Claudia de Rham - Crete Center for Theoretical Physics

The BD Ghost inmassive GravityCrete Center for Theoretical PhysicsApril, 15 th 2011Claudia de RhamWork withGregory Gabadadze and Andrew Tolley

Why Massive Gravity ?PhenomenologySelf-accelerationC.C. Problem

Why Massive Gravity ?Phenomenologywhat are the theoretical and observational bounds on gravity in the IR ?mass of the photon is bounded to m g < 10 -25 GeV,how about the graviton?Can we construct a consistent theory fora massive spin-2 field ?

Why Massive Gravity ?Phenomenologywhat are the theoretical and observational bounds on gravity in the IR ?mass of the photon is bounded to m g < 10 -25 GeV,how about the graviton?Self-accelerationCould dark energy be due to an IR modification of gravity?with no ghosts ... ?C.C. ProblemDeffayet, Dvali, Gabadadze, „01Koyama, „05

Why Massive Gravity ?Phenomenologywhat are the theoretical and observational bounds on gravity in the IR ?mass of the photon is bounded to m g < 10 -25 GeV,how about the graviton?Self-accelerationCould dark energy be due to an IR modification of gravity?with no ghosts ... ?C.C. ProblemIs the cosmological constant small ?ORdoes it have a small effect on the geometry ?Arkani-Hamed, Dimopoulos, Dvali &Gabadadze, „02Dvali, Hofmann & Khoury, „07

Massive GravityA massless spin-2 field in 4d, has 2 dofA massive spin-2 field, has 5 dof

Strong Coupling5 th force constraints in the solar system imply that theextra degrees of freedom must be strongly coupled at ascaler Sr *

Boulware-Deser GhostNon-linearities are fundamental for the survival of the theory.But non-linearly, the theory seems to contain a ghost, whichhas been shown explicitly1. In the ADM formalism,counting constraints in Hamiltonian2. In the Stückelberg language,- In the decoupling limit (ghost scale ~ L)- At higher scales

1. ADM in GRThe ghost of massive gravity was originally pointed outby Boulware and Deser, using the ADM decompositionIn GR, both the lapse and shifts play the role of Lagrangemultipliers, propagating 4 constraints

1. ADM in GRThe ghost of massive gravity was originally pointed outby Boulware and Deser, using the ADM decompositionIn GR, the lapse and shifts play the role of Lagrange mult.symmetryconstraints

1. BD Ghost in ADMIn massive gravity, both the lapse and shifts enter non-linearlyThe Fierz-Pauli combination, ensures that the lapse remainslinear at the quadratic order,

1. BD Ghost in ADMIn massive gravity, both the lapse and shifts enter non-linearlyThe Fierz-Pauli combination, ensures that the lapse remainslinear at the quadratic order, but not beyond…

1. BD Ghost in ADMIn massive gravity, both the lapse and shifts enter non-linearlyThere is no possible mass term for which the lapse remains aLagrange multiplierBoulware & Deser,1972Creminelli et. al. hep-th/0505147

1. BD Ghost in ADMIn massive gravity, both the lapse and shifts enter non-linearlyThere is no possible mass term for which the lapse remains aLagrange multipliersymmetry6 dof propagating non-linearlyconstraintsBoulware & Deser,1972Creminelli et. al. hep-th/0505147

1. BD Ghost in ADMIn massive gravity, both the lapse and shifts enter non-linearlyThere is no possible mass term for which the lapse remains aLagrange multipliersymmetry6 dof propagating non-linearlyconstraintsBoulware & Deser,1972Creminelli et. al. hep-th/05051475+1 dofghost ...

1. BD Ghost in ADMIn massive gravity, both the lapse and shifts enter non-linearlyIn massive gravity,both the lapse and shifts enter non-linearlyIs that really the right criteria ???symmetry6 dof propagating non-linearly

1. BD Ghost in ADMWhether or not there is a constraint,simply depends on the Hessian,

Toy ModelAs an instructive toy example, we can take2

Toy ModelAs an instructive toy example, we can take2Despite being non-linear in the lapse, there is a constraint:

Toy ModelAs an instructive toy example, we can take2We could have simply redefined the shiftto make the constraint transparent:2

Boulware-Deser GhostNon-linearities are fundamental for the survival of the theory.But non-linearly, the theory seems to contain a ghost, whichhas been shown explicitly1. In the ADM formalism,counting constraints in Hamiltonian2. In the Stückelberg language,- In the decoupling limit (ghost scale ~ L)- At higher scales

2. Stückelberg languageTo give the graviton a mass, include the interactionsMass for the fluctuations around flat space-time

2. Stückelberg languageTo give the graviton a mass, include the interactionsMass for the fluctuations around flat space-timeghost ...

2. Stückelberg languageTo give the graviton a mass, include the interactionsMass for the fluctuations around flat space-time

Decoupling limitIn the decoupling limit,withfixed,The ghost can be avoided in that limit,if is a total derivative

Decoupling limitis a total derivative, for instance if

Decoupling limitis a total derivative, for instance ifIn the decoupling limit,orwith

Ghost-free theoryThe mass termwithHas no ghosts in the decoupling limit:CdR, Gabadadze, Tolley, 1011.1232

Ghost-free decoupling limitIn the decoupling limit (keepingfixed)with

Ghost-free decoupling limitIn the decoupling limit (keepingfixed)The Bianchi identity requiresThe decoupling limit stops at 2 nd order.are at most 2 nd order in derivativeNO GHOSTS in the decoupling limit

Ghost-free decoupling limitIn the decoupling limit (keepingfixed)The Bianchi identity requiresThe decoupling limit stops at 2 nd order.are at most 2 nd order in derivativeThese mixings can be removed by a local fieldredefinition

