1. Topics on generalized Galileons I. The simplest Galileon: DGP decoupling limit II. Galileons and Generalized Galileons C.D., G. Esposito-Farese, A. Vikman Phys.Rev. D79 (2009) 084003 C.D., S.Deser, G.Esposito-Farese Phys.Rev. D82 (2010) 061501, Phys.Rev. D80 (2009) 064015. C.D., X.Gao, D. Steer, G. Zahariade Phys.Rev. D84 (2011) 064039 C.D., E.Gümrükçüoglu, S Mukohyama, Y. Wang JHEP 1404 (2014) 082 C.D., G.Esposito-Farese, D, Steer, PRD D92 (2015) 084013 C.D., S. Mukohyama, V. Sivanesan, 1601.01287 [hep-th] I.1. DGP model I.2. DGP decoupling limit II.1. Flat space-time Galileon in 4D II.2. Flat space-time Galileon in arbitrary D II.3. Curved space-time Galileon II.4. Generalization to p-forms II.5. From k-essence to generalized Galileons II.6. Some previous and recent results and other approaches (Horndeski theories) II.7. Dof counting in « Beyond Horndeski » theories FP7/2007-2013 « NIRG » project no. 307934 Cédric Deffayet (IAP and IHÉS, CNRS Paris) Stella Maris, Jan the 11th 2016

2. I. The simplest Galileon: DGP decoupling limit I.1 The DGP model Standard model Peculiar to DGP model Usual 5D brane world action Brane localized kinetic term for the graviton Will generically be induced by quantum corrections A special hierarchy between M (5) and M P is required to make the model phenomenologically interesting Dvali, Gabadadze, Porrati, 2000 Leads to the e.o.m.

3. Phenomenological interest A new way to modify gravity at large distance, with a new type of phenomenology … The first framework where cosmic acceleration was proposed to be linked to a large distance modification of gravity (C.D. 2001; C.D., Dvali, Gababadze 2002) (Important to have such models, if only to disentangle what does and does not depend on the large distance dynamics of gravity in what we know about the Universe) Theoretical interest Consistent (?) non linear massive gravity … DGP model Intellectual interest Lead to many subsequent developments (massive gravity, Galileons, …)

4. Energy density of brane localized matter Homogeneous cosmology of DGP model One obtains the following modified Friedmann equations (C.D. 2001) Analogous to standard (4D) Friedmann equations In the early Universe (small Hubble radii ) Deviations at late time (self-acceleration) Two branches of solutions Light cone Brane Cosmic time Equal cosmic time Big Bang

5. Newtonian potential on the brane behaves as 4D behavior at small distances 5D behavior at large distances The crossover distance between the two regimes is given by This enables to get a “4D looking” theory of gravity out of one which is not, without having to assume a compact (Kaluza-Klein) or “curved” (Randall-Sundrum) bulk. But the tensorial structure of the graviton propagator is that of a massive graviton (gravity is mediated by a continuum of massive modes) Leads to the « van Dam-Veltman-Zakharov discontinuity » on Minkowski background (i.e. the fact that the linearized theory differs drastically – e.g. in light bending - from linearized GR at all scales)! In the DGP model : the vDVZ discontinuity, is believed to disappear via the « Vainshtein mechanism » (taking into account of non linearities) C.D.,Gabadadze, Dvali, Vainshtein, Gruzinov; Porrati; Lue; Lue & Starkman; Tanaka; Gabadadze, Iglesias;…

6. A good description of many DGP key properties is given by the action Luty, Porrati, Rattazzi, 2003 Scalar sector of the model Energy scale Yielding the e.o.m.Leads in vacuum to two branches of solutions, and representing the two branches of solutions of the original model… Around massive body, the cubic self interaction of ¼ becomes of the same order as the quadratic term at the « Vainshtein radius » I.2 DGP decoupling limit

