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

Traditional scalar-tensor theories • Simplest extensions of GR: add a scalar field Z h i d 4 x √− g F ( φ ) (4) R − Z ( φ ) ∂ µ φ∂ µ φ − U ( φ ) S = + S m [ ψ m ; g µ ν ] • K-essence/ k-inflation : non standard kinetic term  M 2 � Z d 4 x √− g (4) R + P ( X, φ ) P S = 2 X ⌘ r µ φ r µ φ

Higher order scalar-tensor theories • Traditional scalar-tensor theories : L ( r λ φ , φ ) • Generalized theories with second order derivatives L ( r µ r ν φ , r λ φ , φ ) • In general, they contain an extra degree of freedom , expected to lead to Ostrogradsky instabilities L (¨ q, ˙ q, q ) • But there are exceptions...

Horndeski theories Horndeski 74 • Combination of the Lagrangians (a.k.a. Generalized Galileons) L H 2 = G 2 ( φ , X ) with X ⌘ r µ φ r µ φ L H 3 = G 3 ( φ , X ) ⇤ φ φ µ ν ⌘ r ν r µ φ 4 = G 4 ( φ , X ) (4) R − 2 G 4 X ( φ , X )( ⇤ φ 2 − φ µ ν φ µ ν ) L H 5 = G 5 ( φ , X ) (4) G µ ν φ µ ν + 1 3 G 5 X ( φ , X )( ⇤ φ 3 − 3 ⇤ φ φ µ ν φ µ ν + 2 φ µ ν φ µ σ φ ν L H σ ) • 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 !

Beyond Horndeski & DHOST theories • Extensions “beyond Horndeski” Gleyzes, DL, Piazza &Vernizzi ’14 σ ✏ µ 0 ν 0 ρ 0 σ � µ � µ 0 � νν 0 � ρρ 0 L bH ≡ F 4 ( � , X ) ✏ µ νρ 4 ≡ F 5 ( � , X ) ✏ µ νρσ ✏ µ 0 ν 0 ρ 0 σ 0 � µ � µ 0 � νν 0 � ρρ 0 � σσ 0 L bH 5 leading to third order equations of motion. • Earlier hint: disformal transformation of Einstein-Hilbert Zumalacarregui & Garcia-Bellido ‘13 • Even if EOM are higher order, no extra DOF if the Lagrangian is “ degenerate”. DL & K. Noui ‘15 DHOST theories (Degenerate Higher-Order Scalar-Tensor)

Higher order scalar-tensor theories • Traditional theories: L ( r λ φ , φ ) • Generalized theories: L ( r µ r ν φ , r λ φ , φ ) Extra DOF Horndeski Degenerate Higher-Order Beyond Horndeski (GLPV) DHOST Scalar-Tensor

Degenerate Lagrangians DL & K. Noui ‘1510 • Scalar-tensor theories : scalar field + metric • Simple toy model: φ ( x λ ) → φ ( t ) , g µ ν ( x λ ) → q ( t ) • Lagrangian L = 1 q + 1 q 2 + 1 φ 2 + b ¨ 2 a ¨ ˙ φ 2 − V ( φ , q ) φ ˙ 2 c ˙ 2 • Equations of motion are higher order (4th order if a nonzero, 3rd order if a=0)

Degrees of freedom Q ≡ ˙ • Introduce the auxiliary variable φ L = 1 q + 1 q 2 + 1 Q 2 + b ˙ 2 Q 2 − V ( φ , q ) − λ ( Q − ˙ 2 a ˙ Q ˙ 2 c ˙ φ ) • Equations of motion a ¨ ˙ ˙ Q + b ¨ q = Q − λ φ = Q , λ = − V φ b ¨ Q + c ¨ q = − V q ✓ a ∂ 2 L • If the Hessian matrix ✓ ◆ ◆ b M ≡ = b c ∂ v a ∂ v b is invertible , one finds 3 DOF. [6 initial conditions]

