pacific 2018 massive and moorea, sept the 3rd …...pacific 2018 moorea, sept the 3rd 2018. massive...

PACIFIC 2018

Moorea, Sept the 3rd 2018.

Massive and

Partially Massless (PM)

gravitons

on curved space-times

1. A short review on Massive

Gravity

2. Massive graviton on arbitrary

spacetimes.

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., K. Hinterbichler and M. von Strauss

arXiv:1703.02538 (PRD)

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

(2015-2016, PRD, JCAP)

3 (good ?) reasons to give this talk here


1. Arkady is in the room !



2. Massive gravity is cool and 2/3 of you were

not there in the PACIFIC.2 Japanese edition !



2. Massive gravity is cool and 2/3 of you were

not there in the PACIFIC.2 Japanese edition !

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

arXiv:1703.02538 (PRD)

3. Erratum necessary for some technicalities in

(thanks to Charles Mazuet)

1. A short review on Massive Gravity

1.1. Why massive gravity ?

One way to modify gravity at « large distances »

… and get rid of dark energy (or dark matter) ?

Changing the dynamics

of gravity ?

Dark matter

dark energy ?






of gravity ?

Historical example the

success/failure of both

approaches: Le Verrier and

• The discovery of Neptune

• The non discovery of Vulcan…

but that of General Relativity

Dark matter

dark energy ?



for this idea to work…

I.e. to « replace » the cosmological constant by a

non vanishing graviton mass…


One obviously needs

a very light graviton

(of Compton length

of order of the size of

the Universe)






NB: It seems one of the

Einstein’s motivations to

introduce the cosmological

constant was to try to « give a

mass to the graviton »

(see « Einstein’s mistake and the

cosmological constant »

by A. Harvey and E. Schucking,

Am. J. of Phys. Vol. 68, Issue 8 (2000))



Theoretical challenge to give a mass to the graviton

(here we will rather stay on this « abstract » side

rather than discussing real world applications)




of gravity ?

Dark matter

dark energy ?

Consider an Einstein space-time obeying

Fierz-Pauli theory (1939)

is the (only correct) theory of a

massive graviton which

propagates on this space-time


1.2. Fierz-Pauli theory on Einstein space-times

Consider an Einstein space-time obeying

Fierz-Pauli theory is defined by

Fierz-Pauli (1939), Deser Nepomechie (1984), Higuchi (1987),

Bengtsson (1995), Porrati (2001)

Field equations

with on shell

Kinetic

operator

Mass

term

Cosmological

constant


1.2. Fierz-Pauli theory on Einstein space-times

Kinetic

operator

Mass

term

Cosmological

constant

Comes from expanding the

Einstein-Hilbert action

Kinetic

operator

Mass

term

Cosmological

constant



on shell

ana

Analogous to Proca equations

for a massive photon :

Kinetic

operator

Mass

term

Cosmological

constant



Interest: Cosmology ?

The mass term leads to « self

acceleration » (C.D., Dvali, Gabadadze 2001)

DOF (= « polarizations » ) counting

The Fierz Pauli theory for a massive graviton

of mass m propagates

• 2 DOF if m = 0

Massless graviton (of GR)



• 2 DOF if m = 0

• 5 DOF if m 0 and m2 2 ¤ /3

Generic massive graviton





• 2 DOF if m = 0

• 5 DOF if m 0 and m2 2 ¤ /3




ana the Proca « photon »

has 3 polarizations



• 2 DOF if m = 0

• 5 DOF if m 0 and m2 2 ¤ /3






• 2 DOF if m = 0

• 5 DOF if m 0 and m2 2 ¤ /3

• 4 DOF if m2 = 2 ¤ /3


Partially Massless

graviton



How to count DOF ?

Kinetic

operator

Mass

term

Cosmological

constant



This implies the (Bianchi) offshell

identities

How to count DOF ?

Kinetic

operator

Mass

term

Cosmological

constant



This implies the (Bianchi) offshell

identities

How to count DOF ?

Analogous to

In Maxwell and Proca theories

Results in an the off-shell identity


And the on-shell relation

4 vector

constraints

Kills 4 out of 10 DOF of


And the on-shell relation

4 vector

constraints

Kills 4 out of 10 DOF of

Analogous to the constraint

of Proca theory

Taking an extra derivative of the field equation

operator yields (off shell)



While tracing it with the metric gives



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


As is

However, if


As is

Shows the existence of a gauge symmetry

Hence, if

one has 6 - 2 = 4 DOF

The massive graviton is

said to be

“Partially massless” (PM)


1.3. Non linear completions of Fierz-Pauli theory

In the generic case (with 5 DOF) the scalar polarization

of the Fierz-Pauli graviton is an obstacle to real world

applications.





applications.

