Rencontres du Vietnam Aug 19 th 2015 Claudia de Rham Modified Gravity LorentzViolating LorentzInvariant Massless spin2 Nonminimally coupled ScalarTensor Gravity as a spin2 ID: 228825
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Slide1
Modified Gravity
Rencontres du Vietnam
Aug. 19th 2015
Claudia de RhamSlide2
Modified Gravity
Lorentz-Violating
Lorentz-Invariant
Massless spin-2
Non-minimally coupled
(Scalar-Tensor)
Gravity as a spin-2
higher spin
Cascading
Gravity
Bi/Multi
Gravity
f(R)
Ghost
Condensate
Horava
-
Lifshitz
Extended
HL
Cuscuton
partially-massless
spin-3
Chameleon
Symmetron
Brans-
Dicke
Multi-
Galileon
Massive Spin-2
DGP
Galileon
Massive
Gravity
Massive
Graviton-Galileon
DBI-
Galileon
Quasi-
dilaton
LV-Massive
Gravity
+ see Martin’s talk for everything not covered hereSlide3
Modified Gravity
Lorentz-Violating
Lorentz-Invariant
Massless spin-2
Non-minimally coupled
(Scalar-Tensor)
Gravity as a spin-2
higher spin
Cascading
Gravity
Bi/Multi
Gravity
f(R)
Ghost
Condensate
Horava
-
Lifshitz
Extended
HL
partially-massless
spin-3
Chameleon
Symmetron
Brans-
Dicke
Multi-
Galileon
Massive Spin-2
DGP
Galileon
Massive
Gravity
Massive
Graviton-Galileon
DBI-
Galileon
Quasi-
dilaton
Cuscuton
New
dof
usually couple weakly to matter
(no direct screening)
LV-Massive
GravitySlide4
Modified Gravity
Lorentz-Violating
Lorentz-Invariant
Massless spin-2
Non-minimally coupled
(Scalar-Tensor)
Gravity as a spin-2
higher spin
Cascading
Gravity
Bi/Multi
Gravity
f(R)
Ghost
Condensate
Horava
-
Lifshitz
Extended
HL
Cuscuton
partially-massless
spin-3
Chameleon
Symmetron
Multi-
Galileon
Massive Spin-2
DGP
Galileon
Massive
Gravity
Massive
Graviton-Galileon
DBI-
Galileon
Quasi-
dilaton
Brans-
Dicke
New
dof
screened via the Chameleon mechanism
LV-Massive
GravitySlide5
partially-massless
spin-3
Massless spin-2
Non-minimally coupled
(Scalar-Tensor)
Modified Gravity
Lorentz-Violating
Lorentz-Invariant
Gravity as a spin-2
higher spin
Cascading
Gravity
f(R)
Ghost
Condensate
Horava
-
Lifshitz
Extended
HL
Cuscuton
Chameleon
Symmetron
Brans-
Dicke
Multi-
Galileon
Massive Spin-2
DGP
Galileon
Massive
Gravity
Massive
Graviton-Galileon
DBI-
Galileon
Bi/Multi
Gravity
New
dof
screened via the Vainshtein mechanism
LV-Massive
Gravity
Quasi-
dilatonSlide6
DGP
Dvali, Gabadadze &
Porrati ’00
gravity
gravity
Extra dimension
Model with an infinite extra dimensionSlide7
DGP
Dvali, Gabadadze &
Porrati ’00
gravity
gravity
Extra dimension
Model with an infinite extra dimension
Effective Friedman equation on the
brane
Deffayet
,
PLB 502 (2001) 199
Corresponds to a IR modification of Gravity
Allows for “self-accelerating” solutions
With a ghost …Slide8
DGP is a model of soft massive gravity
hard mass realization?
Extension of DGPPossibility to screen a large CC ???Slide9
Massive spin-2
Fierz & Pauli, 1939
Breaking of linear diffSlide10
Massive spin-2
Fierz & Pauli, 1939
Breaking of non-linear diff
Breaking of linear diffSlide11
Massive spin-2
Fierz & Pauli, 1939
Breaking of non-linear diff
Breaking of linear diff
Generic potentials (almost all) have a ghost at an unacceptable low-scale…
Boulware
-Deser 1972,
Arkani-Hamed
et al. 2003
Creminelli et.