7. The action Is obtained taking the « decoupling limit » Luty, Porrati, Rattazzi; Nicolis, Rattazzi It can be obtained from the (5D) « Hamiltonian » constraint Where one substitutes the Israel junction condition To obtain A last substitution Yields the e.o.m. for  deduced from above action : second order e.o.m. The first « Galileon » ( No « Boulware-Deser » ghost, C.D., Rombouts, 2005 )

8. II. Generalizations Galileon Originally (Nicolis, Rattazzi, Trincherini 2009) defined in flat space-time as the most general scalar theory which has (strictly) second order fields equations II. 1 Flat space-time Galileon in 4 D In 4D, there is only 4 non trivial such theories ( with )

9. Simple rewriting of those Lagrangians with epsilon tensors (up to integrations by part): (C.D., S.Deser, G.Esposito-Farese, 2009) This leads to (exactly) second order field equations

10. Indeed, consider e.g. Varying this Lagrangian with respect to ¼ yields (after integrating by part) Second order derivative Third order derivative… … killed by the contraction with epsilon tensor Similarly, one also have in the field equations Yields third and fourth order derivative… killed by the epsilon tensor Hence the field equations are proportional to Which does only contain second derivatives

11. The field equations, containing only second derivatives, Have the « Galilean » symmetry They also can be written using 4 (and lower) dimensional determinants and are in fact generalized Monge Ampere equation of the form det ( ¼ ij ) = 0 (Monge Ampere equation have interesting integrability properties - Fairlie) The field equations read Linear in second time derivative ( Good Cauchy problem ?).

12. II. 2 Flat space-time Galileon in arbitrary Dimension In D dimensions, D non trivial Galileons can be defined as Only the Lagrangians with are non vanishing. One has All free indices are contracted with those of Using the tensors Or to alleviate notations is antisymmetric (separately) in odd and even indices

13. Up to total derivatives, the following Lagrangians are equivalent One has the exact relation

14. II. 3 Curved space-time Galileon A naive covariantization leads to the loss of the distinctive properties of the Galileon Indeed, consider now in curved space-time (with ¼ ¹ = r ¹ ¼ and ¼ ¹ º = r ¹ r º ¼ ) Indeed the (naively covariantized) has the field equations Third derivatives generate now Riemann tensors … and fourth derivatives, derivatives of the Riemann Variation yields in particular Kinetic mixing

15. Similarly, varying w.r.t. the metric Yields third order derivatives of the scalar ¼ in the energy momentum tensor Do not vanish in flat space-time !

16. This can be cured by a non minimal coupling to the metric Adding to The Lagrangian Yields second order field equations for the scalar and the metric (but loss of the « Galilean » symmetry) e.g. one has now the energy momentum tensor

17. This can be generalized to arbitrary Galileons (arbitrary number of fields and dimensions) Introducing With The action with Yields second order field equations. Heuristically, one needs to replace sucessively pairs of twice differentiated ¼ by Riemanns

18. This can be understood as follows. Considers e.g. Vary w.r.t. ¼ Only « dangerous » term (i.e. term leading to higher derivatives) Integrating by part Using the antisymmetry of and the Riemann Bianchi identity Can be cancelled by varying

19. Note that the extra term Does not generate unwanted derivatives of the curvature thanks to Bianchi identity Indeed one has Similarly to Yields an easy generalization to p-forms

20. II. 4 Generalization to p-forms E.g. consider With a p-form of field strength In the field equations, Bianchi identities annihilate any E.o.m. are (purely) second order E.g. for a 2-form Note that one must go to 7 dimensions (in general one has ) and that this construction does not work for odd p as we show now C.D., S.Deser, G.Esposito-Farese, arXiv 1007.5278 [gr-qc] (PRD)

21. II.4.2. The case of (odd p)-forms For odd p the previous construction does not work Indeed, the action With an (odd p)-form of field strength Yields vanishing e.o.m. (the action is a total derivative) Integration by part Renumbering of an even (for odd p) number of indices

22. Is there an (odd p) Galileon ? C. D., A. E. Gumrukcuoglu, S. Mukohyama and Y.Wang, [arXiv:1312.6690 [hep-th]], published in PRD C.D., S. Mukohyama, V. Sivanesan [arXiv:1601.01287[hep-th]] C.D., S. Mukohyama, V. Sivanesan In preparation

