Massive Gravity and the Galileon

Massive Gravity and the Galileon
The Return of de Sitter March, 7th 2011 Massive Gravity and the Galileon Thank for the invitation, it’s a pleasure to be here. Today I talk about the 2 hierarchy problems that exist at the interface between particle physics and cosmology, more precisely when putting together the 3 standard forces with gravity and show how extra dimensions, and in particular 2 or more large extra dimensions provide very fruitful new directions to tackle them Work with Gregory Gabadadze 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 < GeV, how about the graviton?

what are the theoretical and observational bounds on gravity in the IR ? mass of the photon is bounded to mg < GeV, how about the graviton? Could dark energy be due to an IR modification of gravity? with no ghosts ... ? Deffayet, Dvali, Gabadadze, ‘01 Koyama, ‘05

what are the theoretical and observational bounds on gravity in the IR ? mass of the photon is bounded to mg < 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 ? Arkani-Hamed, Dimopoulos, Dvali &Gabadadze, ‘02 Dvali, Hofmann & Khoury, ‘07

Degravitation Can Gravity be modified at Large Distances such that the CC gravitates more weakly? One naïve way to modify gravity is to promote the Newton’s constant GN to a high pass filter operator, k: 4d momentum L-2 Arkani-Hamed, Dimopoulos, Dvali &Gabadadze, ‘02 Dvali, Hofmann & Khoury, ‘07

Massive Gravity Filtering gravity is effectively a theory of massive gravity To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like k2 m2

Worries How does it help with the tuning issue?
To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like

Tuning / Fine-tuning From naturalness considerations, we expect a vacuum energy of the order of the cutoff scale (Planck scale). But observations tell us For the degravitation mechanism to work, the mass of the graviton should be To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like

but technically natural
Tuning / Fine-tuning The amount of tuning is the same But the graviton mass is expected to remain stable against quantum corrections we recover a symmetry in the limit m To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like The theory is tuned but technically natural ‘t Hooft naturalness argument

but technically natural
Worries How does it help with the tuning issue? How many degrees of freedom is there in massive gravity ? Are they stable ??? The theory is tuned but technically natural To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like

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

Massive Gravity A massless spin-2 field in 4d, has 2 dof
A massive spin-2 field, has 5 dof To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like

Degrees of freedom In GR, Gauge invariance Constraints

Degrees of freedom In GR, Gauge invariance Constraints
In massive gravity, Gauge invariance Constraints To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like

Remaining degrees of freedom
In GR, In massive gravity, Gauge invariance Constraints To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like Remaining degrees of freedom

Boulware-Deser Ghost The ghost of massive gravity was originally pointed out by Boulware and Deser, using the ADM decomposition In GR, the lapse plays the role of a Lagrange multiplier, propagating a constraint Finally a last worry we might have in this theories of massive gravity is related to the stability of the theory. Typically if we analyze a theory with a hard mass (a constant mass) beyond the linearized level, or around an arbitrary background, there is yet another mode that comes in and the theory has actually 6 degrees of freedom. The way we can see that is by looking at the interactions and observe that they typically include terms which are higher order in derivative, and so we will need more than 2 initial conditions to solve the system completely. This extra freedom corresponds to an extra mode which typically comes in with a negative kinetic term and the theory is then unstable. That corresponds to the Boulware Deser ghost. Boulware & Deser,1972 Creminelli et. al. hep-th/

Boulware-Deser Ghost The ghost of massive gravity was originally pointed out by Boulware and Deser, using the ADM decomposition In GR, the lapse plays the role of a Lagrange multiplier, propagating a constraint If the graviton has a constant (hard) mass, the lapse and shift enter non-linearly Finally a last worry we might have in this theories of massive gravity is related to the stability of the theory. Typically if we analyze a theory with a hard mass (a constant mass) beyond the linearized level, or around an arbitrary background, there is yet another mode that comes in and the theory has actually 6 degrees of freedom. The way we can see that is by looking at the interactions and observe that they typically include terms which are higher order in derivative, and so we will need more than 2 initial conditions to solve the system completely. This extra freedom corresponds to an extra mode which typically comes in with a negative kinetic term and the theory is then unstable. That corresponds to the Boulware Deser ghost. Boulware & Deser,1972 Creminelli et. al. hep-th/

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…) Now for the theory to make sense, we need to make sure this ghost does not appear and we can follow 2 paths to do that. Either we rely on a larger symmetry group than 4d diff invariance to kill this extra degree of freedom, and this why going through extra dimensions will be useful. A massless graviton in 5d has 5 degrees of freedom and that corresponds exactly to the number of dof we want to have for a massive graviton in 4d. In this kind of setups, however, the graviton has a soft mass. 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 Now for the theory to make sense, we need to make sure this ghost does not appear and we can follow 2 paths to do that. Either we rely on a larger symmetry group than 4d diff invariance to kill this extra degree of freedom, and this why going through extra dimensions will be useful. A massless graviton in 5d has 5 degrees of freedom and that corresponds exactly to the number of dof we want to have for a massive graviton in 4d. In this kind of setups, however, the graviton has a soft mass.

