Topics on generalized - PowerPoint Presentation

Slide1

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) 084003C.D., S.Deser, G.Esposito-FaresePhys.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) 082C.D., G.Esposito-Farese, D, Steer, PRD D92 (2015) 084013C.D., S. Mukohyama, V. Sivanesan,1601.01287 [hep-th]

I.1. DGP modelI.2. DGP decoupling limit

II.1. Flat space-time Galileon in 4DII.2. Flat space-time Galileon in arbitrary DII.3. Curved space-time GalileonII.4. Generalization to p-formsII.5. From k-essence to generalized Galileons II.6. Some previous and recent results andother approaches (Horndeski theories)II.7. Dof counting in « Beyond Horndeski » theories

FP7/2007-2013

« NIRG » project no. 307934

Cé

dric Deffayet

(IAP and IHÉS, CNRS Paris)

Stella Maris, Jan the 11th 2016Slide2

I. The simplest Galileon: DGP decoupling limit

I.1 The DGP model

Standard modelPeculiar 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. Slide3

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 interestConsistent (?) non linear massive gravity …DGP modelIntellectual interestLead to many subsequent developments (massive gravity, Galileons, …)Slide4

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 BangSlide5

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;… Slide6

A good description of many DGP key properties is given by the action

Luty, Porrati, Rattazzi, 2003

Scalar sector of the modelEnergy 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 limitSlide7

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 conditionTo 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

)Slide8

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 equationsII. 1 Flat space-time Galileon in 4 DIn 4D, there is only 4 non trivial such theories

( with )Slide9

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 Slide10

Indeed, consider e.g.

Varying this Lagrangian with respect to

¼ yields (after integrating by part) Second order derivativeThird 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 Slide11

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 (¼ij) = 0 (Monge Ampere equation have interesting integrability properties - Fairlie)The field equations read

Linear in second time derivative ( Good Cauchy problem ?).Slide12

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 indicesSlide13

Up to total derivatives, the following Lagrangians are equivalent

One has the exact relationSlide14

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 ¼¹ = r¹ ¼ and ¼¹ º = r¹ rº 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 mixingSlide15

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 !Slide16

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 Slide17

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 RiemannsSlide18

This can be understood as follows. Considers e.g.

Vary w.r.t.

¼Only « 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 Slide19

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-formsSlide20

II. 4 Generalization to p-forms

E.g. consider

With a p-form of field strengthIn the field equations, Bianchi identities annihilate any

E.o.m. are (purely) second order

E.g. for a 2-formNote 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)Slide21

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 strengthYields vanishing e.o.m. (the action is a total derivative)Integration by part

Renumbering of an even (for odd p) number of indicesSlide22

Is there an (odd p) Galileon ?

C. D. , A. E. Gumrukcuoglu, S. Mukohyama and Y.Wang, [arXiv:1312.6690 [hep-th]], published in PRDC.D., S. Mukohyama, V. Sivanesan[arXiv:1601.01287[hep-th]]C.D., S. Mukohyama, V. SivanesanIn preparationSlide23

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 Slide24

Hypothesis (i) (that field equations derive from an action S) leads toWhich upon

using Leads to the « integrability conditions » Where Vanish as a consequence of Hypothesis (ii) (that field equations only

depend on second order derivatives) Vanish as a consequence of Hypothesis (ii) (that field equations only depend on second order derivatives) Slide25

Hypothesis (iii) (gauge invariance of the field equations)Lead to

These symmetries extend in particular to derivatives of the field equationsSlide26

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 Bi1.2. Invariance under interchange of groups of indices Bi and Bj, as well as A and any Bi2.1. Symmetry of each pair of indices (ci, di)2.2. Invariance under interchange (ci, di) and (cj, dj)3. Symmetrizing over any 3 indices yields zeroSlide27

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)Slide28

For (odd) p > 1 ? C.D., S. Mukohyama, V. Sivanesan,1601.01287 [hep-th]

Decomposeinto components belonging to irreducible representations of the symmetric group Spm+2(m-1) = SnUsing 1.1. Antisymmetry within each group of p indices A and Bi1.2. Invariance under interchange of groups of indices

Bi and Bj, as well as A and any Bi2.1. Symmetry of each pair of indices (ci, di)2.2. Invariance under interchange (ci, di) and (cj, dj)3. Symmetrizing over any 3 indices yields zeroThis amounts to act with the Young symmetrizers / appearing in the decomposition of the group algebra of Sn into irreducibles given by Partitions of n Standard Young tableauxYoung symmetrizers

Condition 3. implies that only Young Tableaux with 1 or 2 column will yield a non zero component of Slide29

1.1. Antisymmetry within each group of p indices A and Bi1.2. Invariance under interchange of groups of indices Bi and Bj, as well as A and any Bi2.1. Symmetry of each pair of indices (ci, d

i)2.2. Invariance under interchange (ci, di) and (cj, dj)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 Sn indexed by up-to-two columns Young diagrams Slide30

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. Slide31

Next step: try to construct explicit theoriesE.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 Slide32

Start with the fillingA 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 Slide33

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)Slide34

This can be generalized ….. …… classification on the way Slide35

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)Slide36

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 ,… )Slide37

Answer: the most general such theory is given by a linear combination of the Lagrangians

defined by

where

Free function of ¼ and XSlide38

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 Slide39

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 W0=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))Slide40

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 theorySlide41

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]Slide42

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]Slide43

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)Slide44

Our works aims at

Provides a first step toward a proper Hamiltonian treatment of Horndeski-like and beyond Horndeski theories

Along two directionsII.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)Slide45

Consider one single Galileon « counterterm »:

With Slide46

Consider one single Galileon « counterterm »:

With

The field equations have the following structure (as seen before)Energy momentum tensor

Scalar field eomSlide47

Concentrate on the

Energy momentum tensorSlide48

Concentrate on the

Energy momentum tensor

Where Slide49

Concentrate on the

Energy momentum tensor

and 1/ Do not contain any 3/ Do not contain any

2/ Do contain the same combination of Slide50

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 Slide51

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 Slide52

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 tensorTake 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 timeSlide53

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 Slide54

and

1/ Do not contain any

3/ Do not contain any 2/ Do contain the same combination of

NB: in Slide55

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 eomAnd (from invariance under diffeo)

NB: in Slide56

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.Slide57

II.7.2. Hamiltonian analysis of the

quartic Galileon

With ConsiderSlide58

Where

In the ADM parametrization, the action S becomes (in an arbitrary gauge) Slide59

Where

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

Generates third order time derivativesAbsent in the unitary gauge (used by GLPV)Slide60

In the ADM parametrization, the action

S becomes (in an arbitrary gauge) Where

Generates third order time derivativesAbsent in the unitary gauge (used by GLPV) but depends on , and non linearly on second derivatives of Slide61

More convenient to work with

31 canonical (Lagrangian) fields(where )

23 primary constraints 23 secondary constraintAt least 8 of them are first classAt 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. Slide62

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…Slide63

In generalStill many open and interesting questions on the « formal side » of the GalileonsSlide64

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 issuesKhoury, 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]Slide65

Pure Galileons interactions obey a non renormalization theorem

Luty, Porrati, Rattazzi hep-th/0303116Hinterbichler, Trodden, Wesley 1008.1305 [hep-th]

Recently discussed Galileons duality De Rham, Fasiello, Tolley 1308.2702 [hep-th]invert

Some linear combination of

Some linear combination of equivalentSlide66

In generalStill many open and interesting questions on the « formal side » of the GalileonsA lot of interesting phenomenology (not discussed here)Slide67

Thank you for your attention