Massless Gravity Claudia de Rham July 5 th 2012 Work in collaboration with Sébastien RenauxPetel 12063482 Massive Gravity Massive Gravity The notion of mass requires a reference ID: 403122
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Slide1
(Partially)
Massless
Gravity
Claudia de Rham
July 5
th
2012
Work in collaboration with Sébastien Renaux-Petel 1206.3482 Slide2
Massive GravitySlide3
Massive Gravity
The notion of mass requires a
reference !Slide4
Massive Gravity
The notion of mass requires a
reference !Slide5
Massive Gravity
The notion of mass requires a
reference !
Flat Metric
MetricSlide6
Massive Gravity
The notion of mass requires a
reference ! Having the flat Metric as a Reference breaks Covariance !!! (Coordinate transformation invariance)Slide7
Massive Gravity
The notion of mass requires a
reference ! Having the flat Metric as a Reference breaks Covariance !!! (Coordinate transformation invariance) The loss in symmetry generates new
dof
GR
Loss of 4 symSlide8
Massive Gravity
The notion of mass requires a
reference ! Having the flat Metric as a Reference breaks Covariance !!! (Coordinate transformation invariance) The loss in symmetry generates new
dof
Boulware
& Deser, PRD6, 3368 (1972)Slide9
Fierz
-Pauli Massive Gravity
Mass term for the fluctuations around flat space-time
Fierz & Pauli, Proc.Roy.Soc.Lond.A
173, 211 (1939)Slide10
Fierz
-Pauli Massive Gravity
Mass term for the fluctuations around flat space-timeTransforms under a change of coordinateSlide11
Fierz
-Pauli Massive Gravity
Mass term for the ‘covariant fluctuations’
Does not transform under that change of coordinateSlide12
Fierz
-Pauli Massive Gravity
Mass term for the ‘covariant fluctuations’
The potential has higher derivatives...
Total derivativeSlide13
Fierz
-Pauli Massive Gravity
Mass term for the ‘covariant fluctuations’
The potential has higher derivatives...
Total derivative
Ghost reappears at
the non-linear levelSlide14
Ghost-free Massive GravitySlide15
Ghost-free Massive GravitySlide16
Ghost-free Massive Gravity
With
Has no ghosts at leading order in the
decoupling limit
CdR
, Gabadadze, 1007.0443
CdR
, Gabadadze, Tolley, 1011.1232 Slide17
Ghost-free Massive Gravity
In 4d, there is a
2-parameter family of ghost free theories of massive gravity
CdR, Gabadadze, 1007.0443
CdR, Gabadadze, Tolley, 1011.1232 Slide18
Ghost-free Massive Gravity
In 4d, there is a
2-parameter family of ghost free theories of massive gravity
Absence of ghost has now been proved fully non-perturbatively in many different languages
CdR
, Gabadadze, 1007.0443CdR
, Gabadadze, Tolley, 1011.1232Hassan & Rosen, 1106.3344
CdR, Gabadadze, Tolley, 1107.3820CdR, Gabadadze, Tolley, 1108.4521Hassan & Rosen, 1111.2070 Hassan, Schmidt-May & von Strauss, 1203.5283 Slide19
Ghost-free Massive Gravity
In 4d, there is a
2-parameter family of ghost free theories of massive gravity
Absence of ghost has now been proved fully non-perturbatively in many different languagesAs well as around
any reference metric, be it dynamical or not BiGravity !!!
Hassan, Rosen & Schmidt-May, 1109.3230
Hassan & Rosen,
1109.3515Slide20
Ghost-free Massive Gravity
One can construct a consistent theory of massive gravity around any reference metric which
- propagates 5 dof
in the graviton (free of the BD ghost)- one of which is a helicity-0 mode
which behaves as a scalar field
couples to matter
- “hides” itself via a Vainshtein mechanism
Vainshtein, PLB39, 393 (1972)Slide21
But...
The Vainshtein mechanism always comes hand in hand with
superluminalities
...This doesn’t necessarily mean CTCs,but - there is a family of preferred
frames - there is no absolute notion of light-cone.
Burrage,
CdR, Heisenberg & Tolley, 1111.5549 Slide22
But...
The Vainshtein mechanism always comes hand in hand with
superluminalities
...
The presence of the helicity-0 mode puts
strong bounds on the graviton massSlide23
But...
The Vainshtein mechanism always comes hand in hand with
superluminalities
...
The presence of the helicity-0 mode puts
strong bounds on the graviton mass
Is there a different region in
parameter space where thehelicity-0 mode could also beabsent ???Slide24
Change of Ref. metric
Hassan & Rosen, 2011
Consider massive gravity around
dS
as a
reference
!
