Cascading gravity and de gravitation Claudia de Rham Perimeter Institute/McMaster Miami 2008 Dec, 18 th 2008.

Cascading gravityCascading gravityandand

dedegravitatiogravitationn

Claudia de RhamClaudia de RhamPerimeter Institute/McMasterPerimeter Institute/McMaster

Miami 2008Dec, 18th 2008

Based on work on collaboration withBased on work on collaboration with

“Cascading Gravity and Degravitation”, JCAP02(2008)011

“Cascading DGP”, PRL 100 (251603), 2008

“Tensing the ghost in 6D cascading gravity”, to appear

“Towards Cosmology in theories of massive gravity”, to appear

• Stefan Hofmann, Nordita, Stockholm• Justin Khoury, Perimeter, Waterloo• Andrew Tolley, Perimeter, Waterloo• Oriol Pujolas, CERN• Gia Dvali, NYU, New York &CERN• Michele Redi, EPFL, Lausanne

The c.c. problemThe c.c. problem The current acceleration of the The current acceleration of the

Universe is well described by a c.c., Universe is well described by a c.c., /M/Mplpl

22, , with with (10(10-2-2 eV) eV)44 whilewhile mmee

44// ~ 10 ~ 103636 and and MMplpl44// ~ 10 ~ 10120120

Why is the vacuum energy Why is the vacuum energy so smallso small when when quantum effects lead to much bigger quantum effects lead to much bigger corrections?corrections?

The c.c. problemThe c.c. problem The current acceleration of the The current acceleration of the

Universe is well described by a c.c., Universe is well described by a c.c., /M/Mplpl

22, , with with (10(10-2-2 eV) eV)44 whilewhile mmee

44// ~ 10 ~ 103636 and and MMplpl44// ~ 10 ~ 10120120

Why is the vacuum energy Why is the vacuum energy so smallso small when when quantum effects lead to much bigger corrections?quantum effects lead to much bigger corrections?

Is the vacuum energy actually small or does it Is the vacuum energy actually small or does it simply simply gravitate very littlegravitate very little ? ?

idea behind degravitationDvali, Hofmann&Khoury, hep-th/0703027

Small c.c. / weakly Small c.c. / weakly gravitatinggravitating

In GR, gravity is mediated by a In GR, gravity is mediated by a massless massless spin-2spin-2 particle and particle and gauge invariancegauge invariance makes both questions equivalent. makes both questions equivalent. (universality of graviton coupling)(universality of graviton coupling)

If gravity was mediated by an effectively If gravity was mediated by an effectively massive gravitonmassive graviton, gravity would be , gravity would be weaker in the IRweaker in the IR the vacuum energy (and other IR the vacuum energy (and other IR sources) would gravitate differently sources) would gravitate differently

Dvali, Hofmann&Khoury, hep-th/0703027

Filtering gravityFiltering gravity

In Einstein’s gravity, the c.c. is In Einstein’s gravity, the c.c. is bound to gravitate as any other bound to gravitate as any other sourcesource

The idea behind degravitation is to The idea behind degravitation is to promote the Newton’s constant Gpromote the Newton’s constant GNN to a to a filter operatorfilter operator, ,

Filtering gravityFiltering gravity

At short wavelengths compared to L, if At short wavelengths compared to L, if >0 >0 GGNN G G00

NN there is no filter and sources there is no filter and sources gravitate normallygravitate normally,,

While at long distances, GWhile at long distances, GNN 0, so 0, so sources with large wavelengths, (such sources with large wavelengths, (such as the c.c.) are as the c.c.) are filtered out and filtered out and effectively gravitate very weaklyeffectively gravitate very weakly. .

with

Filtering and graviton Filtering and graviton massmass

As such, the theory would not satisfy As such, the theory would not satisfy the the Bianchi identityBianchi identity,,

This cannot represent a consistent This cannot represent a consistent theory of massless spin-2 gravitons theory of massless spin-2 gravitons (with only 2 degrees of freedom)(with only 2 degrees of freedom)

Instead the theory should be Instead the theory should be understood as the understood as the limit of a theory of limit of a theory of massive gravitymassive gravity, , with mass ~1/L.with mass ~1/L.