Degrees of freedom ✓ a • If the Hessian matrix ∂ 2 L ✓ ◆ ◆ b M ≡ = is degenerate, i.e. ∂ v a ∂ v b b c ac − b 2 = 0 then only 2 DOF (at most) . q + b ¨ ¨ [ can be absorbed in ] φ x ≡ ˙ ˙ φ c • Hamiltonian analysis : primary constraint and secondary constraint p a = ∂ L [ cannot be inverted ] ∂ v a ( v )

Generalization (classical mechanics) Motohashi, Noui, Suyama, Yamaguchi & DL 1603 [See also Klein & Roest 1604] • Consider a general Lagrangian L (¨ φ α , ˙ q i , q i ) φ α , φ α ; ˙ α = 1 , · · · , n ; i = 1 , · · · , m In general, 2n+m DOF . But the n extra DOF can be eliminated by requiring: 1. Primary conditions (n primary constraints ) q i ˙ q j L ˙ q i ( L − 1 ) ˙ Q β = 0 L ˙ Q α − L ˙ Q α ˙ Q α ˙ q j ˙ 2. Secondary condition s (n secondary constraints) L ˙ φ β − L ˙ φ α = 0 if m = 0 Q α ˙ Q β ˙ • Third-order time derivatives... Motohashi, Suyama, Yamaguchi 1711

Quadratic DHOST theories DL & Noui ’1510 • Consider all theories of the form d 4 x p� g Z h i (4) R + C µ νρσ S [ g, φ ] = f 2 r µ r ν φ r ρ r σ φ (2) where depends only on and . and C µ νρσ φ f 2 = f 2 ( X, φ ) r µ φ (2) • All possible contractions of ? φ µ ν φ ρσ e.g. g µ ν g ρσ φ µ ν φ ρσ = ( ⇤ φ ) 2 φ µ φ ν φ ρ φ σ φ µ ν φ ρσ = ( φ µ φ µ ν φ ν ) 2 or a A ( X, φ ) L (2) C µ νρσ X In summary: φ µ ν φ ρσ = A (2) = φ µ ν φ µ ν , = ( ⇤ φ ) 2 , L (2) L (2) L (2) = ( ⇤ φ ) φ µ φ µ ν φ ν 1 2 3 L (2) L (2) = ( φ µ φ µ ν φ ν ) 2 = φ µ φ µ ρ φ ρν φ ν , 4 5

Quadratic DHOST theories • Lagrangians of the form DL & Noui ’1510 5 a A ( X, φ ) L (2) L = f 2 ( X, φ ) (4) R + X A A = I which depend on 6 arbitrary functions. • Degeneracy yields three conditions on the 6 functions. • Classification: 7 subclasses (4 with , 3 with ) f 2 6 = 0 f 2 = 0 [See also Crisostomi et al ‘1602; Ben Achour, DL & Noui ’1602; de Rham & Matas ‘1604] L H • This includes, in particular, and L bH 4 4 f 2 = G 4 , a 1 = − a 2 = 2 G 4 X + XF 4 , a 3 = − a 4 = 2 F 4

Cubic DHOST theories [Ben Achour, Crisostomi, Koyama, DL, Noui & Tasinato ’1608] • Action of the form Z h i d 4 x √− g f 3 G µ ν φ µ ν + C µ νρσαβ S [ g, φ ] = φ µ ν φ ρσ φ αβ (3) 10 b i ( X, φ ) L (3) C µ νρσαβ X depends on eleven functions: φ µ ν φ ρσ φ αβ = (3) i i =1 L H L bH • This includes the Lagrangians and . 5 5 • 9 degenerate subclasses: 2 with , 7 with f 3 6 = 0 f 3 = 0 • 25 combinations of quadratic and cubic theories (out of 7x9) are degenerate.

