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