23. We start from the field equations with For a p-form with components with p antisymmetric indices We ask these field equations (i)To derive from an action (ii)To depend only on second derivatives (iii)to be gauge invariant

24. Hypothesis (i) (that field equations derive from an action S) leads to Which upon using Leads to the « integrability conditions » Where Vanish as a consequence of Hypothesis (ii) (that field equations only depend on second order derivatives)

25. Hypothesis (iii) (gauge invariance of the field equations) Lead to These symmetries extend in particular to derivatives of the field equations

26. Defining then Hypothesis (i) (ii) and (iii) lead to the following symmetries for the tensor 1.1. Antisymmetry within each group of p indices A and B i 1.2. Invariance under interchange of groups of indices B i and B j, as well as A and any B i 2.1. Symmetry of each pair of indices (c i, d i ) 2.2. Invariance under interchange (c i, d i ) and (c j, d j ) 3. Symmetrizing over any 3 indices yields zero

27. NB: Is a (pm + 2(m-1)) tensor C. D., A. E. Gumrukcuoglu, S. Mukohyama and Y.Wang, [arXiv:1312.6690 [hep-th]]. For p=1, Conditions 1.1, 1.2, 2.1, 2.2 and 3 are enough to show that vanishes identically (i.e. field equations are at most linear into the second derivatives) No vector Galileons (with gauge invariance)

28. For (odd) p > 1 ? C.D., S. Mukohyama, V. Sivanesan, 1601.01287 [hep-th] Decompose into components belonging to irreducible representations of the symmetric group S pm+2(m-1) = S n Using 1.1. Antisymmetry within each group of p indices A and B i 1.2. Invariance under interchange of groups of indices B i and B j, as well as A and any B i 2.1. Symmetry of each pair of indices (c i, d i ) 2.2. Invariance under interchange (c i, d i ) and (c j, d j ) 3. Symmetrizing over any 3 indices yields zero This amounts to act with the Young symmetrizers / appearing in the decomposition of the group algebra of S n into irreducibles given by Partitions of n Standard Young tableaux Young symmetrizers Condition 3. implies that only Young Tableaux with 1 or 2 column will yield a non zero component of

29. 1.1. Antisymmetry within each group of p indices A and B i 1.2. Invariance under interchange of groups of indices B i and B j, as well as A and any B i 2.1. Symmetry of each pair of indices (c i, d i ) 2.2. Invariance under interchange (c i, d i ) and (c j, d j ) 3. Symmetrizing over any 3 indices yields zero Conditions 1.1,1.2., 2.1. and 2.2. then imply that belongs to the « Plethysm » Next step: find out the content of this Plethysm. in the representations of S n indexed by up-to-two columns Young diagrams

30. By the use of Schur functions, the multiplicity (which allows to count the max number of possibly non trivial theories fixing and allowing to vary) of the representations indexed by Inside Is given by Where is the number of partitions of r into s number within (0,…,p) with repetition allowed. is the same with repetitions not allowed.

31. Next step: try to construct explicit theories E.g. for m=4, p=3 ) mp + 2(m-1) = 18 We look at Young diagrams with 18 boxes e.g. has multiplicity 1 inside No clear method to constuct the corresponding theory

32. Start with the filling A tensor with these symmetries can be constructed from the metric corresponding to Then act on it with projectors corresponding to the symmetries of This gives a non trivial

33. This can be integrated to yield the following action density for a 3 form (in D= 9 dimensions) To be contrasted with the p-form action constructed in C.D., S.Deser, G.Esposito-Farese, arXiv 1007.5278 [gr-qc] (PRD)

34. This can be generalized ….. …… classification on the way

35. Single p-form case can be generalized to multi p-forms (different species) in which case one can have [odd p]-forms (labels different p-forms) One simple example: bi-galileon, e.g. with ¼ and  being two scalar fields Padilla, Saffin, Zhou (bi-galileon); Hinterbichler,Trodden, Wesley (multi-scalar galileons) (NB: this can/must also be « covariantized » using previously introduced technique) C.D., S.Deser, G.Esposito-Farese, arXiv 1007.5278 [gr-qc] (PRD)