The theory only makes sense up to a new cutoff 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 The theory only makes sense up to a new cutoff scale Now for the theory to make sense, we need to make sure this ghost does not appear and we can follow 2 paths to do that. Either we rely on a larger symmetry group than 4d diff invariance to kill this extra degree of freedom, and this why going through extra dimensions will be useful. A massless graviton in 5d has 5 degrees of freedom and that corresponds exactly to the number of dof we want to have for a massive graviton in 4d. In this kind of setups, however, the graviton has a soft mass.

Now for the theory to make sense, we need to make sure this ghost does not appear and we can follow 2 paths to do that. Either we rely on a larger symmetry group than 4d diff invariance to kill this extra degree of freedom, and this why going through extra dimensions will be useful. A massless graviton in 5d has 5 degrees of freedom and that corresponds exactly to the number of dof we want to have for a massive graviton in 4d. In this kind of setups, however, the graviton has a soft mass.

Is that what really happens in NMG ???
Now for the theory to make sense, we need to make sure this ghost does not appear and we can follow 2 paths to do that. Either we rely on a larger symmetry group than 4d diff invariance to kill this extra degree of freedom, and this why going through extra dimensions will be useful. A massless graviton in 5d has 5 degrees of freedom and that corresponds exactly to the number of dof we want to have for a massive graviton in 4d. In this kind of setups, however, the graviton has a soft mass. Is that what really happens in NMG ???

New Massive Gravity Has a hard mass the graviton is no resonance
Now for the theory to make sense, we need to make sure this ghost does not appear and we can follow 2 paths to do that. Either we rely on a larger symmetry group than 4d diff invariance to kill this extra degree of freedom, and this why going through extra dimensions will be useful. A massless graviton in 5d has 5 degrees of freedom and that corresponds exactly to the number of dof we want to have for a massive graviton in 4d. In this kind of setups, however, the graviton has a soft mass. Cf. Andrew’s talk Deser, Jackiw & Templeton, ’82, Bergshoe, Hohm & Townsend, ’09 ( )

New Massive Gravity Has a hard mass the graviton is no resonance
Yet the theory has no ghosts, and is valid till the Planck scale. What really happens ??? Now for the theory to make sense, we need to make sure this ghost does not appear and we can follow 2 paths to do that. Either we rely on a larger symmetry group than 4d diff invariance to kill this extra degree of freedom, and this why going through extra dimensions will be useful. A massless graviton in 5d has 5 degrees of freedom and that corresponds exactly to the number of dof we want to have for a massive graviton in 4d. In this kind of setups, however, the graviton has a soft mass. Cf. Andrew’s talk Deser, Jackiw & Templeton, ’82, Bergshoe, Hohm & Townsend, ’09 ( )

Hamiltonian Constraint in NMG
We know there are only 2 dof freedom there Now for the theory to make sense, we need to make sure this ghost does not appear and we can follow 2 paths to do that. Either we rely on a larger symmetry group than 4d diff invariance to kill this extra degree of freedom, and this why going through extra dimensions will be useful. A massless graviton in 5d has 5 degrees of freedom and that corresponds exactly to the number of dof we want to have for a massive graviton in 4d. In this kind of setups, however, the graviton has a soft mass. CdR, Gabadadze, Pirtskhalava, Tolley & Yavin, on its way

We know there are only 2 dof freedom there Yet if the Hamiltonian constraint is not present, we would have Now for the theory to make sense, we need to make sure this ghost does not appear and we can follow 2 paths to do that. Either we rely on a larger symmetry group than 4d diff invariance to kill this extra degree of freedom, and this why going through extra dimensions will be useful. A massless graviton in 5d has 5 degrees of freedom and that corresponds exactly to the number of dof we want to have for a massive graviton in 4d. In this kind of setups, however, the graviton has a soft mass. CdR, Gabadadze, Pirtskhalava, Tolley & Yavin, on its way

After integrating out over the usual “metric”, Now for the theory to make sense, we need to make sure this ghost does not appear and we can follow 2 paths to do that. Either we rely on a larger symmetry group than 4d diff invariance to kill this extra degree of freedom, and this why going through extra dimensions will be useful. A massless graviton in 5d has 5 degrees of freedom and that corresponds exactly to the number of dof we want to have for a massive graviton in 4d. In this kind of setups, however, the graviton has a soft mass.