Filtering and graviton Filtering and graviton massmass

Any degravitating (filter) theory Any degravitating (filter) theory must reduce at the linearized level must reduce at the linearized level to a theory of to a theory of massive gravitymassive gravity

Corresponding to the filter theory Corresponding to the filter theory

Filtering and graviton Filtering and graviton massmass

To be a satisfying ghost-free To be a satisfying ghost-free degravitating theory, the mass should degravitating theory, the mass should satisfysatisfy

with 0 with 0 < 1.< 1. == corresponds to the effective corresponds to the effective

4d theory arising from the 5d 4d theory arising from the 5d DGP modelDGP model..

R(4)

Dvali, Gabadadze & Porrati, hep-th/0005016

R(5)

DGP – eg. of massive DGP – eg. of massive gravitygravity

Extra dof arise from 5d nature of Extra dof arise from 5d nature of theory.theory.

We live in a (3+1)-brane embedded We live in a (3+1)-brane embedded in an infinite flat extra dimensionin an infinite flat extra dimension

Dvali, Gabadadze & Porrati, hep-th/0005016

R(5)

DGP – eg. of massive DGP – eg. of massive gravitygravity

Extra dof arise from 5d nature of Extra dof arise from 5d nature of theory.theory.

We live in a (3+1)-brane embedded in We live in a (3+1)-brane embedded in an infinite flat extra dimensionan infinite flat extra dimension

In the UV, the 4d curvatureIn the UV, the 4d curvatureterm dominates, gravity looks 4dterm dominates, gravity looks 4d

In the IR, gravity is 5d.In the IR, gravity is 5d.Dvali, Gabadadze & Porrati, hep-th/0005016

R(4)

R(5)

DGP – eg. of massive DGP – eg. of massive gravitygravity

Effective 4d propagator for DGPEffective 4d propagator for DGP

This corresponds to a degravitating theory with This corresponds to a degravitating theory with =1/2 =1/2

with induced Friedmann eq.with induced Friedmann eq. =1/2 is too large ! Is there an extension with =1/2 is too large ! Is there an extension with

<1/2 ??? <1/2 ???

k: 4d momentum m5=M5

3/M42

Cf. Ghazal Geshnizjani ’s talk

Gravity in higher Gravity in higher dimensionsdimensions

For a given spectral representationFor a given spectral representation , , wwe have the “Newtonian potential”e have the “Newtonian potential”

In a In a (4+n)-dimensional spacetime(4+n)-dimensional spacetime, the , the gravitational gravitational

potential goes as potential goes as ie. ie.

If n=1 (DGP), in the IR If n=1 (DGP), in the IR GG~p~p-1-1 =1/2=1/2 If n=2, in the IR If n=2, in the IR GG~ log p~ log p =0=0 Any higher dim DGP model corresponds to Any higher dim DGP model corresponds to =0=0..

(s)~s(s)~sn/2-1n/2-1

Higher-codimension Higher-codimension sourcessources

Cod-1 or pure tension cod-2 are the only Cod-1 or pure tension cod-2 are the only meaningful distributional sources. meaningful distributional sources. (Geroch&Traschen)(Geroch&Traschen)

Arbitrary matter on cod-2 and higher Arbitrary matter on cod-2 and higher distributions lead to metric divergences distributions lead to metric divergences on the defect.on the defect.

The defect should be regularized. The defect should be regularized.Geroch & Traschen, 1987

Cod-2 sourcesCod-2 sources Cod-1 exampleCod-1 example

Cod-2 divergencesCod-2 divergences

Regularizing Cod-2 Regularizing Cod-2 sourcessources

If we had insteadIf we had instead

the solution is the solution is regularregular (easier to see in (easier to see in momentum space)momentum space)

The new kinetic term plays the role of a The new kinetic term plays the role of a regulatorregulator. Effectively represents a . Effectively represents a brane brane localized kinetic termlocalized kinetic term. .