Galileon in disguise

Galileon in disguiseFor a stable theory of massive gravity, the decouplinglimit isThe interactions have3 special features:1. They are local2. They possess a Shiftand a Galileon symmetry3. They have a well-defined Cauchy problem(eom remain 2 nd order)Corresponds to the Galileon family of interactionsCoupling to matter

Galileon in disguiseFor a stable theory of massive gravity, the decouplinglimit isThe interactions have3 special features:1. They are local2. They possess a Shiftand beyond a Galileonthe symmetry scale L 33. They have a well-defined Cauchy problem(eom remain 2 nd order)The BD ghost can be pushedCorresponds to the Galileon family of interactionsCoupling to matter

Ghost-free theoryThere exist actually a 2-parameter family of theories:withLeading to the entire family of ghost-free Galileoninteractions in the decoupling limit.CdR, Gabadadze, Tolley, 1011.1232

Ghost-free theoryThere exist actually a 2-parameter family of theories:withLeading to the entire family of ghost-free Galileoninteractions in the decoupling limit.Is that enough ???

Beyond de decoupling limitConsider a 2d toy-model,for simplicity we work in the LIF,Both and propagate dynamical equations…However they are not independentThere is stilla constraint !

Back to the BD ghost…We now set unitary gauge,. In ADM split,withThe lapse enters quadratically in the Hamiltonian,Does it really mean that the constraint is lost ?Boulware & Deser,1972Creminelli et. al. hep-th/0505147

Back to the BD ghost…The constraint is manifest after integrating over the shiftThis can be shown- at least up to 4 th order in perturbations- completely non-linearly in simplified cases- in 2d- for conformally flat spatial metric

Summary of BD ghostWe can construct an explicit theory of massive gravity which:1. Exhibits the Galileon interactions in the decouplinglimit ( has no ghost in the decoupling limit)2. Propagates a constraint perturbations (does not excitethe 6 th BD mode to that order)at least up to 4 th order in and indicates that the same remainstrue to all ordersto all orders for a conformally flat spatial metric3. Whether or not the constraint propagates is yetunknown. secondary constraint ?4. Symmetry ???CdR, Gabadadze, Tolley, in progress…

Consequences for Cosmology1. For late time acceleration2. Inflation ?

C.C. Problem

DegravitationScreening the CCltimeH 21/mtimeCould relax towards a flatgeometry even with a large CC

Dark EnergyScreening the CCSelf-accelerationCould relax towards a flatgeometry even with a large CCSource the late time acceleration

Dark EnergyScreening the CCSelf-accelerationWhich branch is possible depends on parametersBranches are stable and ghost-free(unlike self-accelerating branch of DGP)In the screening case, solar systemtests involve a max CC to be screened.CdR, Gabadadze, Heisenberg, Pirtskhalava, 1010.1780

Consequences for Cosmology1. For late time acceleration2. Inflation ?

EFT and relevant operatorsHigher derivative interactions are essential for the viabilityof this class of models.Within the solar system, p reaches the scale L * ,yet, we are still within the regime of validity of the theoryVainshtein, Phys. Lett. B 39 (1972) 393Babichev, Deffayet & Ziour, 0901.0393Luty, Porrati, Rattazzi hep-th/0303116Nicolis & Rattazzi, hep-th/0404159

EFT and relevant operatorsHigher derivative interactions are essential for the viabilityof this class of models.Within the solar system, p reaches the scale L * ,yet, we are still within the regime of validity of the theoryThe breakdown of the EFT is not measured byby itself gradients should be smallbutSo we can trust a regime whereas long asLuty, Porrati, Rattazzi hep-th/0303116Nicolis & Rattazzi, hep-th/0404159

EFT and relevant operatorsCan we use these ideas to builda radiatively stablemodel of inflation ?

Galileon InflationModel of Inflation grounded on the Galileon Symmetry

Galileon InflationModel of Inflation grounded on the Galileon SymmetryShift symmetry guarantees the conservation of ζ outsidethe horizon time independent power spectrumNon-renormalization theorem allows us to consider theseinteractions to be large, without breaking the regime ofvalidity of EFT.Nicolis & Rattazzi, hep-th/0404159

Galileon InflationModel of Inflation grounded on the Galileon SymmetryShift symmetry guarantees the conservation of ζ outsidethe horizon time independent power spectrumNon-renormalization theorem allows us to consider theseinteractions to be large, without breaking the regime ofvalidity of EFT.Remains true with a mass term

Strong couplingWhen the interactions are important,the inflationary phase satisfiesAnd perturbations are given by

Strong couplingWhen the interactions are important,the inflationary phase satisfiesAnd perturbations are given bydim - 6 dim - 7

Non-GaussianitiesThe operator 7 can be of the same order as the others if thedim-6 operators are suppressed by additional H/L.This suppression is stable thanks to the Galileon symmetry.Potentially “large” nG, but with no specific shape.Burrage, CdR, Seery, Tolley, 1009.2497

New shapes of non-GaussianityThe situation is different, if the leading interactions aretuned to vanishLeading interactions then arise fromtermsnG can rely on dim-9 operatorsspecific shapes nG:- 2 operators lead to ~ equilateral triangles- 1 to flattened isocele triangle.Creminelli, d‟Amicob, Musso, Norena and Trincherini, 1011.3004

SummaryGalileon interactions arise naturally- in braneworlds with induced curvature (soft mass gravity)- in hard massive gravity with no ghosts in the dec. limitThe Galileon can play a crucial role in (stable) models of selfacceleration……or provide a framework for the study of degravitationOn different scales, it can provide a radiatively stablemodel of inflation leading to potentially large nG...... Similar in spirit than DBI, but with different signatures