Massive Gravity and the Galileon

Miami 2010 Dec, 18th2010 Massive Gravity and the Galileon Work with Gregory Gabadadze, Lavinia Heisenberg, David Pirtskhalava and Andrew Tolley Claudia de Rham Université de Genève

Why Massive Gravity ? • Phenomenology • Self-acceleration • C.C. Problem

Why Massive Gravity ? • Phenomenology • Self-acceleration • C.C. Problem what are the theoretical and observational bounds on gravity in the IR ? mass of the photon is bounded to mg < 10-25GeV, how about the graviton?

Why Massive Gravity ? • Phenomenology • Self-acceleration • C.C. Problem what are the theoretical and observational bounds on gravity in the IR ? mass of the photon is bounded to mg < 10-25GeV, how about the graviton? Could dark energy be due to an IR modification of gravity? with no ghosts ... ? Deffayet, Dvali, Gabadadze, ‘01 Koyama, ‘05

Why Massive Gravity ? • Phenomenology • Self-acceleration • C.C. Problem what are the theoretical and observational bounds on gravity in the IR ? mass of the photon is bounded to mg < 10-25GeV, how about the graviton? Could dark energy be due to an IR modification of gravity? with no ghosts ... ? Is the cosmological constant small ? ORdoes it have a small effect on the geometry ? Gravity modified in IR Massive gravity

Massive Gravity • A massless spin-2 field in 4d, has 2 dof • A massive spin-2 field, has 5 dof

Degrees of freedom • In GR, Gauge invariance Constraints

Degrees of freedom • In GR, • In massive gravity, Gauge invariance Constraints - Shift does not propagate a constraint remainingdegrees of freedom Shift

Degrees of freedom • In GR, • In massive gravity, Gauge invariance Constraints - Shift does not propagate a constraint - Non-linearly, lapse no longer propagates the Hamiltonian Constraint… remainingdegrees of freedom Boulware & Deser,1972 Creminelli et. al. hep-th/0505147 Shift lapse

Avoiding the Ghost The Ghost can be avoided by • Relying on a larger symmetry group,eg. 5d diff invariance, in models with extra dimensions (DGP, Cascading, ... ) Massive spin-2 in 4d: 5 dof (+ ghost…) Massless spin-2 in 5d: 5 dof The graviton acquires a soft mass resonance

Avoiding the Ghost The Ghost can be avoided by • Relying on a larger symmetry group,eg. 5d diff invariance, in models with extra dimensions (DGP, Cascading, ... ) • Pushing the ghost above an acceptable cutoff scale Typically, the ghost enters at the scale

Avoiding the Ghost The Ghost can be avoided by • Relying on a larger symmetry group,eg. 5d diff invariance, in models with extra dimensions (DGP, Cascading, ... ) • Pushing the ghost above an acceptable cutoff scale Typically, the ghost enters at the scale That scale can be pushed

Graviton mass • To give the graviton a mass, include the interactions • Mass for the fluctuations around flat space-time

Graviton mass • To give the graviton a mass, include the interactions • Mass for the fluctuations around flat space-time Arkani-Hamed, Georgi, Schwartz, hep-th/0210184 Creminelli et. al. hep-th/0505147

Graviton mass • To give the graviton a mass, include the interactions • Mass for the fluctuations around flat space-time

Decoupling limit pl • In the decoupling limit,with fixed, • Which can be formally inverted such thatwith

Decoupling limit • The ghost can usually be seen in the decoupling limit where the mass term is of the formleading to higher order eom • It seems a formidable task to remove these terms to all order in the decoupling limit.

