessence, K-essence, f(R) gravity, massive gravity…

• Generalized framework for scalar-tensor theories,

allowing for 2nd order derivatives in their Lagrangian

• Simplest extensions of GR: add a scalar field
• K-essence/ k-inflation: non standard kinetic term

Traditional scalar-tensor theories

S = Z d4x √−g M 2

P

2

(4)R + P(X, φ)

• X ⌘ rµφ rµφ

S = Z d4x√−g h F(φ)(4)R − Z(φ)∂µφ∂µφ − U(φ) i + Sm[ψm; gµν]

• Traditional scalar-tensor theories :
• Generalized theories with second order derivatives
• In general, they contain an extra degree of freedom,

expected to lead to Ostrogradsky instabilities

• But there are exceptions...

Higher order scalar-tensor theories

L(rµrνφ, rλφ, φ) L(rλφ, φ) L(¨ q, ˙ q, q)

Horndeski theories

• Combination of the Lagrangians
• Second order equations of motion for the scalar field

and the metric

• They contain 1 scalar DOF and 2 tensor DOF.

No dangerous extra DOF !

LH

2 = G2(φ, X)

LH

3 = G3(φ, X) ⇤φ

LH

4 = G4(φ, X) (4)R − 2G4X(φ, X)(⇤φ2 − φµνφµν)

LH

5 = G5(φ, X) (4)Gµνφµν + 1

3G5X(φ, X)(⇤φ3 − 3 ⇤φ φµνφµν + 2 φµνφµσφν

σ)

with

φµν ⌘ rνrµφ

X ⌘ rµφrµφ Horndeski 74

(a.k.a. Generalized Galileons)

Beyond Horndeski & DHOST theories

• Extensions “beyond Horndeski”

leading to third order equations of motion.

• Earlier hint: disformal transformation of Einstein-Hilbert
• Even if EOM are higher order, no extra DOF if the

Lagrangian is “degenerate”. DHOST theories LbH

5

≡ F5(, X)✏µνρσ✏µ0ν0ρ0σ0µµ0νν0ρρ0σσ0 LbH

4

≡ F4(, X) ✏µνρ

σ ✏µ0ν0ρ0σµµ0νν0ρρ0

Gleyzes, DL, Piazza &Vernizzi ’14 Zumalacarregui & Garcia-Bellido ‘13

DL & K. Noui ‘15

(Degenerate Higher-Order Scalar-Tensor)

• Traditional theories:
• Generalized theories:

Higher order scalar-tensor theories

Beyond Horndeski (GLPV) DHOST

L(rµrνφ, rλφ, φ)

Extra DOF

L(rλφ, φ)

Degenerate Higher-Order Scalar-Tensor

Horndeski

Degenerate Lagrangians

• Scalar-tensor theories: scalar field + metric
• Simple toy model:
• Lagrangian
• Equations of motion are higher order

(4th order if a nonzero, 3rd order if a=0) DL & K. Noui ‘1510

φ(xλ) → φ(t) , gµν(xλ) → q(t) L = 1 2a ¨ φ2 + b ¨ φ ˙ q + 1 2c ˙ q2 + 1 2 ˙ φ2 − V (φ, q)

Degrees of freedom

• Introduce the auxiliary variable
• Equations of motion
• If the Hessian matrix

is invertible,

• ne finds 3 DOF.

[6 initial conditions]

Q ≡ ˙ φ L = 1 2a ˙ Q2 + b ˙ Q ˙ q + 1 2c ˙ q2 + 1 2Q2 − V (φ, q) − λ(Q − ˙ φ) a ¨ Q + b ¨ q = Q − λ b ¨ Q + c ¨ q = −Vq ˙ φ = Q , ˙ λ = −Vφ

M ≡ ✓ ∂2L ∂va∂vb ◆ = ✓ a b b c ◆

Degrees of freedom

• If the Hessian matrix

is degenerate, i.e.

then only 2 DOF (at most).

[ can be absorbed in ]

• Hamiltonian analysis: primary constraint and secondary

constraint

M ≡ ✓ ∂2L ∂va∂vb ◆ = ✓ a b b c ◆

ac − b2 = 0

[ cannot be inverted ]

pa = ∂L ∂va (v)

˙ x ≡ ˙ q + b c ¨ φ ¨ φ

Generalization (classical mechanics)

• Consider a general Lagrangian

In general, 2n+m DOF. But the n extra DOF can be eliminated by requiring:

• 1. Primary conditions (n primary constraints )
• 2. Secondary conditions (n secondary constraints)
• Third-order time derivatives...

