Massive Gravity on de Sitter

Galileons & Massive GravityClaudia de Rham Dec. 18th 2013

Why Modify Gravity in the IR ?Late time acceleration& CC problemFirst signs of the breakdown of GRon cosmological scales ?MG from Extra DimensionsBest way to obtain a sensible theory of Modified Gravity is to start with General Relativity5d GravitySoft massive Gravity(DGP)Hard massive GravityGhost-free mGRGalileonsInfinite extra dimensionConfine Gravity on a braneDGP model

5d Gravity4d gravityDvali Gabadadze Porrati, 2000Infinite extra dimensionConfine Gravity on a braneDGP model

5d GravityFinite-size extra dimensionKK or deconstructionHard massive Gravity(or multi-gravity)

4d gravity5d Gravity

MG from Extra DimensionsIn both cases (DGP and hard mass gravity), 5 dof from massless spin-2 in 5d

5 dof for massive spin-2 in 4d (no BD ghost)ensuresMG from Extra DimensionsIn both cases (DGP and hard mass gravity), 5 dof from massless spin-2 in 5d

5 dof for massive spin-2 in 4d (no BD ghost)

In both cases, the scalar dof behaves as a Galileon scalar field (in some limit) Screened via Vainshtein mechanismensuresDeconstructing GravityStart with 5d gravity

Discretize it along the extra dimension (after gauge fixing)

y

Deconstruction: Arkani-Hamed, Cohen & Georgi, PRL 86, 4757 (2001)5d Gravity

Deconstructing GravityStart with 5d gravity

Discretize it along the extra dimension (after gauge fixing)

y

5d Gravity

Deconstructing GravityStart with 5d gravity

Discretize it along the extra dimension (after gauge fixing)

y

5d Gravity

Ghost-free Massive GravityDeconstructing 5d GR naturally leads to a 4d Ghost-free multi-gravity theory

Ghost-free Massive GravityDeconstructing 5d GR naturally leads to a 4d Ghost-free multi-gravity theoryCdR, Matas & Tolley, 1308.4136

Ghost-free Massive GravityDeconstructing 5d GR in a gauge-fixed way (5d lapse=1) leads to Ghost-free massive gravity or multi-gravity

Truncated theory is consistent

Deconstruction is EQUIVALENT to KK (after non-linear field redefinition)

CdR, Matas & Tolley, 1308.4136 mass

Ghost-free Massive GravityStructure of mass term is essential to avoid BD ghostBoulware & Deser, PRD 6, 3368 (1972)CdR & Gabadadze, PRD 82, 044020 (2010)CdR, Gabadadze & Tolley, PRL 106, 231101 (2011)

Ghost-free Massive GravityStructure of mass term is essential to avoid BD ghost

Helicity-0 mode appears as a Galileon (in some limit)Boulware & Deser, PRD 6, 3368 (1972)CdR & Gabadadze, PRD 82, 044020 (2010)CdR, Gabadadze & Tolley, PRL 106, 231101 (2011)

After StckelbergisationGhost-free Massive GravityStructure of mass term is essential to avoid BD ghost

Helicity-0 mode appears as a Galileon (in some limit)Boulware & Deser, PRD 6, 3368 (1972)CdR & Gabadadze, PRD 82, 044020 (2010)CdR, Gabadadze & Tolley, PRL 106, 231101 (2011)

After StckelbergisationOpen Questions in MGViable Cosmological Solutions & Phenomenology

Quantum Corrections- is the small mass technically natural ? - how does the special structure of the potential gets affected by QC ?

Superluminalities

Strong Coupling

Tuning of the MassWe typically consider the graviton mass to be

same tuning as for the CC problem !

Tuning of the MassWe typically consider the graviton mass to be

same tuning as for the CC problem !

However the graviton mass is stable against quantum corrections

Technically natural tuning, unlike the tuning of the CC to zeroNon-renormalizationIn the DL, we recover a Galileon theory which enjoys a non-renormalization theorem

graviton mass does not get renormalized in the DL

Beyond the DL, the graviton mass gets renormalized by

Technically natural tuning, unlike the tuning of the CC to zeroCdR, Gabadadze, Heisenberg & Pirtskhalava, 1212.4128 Open Questions in MGViable Cosmological Solutions & Phenomenology

Quantum Corrections- is the small mass technically natural ? - how does the special structure of the potential gets affected by QC ?