CascadinCascading g

gravitygravity

Cod-2 cascadingCod-2 cascading Consider the 6d actionConsider the 6d action

with couplingswith couplings

L1

L2

z

y

Momentum spaceMomentum space In momentum space, this corresponds to In momentum space, this corresponds to

brane localized couplingsbrane localized couplings=-M=-M55

33(q(q55+k+k22), ), andand 22=-M=-M44

22kk22..

with 2 mass scales mwith 2 mass scales m55=M=M5533/M/M44

22 and m and m66= = MM66

44/M/M5533 . .L1

L2

z

y

Cod-2 propagatorCod-2 propagator Including both couplings, the propagator on Including both couplings, the propagator on

the brane isthe brane is

As mAs m66k, the propagator behaves as in k, the propagator behaves as in 6d 6d ((=0) =0)

As mAs m55kkmm66 it takes a it takes a 5d5d behavior behavior

At small scales, kAt small scales, kmm55, we recover , we recover 4d4d..

log k

log k2G-1

Cascading GravityCascading Gravity::A Naïve approachA Naïve approach

The generalization to gravity is The generalization to gravity is straightforwardstraightforward

The tensor mode behaves precisely The tensor mode behaves precisely as the scalar field toy-model, as the scalar field toy-model,

However one of the scalar modes However one of the scalar modes propagates a propagates a ghostghost. .

Propagating modesPropagating modes

Working around flat space-time,Working around flat space-time,

where the tensor mode behaves as where the tensor mode behaves as expectedexpected

and the scalar field and the scalar field is also is also regularized by the cod-1 braneregularized by the cod-1 brane

source term

Ghost modeGhost mode

is is finite finite on the cod-2 brane,on the cod-2 brane, However in the UV, However in the UV, ~ ~ ++ T T While in the IR, While in the IR, ~ ~ -- T. T. The kinetic term changes sign, The kinetic term changes sign,

signaling the presence of a signaling the presence of a ghostghost. . In the UV, the gravitational In the UV, the gravitational

amplitude isamplitude is

Ghost modeGhost mode

is is finite finite on the cod-2 brane,on the cod-2 brane, However in the UV, However in the UV, ~ ~ ++ T T While in the IR, While in the IR, ~ ~ -- T. T. The kinetic term changes sign, The kinetic term changes sign,

signaling the presence of a signaling the presence of a ghostghost. . In the UV, the gravitational In the UV, the gravitational

amplitude isamplitude is

= -1/3-1/6

Ghost modeGhost mode This ghost is completely independent This ghost is completely independent

to the ghost present in the self-to the ghost present in the self-accelerating branch of DGP.accelerating branch of DGP.

However, it is However, it is generic to any cod-2 generic to any cod-2 and higherand higher framework with localized framework with localized kinetic terms.kinetic terms.

In particular it is present when In particular it is present when considering a considering a pure cod-2 scenariopure cod-2 scenario (no cascading). (no cascading).

L2

Gabadadze&Shifman hep-th/0312289

Curing the ghostCuring the ghost

There are two ways to cure the There are two ways to cure the ghost:ghost:1. Adding a tension on the brane1. Adding a tension on the brane

2. Regularizing the brane. 2. Regularizing the brane.

Curing the ghostCuring the ghost

There are two ways to cure the There are two ways to cure the ghost:ghost:1. Adding a tension on the brane1. Adding a tension on the brane2. Regularizing the brane. 2. Regularizing the brane.

Both approaches lead to a well-Both approaches lead to a well-defined 4d effective theory, with defined 4d effective theory, with gravitational amplitudegravitational amplitude

= 1/3-1/12= 1/2-1/6-1/12

CosmologyCosmology

Cf. Ghazal Geshnizjani ’s talk

de Sitter solutionsde Sitter solutions

To find some de Sitter solution, can To find some de Sitter solution, can slice the 6d Minkowski bulk asslice the 6d Minkowski bulk as

and take the cod-1 brane located atand take the cod-1 brane located atthe cod-2 at . the cod-2 at .