dS
Metric
Metric
dS
is still a maximally symmetric ST
Same amount of symmetry as massive gravity around
Minkowski
!Slide25
Massi
ve Gravity in de Sitter
Only the helicity-0 mode gets ‘seriously’ affected by the dS reference metricSlide26
Massi
v
e Gravity in de SitterOnly the helicity-0 mode
gets ‘seriously’ affected by the dS reference metric
Healthy scalar field(Higuchi bound)
Higuchi, NPB
282, 397 (1987)Slide27
Massi
v
e Gravity in de SitterOnly the helicity-0 mode
gets ‘seriously’ affected by the dS reference metric
Higuchi, NPB
282, 397 (1987)
Healthy scalar field
(Higuchi bound)Slide28
Massi
ve Gravity in de Sitter
Only the helicity-0 mode gets ‘seriously’ affected by the dS reference metric
The helicity-0 mode disappears at the linear level
whenSlide29
Massi
ve Gravity in de Sitter
Only the helicity-0 mode gets ‘seriously’ affected by the dS reference metric
The helicity-0 mode disappears at the linear level
whenRecover a symmetry
Deser & Waldron, 2001Slide30
Partially
massless
Is different from the minimal modelfor which all the interactions cancel in the usual DL, but the kinetic term is still presentSlide31
Partially
massless
Is different from the minimal modelfor which all the interactions cancel in the usual DL, but the kinetic term is still present
Is different from FRW models where the kinetic term disappearsin this case the fundamental theory has a helicity-0 mode but it cancels on a specific backgroundSlide32
Partially
massless
Is different from the minimal modelfor which all the interactions cancel in the usual DL, but the kinetic term is still present
Is different from FRW models where the kinetic term disappearsin this case the fundamental theory has a helicity-0 mode but it cancels on a specific background
Is different from Lorentz violating MG
no Lorentz symmetry around dS, but still have same amount of symmetry.Slide33
(Partially)
massless limitMassless
limit GR + mass term
Recover 4d diff invariance
GR
in 4d:
2
dof (helicity 2)Slide34
(Partially)
massless limitMassless
limit Partially Massless limit
GR + mass term
Recover 4d diff invariance
GR
GR + mass term
Recover 1 symmetryMassive GR4 dof
(
helicity
2 &1)
Deser & Waldron, 2001
in 4d:
2
dof
(
helicity
2)Slide35
Non-linear partially
masslessSlide36
Non-linear partially
masslessLet’s start with ghost-free theory of MG,
But around dS
dS
ref metricSlide37
Non-linear partially
masslessLet’s start with ghost-free theory of MG,
But around dSAnd derive the ‘decoupling limit’
(ie leading interactions for the helicity-0 mode)
dS
ref metric
But we need to properly identify the helicity-0 mode first....Slide38
Helicity-0 on
dS
To identify the helicity-0 mode on de Sitter, we copy the procedure
on Minkowski.Can embed d-dS into (d+1)-Minkowski:
CdR
& Sébastien Renaux-Petel, arXiv:1206.3482 Slide39
Helicity-0 on
dS
To identify the helicity-0 mode on de Sitter, we copy the procedure
on Minkowski.Can embed d-dS into (d+1)-Minkowski:
CdR
& Sébastien Renaux-Petel, arXiv:1206.3482 Slide40
Helicity-0 on
dS
To identify the helicity-0 mode on de Sitter, we copy the procedure
on Minkowski.Can embed d-dS into (d+1)-Minkowski:
behaves as a scalar in the
dec
. limit and captures the physics of the helicity-0 mode
CdR
& Sébastien Renaux-Petel, arXiv:1206.3482 Slide41
Helicity-0 on
dS
To identify the helicity-0 mode on de Sitter, we copy the procedure
on Minkowski. The covariantized metric fluctuation is expressed in terms of the helicity-0 mode as
CdR & Sébastien Renaux-Petel, arXiv:1206.3482
in any dimensions...Slide42
Decoupling limit on
dS
Using the properly identified helicity-0 mode, we can derive the decoupling limit on dS Since we need to satisfy the Higuchi bound,
CdR
& Sébastien Renaux-Petel, arXiv:1206.3482 Slide43
Decoupling limit on
dS
Using the properly identified helicity-0 mode, we can derive the decoupling limit on dS Since we need to satisfy the Higuchi bound,
The resulting DL resembles that in
Minkowski (Galileons), but with specific coefficients...
CdR
& Sébastien Renaux-Petel, arXiv:1206.3482 Slide44
DL on
dS
CdR & Sébastien Renaux-Petel, arXiv:1206.3482
+ non-diagonalizable terms
mixing h and p.
d terms + d-3 terms
(d-1) free parameters (m
2 and
a
3,...,d)Slide45
DL on
dS
The kinetic term vanishes ifAll the other interactions vanish simultaneously if
CdR & Sébastien Renaux-Petel, arXiv:1206.3482
+ non-diagonalizable terms
mixing h and p.
d terms + d-3 terms
(d-1) free parameters (m
2
and
a
3,...,d)Slide46
Massless
limit
In the
massless
limit, the helicity-0 mode still couples to matter
The Vainshtein mechanism is active to decouple this modeSlide47
Partially
massless limit
Coupling to matter
eg
. Slide48
Partially
massless limit
The symmetry cancels the coupling to matter
There is no Vainshtein mechanism, but there is no
vDVZ
discontinuity...Slide49
Partially
massless limit
Unless we take the limit without considering
the PM parameters
a
.
In this case the standard Vainshtein mechanism applies.Slide50
Partially
massless
We uniquely identify the non-linear candidate for the Partially Massless theory to all orders.
In the DL, the helicity-0 mode entirely disappear in any dimensionsWhat happens beyond the DL is still to be worked out
As well as the non-linear realization of the symmetry...
Work in progress with Kurt Hinterbichler, Rachel Rosen & Andrew Tolley
See Deser&Waldron Zinoviev