aspects of massive gravitymoriond.in2p3.fr/j15/transparencies/1_sunday/1_morning/5...vainshtein...
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Aspects of massive gravity
Sébastien Renaux-PetelCNRS - IAP Paris
Rencontres de Moriond, 22.03.2015
Motivations for modifying General Relativity in the infrared
Present acceleration of the Universe:
• Dark Energy
• Infra-Red modification of GR
|{z
}
or
Way out of the (old) cosmological constant problem?
Better understand GR
Modifying GR means addingnew degrees of freedom (dof)
Weinberg, QFT
New degrees of freedom
• General Relativity: the only theory of an interactive massless helicity-2 field
• Examples : - scalar tensor: GR + explicit scalar field
- f(R): GR + a scalar field in disguise.
• Identifying the new dof may be non-trivial. Today: example of massive gravity
Massive gravity
• Several interesting pathologies to cure/phenomena with possible applications in other areas: strong classical non-linearities, ghost instability, low cut-off EFT, IR/UV interplay, screening of long-range scalars...
Reviews: Hinterbichler 1105.3735de Rham 1401.4173
• One interesting IR modification: giving a mass to the graviton (of order the present Hubble scale)
• Motivation: weakens gravity on large scales and possibly degravitates the cosmological constant
• Massive gravity: a theory of an interactivemassive spin-2 field. What is it?
✓V ⇠ e�mr
r
◆
⇤obs
⌧ ⇤bare
Complex history
1939 Fierz-Pauli
1970 van Dam, Veltman, Zakharov
1972 Vainshtein
1972 Boulware, Deser
2003 Arkani-Hamed, Georgi, Schwartz
2010 de Rham, Gabadadze, Tolley
Linear ghost-free MG
Discontinuity...
...resolved by non-linearities?
Ghost at non-linear level
Stückelberg method
Non-linear ghost-free MG
2009 Babichev, Deffayet, Ziour Vainshtein mechanism works!
Ghost-free massive and bimetric gravity
• In 4d (d dim), there is a 2-parameter family (d-2) of ghost-free theories of Lorentz-invariant massive gravity
de Rham, Gabadadze, 10de Rham, Gabadadze, Tolley, 10
• Absence of Boulware-Deser ghost has been proved exactly in many different languages
LMG =M2
Pl
2
⇥�g
�R+m2
�K2 �K2
µ� + �3(K3 + . . .) + �4(K4 + . . .)��
Kµ⇥ = �µ⇥ �
⇥gµ�⇥�⇥
de Rham, Gabadadze, Tolley, 10, 11, Hassan, Rosen, 11Mirbabay, 11, Hassan, Schmidt-May, von Strauss, 12Deffayet, Mourad, Zahariade, 12
• And around any reference metric, even dynamical: bigravityHassan, Rosen, Schmidt-May, 11Hassan, Rosen, 11
�µ� ! fµ�
• Reformulation in terms of vielbeins and extension to multi-metricHinterbichler, Rosen, 12, Noller et al 14
See talk by Blanchet
Degrees of freedom in massive gravity• Mass term breaks diffeomorphism invariance of GR
2 ! 5 degrees of freedom
1 massive spin-2:1 massless spin-2:
General Relativity Massive Gravity
- 2 helicity-2
- 2 helicity-2
- 2 helicity-1
- 1 helicity-0
The details are really only well known for a Minkowski reference metric, and around Minkowski space
Stückelberg trick
breaks the U(1) gauge invariance
• We restore gauge invariance with the replacement:
gauge-invariant under Stückelberg, 1938
L = �1
4Fµ⌫F
µ⌫ � 1
2m2AµA
µ +AµJµ
Vector example:
• The original theory is simply a gauge-fixed version of the gauge-invariant theory. Same physical content and same predictions.
• Central idea: gauge-invariance is a redundancy of description. So any theory can be made gauge-invariant, with any gauge-invariance we like.
Aµ ! Aµ + @µ⇡
L̃ = �1
4Fµ⌫F
µ⌫ � 1
2m2(Aµ + @µ⇡)
2 +AµJµ � �@µJ
µ
�Aµ = @µ⇤, �⇡ = �⇤
• Gauge-invariance of GR: full diffeomorphism invariance:
• We restore gauge-invariance in MG with the introduction of 4 scalar fields and the replacement:
transforms like a metric tensor under diff
Reference metric
transforms like a scalar under diff
|{z
}
Y
↵ = x
↵• Original Lagrangian recovered in the ‘unitary’ gauge
fµ⌫(x) !@g
↵
@x
µ
@g
�
@x
⌫f↵� (g(x))
fµ⌫(x) ! f̃µ⌫(x) = f↵� (Y (x)) @µY↵@⌫Y
�
f̃µ⌫
gµ⌫ f̃µ⌫
Stückelberg for massive gravity
Arkani-Hamed et al, 03
Y
↵ = x
↵ �A
↵• Expansion with reference metric fµ⌫ = ⌘µ⌫
hµ⌫ ⌘ gµ⌫ � ⌘µ⌫ ! Hµ⌫ = hµ⌫ + @µA⌫ + @⌫Aµ � @µA↵@⌫A↵
• Metric fluctuation:
• Additional replacement:
Stückelberg for massive gravity
Aµ ! Aµ + @µ⇡
Hµ⌫ = hµ⌫ + @µA⌫ + @⌫Aµ + 2@µ@⌫⇡ � @µA↵@⌫A↵
� @µA↵@⌫@↵⇡ � @µ@
↵⇡@⌫A↵ � @µ@↵⇡@⌫@↵⇡
• Decoupling limit: concentrates on the new interactions beyond GR at the lowest energy scale
• In the decoupling limit, the massive gravity Lagrangian is invariant under
GR-like
Maxwell-like
encode the helicity-2, helicity-1 and scalar dofs of the theory
Stückelberg for massive gravity
• In the decoupling limit,
�hµ⌫ = @µ⇠⌫ + @⌫⇠µ
�Aµ = @µ⇤
�⇡ = 0
hµ⌫ , Aµ,⇡
• Reference metric: Minkowski de Sitter. Still a maximally symmetric spacetime (same amount of symmetry).Study the theory around de Sitter spacetime
Massive gravity on de Sitter
• How the dS reference metric affects the helicity-0 mode is well-known at linear order:
Higuchi, 87, Deser, Waldron, 01Grisa, Sorbo 09
�m4(⇥�)2 �m2�m2 � (d� 2)H2
�(⇥�)2
m2 > (d� 2)H2Higuchi bound:
• The helicity 0-mode disappears completely, at the linear level, for
m2 = (d� 2)H2 (partially massless)
Strategy
Y
Y = 0
• Embed d-dS into (d+1)-Minkowski ...