36. II. 5 From k-essence to generalized Galileons C.D. Xian Gao, Daniele Steer, George Zahariade arXiv:1103.3260 [hep-th] (PRD) What is the most general scalar theory which has (not necessarily exactly) second order field equations in flat space ? Specifically we looked for the most general scalar theory such that (in flat space-time) i/ Its Lagrangian contains derivatives of order 2 or less of the scalar field ¼ ii/ Its Lagrangian is polynomial in second derivatives of ¼ (can be relaxed: Padilla, Sivanesan; Sivanesan ) iii/ The field equations are of order 2 or lower in derivatives (NB: those hypothesis cover k-essence, simple Galileons,… )

37. Answer: the most general such theory is given by a linear combination of the Lagrangians defined by where Free function of ¼ and X

38. Our most general flat space-time theories can easily be « covariantized » using the previously described technology The covariantized theory is given by a linear combination of the Lagrangians with Specifically the covariantized theory is given by with

39. II. 6 Some previous and recent results and other approaches Flat space-time Galileons and flat-space time generalized Galileons (in the shift symmetric case) have been obtained previously by Fairlie, Govaerts and Morozov (1992) by the « Euler hierarchy » construction : Start from a set of arbitrary functions Then define the recursion relation being the Euler-Lagrange operator (and W 0 =1) Hence is the field equation of the Lagrangian (« Euler hierarchy ») The hierarchy stops after at most D steps Choosing F k = ¼ ¹ ¼ ¹ /2, one has (see e.g. the review by Curtright and Fairlie (2012))

40. Horndeski (1972) obtained the most general scalar tensor theory in 4D which has second order field equation for the scalar and the metric Using our notations it is given by and is parametrized by four free functions of X and ¼ : and one constraint First it is clear that the flat space-time restriction of Horndeski theory must be included in our generalized flat-space time Galileon Conversely, our covariantized generalized Galileons must be included into Horndeski theory

41. In fact, one can show that the two sets of theories (Horndeski and covariantized generalized Galileons) are identical in 4D (even though they start from different hypothesis) C.D. Xian Gao, Daniele Steer, George Zahariade T.Kobayashi, M.Yamaguchi, J. Yokoyama arXiv:1103.3260 [hep-th] (PRD) arXiv 1105.5723 [hep-th]

42. II.7. D.o.f. counting in Galileons and generalized Galileons theories Horndeski-like theories: Cauchy problem ? Numerical studies (e.g. adressing the Vainshtein mechanism in grav. Collapse) ? A priori 2 (graviton) + 1 (scalar) d.o.f. No Hamiltonian analysis so far ! Claimed to be true in an even larger set of theories (« Beyond Horndeski » theories) ! Horndeski-like theories: Scalar tensor theories with second order field equations + diffeo invariance J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, (GLPV) arXiv:1404.6495, arXiv:1408.1952 C.D., G. Esposito-Farèse, D. Steer, arXiv:1506.01974 [gr-qc]

43. Our works aims at Provides a first step toward a proper Hamiltonian treatment of Horndeski-like and beyond Horndeski theories Rexamine the GLPV claim (arguments of GLPV being not convincing to us)

44. Our works aims at Provides a first step toward a proper Hamiltonian treatment of Horndeski-like and beyond Horndeski theories Along two directions II.7.1. Show how in a (large) set of beyond Horndeski theories (matching the one considered by GLPV), the order in time derivatives of the field equations can indeed be reduced. II.7.2. Analyze in details via Hamiltonianian formalism a simple (the simplest ?) non trivial beyond Horndeski theory Rexamine the GLPV claim (arguments of GLPV being not convincing to us)

45. Consider one single Galileon « counterterm »: With

46. Consider one single Galileon « counterterm »: With The field equations have the following structure (as seen before) Energy momentum tensor Scalar field eom