dS solutions in 6ddS solutions in 6d

The Cod-1 is not flatThe Cod-1 is not flat

But the brane adapts its position But the brane adapts its position to balance the to balance the extrinsic curvatureextrinsic curvature and and the the Einstein tensorEinstein tensor on the brane for y on the brane for y

>> 00

RR55

forfor

this configuration can only this configuration can only support a support a minimal Hminimal H

dS solutions in 6ddS solutions in 6d The Friedmann eq. on the brane is The Friedmann eq. on the brane is

thenthen

from brane EH Rfrom brane EH R44

dS solutions in 6ddS solutions in 6d The Friedmann eq. on the brane is thenThe Friedmann eq. on the brane is then

Solution only makes sense for minimal Solution only makes sense for minimal tensiontension

from brane EH Rfrom brane EH R44

dS solutions in 6ddS solutions in 6d The Friedmann eq. on the brane is thenThe Friedmann eq. on the brane is then

Solution only makes sense for Solution only makes sense for minimal tensionminimal tension

which is the same bound as the which is the same bound as the no-ghost no-ghost conditioncondition in the deficit angle solution. in the deficit angle solution.

from brane EH Rfrom brane EH R44

Properties of the solutionProperties of the solution Away for the source, the cod-1 brane Away for the source, the cod-1 brane

asymptotes to a constant positionasymptotes to a constant position

The 6d bulk is Minkowski The 6d bulk is Minkowski (in non trivial coordinates)(in non trivial coordinates) volume of the extra dimensions is infinite, volume of the extra dimensions is infinite, there are no separate massless zero mode.there are no separate massless zero mode.

0.1 0.2 0.3 0.4m 6 Fy

-2

-1

1

2

m 6 y

Asymptotically, the 5d brane is flatAsymptotically, the 5d brane is flat

Properties of the Properties of the Friedmann eq.Friedmann eq.

Does correspond to a IR modification Does correspond to a IR modification of gravityof gravity

Could in principle have a large Could in principle have a large with with a small H a small H

BUT still a local expression…BUT still a local expression…

Properties of the Properties of the Friedmann eq.Friedmann eq.

Does correspond to a IR modification Does correspond to a IR modification of gravityof gravity

Could in principle have a large Could in principle have a large with with a small H a small H

BUT still a local expression…BUT still a local expression… In the absence of brane EH term, In the absence of brane EH term,

there is a self-accelerating solution there is a self-accelerating solution ghost?? ghost??

Properties of the Properties of the Friedmann eq.Friedmann eq.

Does correspond to a IR modification of Does correspond to a IR modification of gravitygravity

Could in principle have a large Could in principle have a large with a with a small H small H

BUT still a local expression…BUT still a local expression… In the absence of brane EH term, there is In the absence of brane EH term, there is

a self-accelerating solution ghost?? a self-accelerating solution ghost?? although different from the although different from the

“standard self-acceleration’’“standard self-acceleration’’

Properties of the Properties of the Friedmann eq.Friedmann eq.

Does correspond to a IR modification of Does correspond to a IR modification of gravitygravity

Could in principle have a large Could in principle have a large with a with a small H small H

BUT still a local expression…BUT still a local expression… In the absence of brane EH term, there is In the absence of brane EH term, there is

a self-accelerating solution ghost?? a self-accelerating solution ghost?? If the solution was unstable, would be If the solution was unstable, would be

interesting to see interesting to see where it decays towhere it decays to… …

ConclusionsConclusions

Models of Models of massive gravitymassive gravity represent a represent a novel framework to understand the c.c. novel framework to understand the c.c. problemproblem

There is to date only one known There is to date only one known ghost-ghost-freefree non-perturbativenon-perturbative theory capable of theory capable of exhibiting a model of massive gravity exhibiting a model of massive gravity that does not violate Lorentz invariance:that does not violate Lorentz invariance:

that is DGP that is DGP and its Cascading extension.and its Cascading extension.

ConclusionConclusion

In 6d cascading gravity, there are at In 6d cascading gravity, there are at least 2 kind of different solutions for least 2 kind of different solutions for a pure tension source:a pure tension source:static,static,

““wedge solution”wedge solution”

de Sitterde Sittersolutionsolution