Post on 31-Dec-2015
description
Massive Gravity and the Galileon
Claudia de RhamUniversité de Genève
Miami 2010Dec, 18th 2010
Work with Gregory Gabadadze, Lavinia Heisenberg,
David Pirtskhalava and Andrew Tolley
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 < 10-25 GeV,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 < 10-25 GeV,how about the graviton?
Could dark energy be due to an IR modification of gravity?
with no ghosts ... ? Deffayet, Dvali, Gabadadze, ‘01Koyama, ‘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 < 10-25 GeV,how about the graviton?
Could dark energy be due to an IR modification of gravity?
with no ghosts ... ?
Is the cosmological constant small ? OR
does it have a small effect on the geometry ?
Gravity modified in IRMassive gravity
A massless spin-2 field in 4d, has 2 dof
A massive spin-2 field, has 5 dof
Massive Gravity
In GR,
Degrees of freedom
Gauge invariance Constraints
In GR,
In massive gravity,
Degrees of freedom
Gauge invariance Constraints
Shift
- Shift does not propagate a constraint
remainingdegrees of
freedom
In GR,
In massive gravity,
Degrees of freedom
Gauge invariance Constraints
- Shift does not propagate a constraint- Non-linearly, lapse no longer propagates the Hamiltonian Constraint…
remainingdegrees of
freedom
Shift lapse Boulware & Deser,1972Creminelli et. al. hep-th/0505147
Avoiding the Ghost
Relying on a larger symmetry group,eg. 5d diff invariance, in models with extra dimensions (DGP, Cascading, ...
)
The Ghost can be avoided by
Massless spin-2 in 5d: 5 dof
Massive spin-2 in 4d: 5 dof (+ ghost…)
The graviton acquires a soft massresonance
Avoiding the Ghost
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
The Ghost can be avoided by
Typically, the ghost enters at the scale
Avoiding the Ghost
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
The Ghost can be avoided by
Typically, the ghost enters at the scale
That scale can be pushed
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
Graviton mass
Arkani-Hamed, Georgi, Schwartz, hep-th/0210184Creminelli et. al. hep-th/0505147
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
Graviton mass
In the decoupling limit,
with fixed,
Decoupling limit
pl
Which can be formally inverted such that
with
The ghost can usually be seen in the decoupling limit where the mass term is of the form
leading 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 form
leading 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 form
leading 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
with
The ghost can usually be seen in the decoupling limit where the mass term is of the form
leading 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
with
CdR, Gabadadze, Tolley, 1011.1232
That potential ensures that the problematic terms cancel in the decoupling limit
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
Ghost-free decoupling limit
The Galileon For a stable theory of massive gravity, the decoupling limit is
CdR, Gabadadze, 1007.0443
The interactions have 3 special features:1. They are local2. They possess a Shift
and a Galileon symmetry3. They have a well-defined Cauchy problem
(eom remain 2nd order)
Nicolis, Rattazzi and Trincherini, 0811.2197
Corresponds to the Galileon family of interactions
The BD ghost can be pushedbeyond the scale L3
Coupling to matter
Back to the BD ghost…
In the ADM decomposition,
with
The lapse enters quadratically in the Hamiltonian,
Boulware & Deser,1972Creminelli 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,1972Creminelli 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:
1. Exhibits the Galileon interactions in the decoupling limit ( has no ghost in the decoupling limit)
2. 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
3. Whether or not the constraint propagates is yet unknown. secondary constraint ?
4. 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
Arkani-Hamed et. al., ‘02Dvali, Hofmann & Khoury, ‘07
k: 4d momentum
The degravitation mechanism is a causal process.
1/mH2
time
time
Phase transitionL
Degravitation
The degravitation mechanism is a causal process.
1/mH2
time
time
Phase transitionL
Degravitation
In Massive gravity,
Degravitation
Degravitation
Screening the CC
1/mH2
time
time
l
Relaxes towards a flat geometry even with a large CC
Dark Energy
Screening the CC
Relaxes towards a flat geometry even with a large CC
Self-acceleration
Source the late time acceleration
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)