Decoupling limit • The ghost can usually be seen in the decoupling limit where the mass term is of the formleading to higher order eom • But we can attack the problem by the other end: starting with what we want in the decoupling limit

Decoupling limit • The ghost can usually be seen in the decoupling limit where the mass term is of the formleading to higher order eom • But we can attack the problem by the other end: starting with what we want in the decoupling limit with

Decoupling limit • The ghost can usually be seen in the decoupling limit where the mass term is of the formleading to higher order eom • But we can attack the problem by the other end: starting with what we want in the decoupling limit with CdR, Gabadadze, Tolley, 1011.1232

Decoupling limit • That potential ensures that the problematic terms cancel in the decoupling limit

Ghost-free decoupling limit • In the decoupling limit (keeping fixed)with

Ghost-free decoupling limit • In the decoupling limit (keeping fixed) • The Bianchi identity requires

Ghost-free decoupling limit • In the decoupling limit (keeping fixed) • The Bianchi identity requires • The decoupling limit stops at 2nd order.

Ghost-free decoupling limit • In the decoupling limit (keeping fixed) • The Bianchi identity requires • The decoupling limit stops at 2nd order. • are at most 2nd order in derivative • These mixings can be removed by a local field redefinition

The Galileon • For a stable theory of massive gravity, the decoupling limit is • The interactions have 3 special features: The BD ghost can be pushedbeyond the scale L3 They are local They possess a Shift and a Galileonsymmetry They have a well-defined Cauchy problem(eom remain 2nd order) • Corresponds to the Galileon family of interactions Coupling to matter CdR, Gabadadze, 1007.0443 Nicolis, Rattazzi and Trincherini, 0811.2197

Back to the BD ghost… • In the ADM decomposition, • with • The lapse enters quadratically in the Hamiltonian, Boulware & Deser,1972 Creminelli et. al. hep-th/0505147

Back to the BD ghost… • In the ADM decomposition, • with • The lapse enters quadratically in the Hamiltonian, • Does it really mean that the constraint is lost ? Boulware & Deser,1972 Creminelli et. al. hep-th/0505147

Back to the BD ghost… • In the ADM decomposition, • with • The lapse enters quadratically in the Hamiltonian, • Does it really mean that the constraint is lost ? • The constraint is manifest after integrating over the shift • This can be shown - at least up to 4th order in perturbations - completely non-linearly in simplified cases

Massive gravity - Summary We can construct an explicit theory of massive gravity which: Exhibits the Galileon interactions in the decoupling limit (has no ghost in the decoupling limit) Propagates a constraint at least up to 4th order in perturbations (does not excite the 6th BD mode to that order) and indicates that the same remains true to all orders Whether or not the constraint propagates is yet unknown. secondary constraint ? Symmetry ??? CdR, Gabadadze, Tolley, in progress…

Consequences for Cosmology

Degravitation • From naturalness considerations, we expect a vacuum energy of the order of the cutoff scale (Planck scale). • But observations tell us

Degravitation • From naturalness considerations, we expect a vacuum energy of the order of the cutoff scale (Planck scale). • But observations tell us • Idea behind degravitation: Gravity modified on large distances such that the vacuum energy gravitates more weakly k: 4d momentum Arkani-Hamed et. al., ‘02 Dvali, Hofmann & Khoury, ‘07

Degravitation L Phase transition • The degravitation mechanism is a causal process. time H2 1/m time

Degravitation In Massive gravity,

Degravitation l time H2 Screening the CC 1/m time Relaxes towards a flat geometry even with a large CC

Dark Energy Screening the CC Self-acceleration Source the late time acceleration Relaxes towards a flat geometry even with a large CC

Dark Energy Screening the CC Self-acceleration • Which branch is possible depends on parameters • Branches are stable and ghost-free (unlike self-accelerating branch of DGP) • In the screening case, solar system tests involve a max CC to be screened. CdR, Gabadadze, Heisenberg, Pirtskhalava, 1010.1780

Summary • Galileon interactions arise naturally- in braneworlds with induced curvature (soft mass gravity) - in hard massive gravity with no ghosts in the dec. limit • The Galileon can play a crucial role in (stable) models of self-acceleration… • …or provide a framework for the study of degravitation • On different scales, it can provide a radiatively stablemodel of inflation leading to potentially large nG... (Cf. Andrew’s talk - Sunday)

Massive Gravity and the Galileon