1. DOF counting for a Massive FP7/2007-2013 graviton ( la - - PowerPoint PPT Presentation

Gravity and Cosmology 2018 Kyoto, 2018 march 1st.

Massive and Partially Massless (PM) graviton

• n curved space-times
• 1. DOF counting for a Massive

graviton (à la Fierz-Pauli)

• n an Einstein spacetime
• 2. Consistent massive graviton on arbitrary

spacetime.

• 3. PM graviton on non Einstein spacetimes.

Cédric Deffayet (IAP and IHÉS, CNRS Paris)

FP7/2007-2013 « NIRG » project no. 307934

• L. Bernard, C.D., M. von Strauss + A. Schmidt-May

(2015-2016, PRD, JCAP)

• L. Bernard, C.D., K. Hinterbichler and M. von Strauss

arXiv:1703.02538 (PRD)

• 1. DOF counting for a massive graviton on an Einstein

space-time (Fierz-Pauli linear theory).

Consider an Einstein space-time obeying

• 1. DOF counting for a massive graviton on an Einstein

space-time (Fierz-Pauli linear theory).

Consider an Einstein space-time obeying Fierz-Pauli theory can be defined by

Fierz-Pauli (1939), Deser Nepomechie (1984), Higuchi (1987), Bengtsson (1995), Porrati (2001)

• 1. DOF counting for a massive graviton on an Einstein

space-time (Fierz-Pauli linear theory).

Consider an Einstein space-time obeying Field equations

• n shell

Fierz-Pauli theory can be defined by

Fierz-Pauli (1939), Deser Nepomechie (1984), Higuchi (1987), Bengtsson (1995), Porrati (2001)

• 1. DOF counting for a massive graviton on an Einstein

space-time (Fierz-Pauli linear theory).

Consider an Einstein space-time obeying Field equations with

• n shell

Kinetic

• perator

Mass term Cosmological constant

Fierz-Pauli theory can be defined by

Fierz-Pauli (1939), Deser Nepomechie (1984), Higuchi (1987), Bengtsson (1995), Porrati (2001)

Kinetic

• perator

Mass term Cosmological constant

Comes from expanding the Einstein-Hilbert action

Kinetic

• perator

Mass term Cosmological constant

Only consistent mass term among

DOF counting

The fierz Pauli theory for a massive graviton

• f mass m propagates
• 2 DOF if m = 0

Massless graviton

DOF counting

The fierz Pauli theory for a massive graviton

• f mass m propagates
• 2 DOF if m = 0
• 5 DOF if m  0 and m2  2 ¤ /3

Massless graviton Generic massive graviton

DOF counting

The fierz Pauli theory for a massive graviton

• f mass m propagates
• 2 DOF if m = 0
• 5 DOF if m  0 and m2  2 ¤ /3
• 4 DOF if m2 = 2 ¤ /3

Massless graviton Generic massive graviton Partially Massless graviton

How to count DOF ?

Kinetic

• perator

Mass term Cosmological constant

Comes from expanding the Einstein-Hilbert action This implies the (Bianchi) offshell identities

How to count DOF ?

Results in the off-shell identity

Results in the off-shell identity i.e.

• n shell

NB: in this talk Equals off-shell up to undifferentiated or (only) once differentiated h¹ º but NOT twice differentiated h¹ º

Results in an the off-shell identity And the on-shell relation

4 vector constraints

Kills 4 out of 10 DOF of i.e.

Taking an extra derivative of the field equation

• perator yields (off shell)

Taking an extra derivative of the field equation

• perator yields (off shell)

While tracing it with the metric gives

Taking an extra derivative of the field equation

• perator yields (off shell)

While tracing it with the metric gives Hence we have the identity Yielding on shell

Generically: yields i.e. a “scalar constraint” reducing from 6 to 5 the number of propagating DOF

However, if Then this vanishes identically

However, if Then this vanishes identically As is

However, if Then this vanishes identically As is Shows the existence of a gauge symmetry

Hence, if

• ne has 6 - 2 = 4 DOF (and a gauge symmetry)

The massive graviton is said to be “Partially massless” (PM)

Two questions: Can a fully non linear PM theory exist ? Can a PM graviton exist on non Einstein space-times ?

