Dark energy & Modified gravity in scalar-tensor theories. David Langlois (APC, Paris)

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1 Dark energy & Modified gravity in scalar-tensor theories David Langlois (APC, Paris)

2 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

3 Traditional scalar-tensor theories Simplest extensions of GR: add a scalar field S = Z d 4 x p g h i F ( ) (4) R Z( )@ µ U( ) + S m [ m ; g µ ] K-essence/ k-inflation: non standard kinetic term Z S = d 4 x p apple M 2 g P (4) R + P (X, ) 2 X r µ r µ

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

5 Horndeski theories Combination of the Lagrangians Horndeski 74 (a.k.a. Generalized Galileons) L H 2 L H 3 = G 2 (,X) with = G 3 (,X) L H 4 = G 4 (,X) (4) R 2G 4X (,X)( 2 µ µ ) X r µ r µ µ r r µ L H 5 = G 5 (,X) (4) G µ µ G 5X(,X)( 3 3 µ µ +2 µ µ ) 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!

6 Beyond Horndeski & DHOST theories Extensions beyond Horndeski L bh 4 F 4 (,X) µ µ0 0 0 µ µ leading to third order equations of motion. Gleyzes, DL, Piazza &Vernizzi 14 L bh 5 F 5 (,X) µ µ µ µ Earlier hint: disformal transformation of Einstein-Hilbert Even if EOM are higher order, no extra DOF if the Lagrangian is degenerate. DHOST theories (Degenerate Higher-Order Scalar-Tensor) Zumalacarregui & Garcia-Bellido 13 DL & K. Noui 15

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

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

9 Degrees of freedom Introduce the auxiliary variable Q L = 1 2 a Q 2 + b Q q c q Q2 V (,q) (Q ) Equations of motion a Q + b q = Q = Q, = V b Q + c q = If the Hessian matrix is invertible, one finds 3 DOF. V q 2 b [6 initial conditions] = a b b c

10 Degrees of freedom If the Hessian matrix is degenerate, i.e. 2 b = a b b c ac b 2 =0 then only 2 DOF (at most). [ can be absorbed in ẋ q + b ] c Hamiltonian analysis: primary constraint and secondary constraint [ p a cannot be inverted a (v)

11 Generalization (classical mechanics) Consider a general Lagrangian Motohashi, Noui, Suyama, Yamaguchi & DL 1603 [See also Klein & Roest 1604] L(,, ; q i,q i ) =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 ) L Q Q L Q q i (L 1 ) qi q j L q j Q =0 2. Secondary conditions (n secondary constraints) L Q L Q =0 if m =0 Third-order time derivatives... Motohashi, Suyama, Yamaguchi 1711

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

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

14 Action of the form Z S[g, ]= d 4 x p Cubic DHOST theories depends on eleven functions: [Ben Achour, Crisostomi, Koyama, DL, Noui & Tasinato 1608] g h f 3 G µ µ + C µ (3) µ C µ (3) µ = L bh 5 This includes the Lagrangians and. L H 5 X10 i=1 i b i (X, ) L (3) i 9 degenerate subclasses: 2 with f 3 6=0, 7 with f 3 =0 25 combinations of quadratic and cubic theories (out of 7x9) are degenerate.

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

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

17 Cosmology: Effective description of Dark Energy & Modified Gravity

18 Theories Parametrized Effective Description Observational constraints

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

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

21 ADM formulation ADM decomposition of spacetime ds 2 = N 2 dt 2 + h ij dx i + N i dt dx j + N j dt Nn µ N i h ij Extrinsic curvature: K ij = 1 2N ḣij D i N j D j N i Intrinsic curvature: R ij X g µ r µ r = 2 (t) N 2 Generic Lagrangians of the form Z S g = d 4 xn p hl(n,k ij,r ij ; t)

22 Background Homogeneous background & linear perturbations L(a, ȧ, N) L ds 2 = apple K i j = N 2 (t) dt 2 + a 2 (t) ij dx i dx j ȧ Na i j,r i j =0,N = N(t) Perturbations: N N N, K i j K i j H i j, R i j R j i Expanding the Lagrangian L(q A ) with q A {N,Kj,R i j} i yields L(q A )= A q A B q A q B +... The quadratic action describes the dynamics of linear perturbations

23 Horndeski & beyond Horndeski Quadratic action Z S (2) = M d ln M 2 Hdt dx 3 dt a 3 M 2 apple 2 K i j K j i K 2 + K H 2 N 2 +4 B H K N p h +(1+ T ) 2 a 3 R Gleyzes, DL, Piazza & Vernizzi 13, [notation: Bellini & Sawicki 14] +(1+ H )R N Quintessence, K-essence K B M T H 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

24 Scalar degree of freedom Scalar perturbations: N, N i, h ij = a 2 (t)e 2 ij Quadratic action for the physical degree of freedom: S (2) = 1 2 Z dx 3 dt a 3 apple K t 2 + K s (@ i ) 2 a 2 K t K +6 2 B (1 + B ) 2, K s 2M 2 ( 1+ H Ḣ 1+ T 1+ M 1+ B H 2 1 H d dt 1+ H 1+ B ) Stability (neither ghost nor gradient instability) K t > 0 c 2 s K s K t > 0

