The The galileon as galileon as a a locallocalmodification modification of of gravitygravity

with Alberto Nicolisand Riccardo Rattazzi

Phys. Rev. D 76 (2009) 064036

Enrico Trincherini(SNS, Pisa)

MotivationsMotivationsExperimental finding: the universe is undergoing a phase ofaccelerated expansion Cosmological constant

Modify GR at cosmological distances to have acceleration without cc

“Don’t modify gravity: understand it!”Nima @ TGGP 09

extraordinary ambitous task!

totally foolish

But maybe it isn’t a dichotomy…

We realized that the galileon allows for stable NEC solutions

or/and

Nicolis, Rattazzi, ET, to appear

If NEC, present expansion may be due to a bounce from anearlier contracting phase (cyclic universe)

Maybe relevant for an entirely different epoch of our universeAlternative to inflation? Nicolis, Rattazzi, ET, in (slow) progress

Fierz-Pauli massive gravity, DGP,Lorentz violating massive gravity,

the ghost condensate, …

O(1)O(1)

rrIRIR~ ~ HH00-1-1

10102828 cm cmintroduce a light scalar d.o.ff(R), Brans-Dicke, Fierz-Pauli massive gravity,DGP, Lorentz-violating massive gravity,…

O(1)O(1)

rrIRIR~ ~ HH00-1-1

10102828 cm cm

scalar-tensorscalar-tensor

Van Dam, Veltman, Zacharov 70

vDVZ discontinuity

O(1)O(1)≤≤ O(10 O(10-3-3))

rrIRIR~ ~ HH00-1-1rrPlutoPluto

10101414 cm cm 10102828 cm cm

almost almost GRGR

screening mechanismf(R)-like: chameleon mechanism Khoury, Weltman 03

the scalar mass depends on the local density

scalar-tensorscalar-tensor

O(1)O(1)≤≤ O(10 O(10-3-3))

rrIRIR~ ~ HH00-1-1rrPlutoPluto

10101414 cm cm 10102828 cm cm

almost almost GRGR

screening mechanismVainshtein effect: non-linear dynamics suppresses the scalar contribution

Vainshtein 72

non-linear non-linear regimeregime linear linear regimeregime

scalar-tensorscalar-tensor

O(1)O(1)≤≤ O(10 O(10-3-3))

rrIRIR~ ~ HH00-1-1rrPlutoPluto

10101414 cm cm 10102828 cm cm

almost almost GRGR

non-linear non-linear regimeregime linear linear regimeregime

scalar-tensorscalar-tensor

ΛΛ-1-1 rrVV10102121 cm cm

MMPP-1-1

O(1)O(1)≤≤ O(10 O(10-3-3))

rrIRIR~ ~ HH00-1-1rrPlutoPluto

10101414 cm cm 10102828 cm cm

almost almost GRGR

non-linear non-linear regimeregime linear linear regimeregime

scalar-tensorscalar-tensor

ΛΛ-1-1 rrVV10102121 cm cm

rrSS

101055 cm cmlinearized gravitylinearized gravity

FRW FRW universeuniverse: : conformal conformal deformation deformation of of MinkowskiMinkowski

O(1)O(1)≤≤ O(10 O(10-3-3))

rrIRIR~ ~ HH00-1-1rrPlutoPluto

10101414 cm cm 10102828 cm cm

almost almost GRGR

non-linear non-linear regimeregime linear linear regimeregime

scalar-tensorscalar-tensor

ΛΛ-1-1 rrVV10102121 cm cm

rrSS

101055 cm cm

??

linearized gravitylinearized gravity

locally de Sitter

the the galileongalileon: a : a local modification local modification of of gravitygravity

IR completion

coupling to gravity

O(1)O(1)≤≤ O(10 O(10-3-3))

rrIRIR~ ~ HH00-1-1rrPlutoPluto

10101414 cm cm 10102828 cm cm

almost almost GRGR scalar-tensorscalar-tensor

ΛΛ-1-1 rrVV10102121 cm cm

rrSS

101055 cm cm

DGPDGP

5D 5D gravitygravity

full non-linear 5D “self-accelerating” solutionIs plagued by a ghost instability

Dvali, Gabadadze, Porrati 00

Long-distance modification of GR

• locally and for weak gravitational fields due to a light scalard.o.f. π

• π kinetically mixed with the metric:

We allow for generic non-linarities in the π sector(while we treat the gravitational field and π’s contribution to it at linear order)

Long-distance modification of GR

• locally and for weak gravitational fields due to a light scalard.o.f. π

• π kinetically mixed with the metric:

• π does not couple directly to matter (minimally coupled to )

Demix π and the metric by performing the Weyl rescaling:

Towards a realistic modification of gravity, the structure of

3) π e.o.m. must be of second order

F is a non-linear Lorentz invariant function of

Generalization of the π Lagrangian of the DGP model

Give rise to on O(1) modification of the Hubble flow butno deviation larger than O(10-3) at solar system distances:1) non-linearities in the π dynamics must play a crucial role

the quadratic approximation at subhorizon distances is invariant under the combined action of a spacetime translation and a shift

2) the equations of motion should be invariant under this “galilean” symmetry

this is equivalent as demanding at least 2 derivatives per π in the e.o.m. ⇒ generically ghost degrees of freedom appear (ex. Fierz Pauli massive gravity)

Introduce an auxiliary scalar field and a new lagrangian:

It reduces to the previous one when χ is integrated out.