Two questions: Can a fully non linear PM theory exist ? Can a PM graviton exist on non Einstein space-times ? Here we address the second one…

• 2. Consistent massive graviton h¹ º

º on an

arbitrary background specified by some given (non dynamical) metric g¹ º First, we need to introduce the theory of a massive graviton on arbitrary backgrounds

• 2. Consistent massive graviton h¹ º

º on an

arbitrary background specified by some given (non dynamical) metric g¹ º

Einstein-Hilbert kinetic operator Mass term

First, we need to introduce the theory of a massive graviton on arbitrary backgrounds

The theory has been obtained in

• ut of the dRGT theory

L.Bernard, CD, M. von Strauss 1410.8302 + 1504.04382 + 1512.03620 (with A. Schmidt-May) de Rahm, Gabadadze; de Rham, Gababadze, Tolley, 2010, 2011

Our massive graviton theory is defined by

Our massive graviton theory is defined by

• 1. A symmetric tensor obtained from the

background curvature solving with ¯0, ¯1 and ¯2 dimensionless parameters and m the graviton mass, en the symmetric polynomials

• 2. The following (linear) field equations

• 2. The following (linear) field equations

Linearized Einstein operator

• 2. The following (linear) field equations

With

Where with Has been shown to propagate 5 DOF in a fully non linear way (dRGT, Hassan, Rosen )… … evading Boulware-Deser no-go « theorem » How we got it out of dRGT theory ? Uses two metrics: g and f

Idea: expand dRGT theory around arbitrary backgrounds However

• The « square root » tensor S ¹ º

makes things unpleasant !

• Two metrics in the game !

Linearize around some background geometry (e.g. here for the « ¯1 » models)

Expand the dynamical metric g¹ º around arbitrary backgrounds for f¹ º and g¹ º A technical difficulty: expand the matrix square root S¹

º

from One has

Sylvester (Matrix) equation (A X + X B = C) Easy to get in terms of the linear perturbation h¹ º of g¹ º

Solving the Sylvester equation (which is possible iff the spectra of S and -S do not intersect) we get with

One thus obtains the linearized field equations

Linearized Einstein tensor (curvature dependent) Mass term

I.e. the field equations shown previously (where previous E¹ º is denoted above as ± E¹ º):

In these field equations … We got rid of the second metric f¹ º and … … expressed everything in terms

• f g¹ º and its curvature
• Using the background equations of motion:

• Expressions have been simplified using Cayley-

Hamilton theorem stating that S ¹ º obeys 4th power of S in the matricial sense Allowing to replace any power of S with i ¸ 4 by linear combinations of lower powers of S In these field equations …

This provides consistent (as we now show) field equations (and action) for a massive graviton on a background simply defined by just

• ne arbitrary metric g¹ º

NB: We cheked that (in the special cases of diagonal

metrics g and f) our equations (before getting rid

• f the non dynamical metric f) match those
• btained by Guarato & Durrer 2014 (which are

not fully general and not explicitly covariant)

DOF counting (consistency check) Start from the (linear) field equations

DOF counting (consistency check) Start from the (linear) field equations The linearized Bianchi identities Reading i.e.

DOF counting (consistency check) Start from the (linear) field equations The linearized Bianchi identities Reading i.e. Yield the (off-shell) identities

DOF counting (consistency check) Start from the (linear) field equations The linearized Bianchi identities Reading i.e. Yield the (off-shell) identities Yielding the four on shell vector constraints

Can one get an extra scalar constraint in a way analogous to Fierz-Pauli on Einstein space-time ?

Can one get an extra scalar constraint in a way analogous to Fierz-Pauli on Einstein space-time ? Need to trace over the field equations and their second derivatives and look for a linear combination of these traces yielding a constraints

Can one get an extra scalar constraint in a way analogous to Fierz-Pauli on Einstein space-time ? However, we can trace both with the metric g¹ º and with its Ricci curvature R¹ º (or equivalently S¹ º) Need to trace over the field equations and their second derivatives and look for a linear combination of these traces yielding a constraints

So we look for linear combinations of the scalars

So we look for linear combinations of the scalars Thanks again to Cayley Hamilton theorem, only 8 of them are independent … … I.e. we look for 4 ui and 4 vi such that

After some algebra we find : 26 « irreducible » scalars made by contracting with powers of S, e.g. … . …

After some algebra we find Are scalar functions of

After some algebra we find Are scalar functions of

After some algebra we find … . Are scalar functions of That should all vanish … ….i.e. 26 equations for the 8 unknown

After some algebra we find … . Are scalar functions of That should all vanish … ….i.e. 26 equations for the 8 unknown

After some algebra we find … . Are scalar functions of That should all vanish … ….i.e. 26 equations for the 8 unknown However !