Wrong light bending





applications.

Wrong light bending

Stays sending the mass of the

graviton smoothly to zero (van Dam,

Veltman, Zakharov, Iwasaki 1970)





applications.

Wrong light bending

Stays sending the mass of the

graviton smoothly to zero (van Dam,

Veltman, Zakharov, Iwasaki 1970)

A way out relying on non linearities

was suggested by Arkady in 1972

(« Vainshtein mechanism »)

Arkady’s suggestion was criticized by

Boulware and Deser (BD) in 1972

BD pointed out that one needed to find non singular

solution of an existing massive gravity theory

featuring the behaviour conjectured by Arkady






It was only recently that such solutions were found

(C.D., Dvali, Gabadadze, Vainshtein 2002,

Babichev, C.D., Ziour, 2009-2010)









In the same paper BD also pointed out another generic (they

think) pathology of non linear massive gravity: the « BD ghost »









In the same paper BD also pointed out another generic (they

think) pathology of non linear massive gravity: the « BD ghost »

This was overcome in 2010-2011 by de Rham,

Gababadze, Tolley (« dRGT theory »)



Where

with

Has been shown to propagate 5 (or less d.o.f.)

in a fully non linear way …

… evading Boulware-Deser no-go « theorem »

dRGT theory de Rahm, Gabadadze; de Rham, Gababadze, Tolley;

Hassan, Rosen 2010, 2011


1.4. Non linear completions of Partially Massless theory ?

Finding a non linear completion of the PM theory would

be very interesting




be very interesting

Open question with some attempts and

no-go results (de Rham, Hinterbichler, Rosen, Tolley; Deser, Waldron…)




be very interesting

Open question with some attempts and

no-go results

A related question: can a PM graviton

exist on a non Einstein space-time ?

Addressed here…

(de Rham, Hinterbichler, Rosen, Tolley; Deser, Waldron…)

2. Consistent massive graviton on

arbitrary backgrounds

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

out of the dRGT theory

L.Bernard, CD, M. von Strauss

1410.8302 + 1504.04382

+ 1512.03620 (with A. Schmidt-May)

(see also C. Mazuet, M. Volkov 2015)

Can be compared with Buchbinder, Gitman,

Krykhtin, Pershin (2000)

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, ei the symmetric polynomials

2. The following (linear) field equations


Linearized Einstein operator

We follow the analysis for the case of Einstein

space-times:

DOF counting ?


space-times:

The linearized Bianchi identity

Yields 4 vector constaints reducing from 10 to 6

the number of propagating DOF

DOF counting ?


space-times:

The linearized Bianchi identity

Yields 4 vector constaints reducing from 10 to 6

the number of propagating DOF

A scalar constraint reduces to 5 the

number of DOF

DOF counting ?

We look for (non Einstein) space-times where the

scalar constraint

Identically vanishes

3. PM graviton on non Einstein space-times ?

Yielding the gauge symmetry with

We look for (non Einstein) space-times where the

scalar constraint

Identically vanishes

3. PM graviton on non Einstein space-times ?

Yielding the gauge symmetry with

We need to look in detail at the

structure of the constraint

The scalar constraint reads

With


With

Missing terms in v1


With

To get a PM theory we need to look for space-times

where and vanish identically.



Most general solution ?



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 covariantly constant tensor is the Ricci

tensor…

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)


with and constant



In order to get a vanishing


with and constant



From the definition of


with and constant



NB: this implies that the Ricci tensor

obeyes indeed the required relation

Explicit solutions (I)

with

dimensionless dimensionful

Analogous

to

For all these solutions one has

There also exist another set of Petrov

type D solutions for which

Explicit solutions (II)

with

No type O solutions

(Einstein static Universe is gone as a

solution in our v2)

Explicit solutions (II)

with

No type O solutions (if one chooses the simplest kind of matrix square root

solving for S¹º)

(Einstein static Universe is gone as a

solution in our v2)

Conclusions

PM exists on non Einstein spacetimes ! (in contrast with previous no-go claim by Deser, Joung, Waldron

In 1208.1307 [hep-th]…)

Solution for the vanishing

Of and

are not known in full generality !

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