Diagonalize it by the substitution

Clearly signals the presence of a ghost, independently of the sign of

The galileon Lagrangian

classify all the operators in the Lagrangian:• n powers of π and 2n-2 derivatives• galilean invariant

each derivative term is associated with one invariant of the matrix Πnumber of invariant = rank of Π = number of spacetime dimensions

There are only 4+1 Lagrangian terms in 4D!

The galileon Lagrangian

classify all the operators in the Lagrangian:• n powers of π and 2n-2 derivatives• galilean invariant

is the DGP π LagrangianLuty, Porrati, Rattazzi 03

The coupling to matter

π is dimensionless

The scale controlling higher order terms must be so that the πcontribution affects at O(1) the present expansion

Lower bound: direct contribution of π to the geometry percieved by matter

is the Riemann curvature

Strongest bound on comes from the solar system but

In galaxy superclusters and GR is valid at O(1)

measures the coupling between π quanta and matter

The coupling to matter

π is dimensionless

The scale controlling higher order terms must be so that the πcontribution affects at O(1) the present expansion

measures the coupling between π quanta and matter

Spherically symmetric solution and the Vainshtein effect

localized source

spherical symm + shift and galilean symm ⇒ drastical simplification

e.o.m. is an algebraic equation for

Spherically symmetric solution and the Vainshtein effect

rewrite the equation of motion as

the non-linear terms become relevant at the scale

in the linear regime, far from the source, the kinetic term dominates and wehave the asymptotic behaviour

Results

The existence of only 5 possible terms makes possible a thorough analysisof very symmetric solutions. The sign of the coefficients cn can be chosensuch that:

1. the galileon admits a local deSitter solution in the absenceof a cosmological constant;

2. it is ghost free;

3. about this dS configuration there exist sphericallysymmetric solutions that describe the π field generated bycompact sources;

4. these configurations are also stable against smallperturbations;

5. the Vainshtein effect is implemented.

In the subclass of Lagrangians c4=c5=0, the large kinetic term supresses quantumfluct and the effective cutoff increases above the naive esitmate Λ ~ (1000 km)-1

i) Small excitations are forced to be superluminal at large distances fromcompact sources, even if we do not require the existence of dS solutions;

ii) Very low strong-coupling scale for ππ scattering;

On a non-Lorentz invariant background this is not necessarily an inconsistency, but UV completion cannot be a Lorentz invariant, local QFT

Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi 06

Nicolis, Rattazzi 04

2. Common to DGP:

1. What about the existence and stability of more complicated (less symmetric) solutions, e.g. with several compact sources in generic positions?

Problems and open questions

3. What is the fate of the galileon in the IR? How do we couple it to gravity?

The IR completion

Is there a “global completion” of the local dynamics of the galileon?

Maybe not: in DGP the 4D description in terms of π breaks down at

Consider first the limit of decoupled gravity

Conformally invariant 4D lagrangian

corresponds to Poincaré (translation and boost unbroken)

The IR completion

Is there a “global completion” of the local dynamics of the galileon?

Maybe not: in DGP the 4D description in terms of π breaks down at

Consider first the limit of decoupled gravity

Conformally invariant 4D lagrangian

Generically we may expect one to describe the vacuum; which will be the solution to the dynamics depends on the Lagrangian

Fubini 76

The IR completion (2)

How can we write conf. invariant Lagrangian that reduces to the galileon?

Conformal group on π corresponds to a subgroup of diff. on

Consider the fictitious metric

All the curvature invariants written in terms ofπ are conformally invariant

Terms with fewer derivatives per π are less important at short distances

The IR completion (2)

How can we write conf. invariant Lagrangian that reduces to the galileon?

Conformal group on π corresponds to a subgroup of diff. on

Consider the fictitious metric

All the curvature invariants written in terms ofπ are conformally invariant

Find combinations of curvature invariants for which higher derivative contributions (that lead to ghosts) vanish

We can reproduce any galilean-invariant Lagrangian term as the short distance limit of a globally defined conformally invariant lagrangian for πbecause of the symmetries of the full Lagrangian the local deSitter solution can be promoted to a global one

Gravitational backreactionViable gravitational interactions of π cannot preserve conformal invariance

the gravitational backreaction spoils thedeSitter geometry at Hubble scales

• Covariantize the π action by contracting Lorentz indices with• Raplace ordinary derivatives with covariant ones

use coordinates where the solution only depends on conformal time

The stress energy tensor corresponding to our the deSitter solution

Curvature perceived by matter is sum of that due to its conformal coupling to π, and that of , which in turn is sourced by the π stress energy tensor

Minimal coupling to

Conclusions

We have considered IR modifications of gravity that, at least in a local patch smaller than the cosmological horizon, are due to a relativistic scalar d.o.f.kinetically mixed with the metric;

the dynamics of π enjoys an internal galilean symmetry (only 5 possibleLagrangian terms) and the e.o.m. is of second order;

π decouples from matter at short scales thanks to derivative self-interactions.

We found stable deSitter solutions in the absence of a c.c. and stable spherically symmetric Vainshtein-like background around compact sources Small excitations are forced to be superluminal at large distances from compact sources

The UV cutoff around flat space is

Conclusions

We considered if the galileon can be promoted to a global scalar d.o.f. coupled someway to 4D GR;

neglecting gravity, we promoted the Poincaré group and our galilean invariance to the conformal group, then the existence of deSitter-like Solutions may be expected by symmetry considerations,

our local galilean Invariant terms are short-distance limit of globally defined conformally invariant Lagrangian terms.

Coupling the system to gravity breaks conformal invariance and spoils the deSitter geometry at times of order Hubble

not necessarily a phenomenological problem, but the symmetry motivation for this “conformal completion” is somewhat lost.