a
l. 2005, … Slide12
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 languagesCdR
, Gabadadze, 1007.0443CdR, Gabadadze, Tolley, 1011.1232Hassan & Rosen, 1106.3344
CdR, Gabadadze, Tolley, 1107.3820
CdR
, Gabadadze, Tolley, 1108.4521
Hassan & Rosen, 1111.2070
Mirbabayi
, 1112.1435
Hassan, Schmidt-May & von Strauss, 1203.5283
Deffayet,
Mourad
& Zahariade, 1207.6338
… Slide13
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 languagesCdR
, Gabadadze, 1007.0443CdR, Gabadadze, Tolley, 1011.1232Hassan & Rosen, 1106.3344
CdR, Gabadadze, Tolley, 1107.3820
CdR
, Gabadadze, Tolley, 1108.4521
Hassan & Rosen, 1111.2070
Mirbabayi
, 1112.1435
Hassan, Schmidt-May & von Strauss, 1203.5283
Deffayet,
Mourad
& Zahariade, 1207.6338
…
See Laura Bernard’s talkSlide14
Degrees of FreedomMassive Gravity
1 massive spin-2 - 2 helicity-2 - 2 helicity-1 - 1 helicity-0
5 dofs
CdR
, Gabadadze, Tolley 2010Slide15
Degrees of FreedomMassive Gravity
1 massive spin-2 - 2 helicity-2 - 2 helicity-1 - 1 helicity-0
- 2 dof in metric (after gauge fixing) - 3 Stückelberg fields
5 dofs
Restore diff invariance
CdR
, Gabadadze, Tolley 2010Slide16
Degrees of FreedomMassive Gravity
1 massive spin-2 - 2 helicity-2 - 2 helicity-1 - 1 helicity-0
- 2 dof in metric (after gauge fixing) - 3 Stückelberg fields
5 dofs
Restore diff invariance
Internal space metric
(here reference metric for massive graviton)
Could
be made dynamical
bi-gravity
Hassan, Rosen, 2011
CdR
, Gabadadze, Tolley 2010Slide17
Degrees of Freedom
Massive Gravity
Bi-Gravity1 massive spin-2 - 2 helicity-2 - 2 helicity-1 - 1 helicity-0 - 2 dof in metric (after gauge fixing) -
3 Stückelberg fields1 massive & 1 massless
spin-2 - 2x2 helicity-2 - 2 helicity-1
- 1 helicity-0
- 2
x2
dof
in both metrics
(after gauge fixing) - 3 Stückelberg
fields
5
dofs
7
dofs
Restore diff invariance
Restore 2
nd
copy of diff invarianceSlide18
Degrees of Freedom
Bi-Gravity
1 massive & 1 massless spin-2 - 2x2 helicity-2 - 2 helicity-1 - 1 helicity-0 - 2x2
dof in both metrics (after gauge fixing) - 3 Stückelberg fields
7
dofs
Restore 2
nd
copy of diff invariance
Bi-Gravity = GR + exotic matter
0Slide19
Uniqueness ???At the linear level, massive spin-2 fields can have new kinetic interactions
Folkerts,
Pritzel & Wintergerst 1107.3157Hinterbichler, 1305.7227
5
dof
in the gravitonSlide20
Uniqueness ???Non-linearly, the only acceptable kinetic term is the EH one
5
dof in the gravitonCdR, Matas
& Tolley, 1311.6485
No non-linear completion without a ghost at the same scaleSlide21
Uniqueness ???Matter could also couple to the other metric
no ghost (still 5 dof in the graviton g).
5 dof in the gravitonSlide22
Uniqueness ???Matter could also couple to the other metric
Or to a combination of both
5 dof in the graviton
CdR, Heisenberg, Ribeiro, 2014
Ghost in the full theory, but no ghost in the decoupling limit
In
vielbeinSlide23
Uniqueness ???The combinations obtained
from deconstruction are the only ones without a ghost.
But typically the internal space metric could depend on the fieldsCdR, Keltner, Tolley, 2010
5
dof
in the gravitonSlide24
Generalized MGT
he mass term can be generalized Lorentz invariant !
Same number of constraints as Ghost-free MG 5 propagating dofs.In unitary gauge translation invariance broken Slide25
Massive spin-2 fields & HolographySpin-2
field may be useful in condensed matter applications of the AdS/CFT correspondence‘realistic’ materials with momentum
relaxation (lattice) are dual to massive gravity – momentum relaxation prevents infinite conductivity in far IRNew dofs in graviton encodes the phonon dynamicsVegh, arXiv:1301.0537,
Blake, Tong, Vegh, arXiv:1310.3832Baggioli, Pujolas, arXiv:1411.1003,… Slide26
Consequences for CosmologySlide27
Cosmology in Massive Gravity
MG has a flat Minkowski
metric as a reference metric Which has a fundamentally different topology than dS
there is
NO spatially flat
homogeneous
FLRW
solutions.