After integrating out over the usual “metric”, Now for the theory to make sense, we need to make sure this ghost does not appear and we can follow 2 paths to do that. Either we rely on a larger symmetry group than 4d diff invariance to kill this extra degree of freedom, and this why going through extra dimensions will be useful. A massless graviton in 5d has 5 degrees of freedom and that corresponds exactly to the number of dof we want to have for a massive graviton in 4d. In this kind of setups, however, the graviton has a soft mass. With A naïve argument would imply the loss of the Ham Constraint

After integrating out over the usual “metric”, Now for the theory to make sense, we need to make sure this ghost does not appear and we can follow 2 paths to do that. Either we rely on a larger symmetry group than 4d diff invariance to kill this extra degree of freedom, and this why going through extra dimensions will be useful. A massless graviton in 5d has 5 degrees of freedom and that corresponds exactly to the number of dof we want to have for a massive graviton in 4d. In this kind of setups, however, the graviton has a soft mass. However whether or not there is a constraint, depends on With with A naïve argument would imply the loss of the Ham Constraint In NMG,

Toy Model As an instructive toy example, we can take
Now for the theory to make sense, we need to make sure this ghost does not appear and we can follow 2 paths to do that. Either we rely on a larger symmetry group than 4d diff invariance to kill this extra degree of freedom, and this why going through extra dimensions will be useful. A massless graviton in 5d has 5 degrees of freedom and that corresponds exactly to the number of dof we want to have for a massive graviton in 4d. In this kind of setups, however, the graviton has a soft mass.

Despite being non-linear in the lapse, there is a constraint: Now for the theory to make sense, we need to make sure this ghost does not appear and we can follow 2 paths to do that. Either we rely on a larger symmetry group than 4d diff invariance to kill this extra degree of freedom, and this why going through extra dimensions will be useful. A massless graviton in 5d has 5 degrees of freedom and that corresponds exactly to the number of dof we want to have for a massive graviton in 4d. In this kind of setups, however, the graviton has a soft mass.

We could have simply redefined the shift to make the constraint transparent: Now for the theory to make sense, we need to make sure this ghost does not appear and we can follow 2 paths to do that. Either we rely on a larger symmetry group than 4d diff invariance to kill this extra degree of freedom, and this why going through extra dimensions will be useful. A massless graviton in 5d has 5 degrees of freedom and that corresponds exactly to the number of dof we want to have for a massive graviton in 4d. In this kind of setups, however, the graviton has a soft mass.

BD ghost - Summary 4d massive gravity seems to suffer from an inevitable BD ghost If that argument were true, only a soft mass would make sense The same argument applied to 3d NMG, would imply the presence of a ghost … … which we know is absent. Let’s reopen the trial of 4d hard mass gravity… Can we construct a ghost-free theory for a massive spin-2 field in 4d ? (or arbitrary dims) What is interesting to notice in this model is that we are working in a regime where the interactions are important, but quantum corrections are still well under control, and so the theory is not breaking down. In particular the field can have a large velocity but the amplitude of the quantum corrections small and still remain under control. This is very important to be able to trust the theory at the scale \Lambda_\star. And actually this is very similar to what happens in another model which is to due this time with the very early Universe.

Graviton mass To give the graviton a mass, include the interactions
Mass for the fluctuations around flat space-time To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like

Mass for the fluctuations around flat space-time To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like Arkani-Hamed, Georgi, Schwartz, hep-th/ Creminelli et. al. hep-th/

Mass for the fluctuations around flat space-time To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like ghost ...

Mass for the fluctuations around flat space-time To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like

Decoupling limit In the decoupling limit, with fixed,
Which can be formally inverted such that with To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like

Ghost-free theory The mass term Has no ghosts in the decoupling limit:
with To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like CdR, Gabadadze, Tolley,

Ghost-free decoupling limit
In the decoupling limit (keeping fixed) with To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like

In the decoupling limit (keeping fixed) The Bianchi identity requires To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like

In the decoupling limit (keeping fixed) The Bianchi identity requires The decoupling limit stops at 2nd order. To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like

In the decoupling limit (keeping fixed) The Bianchi identity requires The decoupling limit stops at 2nd order. are at most 2nd order in derivative To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like NO GHOSTS in the 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 To see that, we can simply look at the linearized Einstein equation. If I omit the tensor structure for a second, the Einstein equation looks like

Galileon in disguise Before looking at how these interactions affect observations, let me emphasize 3 very important characteristics of this model. First of all the interactions are local, are derive from an action. Second they have a specific symmetry which is a shift symmetry inheritated from 5d diff invariance, And finally all the terms in the eom are at most (and actually exactly) 2nd order in derivative, which means that the system will have a well-defined cauchy problem and no ghost like instabilities.