The scalars are not all independent from each other thanks to identities (Syzygies) … … that we derived using again the Cayley- Hamilton Theorem (or the « second fundamental

theorem » of the theory of invariants)

i.e. we define a matrix M by With

i.e. we define a matrix M by With Then each term with a given homogeneity in the In the Cayley Hamilton identity: Yields some identity betwen the matrices A,B,C,D

Picking out the identities linear in h¹ º and acting on it with two derivatives we get non trivial identities between the

e.g. with …

At the end of the day, demanding that

At the end of the day, demanding that We get a unique solution for ui and vi yielding the off-shell identity

At the end of the day, demanding that We get a unique solution for ui and vi yielding the off-shell identity and the (on-shell scalar) constraint

To conclude part 2.

• We have obtained a theory for a massive

graviton with 5 (or less) d.o.f. on an arbitrary background (agrees at first order in curvature with

Buchbinder, Gitman, Krykhtin 1999)

• Can be used independently of dRGT as a

well behaved theory of a massive graviton in just one background metric

• Various applications, e.g. : the correct theory of a

massive graviton in FLRW space time

(does not agree with Guarato-Durrer 2014)

• See also the recent Mazuet-Volkov (2017) for the

same game played with vierbeins

• 3. PM graviton on non Einstein space-times ?

We use the theory introduced in part 2.

Hence we look for (non Einstein) space-times where the scalar constraint (here written with ¯1 and ¯2 non vanishing) Identically vanishes

Yielding the gauge symmetry with

Hence we look for (non Einstein) space-times where the scalar constraint (here written with ¯1 and ¯2 non vanishing) Identically vanishes

Yielding the gauge symmetry with

We need to look in detail at the structure of the constraint

The scalar constraint reads With

The scalar constraint reads With

The scalar constraint reads With

The scalar constraint reads With

To get a PM theory we need to look for space-times where and vanish identically.

To get a PM theory we need to look for space-times where and vanish identically. Most general solution ?

To get a PM theory we need to look for space-times where and vanish identically. Most general solution ? Assume (i.e. covariantly constant) makes and vanish.

Space-times possessing a covariantly constant tensor H¹ º are severely restricted… Non trivial integrability conditions

Space-times possessing a covariantly constant tensor are classified as (provided is not proportional to the metric)

• 1. Spacetime is decomposable

and

• 2. The spacetime admits a covariantly constant

vector and

Note further that the definition Imposes that Hence the space-time must be “Ricci Symmetric” …i.e. the Ricci tensor is covariantly constant …

The spacetimes of interest for us here will all have with and constant as a consequence of the integrability conditions And have to solve (in order to get a PM graviton)

The spacetimes of interest for us here will all have with and constant as a consequence of the integrability conditions And have to solve (in order to get a PM graviton) In order to get a vanishing

The spacetimes of interest for us here will all have with and constant as a consequence of the integrability conditions And have to solve (in order to get a PM graviton) From the definition of

The spacetimes of interest for us here will all have with and constant as a consequence of the integrability conditions And have to solve (in order to get a PM graviton) NB: this implies that the Ricci tensor

• beyes indeed the required relation

Explicit solutions (I) with

dimensionless dimensionful Analogous to

Explicit solutions (II) with

dimensionless dimensionful Analogous to

One simple example of the last kind is Einstein static Universe !

with

One simple example of the last kind is Einstein static Universe !

with The gauge symmetry reads

With e.g. one solution being

Conclusions PM exists on non Einstein spacetimes !

(in contrast with previous no-go claim by Deser, Joung, Waldron In 1208.1307 [hep-th]… … in particular some of our solution have a vanishing Bach tensor)

Solution for the vanishing Of and are not known in full generality !

Thank you ! and best wishes to Misao !