... and copy the procedure on Minkowski
�̃µ�dxµdx� = (⇥̃ABdZ
AdZB)|projected
=�⇥MN⌅A⇤
M⌅B⇤NdZAdZB
�|projected
= ⇥MN⌅µ⇤M⌅�⇤
Ndxµdx�
• Introduction of Stückelberg fields:
⇤M = ZM � �MN⌅N⇥
behaves as a scalar field in the decoupling limit and captures the physics of the helicity-0 mode
⇡
ds2d+1 = e�2HY⇣dY 2 + �(dS)
µ� dxµdx�⌘
= ⇥ABdZAdZB
de Rham, RP, 1206.3482
dS
Minkowski
Decoupling limit
• Covariantized metric fluctuation in terms of the helicity-0 mode
Hµ⇤ = hµ⇤ + 2�µ⇤ ��2µ⇤
+ H2�(⇤⇥)2 (�µ⇤ ��µ⇤)� (�µ�⇤
�⇥ � ⇤µ⇥)��⇤⇥⇤
⇥⇥ � ⇤⇤⇥��
+O(H4)
�µ� = rµr�⇡with
• Decoupling limit:
MPl ! 1
H ! 0
m ! 0 , � ⌘ (m2M (d�2)/2Pl )2/(d+2) fixed
Suppress non-linearities of GR
Keep finite the lowest energy interactions of the
helicity-0 mode
Satisfy the Higuchi bound
Study the Partially
Massles case
H
mfixed
Decoupling limit
�3 = �1
3
d� 1
d� 2�n = � 1
n�n�1 for n � 4
• Remarkably, all the other interactions vanish simultaneously for
• The kinetic term vanishes for m2 = (d� 2)H2 (known)
non-diagonalizable terms mixing h and ⇡+L(dec)
� =X
n
cn(�n, d,H2/m2)L(n)
Gal
⇠ (@⇡)2✓(@2⇡)n�2
�(d+2)/2
◆n�2
de Rham et al 13
Helicity-0 mode disappears completely in the DL!Unique candidate theory for partially massless gravity.
Prompted lots of studies: the helicity-0 mode reappears at higher energy
Another limitation of the decoupling limit, in screening mechanisms
General Relativity
Modified Gravity
⇠
Modifications of GR on cosmological scales
butrecovery of GR in the Solar
System
Challenge:
Screening mechanisms are necessary,
which hide the additional dof.
Chameleon, Symmetron, Vainshtein screening etc
See talks by Babichev, Minazzoli and Mota
Vainshtein mechanism and decoupling limit
• Most studies of the Vainshtein mechanism consider static and spherically symmetric configurations, and rely on the decoupling limit
e.g. Koyama, Niz & Tasinato
• Particular model: minimal model, with ↵3 = �1
3,↵4 =
1
12has no interactions in the decoupling limit. No non-linearities, so no Vainshtein mechanism ?
• Not necessarily. Simply decoupling limit is not enough here.
• Need to study the structure of interactions at higher energies
Generic Static and spherically symmetric
RP, 1401.0947
Energy scales of interactions in the minimal model of massive gravity
⇣MPl m
2nn+1
⌘ n+13n+1 ����!
n!1
�MPl m
2�1/3MPl
⇤ =�MPlm
2�1/3
Study of structure of solutions: no recovery of GR
• Hard to decipher whether the Vainshtein mechanism is effective
• Tower of interactions, of energy
• Static spherically symmetric configurations can be misleading due to their high degree of symmetry
Vainshtein mechanism and decoupling limit
• Decoupling limit is not enough
+
Conclusion
• Identification of degrees of freedom is not known in massive gravity with a general reference metric and background metric
See talk by Bernard
• Some insights for maximally symmetric reference metrics: (A)dS
• Importance of studying screening mechanisms beyond static spherically symmetric configurations (small breaking in the Solar System, and walls an filaments in Large Scale Structure)
Time-dependence: Babichev et al 11Shape-dependence: Bloomfield et al 14, Bloomfield, Burrage & RP 15, to appear
• Identifying the new degrees of freedom in modified gravity can be non-trivial
de Rham, RP