47. Concentrate on the Energy momentum tensor

48. Concentrate on the Energy momentum tensor Where

49. Concentrate on the Energy momentum tensor and 1/ Do not contain any 3/ Do not contain any 2/ Do contain the same combination of

50. Concentrate on the Energy momentum tensor and 1/ Do not contain any 3/ Do not contain any 2/ Do contain the same combination of Contains only as second time derivatives

51. Hence, the combination of the field equations Can be used on shell to extract as function of time derivatives of order · 2 of the scalar field and the metric

52. Hence, the combination of the field equations Can be used on shell to extract as function of time derivatives of order · 2 of the scalar field and the metric Energy momentum tensor Take then a time derivative of the obtained expression and insert back into Reduces the Einstein equations to a system od PDE of second order in time

53. Similarly, another linear combination of the time derivatives of the field equations and can be used to reduce the order of the time derivatives of the scalar field equation by extracting as function of lower time derivatives

54. and 1/ Do not contain any 3/ Do not contain any 2/ Do contain the same combination of NB: in

55. and 1/ Do not contain any 3/ Do not contain any 2/ Do contain the same combination of The crucial 1/ and 2/ are just consequences of Energy momentum tensor Scalar field eom And (from invariance under diffeo) NB: in

56. The found « reduction » of the order in time derivatives of the field equations can be generalized to an arbitrary theory of the type Each theory of this type should propagate 2 (graviton) + 1 (scalar) d.o.f.

57. II.7.2. Hamiltonian analysis of the quartic Galileon With Consider

58. Where In the ADM parametrization, the action S becomes (in an arbitrary gauge)

59. Where In the ADM parametrization, the action S becomes (in an arbitrary gauge) Generates third order time derivatives Absent in the unitary gauge (used by GLPV)

60. In the ADM parametrization, the action S becomes (in an arbitrary gauge) Where Generates third order time derivatives Absent in the unitary gauge (used by GLPV) but depends on, and non linearly on second derivatives of

61. More convenient to work with 31 canonical (Lagrangian) fields (where ) 23 primary constraints 23 secondary constraint At least 8 of them are first class At most Hamiltonian d.o.f. Further analysis shows that there exist a tertiary (and likely also a quaternary) second class constraint, hence less than 8 d.o.f.

62. Conclusions (of part II.7.) Have shown how the e.o.m. of beyond Horndeski theory can indeed be reduced in agreement with GLPV claim (but correcting a flaw in GLPV proof). Provide a first step toward a proper Hamiltonian treatment of these theories (also supporting GLPV claim). Various possible follow up: classification of these theories, Cauchy problem etc…

63. In general Still many open and interesting questions on the « formal side » of the Galileons

64. Galileons and generalized Galileons have also been obtained more recently by other constructions Kaluza-Klein compactifications of Lovelock Gravity Van Acoleyen, Van Doorsselaere 1102.0847 [gr-qc] Brane world constructions De Rahm, Tolley 1003.5917 [hep-th] Padilla, Saffin, Zhou 1007.5424 [hep-th] Hinterbichler, Trodden, Wesley 1008.1305 [hep-th] Galileons can be supersymmetrized but stability issues Khoury, Lehners, Ovrut 1103.0003 [hep-th] Koehn, Lehners, Ovrut 1302.0840 [hep-th] Farakos, Germani, Kehagias 1306.2961 [hep-th] Construction à la Palatini R. Klein, M. Ozkan, D. Roest, 1510.08864 [hep-th]

65. Pure Galileons interactions obey a non renormalization theorem Luty, Porrati, Rattazzi hep-th/0303116 Hinterbichler, Trodden, Wesley 1008.1305 [hep-th] Recently discussed Galileons duality De Rham, Fasiello, Tolley 1308.2702 [hep-th] invert Some linear combination of equivalent

66. In general Still many open and interesting questions on the « formal side » of the Galileons A lot of interesting phenomenology (not discussed here)

67. Thank you for your attention