Healthy theories beyond Horndeski
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1 Healthy theories beyond Horndeski Jérôme Gleyzes, IPhT CEA Saclay with D. Langlois, F. Piazza and F. Vernizzi, arxiv: , arxiv: ITP Heidelberg 26/11/14
2 Introduction to Horndeski Going safely beyond Horndeski Phenomenology
3 Our Universe so far Cosmology Standard Model Well tested
4 Beyond ΛCDM Why? Size of Λ : more than a parameter? How? New dynamical field(s) Background mimics Λ + perturbations Behaviour on small scales New missions like Euclid
5 Cosmology and scalar fields Simple Only 1 degree of freedom (DOF) Universal behaviour Massive gravity: 5 DOF reduce to only scalar DOF in decoupling limit Background field breaks time reparametrization invariance low energy perturbations described by scalar goldstone boson
6 Cosmology and scalar fields Acceleration of the Universe Extra scalar degree of freedom (DOF) Cosmological constant Equation of state No perturbations
7 Cosmology and scalar fields Acceleration of the Universe Extra scalar degree of freedom (DOF) K-essence Equation of state Perturbations with non trivial evolution
8 Cosmology and scalar fields Acceleration of the Universe Extra scalar degree of freedom (DOF) Additional ghost DOF (Ostrogradski) In Minkowski Galileons A. Nicolis, R. Rattazzi and E. Trincherini, Phys. Rev. D 79, (2009) e.g. No ghost DOF
9 Galileons : G The original Galileons Minkowski General metric Naive covariantization Three derivatives terms Need to add non minimally coupled terms
10 Horndeski/Generalised Galileons G 2 Horndeski G. W. Horndeski, Int. J. Theor. Phys. 10, 363 (1974) Generalised Galileons C. Deffayet, X. Gao, D. A. Steer and G. Zahariade, Phys. Rev. D 84, (2011) T. Kobayashi, M. Yamaguchi and J. Yokoyama, Int.J.Mod.Phys.Conf.Ser. 10 (2012) Most general scalar-tensor theory with 2nd order EOM : Non minimal
11 Geometrical Horndeski with D. Langlois, F. Piazza, and F. Vernizzi JCAP 1308 (2013) 025
12 Generalised Generalised Galileons : G 3 Horndeski with D. Langlois, F. Piazza, and F. Vernizzi arxiv: G 3 Two new functions,
13 Covariant Lagrangians The dictionary First non trivial example : Horndeski G 3 No higher derivatives
14 A special case G 3 Covariant form Minimally coupled original Galileons are ghost free
15 Decoupling limit The limit G 3 with D. Langlois, F. Piazza and F. Vernizzi, arxiv: No new terms in the decoupling limit
16 What s new so far? Horndeski G 3 Lagrangians in terms of the scalar field Lagrangians in terms of 3-geometrical quantities 4 free functions of 6 free functions of Derivatives 2 in the EOM Up to 3 derivatives in the EOM
17 Caveat for Ostrogradski Second order EOM Well behaved theory Well behaved theory Second order EOM Examples see also Miguel s paper Phys.Rev. D 89 (2014)
18 3+1 decomposition Time slicing Space and time derivatives disentangled ADM metric
19 Unitary gauge G 3
20 Hamiltonian and number of DOF GR The Hamiltonian Counting the DOF DOF : gauge constraints, remove 1 DOF 2 polarisations DOF
21 Hamiltonian and number of DOF G 3 The Hamiltonian Counting the DOF DOF : gauge constraints, remove 1 DOF : regular constraints, remove 1/2 DOF 2 polarisations +1 extra scalar DOF
22 How to have extra DOF? A simple example Unitary Gauge Counting the DOF constraints DOF
23 The quadratic action 2 DOF 1 DOF No-ghost conditions 3 healthy DOF No gradient instability
24 What s new so far? Horndeski G 3 Lagrangians in terms of the scalar field Lagrangians in terms of 3-geometrical quantities 4 free functions of 6 free functions of Derivatives 2 in the EOM Up to 3 derivatives in the EOM 3 propagating DOF 3 propagating DOF
25 Matter matters Ostrogradski and field redefinitions 3 What about G? with D. Langlois, F. Piazza and F. Vernizzi, arxiv: OR with Non minimal see also Bettoni and Liberati Phys.Rev. D 88 (2013) 8, and Miguel s paper Phys.Rev. D 89 (2014)
26 Matter and Gravity : the GR case On sub horizon scales Fourier space Poisson equation Matter conservation + Euler Sound speed Jeans scale
27 Horndeski GR Linear equations Horndeski Decoupling
28 G 3 GR Linear equations G 3 No decoupling Scalar field cannot be ignored at small scales
29 What s new so far? Horndeski G 3 Lagrangians in terms of the scalar field Lagrangians in terms of 3-geometrical quantities 4 free functions of 6 free functions of Derivatives 2 in the EOM Up to 3 derivatives in the EOM 3 true DOF 3 true DOF Matter sound speed unaffected Scalar changes matter sound speed
30 Conclusions Higher derivatives Ghost DOF Horndeski G 3 Healthy theories beyond Horndeski Linear G 3 : second order EOM unusual small scales behaviour Horndeski Non linear behaviour? Field redefinitions?
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