The BD ghost can be pushed beyond the scale L3
Galileon in disguise For a stable theory of massive gravity, the decoupling limit is The interactions have 3 special features: The BD ghost can be pushed beyond the scale L3 They are local They possess a Shift and a Galileon symmetry They have a well-defined Cauchy problem (eom remain 2nd order) Before looking at how these interactions affect observations, let me emphasize 3 very important characteristics of this model. First of all the interactions are local, are derive from an action. Second they have a specific symmetry which is a shift symmetry inheritated from 5d diff invariance, And finally all the terms in the eom are at most (and actually exactly) 2nd order in derivative, which means that the system will have a well-defined cauchy problem and no ghost like instabilities. Corresponds to the Galileon family of interactions Coupling to matter CdR, Gabadadze, Nicolis, Rattazzi and Trincherini,

Back to the BD ghost… We now set unitary gauge, . In ADM split, with
The lapse enters quadratically in the Hamiltonian, What is interesting to notice in this model is that we are working in a regime where the interactions are important, but quantum corrections are still well under control, and so the theory is not breaking down. In particular the field can have a large velocity but the amplitude of the quantum corrections small and still remain under control. This is very important to be able to trust the theory at the scale \Lambda_\star. And actually this is very similar to what happens in another model which is to due this time with the very early Universe. Boulware & Deser,1972 Creminelli et. al. hep-th/

The lapse enters quadratically in the Hamiltonian, Does it really mean that the constraint is lost ? What is interesting to notice in this model is that we are working in a regime where the interactions are important, but quantum corrections are still well under control, and so the theory is not breaking down. In particular the field can have a large velocity but the amplitude of the quantum corrections small and still remain under control. This is very important to be able to trust the theory at the scale \Lambda_\star. And actually this is very similar to what happens in another model which is to due this time with the very early Universe. Boulware & Deser,1972 Creminelli et. al. hep-th/

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 What is interesting to notice in this model is that we are working in a regime where the interactions are important, but quantum corrections are still well under control, and so the theory is not breaking down. In particular the field can have a large velocity but the amplitude of the quantum corrections small and still remain under control. This is very important to be able to trust the theory at the scale \Lambda_\star. And actually this is very similar to what happens in another model which is to due this time with the very early Universe. This can be shown - at least up to 4th order in perturbations - completely non-linearly in simplified cases - in 2d - for conformally flat spatial metric

BD ghost: To be or not to be
In 2d, the Hamiltonian can be computed explicitly The Constraint is manifest after redefinition of the shift, What is interesting to notice in this model is that we are working in a regime where the interactions are important, but quantum corrections are still well under control, and so the theory is not breaking down. In particular the field can have a large velocity but the amplitude of the quantum corrections small and still remain under control. This is very important to be able to trust the theory at the scale \Lambda_\star. And actually this is very similar to what happens in another model which is to due this time with the very early Universe.

More Generally, with What is interesting to notice in this model is that we are working in a regime where the interactions are important, but quantum corrections are still well under control, and so the theory is not breaking down. In particular the field can have a large velocity but the amplitude of the quantum corrections small and still remain under control. This is very important to be able to trust the theory at the scale \Lambda_\star. And actually this is very similar to what happens in another model which is to due this time with the very early Universe.

More Generally, with If the spatial metric is conformally flat, we can use the ansatz What is interesting to notice in this model is that we are working in a regime where the interactions are important, but quantum corrections are still well under control, and so the theory is not breaking down. In particular the field can have a large velocity but the amplitude of the quantum corrections small and still remain under control. This is very important to be able to trust the theory at the scale \Lambda_\star. And actually this is very similar to what happens in another model which is to due this time with the very early Universe.

More Generally, with If the spatial metric is conformally flat, the mass term is simply The constraint is manifest ! What is interesting to notice in this model is that we are working in a regime where the interactions are important, but quantum corrections are still well under control, and so the theory is not breaking down. In particular the field can have a large velocity but the amplitude of the quantum corrections small and still remain under control. This is very important to be able to trust the theory at the scale \Lambda_\star. And actually this is very similar to what happens in another model which is to due this time with the very early Universe.

We can construct an explicit theory of massive gravity which:
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 perturbations (does not excite the 6th BD mode to that order) at least up to 4th order in and indicates that the same remains true to all orders to all orders for a conformally flat spatial metric Whether or not the constraint propagates is yet unknown. secondary constraint ? Symmetry ??? What is interesting to notice in this model is that we are working in a regime where the interactions are important, but quantum corrections are still well under control, and so the theory is not breaking down. In particular the field can have a large velocity but the amplitude of the quantum corrections small and still remain under control. This is very important to be able to trust the theory at the scale \Lambda_\star. And actually this is very similar to what happens in another model which is to due this time with the very early Universe. CdR, Gabadadze, Tolley, in progress…

Massive Gravity and the Galileon

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