A unifying description of Dark Energy (& modified gravity) David Langlois (APC, Paris)

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1 A unifying description of Dark Energy (& modified gravity) David Langlois (APC, Paris)

2 Outline 1. ADM formulation & EFT formalism. Illustration: Horndeski s theories 3. Link with observations Based on J. Gleyzes, D.L., F. Piazza & F. Vernizzi, [hep-th]

3 Introduction & motivations Origin of the present acceleration? Cosmological constant: simplest but difficult to understand Plethora of dark energy models: Dynamical dark energy: quintessence, K-essence Modified gravity Dark energy / modified gravity Goal: effective description as a bridge between models and observations. Assumptions: Single scalar field models All matter fields coupled to the same metric

4 ADM approach The scalar field defines a preferred slicing Constant time hypersurfaces = uniform field hypersurfaces ADM formalism ds = N dt + h ij dx i + N i dt dx j + N j dt Generic action: S = d 4 x gl(t; N,K,S, R, Z,...) K K µ µ, S K µν K µν, R (3) R, Z (3) R µν (3) R µν Note: dependence on (3) R µν K µν can be reabsorbed.

5 Expansion (up to second order) It is convenient to introduce which implies Lagrangian up to first order: but δk K 3H, δk µν K µν Hh µν S =3H +HδK + δk µ νδk ν µ L(N,K,S, R, Z) = L + L N δn + FδK + L R δr +... F δk = F(K 3H) d 4 x g FK = with d 4 x gn µ µ F = F HL S + L K d 4 x g F N

6 Expansion (up to second order) It is convenient to introduce which implies δk K 3H, δk µν K µν Hh µν S =3H +HδK + δk µ νδk ν µ Lagrangian up to first order: L(N,K,S, R, Z) = L F 3HF +( F + L N ) δn + L R δr +...

7 Expansion (up to second order) It is convenient to introduce which implies δk K 3H, δk µν K µν Hh µν S =3H +HδK + δk µ νδk ν µ Lagrangian up to second order: L(N,K,S, R, Z) = L F 3HF +( F + L N ) δn + L R δr + A 1 δk + L S δk µ νδk ν µ + L NN F δn + 1 L RR δr + B δkδn + C δkδr + L NR δnδr + L Z δz F HL S + L K, B HL SN + L KN, A 4H L SS +4HL SK + L KK C HL SR + L KR

8 Background equations FLRW metric: ds = N (t)dt + a (t)δ ij dx i dx j Homogeneous action: S 0 = dtd 3 xna 3 L N,K = 3HN 3H, S =, R =0, Z =0 N Friedmann equations δs 0 = dtd 3 x a 3 L + LN 3HF δn + L 3HF F δ(a 3 ) 3HF L L N =0, F +3HF L =0

9 Quadratic Lagrangian Explicit gauge choice: h ij = a (t)e ζ δ ij, N i = i ψ Quadratic Lagrangian: L 1 a L = L ζ, δn, ψ 1 (A +L S)( ψ) +4C ψ ζ +(4L RR +3L Z )( ζ) One can get rid of higher spatial derivatives by imposing A +L S =0, C =0, 4L RR +3L Z =0 The momentum constraint then reduces to: δn = 4L S B +4HL S ζ

10 Quadratic Lagrangian for the perturbations One finally obtains the quadratic Lagrangian in the form L = a3 L ζ ζ ζ + L i ζ i ζ ( i ζ) a L ζ ζ L i ζ i ζ where the coefficients and are given explicitly in terms of derivatives of the initial Lagrangian in ADM form. Absence of ghosts: L ζ ζ > 0 Effective speed of sound: c s = L i ζ i ζ L ζ ζ

11 Simple example: K-essence K-essence Lagrangian L = P (φ,x), X µ φ µ φ Total Lagrangian in the ADM formulation L = 1 S K + R + P (X, φ) Background Lagrangian and its derivatives: L = P 3H, L N = XP X, L K = 3H, L S = 1, L R = 1 F =HL S + L K = H, A = 1, B =0, C =0 Speed of sound: c s = P X P X +XP XX

12 EFT formalism Requiring no higher derivative terms, the action up to quadratic order can be written in the form L = M f(t)r Λ(t) c(t)g00 + M 4 (t) (δg 00 ) m3 3(t) δkδg 00 m 4(t) δk δk µ ν δk ν m µ + 4(t) (3) R δg 00 Link between the ADM & EFT formulations: R = (3) R + K µν K µν K + ν (n ν µ n µ n µ µ n ν ) g 00 = 1 N c + Λ =3M fh + fh, Λ c = M fḣ +3fH + fh + f Immediate dictionary between the ADM and the EFT forms

13 Generalized Galileons Most general action for a scalar field leading to at most second order equations of motion (Horndeski) Combination of the following four Lagrangians L = G (φ,x), L 3 = G 3 (φ,x)φ, L 4 = G 4 (φ,x)r G 4X (φ,x)(φ φ ;µν φ ;µν ), with X φ ;µ φ ;µ L 5 = G 5 (φ,x)g µν φ ;µν G 5X(φ,X)(φ 3 3 φφ ;µν φ ;µν +φ ;µν φ ;µσ φ ;ν ;σ) ADM formulation? Which operators appear in the quadratic Lagrangian?