L(¨ φα, ˙ φα, φα; ˙ qi, qi)

α = 1, · · · , n; i = 1, · · · , m

Motohashi, Noui, Suyama, Yamaguchi & DL 1603

L ˙

Qα ˙ Qα − L ˙ Qα ˙ qi(L−1) ˙ qi ˙ qjL ˙ qj ˙ Qβ = 0

L ˙

Qα ˙ φβ − L ˙ Qβ ˙ φα = 0

if m = 0

[See also Klein & Roest 1604] Motohashi, Suyama, Yamaguchi 1711

Quadratic DHOST theories

• Consider all theories of the form

where

and depends only on and .

• All possible contractions of

? In summary:

φ rµφ

DL & Noui ’1510

f2 = f2(X, φ)

S[g, φ] = Z d4x pg h f2

(4)R + Cµνρσ (2)

rµrνφ rρrσφ i

Cµνρσ

(2)

L(2)

1

= φµν φµν , L(2)

2

= (⇤φ)2 , L(2)

3

= (⇤φ)φµφµνφν L(2)

4

= φµφµρφρνφν , L(2)

5

= (φµφµνφν)2

Cµνρσ

(2)

φµν φρσ = X aA(X, φ) L(2)

A

φµν φρσ

e.g. gµνgρσφµν φρσ = (⇤φ)2

φµφνφρφσφµν φρσ = (φµφµνφν)2

• r

Quadratic DHOST theories

• Lagrangians of the form

which depend on 6 arbitrary functions.

• Degeneracy yields three conditions on the 6 functions.
• Classification: 7 subclasses (4 with , 3 with )
• This includes, in particular, and

LH

4

LbH

4

DL & Noui ’1510

f2 6= 0 f2 = 0

[See also Crisostomi et al ‘1602; Ben Achour, DL & Noui ’1602; de Rham & Matas ‘1604]

L = f2(X, φ) (4)R +

5

X

A=I

aA(X, φ) L(2)

A

f2 = G4 , a1 = −a2 = 2G4X + XF4 , a3 = −a4 = 2F4

Cubic DHOST theories

• Action of the form

depends on eleven functions:

• This includes the Lagrangians and .
• 9 degenerate subclasses: 2 with , 7 with
• 25 combinations of quadratic and cubic theories (out
• f 7x9) are degenerate.

LbH

5

LH

5

[Ben Achour, Crisostomi, Koyama, DL, Noui & Tasinato ’1608]

Cµνρσαβ

(3)

φµν φρσ φαβ =

10

X

i=1

bi(X, φ) L(3)

i

S[g, φ] = Z d4x √−g h f3 Gµνφµν + Cµνρσαβ

(3)

φµν φρσ φαβ i

f3 = 0

f3 6= 0

Disformal transformations

• Transformations of the metric
• Starting from an action

, one can define the new action

• Disformal transformation of quadratic DHOST theories ?

The structure of DHOST theories is preserved and all seven subclasses are stable. ˜ S [φ, ˜ gµν]

[Ben Achour, DL & Noui ’1602]

S[φ, gµν] ≡ ˜ S [φ, ˜ gµν = C gµν + D φµφν] gµν − → ˜ gµν = C(X, φ) gµν + D(X, φ) ∂µφ ∂νφ

[Bekenstein ’93]

˜ S = Z d4x p −˜ g " ˜ f2

(4)˜

R + X

I

˜ aI ˜ L(2)

I

#

Disformal transformations

• Stability under
• When matter is included (with minimal coupling), two

disformally related theories are physically inequivalent !

Horndeski Beyond Horndeski DHOST

C(φ), D(φ) C(φ), D(X, φ) C(X, φ), D(X, φ)

gµν − → ˜ gµν = C gµν + D ∂µφ ∂νφ

Ben Achour, DL & Noui ’16 Gleyzes, DL, Piazza & Vernizzi ‘14 Bettoni & Liberati ‘13

Cosmology: Effective description of Dark Energy & Modified Gravity

Parametrized Effective Description Observational constraints Theories

Effective description of Dark Energy

• Restriction: single scalar field models
• The scalar field defines a preferred slicing

Constant time hypersurfaces = uniform field hypersurfaces

• All perturbations embodied by the metric only

φ = φ1 φ = φ2 φ = φ3

[ See e.g review: Gleyzes, DL & Vernizzi 1411.3712 ]