Superluminalities

Strong Coupling

See CdR, Heisenberg & Ribeiro, 1307.7169Galileons in trouble (?)

Perturbations about a background

Galileons in trouble (?)Change the kinetic structure !!!

Fluctuations see a different effective metric

Fluctuations can propagate faster than light !

Adams, Arkani-Hamed, Dubovsky, Nicolis & Rattazzi, JHEP 0610 (2006) 014Hinterbichler, Nicolis & Porrati, JHEP 0909, 089 (2009)Burrage, CdR, Heisenberg & Tolley, JCAP 1207, 004 (2012)Deser & Waldron, PRL 110, 111101 (2013)

Galileons in trouble (?)Strong coupling scale gets redressed:

At the surface of the Earth(only from the effect of the Earth)Galileons in trouble (?)Strong coupling scale gets redressed:

After that ? break tree-level unitarity

At the surface of the Earth(only from the effect of the Earth)Galileons in trouble (?)Strong Coupling Issues & Superluminalities

Ubiquitous to Galileon theories& to massive gravity (?)This was the state of Affairs till the end of 2012 2012 Breakthrough July: Higgs Boson discoveredAugust: Curiosity Lands on MarsSummer: Quantum Teleportation Distance Record Broken of more than 50 milesOctober:Earths Exoplanet Twin discovered orbiting Alpha Centauri BDecember 31st 2012 @ 20:55:59 GMT:

2012 Breakthrough July: Higgs Boson discoveredAugust: Curiosity Lands on MarsSummer: Quantum Teleportation Distance Record Broken of more than 50 milesOctober:Earths Exoplanet Twin discovered orbiting Alpha Centauri BDecember 31st 2012 @ 20:55:59 GMT:

Missed the final breakthroughbecause

December 31st 2012, 20:55:59 GMTDual Galileon

In 3dDual GalileonCurtright & Fairlie, 1212.6972 CdR, Fasiello & Tolley, 1308.2702+ in preparation

In 3d

Dual GalileonCurtright & Fairlie, 1212.6972 CdR, Fasiello & Tolley, 1308.2702+ in preparation

In 3d

Dual Galileon

Large x no VainshteinSmall xVainshtein

Large

Large

Small

Vainshtein region inoriginal theory mapsto weak region in dual theory.

Dual Galileon

Canonical normalization of r:

Dual Galileon

Canonical normalization of r:

Strong couplingWeak coupling

Strong Coupling in the Dual theoryThe dual theory is under control as long as

Back to the original theory, this implies

For the EarthCdR, Fasiello & Tolley, 1308.2702+ in preparation

Other eg. of duality

In 4dOther eg. of duality

In 4dOther eg. of dualityThe original theory admits superluminal propagation

in the vacuum ! (ie. no matter or other sources)

Other eg. of dualityThe original theory admits superluminal propagation

in the vacuum ! (ie. no matter or other sources)

plane wave solutions

Other eg. of dualityThe original theory admits superluminal propagation

in the vacuum ! (ie. no matter or other sources)

plane wave solutions

Speed of fluctuations:

Superluminal propagation for

Other eg. of dualityThe original theory admits superluminal propagation

Yet the dual theory is just free !

Trivially causal, unitary, UV complete,

Group vs Front velocityIn the original theory,

What is SL is the GROUP velocity

What counts for causality is the FRONT velocity

here

Group vs Front velocityIn the original theory,

What is SL is the GROUP velocity

What counts for causality is the FRONT velocity

Duality implies that the front velocity is LUMINAL

No acausalityMassive 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 problem

We now have the theoretical formalism to describe a stable theory of massive gravity

It behaves a scalar-tensor Galileon theory in some limit, hand in hand with a Vainshtein mechanismOutlookGalileon theories are plagued with bothSuperluminalities & Strong Issues

These issues could be tackled by a field redefinition DUAL GALILEON ?

Massive Gravity requires tuningThe graviton mass is technically naturalAt 1-loop the potential does not get strongly distabilizedAt higher loops, it still need to be exploredOutlook