Disformal transformations • Transformations of the metric [Bekenstein ’93] → ˜ g µ ν = C ( X, φ ) g µ ν + D ( X, φ ) ∂ µ φ ∂ ν φ g µ ν − ˜ • Starting from an action , one can define the S [ φ , ˜ g µ ν ] new action S [ φ , g µ ν ] ≡ ˜ S [ φ , ˜ g µ ν = C g µ ν + D φ µ φ ν ] • Disformal transformation of quadratic DHOST theories ? " # Z L (2) X d 4 x (4) ˜ ˜ p ˜ a I ˜ S = g f 2 R + − ˜ ˜ I I The structure of DHOST theories is preserved and all seven subclasses are stable. [Ben Achour, DL & Noui ’1602]

Disformal transformations • Stability under g µ ν − g µ ν = C g µ ν + D ∂ µ φ ∂ ν φ → ˜ Ben Achour, DHOST DL & Noui ’16 C ( X, φ ) , D ( X, φ ) Beyond Horndeski Gleyzes, DL, Piazza & C ( φ ) , D ( X, φ ) Vernizzi ‘14 Horndeski Bettoni & Liberati ‘13 C ( φ ) , D ( φ ) • When matter is included (with minimal coupling), two disformally related theories are physically inequivalent !

Cosmology: Effective description of Dark Energy & Modified Gravity

Theories Parametrized Effective Description Observational constraints

Effective description of Dark Energy [ See e.g review: Gleyzes, DL & Vernizzi 1411.3712 ] • Restriction: single scalar field models • The scalar field defines a preferred slicing Constant time hypersurfaces = uniform field hypersurfaces φ = φ 3 φ = φ 2 φ = φ 1 • All perturbations embodied by the metric only

Uniform scalar field slicing • 3+1 decomposition based on this preferred slicing • Basic ingredients – Unit vector normal to the hypersurfaces r µ φ n µ = � p � ( r φ ) 2 – Projection on the hypersurfaces: h µ ν = g µ ν + n µ n ν

ADM formulation • ADM decomposition of spacetime ds 2 = − N 2 dt 2 + h ij dx i + N i dt dx j + N j dt � � � � N i Extrinsic curvature: 1 Nn µ � ˙ � K ij = h ij − D i N j − D j N i 2 N Intrinsic curvature: R ij h ij ˙ φ 2 ( t ) X ⌘ g µ ν r µ φ r ν φ = � N 2 • Generic Lagrangians of the form Z √ d 4 x N S g = h L ( N, K ij , R ij ; t )

Homogeneous background & linear perturbations • Background ds 2 = − ¯ N 2 ( t ) dt 2 + a 2 ( t ) δ ij dx i dx j  ˙ � a ¯ a, ¯ j = 0 , N = ¯ K i Na δ i j , R i L ( a, ˙ N ) ≡ L j = N ( t ) ¯ j ≡ R j • Perturbations: N , δ K i j ≡ K i j − H δ i j , δ R i δ N ≡ N − ¯ i q A ≡ { N, K i j , R i • Expanding the Lagrangian with L ( q A ) j } ∂ 2 L δ q A + 1 L + ∂ L L ( q A ) = ¯ yields δ q A δ q B + . . . 2 ∂ q A ∂ q A ∂ q B • The quadratic action describes the dynamics of linear perturbations

Horndeski & beyond Horndeski Gleyzes, DL, Piazza & Vernizzi ’13, • Quadratic action [notation: Bellini & Sawicki ‘14] dx 3 dt a 3 M 2  Z S (2) = i − δ K 2 + α K H 2 δ N 2 + 4 α B H δ K δ N j δ K j δ K i 2 ✓ √ ◆ � h + (1 + α T ) δ 2 + (1 + α H ) R δ N a 3 R α M ≡ d ln M 2 H dt α H α K α B α M α T Quintessence, X K-essence X X Kinetic braiding, DGP X X X Brans-Dicke, f(R) X X X X Horndeski X X X Beyond Horndeski X X X