Uniform scalar field slicing

• 3+1 decomposition based on this preferred slicing
• Basic ingredients

– Unit vector normal to the hypersurfaces – Projection on the hypersurfaces: hµν = gµν + nµ nν nµ = rµφ p (rφ)2

ADM formulation

• ADM decomposition of spacetime
• Generic Lagrangians of the form

ds2 = −N 2dt2 + hij

• dxi + N idt

dxj + N jdt

• hij

Extrinsic curvature:

Kij = 1 2N ˙ hij − DiNj − DjNi

• Rij

X ⌘ gµνrµφrνφ = ˙ φ2(t) N 2

Intrinsic curvature:

N i Nnµ

Sg = Z d4x N √ h L(N, Kij, Rij; t)

Homogeneous background & linear perturbations

• Background
• Perturbations:
• Expanding the Lagrangian

with yields

• The quadratic action describes the dynamics of linear

perturbations L(qA) qA ≡ {N, Ki

j, Ri j}

L(qA) = ¯ L + ∂L ∂qA δqA + 1 2 ∂2L ∂qA∂qB δqAδqB + . . .

¯ L(a, ˙ a, ¯ N) ≡ L  Ki

j =

˙ a ¯ Na δi

j, Ri j = 0, N = ¯

N(t)

• ds2 = − ¯

N 2(t) dt2 + a2(t) δijdxidxj δN ≡ N − ¯ N , δKi

j ≡ Ki j − Hδi j , δRi j ≡ Rj i

Horndeski & beyond Horndeski

• Quadratic action

S(2) = Z dx3dt a3 M 2 2  δKi

jδKj i − δK2 + αKH2δN 2 + 4 αBH δK δN

+ (1 + αT )δ2 ✓√ h a3 R ◆ + (1 + αH)R δN

• X

αM αK αB αT αH

Quintessence, K-essence

X

Kinetic braiding, DGP

X X

Brans-Dicke, f(R)

X X X

Horndeski Beyond Horndeski

X X X X X X X X X

αM ≡ d ln M 2 H dt

Gleyzes, DL, Piazza & Vernizzi ’13, [notation: Bellini & Sawicki ‘14]

Scalar degree of freedom

• Scalar perturbations:
• Quadratic action for the physical degree of freedom:
• Stability (neither ghost nor gradient instability)

δN , Ni ≡ ∂iψ , hij = a2(t)e2ζδij S(2) = 1 2 Z dx3dt a3  Kt ˙ ζ2 + Ks (∂iζ)2 a2

• Kt ≡ αK + 6α2

B

(1 + αB)2 , Ks ≡ 2M 2 ( 1 + αT − 1 + αH 1 + αB ⇣ 1 + αM − ˙ H H2 ⌘ − 1 H d dt ✓1 + αH 1 + αB ◆)

Kt > 0 c2

s ≡ −Ks

Kt > 0

Tensor degrees of freedom

• Quadratic action for the tensor modes:
• Stability:

and S(2)

γ

= 1 2 Z dt d3x a3 M 2 4 ˙ γ2

ij − M 2

4 (1 + αT )(∂kγij)2 a2

• M 2 > 0

c2

T ≡ 1 + αT > 0

Extension to DHOST theories

• Quadratic action in terms of 9 functions of time
• Degeneracy conditions: 2 categories

Squad = Z d3x dt a3 M 2 2 ⇢ δKijδKij − ✓ 1 + 2 3αL ◆ δK2 + (1 + αT ) ✓ Rδ √ h a3 + δ2R ◆ +H2αKδN 2 + 4HαBδKδN + (1 + αH)RδN + 4β1δKδ ˙ N + β2δ ˙ N

2 + β3

a2 (∂iδN)2

• DL, Mancarella, Noui & Vernizzi ’1703

CII : β1 = −(1 + αL)1 + αH 1 + αT , β2 = −6(1 + αL)(1 + αH)2 (1 + αT)2 , β3 = 2(1 + αH)2 1 + αT CI : αL = 0 , β2 = −6β2

1 , β3 = −2β1 [2(1 + αH) + β1(1 + αT)]

: gradient instability either in the scalar or the tensor sector

CII

9-3=6 independent coefficients

Scalar-tensor theories

Type I Type II DHOST Horndeski Beyond Horndeski DHOST I (Gradient instability !)

Disformal transformations

• Disformal transformations:

D(X) C(X)

DHOST C(X, φ), D(X, φ) Horndeski C(φ), D(φ) Beyond Horndeski C(φ), D(X, φ) DHOST I

˜ gµν = C gµν + D ∂µφ ∂νφ

Type I Type II

Disformal transformations

• Disformal transformations:
• Mimetic gravity & extensions (non-invertible transf) are DHOST

theories of type II (some of type I too) and all unstable.

