Degenerate Higher-Order Scalar-Tensor (DHOST) theories. David Langlois (APC, Paris)

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1 Degenerate Higher-Order Scalar-Tensor (DHOST) theories David Langlois (APC, Paris)

2 Higher order scalar-tensor theories Usual theories (Brans-Dicke, k-essence, ): L(r, ) Generalized theories: L(r µ r, r, ) L( q, q, q) Horndeski Extra DOF (Ostrogradsky instability) Only one scalar DOF!

3 Horndeski theories Horndeski 74 Combination of the following four Lagrangians 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!

4 Higher order scalar-tensor theories Usual theories: L(r, ) Generalized theories: L(r µ r, r, ) Horndeski Extra DOF (Ostrogradsky instability)

5 Higher order scalar-tensor theories Usual theories: L(r, ) Generalized theories: L(r µ r, r, ) Horndeski Extra DOF (Ostrogradsky instability) Beyond Horndeski

6 Higher order scalar-tensor theories Usual theories: L(r, ) Generalized theories: L(r µ r, r, ) Horndeski Extra DOF (Ostrogradsky instability) Beyond Horndeski DHOST Degenerate Higher-Order Scalar-Tensor

7 Beyond Horndeski Two extensions beyond Horndeski [Gleyzes, DL, Piazza & Vernizzi 14] 4 F 4 (,X) µ µ0 0 0 µ µ F 5 (,X) µ µ µ µ leading to third order equations of motion. In contrast with earlier belief, no extra DOF if the total Lagrangian is degenerate (in a generalized sense). L H 4 4 L H L H 4 4 L H L H 5

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 ) If the Hessian matrix M is invertible, one finds 3 2 b = a b b c If the Hessian matrix is degenerate, i.e. ac b 2 =0, then only 2 DOF (at most). Hamiltonian analysis: primary constraint and secondary constraint [ cannot be inverted ] p a (v)

10 Generalization (classical mechanics) [Motohashi, Noui, Suyama, Yamaguchi & DL 1603] Consider a general Lagrangian of the form L(,, ; q i,q i ) =1,,n; i =1,,m In general, one finds 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 [ See also Klein & Roest 1604 ]

11 Higher Order Scalar-Tensor theories Consider all Lagrangians 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 Equivalently: 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 7 degenerate subclasses: 4 with f 2 6=0, 3 with f 2 =0 L H 4 They include and 4 [See also Crisostomi et al 1602; Ben Achour, DL & Noui 1602; de Rham & Matas 1604]

12 Disformal transformations Transformations of the metric 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? Z S = d 4 x p h i g f R + I LI The structure of DHOST theories is preserved and all seven subclasses are stable. [Ben Achour, DL & Noui 1602]

13 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 = 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.

14 Cosmology One can use the unifying effective description of dark energy [ See e.g review: Gleyzes, DL & Vernizzi ] ADM (3+1) 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 with uniform scalar field slicing h ij Lagrangians of the form Z S g = d 4 xn p hl(n,k ij,r ij ; t) K ij = 1 2N ḣij D i N j D j N i

15 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

16 Linear perturbations 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

17 Extension to DHOST theories DL, Mancarella, Noui & Vernizzi 1703 Quadratic action in terms of 9 functions of time Z S quad = d 3 xdta 3 M 2 K ij K 1+ ij 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 + 2 Degeneracy conditions: 2 possible sets Ṅ a 2 (@ i N) 2 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 Static linear regime around Minkowski: apple (1 + 8 G N = M 2 H ) T 2 1 (! 1 for C II )

18 General disformal transformation Disformal transformations g µ = C(X, )g µ + D(X, ) µ. DL, Mancarella, Noui & Vernizzi Y D D D + C/X, X All effective parameters transform, except L DHOST I Beyond Horndeski Horndeski Y X The coupling to matter is also modified: Horndeski vs Jordan frames

19 Stars in beyond Horndeski theories Partial breaking of Vainshtein mechanism inside matter Spherical symmetry & nonrelativistic limit: d dr = G N M r 2 d2 M dr 2 Modified Lane-Emden equation (for P = K 1+ 1 n ) Universal bound < 1/6 Astrophysical constraints on 0.18 < < Saito, Yamauchi, Mizuno, Gleyzes & DL 15 (see also Koyama & Sakstein 15) Kobayashi, Watanabe & Yamauchi 14, M(r) =4 4 [Sakstein 15, Jain et al 15] Χ Ξ Z r 0 r 0 2 (r 0 )dr 0 Ε = -0.5 (orange), -0.3, -0.1, 0, 0.1, 0.15 (blue), 0.3, 0.5, 1 (red) Ξ

20 Neutron stars in beyond Horndeski Babichev, Koyama, DL, Saito & Sakstein 16 Model with S = Z d 4 x p g apple M 2 P R 2 k 2 X + f = X ( ) 2 ( µ ) 2 +2 µ [ µ µ ] Cosmological solution: de Sitter with = v0 6=0, H 6= 0 ds 2 = (1 H 2 r 2 ) dt 2 + dr 2 1 H 2 r 2 + r2 d 2 2 (r, t) =v 0 t + v 0 2H ln 1 H2 r 2

21 Neutron stars in beyond Horndeski Babichev, Koyama, DL, Saito & Sakstein 16 Spherical symmetric solutions ds 2 = e (r) dt 2 + e (r) dr 2 + r 2 d 2 2 with (r) = cosmo + (r), (r) = cosmo + (r) (t, r) = cosmo (t, r)+ (r) External solution: Schwarzschild-de Sitter ds 2 = fdt 2 + f 1 dr 2 + r 2 d 2 2, f 1 (t, r) =v 0 applet Z p 1 dr f f 2G N M r G N 3G H 2 r 2

22 Neutron stars in beyond Horndeski Internal solution System analog to TOV equations Babichev, Koyama, DL, Saito & Sakstein 16 Mass-radius relations For < 0 the maximum mass is larger than in GR. See also Sakstein, Babichev, Koyama, DL & Saito 16

23 Conclusions DHOST theories: systematic classification of degenerate theories that contain a single scalar DOF. They generalize Horndeski and beyond Horndeski theories. General disformal transformations preserve all subclasses of quadratic DHOST theories. These theories of modified gravity can be tested in cosmology, by using the effective description of dark energy and modified gravity. Construction of Newtonian stars and neutron stars in simple beyond Horndeski models.