Claudia de Rham July 5 th 2012 Work in collaboration with Work in collaboration with Sébastien...
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- Slide 1
- Claudia de Rham July 5 th 2012 Work in collaboration with Work
in collaboration with Sbastien Renaux-Petel 1206.3482
- Slide 2
- Massive Gravity
- Slide 3
- The notion of mass requires a reference !
- Slide 4
- Massive Gravity The notion of mass requires a reference !
- Slide 5
- Massive Gravity The notion of mass requires a reference ! Flat
Metric Metric
- Slide 6
- Massive Gravity The notion of mass requires a reference !
Having the flat Metric as a Reference breaks Covariance !!!
(Coordinate transformation invariance)
- Slide 7
- 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 sym
- Slide 8
- 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)
- Slide 9
- Fierz-Pauli Massive Gravity Mass term for the fluctuations
around flat space-time Fierz & Pauli, Proc.Roy.Soc.Lond.A 173,
211 (1939)
- Slide 10
- Fierz-Pauli Massive Gravity Mass term for the fluctuations
around flat space-time Transforms under a change of coordinate
- Slide 11
- Fierz-Pauli Massive Gravity Mass term for the covariant
fluctuations Does not transform under that change of
coordinate
- Slide 12
- Fierz-Pauli Massive Gravity Mass term for the covariant
fluctuations The potential has higher derivatives... Total
derivative
- Slide 13
- Fierz-Pauli Massive Gravity Mass term for the covariant
fluctuations The potential has higher derivatives... Total
derivative Ghost reappears at the non-linear level
- Slide 14
- Ghost-free Massive Gravity
- Slide 15
- Slide 16
- With Has no ghosts at leading order in the decoupling limit
CdR, Gabadadze, 1007.0443 CdR, Gabadadze, Tolley, 1011.1232
- Slide 17
- 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
- Slide 18
- 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, 1007.0443 CdR, Gabadadze, Tolley, 1011.1232 Hassan
& Rosen, 1106.3344 CdR, Gabadadze, Tolley, 1107.3820 CdR,
Gabadadze, Tolley, 1108.4521 Hassan & Rosen, 1111.2070 Hassan,
Schmidt-May & von Strauss, 1203.5283
- Slide 19
- 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
As well as around any reference metric, be it dynamical or
notBiGravity !!! Hassan, Rosen & Schmidt-May, 1109.3230 Hassan
& Rosen, 1109.3515
- Slide 20
- 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)
- Slide 21
- But... The Vainshtein mechanism always comes hand in hand with
superluminalities... This doesnt 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
- Slide 22
- 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
- Slide 23
- 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 the helicity-0 mode could also be absent
???
- Slide 24
- 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 !
- Slide 25
- Massive Gravity in de Sitter Only the helicity-0 mode gets
seriously affected by the dS reference metric
- Slide 26
- Massive Gravity in de Sitter Only the helicity-0 mode gets
seriously affected by the dS reference metric Healthy scalar field
(Higuchi bound) Higuchi, NPB282, 397 (1987)
- Slide 27
- Massive Gravity in de Sitter Only the helicity-0 mode gets
seriously affected by the dS reference metric Higuchi, NPB282, 397
(1987) Healthy scalar field (Higuchi bound)
- Slide 28
- Massive 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 when
- Slide 29
- Massive 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 when Recover a symmetry Deser &
Waldron, 2001
- Slide 30
- Partially massless Is different from the minimal model for
which all the interactions cancel in the usual DL, but the kinetic
term is still present
- Slide 31
- Partially massless Is different from the minimal model for
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 disappears in this case the fundamental theory has a
helicity-0 mode but it cancels on a specific background
- Slide 32
- Partially massless Is different from the minimal model for
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 disappears in 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.
- Slide 33
- (Partially) massless limit Massless limit GR + mass term
Recover 4d diff invariance GR in 4d: 2 dof (helicity 2)
- Slide 34
- (Partially) massless limit Massless limit Partially Massless
limit GR + mass term Recover 4d diff invariance GR GR + mass term
Recover 1 symmetry Massive GR 4 dof (helicity 2 &1) Deser &
Waldron, 2001 in 4d: 2 dof (helicity 2)
- Slide 35
- Non-linear partially massless
- Slide 36
- Lets start with ghost-free theory of MG, But around dS dS ref
metric
- Slide 37
- Non-linear partially massless Lets start with ghost-free theory
of MG, But around dS And 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....
- Slide 38
- Helicity-0 on dS on de Sitter on Minkowski To identify the
helicity-0 mode on de Sitter, we copy the procedure on Minkowski.
Can embed d-dS into (d+1)-Minkowski: CdR & Sbastien
Renaux-Petel, arXiv:1206.3482
- Slide 39
- Helicity-0 on dS on de Sitter on Minkowski To identify the
helicity-0 mode on de Sitter, we copy the procedure on Minkowski.
Can embed d-dS into (d+1)-Minkowski: CdR & Sbastien
Renaux-Petel, arXiv:1206.3482
- Slide 40
- Helicity-0 on dS on de Sitter on Minkowski 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
& Sbastien Renaux-Petel, arXiv:1206.3482
- Slide 41
- Helicity-0 on dS on de Sitter on Minkowski 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 & Sbastien Renaux-Petel, arXiv:1206.3482
in any dimensions...
- Slide 42
- 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 & Sbastien Renaux-Petel,
arXiv:1206.3482
- Slide 43
- 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 &
Sbastien Renaux-Petel, arXiv:1206.3482
- Slide 44
- DL on dS CdR & Sbastien Renaux-Petel, arXiv:1206.3482 +
non-diagonalizable terms mixing h and . d terms + d-3 terms (d-1)
free parameters (m 2 and 3,...,d)
- Slide 45
- DL on dS The kinetic term vanishes if All the other
interactions vanish simultaneously if CdR & Sbastien
Renaux-Petel, arXiv:1206.3482 + non-diagonalizable terms mixing h
and . d terms + d-3 terms (d-1) free parameters (m 2 and
3,...,d)
- Slide 46
- Massless limit In the massless limit, the helicity-0 mode still
couples to matter The Vainshtein mechanism is active to decouple
this mode
- Slide 47
- Partially massless limit Coupling to matter eg.
- Slide 48
- Partially massless limit The symmetry cancels the coupling to
matter There is no Vainshtein mechanism, but there is no vDVZ
discontinuity...
- Slide 49
- Partially massless limit Unless we take the limit without
considering the PM parameters . In this case the standard
Vainshtein mechanism applies.
- Slide 50
- 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 dimensions What
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