D(X) C(X)

DHOST C(X, φ), D(X, φ) Horndeski C(φ), D(φ) Beyond Horndeski C(φ), D(X, φ) DHOST I

˜ gµν = C gµν + D ∂µφ ∂νφ

DL, Mancarella, Noui & Vernizzi ’1802 [see also Takahashi & Kobayashi ‘1708]

DHOST theories after GW170817

DHOST theories after GW170817

• Constraint on the speed of gravitational waves:
• Assuming

holds exactly, this implies

• 1. Quadratic terms:
• 2. No cubic term (for type I theories)
• Remain quadratic DHOST theories of type I with

αT < 10−15

αT = 0

a1 = 0 a1 = 0 LADM = (f − Xa1)KijKij − f (3)R

DHOST theories with cg= c

• Taking into account the degeneracy conditions,
• Total Lagrangian

4 free functions of and (as in Horndeski without ! )

a1 = a2 = 0 , a4 = 1 8f2 ⇥ 48f 2

2X − 8(f2 − Xf2X)a3 − X2a2 3

⇤ , a5 = 1 2f2 (4f2X + Xa3) a3

2 free functions

LDHOST

cg=1

= f2(X, φ) (4)R + P(X, φ) + Q(X, φ) ⇤φ + a3(X, φ)φµφνφµν⇤φ + a4(X, φ)φµφµνφλφλν + a5(X, φ)(φµφµνφν)2 X

φ

cg = 1

Horndeski and Beyond Horndeski with cg= c

• Remaining Beyond Horndeski theories
• Remaining Horndeski theories

G2(X, φ) , G3(X, φ) , G4(φ) a1 = 2G4X + XF4 = 0 = ⇒ F4 = − 2 X G4X

S[g, φ] = Z d4x √−g ⇢ f(φ, X) R − 4 X fX h (⇤φ)φµφµνφν − φµφµνφνρ φρ i

Gravitation in DHOST with cg= c

• Quasi-static approximation on scales
• Equations of motion for

– Scalar equation – Metric equations

• Matter source: spherical body with density

ds2 = −(1 + 2Φ)dt2 + (1 − 2Ψ)δijdxidxj r ⌧ H−1 φ = φc(t) + χ(r)

χ, Φ and Ψ

ρ(r)

DL, Saito, Yamauchi & Noui ’1711 [see also Crisostomi & Koyama ‘1711

and Dima & Vernizzi ‘1712]

Gravitation in DHOST with cg= c

• Gravitational laws

where the coefficients are given in terms of and

• Breaking of the Vainshtein screening inside matter !

already noticed for Beyond Horndeski (GLPV)

dΦ dr = GN M(r) r2 + Ξ1 GN M00(r) , dΨ dr = GN M(r) r2 + Ξ2 GN M0(r) r + Ξ3 GN M00(r)

M(r) ≡ 4π Z r ¯ r2ρ(¯ r)d¯ r

DL, Saito, Yamauchi & Noui ’1711 [see also Crisostomi & Koyama ‘1711 and Dima & Vernizzi ‘1712]

with (8πGN)−1 ≡ 2f (1 + Ξ0)

ΞI

Kobayashi, Watanabe & Yamauchi ’14

f, fX, a3 ˙ φc

Gravitation in DHOST with cg= c

• The four coefficients depend on only 2 parameters
• Constraints on the coefficients

ΞI Ξ0 = −αH − 3β1 , Ξ1 = − (αH + β1)2 2(αH + 2β1) , Ξ2 = αH , Ξ3 = −β1(αH + β1) 2(αH + 2β1) .

Beltran Jimenez, Piazza & Velten 1507

Hulse-Taylor binary pulsar:

− 1 12 < Ξ1 . 0.2 Ξ0 = Ggw GN − 1 |Ξ0| < 10−2

[Saito, Yamauchi, Mizuno, Gleyzes & DL ’15] [Sakstein 15]

Stars: Gravitational lensing for the

• ther coefficients…

Conclusions

• DHOST theories provide a very general framework to

describe scalar-tensor theories with higher derivatives.

Systematic classification of “degenerate” theories that contain a single scalar DOF. They include and extend Horndeski and “beyond Horndeski” theories as particular cases.

• Drastic reduction of viable models after GW170817.
• These theories of modified gravity can be tested &

constrained via cosmology (future LSS observations) and astrophysics.