25 Tensor degrees of freedom Quadratic action for the tensor modes: S (2) = 1 2 Z dt d 3 xa 3 apple M 2 4 2ij M 2 4 (1 + T ) (@ k ij) 2 a 2 Stability: M 2 > 0 and c 2 T 1+ T > 0

26 Extension to DHOST theories DL, Mancarella, Noui & Vernizzi 1703 Quadratic action in terms of 9 functions of time S quad = Z d 3 xdta 3 M 2 2 K ij K 1+ ij 2 3 L K 2 +(1+ T ) R p h a R +H 2 K N 2 +4H B K N +(1+ H )R N +4 1 K Ṅ =6 independent coefficients Ṅ a 2 (@ i N) 2 Degeneracy conditions: 2 categories C I : L =0, 2 = 6 2 1, 3 = 2 1 [2(1 + H )+ 1 (1 + T )] C II : 1 = (1 + L ) 1+ H 1+ T, 2 = 6(1 + L ) (1 + H) 2 (1 + T ) 2, 3 =2 (1 + H) 2 1+ T C II : gradient instability either in the scalar or the tensor sector

27 Scalar-tensor theories DHOST I Beyond Horndeski DHOST Horndeski Type I Type II (Gradient instability!)

28 Disformal transformations Disformal transformations: g µ = Cg µ + DHOST I DHOST C(X) D(X) Beyond Horndeski C( ),D(X, ) Horndeski C( ),D( ) C(X, ),D(X, ) Type I Type II

29 Disformal transformations Disformal transformations: g µ = Cg µ + DHOST I DHOST C(X) D(X) Beyond Horndeski C( ),D(X, ) Horndeski C( ),D( ) C(X, ),D(X, ) Mimetic gravity & extensions (non-invertible transf) are DHOST theories of type II (some of type I too) and all unstable. DL, Mancarella, Noui & Vernizzi 1802 [see also Takahashi & Kobayashi 1708]

30 DHOST theories after GW170817

31 DHOST theories after GW Constraint on the speed of gravitational waves: T < Assuming T =0 holds exactly, this implies 1. Quadratic terms: a 1 =0 L ADM =(f Xa 1 )K ij K ij f (3) R 2. No cubic term (for type I theories) Remain quadratic DHOST theories of type I with a 1 =0

32 DHOST theories with c g = c Taking into account the degeneracy conditions, a 1 = a 2 =0, a 4 = 1 48f 2 8f 2X 2 8(f 2 Xf 2X )a 3 X 2 a 2 3, a 5 = 1 2f 2 (4f 2X + Xa 3 ) a 3 Total Lagrangian L DHOST c g =1 = f 2 (X, ) (4) R + P (X, )+Q(X, ) + a 3 (X, ) µ µ + a 4 (X, ) µ µ µ + a 5 (X, )( µ ) 2 2 free functions X c g =1 4 free functions of and (as in Horndeski without! )

33 Horndeski and Beyond Horndeski with c g = c Remaining Beyond Horndeski theories a 1 =2G 4X + XF 4 =0 =) F 4 = 2 X G 4X S[g, ]= Z d 4 x p g f(,x) R 4 h i X f X ( ) µ µ µ µ Remaining Horndeski theories G 2 (X, ), G 3 (X, ), G 4 ( )

34 Gravitation in DHOST with c g = c DL, Saito, Yamauchi & Noui 1711 [see also Crisostomi & Koyama 1711 and Dima & Vernizzi 1712] Quasi-static approximation on scales r H 1 ds 2 = (1 + 2 )dt 2 +(1 2 ) ij dx i dx j = c (t)+ (r) Equations of motion for Scalar equation Metric equations, and Matter source: spherical body with density (r)

35 Gravitation in DHOST with c g = c DL, Saito, Yamauchi & Noui 1711 [see also Crisostomi & Koyama 1711 and Dima & Vernizzi 1712] Gravitational laws d dr = G M(r) 4 N M(r) r G N M 00 (r), d dr = G N M(r) r G N M 0 (r) r with (8 G N ) 1 2f (1 + 0 ) Z r G N M 00 (r) r 2 ( r)d r I f,f X,a 3 where the coefficients are given in terms of and c Breaking of the Vainshtein screening inside matter! already noticed for Beyond Horndeski (GLPV) Kobayashi, Watanabe & Yamauchi 14

36 Gravitation in DHOST with c g = c The four coefficients I depend on only 2 parameters 0 = H 3 1, 1 = ( H + 1 ) 2 2( H +2 1 ), 2 = H, 3 = Constraints on the coefficients 0 = G gw 1( H + 1 ) 2( H +2 1 ). 1 Hulse-Taylor binary pulsar: 0 < 10 2 G N Beltran Jimenez, Piazza & Velten 1507 Stars: [Saito, Yamauchi, Mizuno, Gleyzes & DL 15] 1 12 < [Sakstein 15] Gravitational lensing for the other coefficients

37 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 GW These theories of modified gravity can be tested & constrained via cosmology (future LSS observations) and astrophysics.