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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)
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3 (good ?) reasons to give this talk here
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3 (good ?) reasons to give this talk here
1. Arkady is in the room !
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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 !
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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 !
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)
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1. A short review on Massive Gravity
1.1. Why massive gravity ?
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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 ?
1. A short review on Massive Gravity
1.1. Why massive gravity ?
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One way to modify gravity at « large distances »
… and get rid of dark energy (or dark matter) ?
Changing the dynamics
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 ?
1. A short review on Massive Gravity
1.1. Why massive gravity ?
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for this idea to work…
I.e. to « replace » the cosmological constant by a
non vanishing graviton mass…
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for this idea to work…
One obviously needs
a very light graviton
(of Compton length
of order of the size of
the Universe)
I.e. to « replace » the cosmological constant by a
non vanishing graviton mass…
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for this idea to work…
I.e. to « replace » the cosmological constant by a
non vanishing graviton mass…
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))
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1. A short review on Massive Gravity
1.1. Why massive gravity ?
Theoretical challenge to give a mass to the graviton
(here we will rather stay on this « abstract » side
rather than discussing real world applications)
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 ?
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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. A short review on Massive Gravity
1.2. Fierz-Pauli theory on Einstein space-times
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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. A short review on Massive Gravity
1.2. Fierz-Pauli theory on Einstein space-times
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Kinetic
operator
Mass
term
Cosmological
constant
Comes from expanding the
Einstein-Hilbert action
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Kinetic
operator
Mass
term
Cosmological
constant
Comes from expanding the
Einstein-Hilbert action
on shell
ana
Analogous to Proca equations
for a massive photon :
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Kinetic
operator
Mass
term
Cosmological
constant
Comes from expanding the
Einstein-Hilbert action
Interest: Cosmology ?
The mass term leads to « self
acceleration » (C.D., Dvali, Gabadadze 2001)
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DOF (= « polarizations » ) counting
The Fierz Pauli theory for a massive graviton
of mass m propagates
• 2 DOF if m = 0
Massless graviton (of GR)
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The Fierz Pauli theory for a massive graviton
of mass m propagates
• 2 DOF if m = 0
• 5 DOF if m 0 and m2 2 ¤ /3
Generic massive graviton
Massless graviton (of GR)
DOF (= « polarizations » ) counting
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The Fierz Pauli theory for a massive graviton
of mass m propagates
• 2 DOF if m = 0
• 5 DOF if m 0 and m2 2 ¤ /3
Generic massive graviton
Massless graviton (of GR)
DOF (= « polarizations » ) counting
ana the Proca « photon »
has 3 polarizations
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The Fierz Pauli theory for a massive graviton
of mass m propagates
• 2 DOF if m = 0
• 5 DOF if m 0 and m2 2 ¤ /3
Generic massive graviton
Massless graviton (of GR)
DOF (= « polarizations » ) counting
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The Fierz Pauli theory for a massive graviton
of mass m propagates
• 2 DOF if m = 0
• 5 DOF if m 0 and m2 2 ¤ /3
• 4 DOF if m2 = 2 ¤ /3
Generic massive graviton
Partially Massless
graviton
Massless graviton (of GR)
DOF (= « polarizations » ) counting
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How to count DOF ?
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Kinetic
operator
Mass
term
Cosmological
constant
Comes from expanding the
Einstein-Hilbert action
This implies the (Bianchi) offshell
identities
How to count DOF ?
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Kinetic
operator
Mass
term
Cosmological
constant
Comes from expanding the
Einstein-Hilbert action
This implies the (Bianchi) offshell
identities
How to count DOF ?
Analogous to
In Maxwell and Proca theories
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Results in an the off-shell identity
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Results in an the off-shell identity
And the on-shell relation
4 vector
constraints
Kills 4 out of 10 DOF of
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Results in an the off-shell identity
And the on-shell relation
4 vector
constraints
Kills 4 out of 10 DOF of
Analogous to the constraint
of Proca theory
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Taking an extra derivative of the field equation
operator yields (off shell)
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Taking an extra derivative of the field equation
operator yields (off shell)
While tracing it with the metric gives
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Taking an extra derivative of the field equation
operator yields (off shell)
While tracing it with the metric gives
Hence we have the identity
Yielding on shell
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Generically: yields
i.e. a “scalar constraint”
reducing from 6 to 5 the
number of propagating DOF
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However, if
Then this vanishes identically
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However, if
Then this vanishes identically
As is
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However, if
Then this vanishes identically
As is
Shows the existence of a gauge symmetry
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Hence, if
one has 6 - 2 = 4 DOF
The massive graviton is
said to be
“Partially massless” (PM)
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1. A short review on Massive Gravity
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.
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1. A short review on Massive Gravity
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.
Wrong light bending
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1. A short review on Massive Gravity
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.
Wrong light bending
Stays sending the mass of the
graviton smoothly to zero (van Dam,
Veltman, Zakharov, Iwasaki 1970)
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1. A short review on Massive Gravity
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.
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 »)
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Arkady’s suggestion was criticized by
Boulware and Deser (BD) in 1972
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BD pointed out that one needed to find non singular
solution of an existing massive gravity theory
featuring the behaviour conjectured by Arkady
Arkady’s suggestion was criticized by
Boulware and Deser (BD) in 1972
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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)
Arkady’s suggestion was criticized by
Boulware and Deser (BD) in 1972
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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 »
Arkady’s suggestion was criticized by
Boulware and Deser (BD) in 1972
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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 »
This was overcome in 2010-2011 by de Rham,
Gababadze, Tolley (« dRGT theory »)
Arkady’s suggestion was criticized by
Boulware and Deser (BD) in 1972
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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
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1. A short review on Massive Gravity
1.4. Non linear completions of Partially Massless theory ?
Finding a non linear completion of the PM theory would
be very interesting
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1. A short review on Massive Gravity
1.4. Non linear completions of Partially Massless theory ?
Finding a non linear completion of the PM theory would
be very interesting
Open question with some attempts and
no-go results (de Rham, Hinterbichler, Rosen, Tolley; Deser, Waldron…)
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1. A short review on Massive Gravity
1.4. Non linear completions of Partially Massless theory ?
Finding a non linear completion of the PM theory would
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…)
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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
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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)
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Our massive graviton theory is defined by
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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
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2. The following (linear) field equations
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2. The following (linear) field equations
Linearized Einstein operator
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2. The following (linear) field equations
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We follow the analysis for the case of Einstein
space-times:
DOF counting ?
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We follow the analysis for the case of Einstein
space-times:
The linearized Bianchi identity
Yields 4 vector constaints reducing from 10 to 6
the number of propagating DOF
DOF counting ?
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We follow the analysis for the case of Einstein
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 ?
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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
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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
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The scalar constraint reads
With
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The scalar constraint reads
With
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The scalar constraint reads
With
Missing terms in v1
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The scalar constraint reads
With
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The scalar constraint reads
With
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To get a PM theory we need to look for space-times
where and vanish identically.
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To get a PM theory we need to look for space-times
where and vanish identically.
Most general solution ?
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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.
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Space-times possessing a covariantly constant
tensor H¹ º are severely restricted…
Non trivial integrability conditions
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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
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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…
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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)
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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
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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
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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
obeyes indeed the required relation
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Explicit solutions (I)
with
dimensionless dimensionful
Analogous
to
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For all these solutions one has
There also exist another set of Petrov
type D solutions for which
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Explicit solutions (II)
with
No type O solutions
(Einstein static Universe is gone as a
solution in our v2)
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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)
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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 !