D’Amico et.al
. arXiv:1108.5231 Slide28
Cosmology in Massive Gravity MG has a flat
Minkowski metric as a reference metric
Which has a fundamentally different topology than dSThe constraint that removes the BD ghost
is also what prevents the theory from having a flat FLRW sol.Slide29
No-FLRW Constraint
St
ückelberg fields enterleads to a constraint prevents FLRW solutions
Already diff invariant
Stückelberg
fields do not enterSlide30
No spatially-flat
FLRW solutions
with Minkowski Reference metric
Open solutions
(unstable)
Additional degrees of freedom
Large Scale
Inhomogeneities
Scalar
Tensor
(Extended) Quasi-
dilaton
Mass-Varying
f(R)
…
Break
Poincar
é
invariance
Break
Lorentz
Break
Translation
New coupling
Alternatives
dS
or FRLW reference metric
(problem with Higuchi ghost)
Bi-Gravity, Multi-GravitySlide31
Alternative 1: Generalized MGGeneralized massive gravity Slide32
Alternative 1: Generalized MGGeneralized massive gravity
Free of any BD ghost (at all scales)Allows for exact FLRW solutions
Stable self-accelerating solutions
CdR, Fasiello, Tolley, 1410.0960 Slide33
Alternative 1: Generalized MGGeneralized massive gravity
Allows for exact FLRW solutions Slide34
From Lorentz invariance to cosmology
Start with Open Universe (could be thought of as local effect from long wavelength inhomogeneity)
Gumrukcuoglu, Lin, Mukohyama, arXiv:1109.3845Slide35
From Lorentz invariance to cosmology
Start with Open Universe (could be thought of as local effect from long wavelength inhomogeneity)
Gumrukcuoglu, Lin, Mukohyama, arXiv:1109.3845Slide36
Exact FLRW solutions
There are exact (self-accelerating) FLRW solutions
Which are stable in the decoupling limit where
fixed
For all the modes
(tensors, vectors, scalar, no tachyon, gradient or ghost instability)
CdR
, Fasiello, Tolley
arXiv:1410.0960
Eg
.Slide37
Validity of DL
Derived a family of DL theories valid for arbitrary timeFails to account for long wavelength modes
Stability analysis only fails to account
for the long-wavelength modes.
Any
instability which arises at the
resp. horizon
scale is harmless
.
DL at
DL at
DL at
DL centered
at
DL at
Slide38
Alternative 2: Inhomogeneous Cosmology
At high densities, our Universe would be composed of homogeneous patches
Locally, each patch
is essentially FRW
Vainshtein mechanism
at work to “hide” the
additional helicity-0
modes
D’Amico et.al
. arXiv:1108.5231 Slide39
Alternative 2: Inhomogeneous Cosmology
At high densities, Vainshtein mechanism at work to “hide” the
additional scalar modeEvolution of the very early Universe is similar to GR(expect it to be undistinguishable) no features at high
l in CMBSlide40
At low densities, start seeing first effects from graviton mass
Cosmology is expected to
differ significantly from GR
Gives a bound on the graviton mass
D’Amico,
CdR
, Dubovsky, Gabadadze, Pirtskhalava & Tolley
, 1108.5231
Inhomogeneous
CosmologySlide41
At low densities, the Vainshtein mechanism stops being efficient
Large scale inhomogeneities
D’Amico, CdR, Dubovsky, Gabadadze, Pirtskhalava & Tolley, 1108.5231
expect
low-
l
CMB
anomalies
…
& suppression
of power
…
Inhomogeneous
CosmologySlide42
B-modes
if ever detected…
would imply the graviton is effectively massless at the time of recombination At late time, the graviton mass is expected to beBut at early time, the effective mass could be much larger, so B-modes could put strong bounds
Dubovsky
, Flauger
,
Starobinsky
&
Tkachev
, (for
Lorentz-breaking MG) PRD81 (2010) 023523
See also Fasiello & Ribeiro JCAP 1507 (2015) 07, 027Slide43
Modes relativistic during recombination
Modes non-relativistic before entering the horizon
Modes relativistic before entering the horizon but turning non-relativistic by the time of recombination
Dubovsky
,
Flauger
,
Starobinsky
&
Tkachev
, (for
Lorentz-breaking MG) PRD81 (2010) 023523
See also Fasiello & Ribeiro
JCAP 1507 (2015) 07, 027Slide44
For region I
effectively masslessm
eff = 10-30 eVmeff = 10-31 eVmeff = 10-32 eV
From
Dubovsky
,
Flauger
,
Starobinsky
&
Tkachev
, 2010
PRD81 (2010) 023523
(for Lorentz-breaking MG
)Slide45
For region II
low-ell plateaufor small mass
m
eff
= 1.8 10
-28
eV
m
eff
= 1.3 10
-28
eV
m
eff = 1.1 10-28 eVmeff = 0.6 10
-28 eV
For larger mass (~ 10
-27 eV) mode are suppressed
no power in massive gravityFrom
Dubovsky,
Flauger, Starobinsky & Tkachev, 2010PRD81 (2010) 023523(for Lorentz-breaking MG)Slide46
Outlook
Massive Gravity is a specific framework to study IR modifications of Gravity It could play a role for -
the late-time acceleration of the Universe - the cosmological constant problemExact FLRW solutions do not exist in massive gravity with Minkowski reference metric and no curvatureMany alternatives are investigated (eg. Bi-gravity, quasi-dilaton
, Lorentz breaking, Poincare breaking, alternative couplings,…)Or the Universe may be inhomogeneous at large distances (beyond our horizon)