14 Generalized Galileons: ADM form Uniform scalar field slicing Unit normal vector n µ = γφ ;µ with γ = 1 X Induced metric h µν = n µ n ν + g µν Using K µν = h σ µ n ν;σ ṅ µ = n ν n µ;ν one can write n ν;µ = K µν n µ ṅ ν φ ;µν = γ 1 (K µν n µ ṅ ν n ν ṅ µ )+ γ φ;λ X ;λ n µ n ν The Gauss-Codazzi relations are also useful.

15 Generalized Galileons: ADM form Lagrangian L 3 = G 3 (φ,x)φ Introducing the auxiliary function such that one finds G 3 F 3 +XF 3X EFT coefficients: F 3 (φ,x) L 3 = γ 1 XF 3X K XF 3φ L = M f(t)r Λ(t) c(t)g00 + M 4 (t) (δg 00 ) m3 3(t) m 4(t) δk δk µ ν δk ν m µ + 4(t) (3) R δg 00 δkδg 00

16 Generalized Galileons: ADM form Lagrangian L 4 = G 4 (φ,x)r G 4X (φ,x)(φ φ ;µν φ ;µν ) Using the Gauss-Codazzi equations, one finally gets L 4 = G 4 (3) R +(XG 4X G 4 )(K K µν K µν ) γ 1 G 4φ K. In the quadratic Lagrangian, all terms are present, but with the restriction m 4 = m 4 L = M f(t)r Λ(t) c(t)g00 + M 4 (t) (δg 00 ) m3 3(t) m 4(t) δk δk µ ν δk ν m µ + 4(t) (3) R δg 00 δkδg 00

17 Generalized Galileons: ADM form L 5 = 1 3 G 5X(φ,X)(φ 3 3 φφ ;µν φ ;µν +φ ;µν φ ;µσ φ ;ν ;σ) +G 5 (φ,x)g µν φ ;µν Using the auxiliary function F 5 (φ,x) one finally gets such that G 5X = F 5X + F 5 X L 5 = 1 X(G 5φ F 5φ ) (3) R + 1 XG 5φ(K K µν K µν ) with γ 1 F 5 K µν (3) R µν 1 K(3) R + γ 1 3 XG 5XK K K 3 3KK µν K µν +K µν K µσ K ν σ = 6H 3 6H K +3HK 3HS + O(3) All operators are present, again with the restriction m 4 = m 4

18 Perturbations in an arbitrary gauge The action for the perturbations in an arbitrary gauge can be obtained via the Stueckelberg trick: t t + π(t, x) The new quadratic action can be derived using the following substitutions: f f + fπ + 1 fπ, g 00 g 00 +g 0µ π + g µν µ π ν π, δk ij δk ij Ḣπh ij i j π, δk δk 3Ḣπ 1 a π, (3) R ij (3) R ij + H( i j π + δ ij π), Note: the 3-dim quantities on the right are defined with respect to the new time hypersurfaces. (3) R (3) R + 4 a H π.

19 Perturbations in the Newtonian gauge Perturbed metric ds = (1 + Φ)dt + a (t)(1 Ψ) δ ij dx i dx j Einstein s equations Generalized Poisson equation k a (fm +4 m 4)Ψ ( fm m Hm 4 4H m 4)π +(6M H f 6Hc ċ Λ +3m 3 3Ḣ)π (c +4M 4 ) π +(3M H f +c +4M 4 )Φ 3M f Ψ +3m 3 3( Ψ + HΦ) =ρ m m Anisotropic equation f(φ Ψ)+ fπ + m 4 π + m 4Hπ +(m 4) π + m 4(Φ π) =0 M Equation for π

20 Deviations from GR Quasi-static approximation for sub-hubble scales (time derivatives are neglected) Effective Newton s constant k a Φ 4π G eff(t, k) ρ m m Ratio between the two gravitational potentials Ψ γ(t, k) Φ

21 Conclusions Unified treatment of single scalar field models of dark energy/modified gravity Models leading to at most second-order equations of motion for linear perturbations can be parametrized by 7 time-dependent functions (3 needed for the background). ADM reformulation of Horndeski s theories; only six operators are needed to describe the background and linear perturbations. Link with observations (effective Newton constant, ratio of the two gravitational potentials)