Florian KühnelLudwig-Maximilians-University Munich &

Excellence Cluster “Universe”(on leave to Nordita, Stockholm)

Massive Gravity

with

Felix Berkhahn,Dennis Dietrich,Stefan Hofmann,Parvin Moyassari

based onF. Berkhahn, D. Dietrich, S. Hofmann,

F.K., P. MoyassariPhys. Rev. Lett. 108, 131102 (2012)

F.K.arXiv:1208.1764

[submitted to Phys. Rev. Lett. (2012)]

Florian KühnelLudwig-Maximilians-University Munich &

Excellence Cluster “Universe”(on leave to Nordita, Stockholm)

Forget about Massive Gravity

with

Felix Berkhahn,Dennis Dietrich,Stefan Hofmann,Parvin Moyassari

based onF. Berkhahn, D. Dietrich, S. Hofmann,

F.K., P. MoyassariPhys. Rev. Lett. 108, 131102 (2012)

F.K.arXiv:1208.1764

[submitted to Phys. Rev. Lett. (2012)]

∂µ∂µφa +

δVeffδφa

= 0

. Motivation ,

Point of View

The technical naturalness challenge is communicated exclusively via gravity

The resolution might originate from the gravitational sector

New degrees of freedom in the infra-red?

Massive gravity is an example of such a theory

The vacuum energy density can only be inferred with gravity

the unique Lagrangian for is the Fierz-Pauli one:

Fierz-Pauli Theory

gµν = ηµν + hµν

This theory describes five degrees of freedom

L = 12

hT · E · h + m2

2

�Tr

�h�2 − Tr

�h2

��hµν

The scalar becomes strongly coupled at “short” distances[Vainshtein]

[vDVZ]m→ 0The limit is discontinuous

For small fluctuations around Minkowski,

General „massive“ Deformations

is the standard Friedmann-Robertson-Walker metricgµν0

M

S =1

2

�

Md4x

�|g0|

�hµν

�Eαβµν(g0,∇) +M(g0)αβµν

�hαβ + T

µνhµν�

Linearised Einstein-Hilbert term

Deformation term

Question: Is there a unique choice forlike for a Minkowski background?

We are attacking cosmological questions

Consider linear theory on a cosmological background:

.Stability Analys! ,

.Naïve Fierz-Pauli ,

Higuchi-Bound

m2> H

2 = const.

Mµναβ = m2�gµν0 g

αβ0 − g

µβ0 g

να0

�

On a de Sitter background, Higuchi has shown that

to guarantee the absence of negative-norm states

On FRW: H → H(t) What are the implications?

The most simple ansatz would be the naïve Fierz-Pauli term

Results of naïve Fierz-Pauli in FRW I

L ⊃ A(t)φ̇2 +B(t)(�∇φ/a)2

2. Sign of implies classical (in)stability B(t)

1. Sign of determines the norm in Fock-space:A(t)

m2> H

2 +1

3ḢStability bound:

�a(k), a†(k’)

�= sign(A)δ(3)(k− k’)

Unitarity bound: m2 > H2 + Ḣ

[Berkhahn et al.]

At high energies the action can be diagonalised.Consider the scalar part:

Results of naïve Fierz-Pauli in FRW II

Valid at all energiesIncorporates all degrees of freedom

Orange:

Classically unstable for high momenta

Unitarity violating

[Berkhahn et al.]

Classically unstablefor zero momentum

Green:

Blue:

Result of a complete cosmological perturbation analysis:

Self-Protection

System self-protects from direct unitarity violation

Violation of stability bound

How to avoid the classical instability?

Large fluctuations

Formation of a new background

m → m(t)Try ... or even more general...

Cosmological Classicalization

The stability bound is stronger than the unitarity bound

.Cova"ant!edFierz-Pauli ,

Deformation Matrix

S =1

2

�

Md4x

�|g0|

�hµν

�Eαβµν(g0,∇) +M(g0)αβµν

�hαβ + T

µνhµν�

+β�Rµ[ν0 g

β]α0 +R

α[β0 g

ν]µ0

�Mµναβ = (m2 + αR0)

�gµν0 g

αβ0 − g

µβ0 g

να0

�

+ γ Rµανβ0

Remember that the action reads

Covariance and symmetry constrain the infra-red leading terms of the deformation matrix:

Introduce Stückelberg fields:

Stability Analysis: Technique I

hµν → hµν +∇(µAν) +∇µ∇νφ

Gives the gauge symmetries:

hµν → hµν +∇(µζν) Aµ → Aµ − ζµ,

Aµ → Aµ +∇µξ φ → φ− ξ,

h0µ = 0 A0 = 0,

Then use the gauge

Stability Analysis: Technique II

(Ψi) = (φ, A1, A2, A3, h11, h12, ...)Kµνij (∇µΨi)(∇νΨj) ,

signal the following instabilities:

1. Spatial directions: Classical instability

2. Time direction: Unitarity violation

System evolves into a new background

Quantum theory loses probabilistic interpretation

Equation of motion becomes singular

Singular points of the resulting kinetic matrix

Bounds in the general case are much more complicated

Stability Analysis: General Case

α �= 0 β �= 0,

0�0.5 �0.3 �0.1 0.1 0.3 0.5

0

�1.5

�1.1

�0.7

�0.3

0.5

Β

Α

Green:

Yellow:

Red:

Black:

m = 0

Classically stable and unitary

Self-Protection

Unitarity violation

No stability or unitarity today

....more #neral ,

(Al)most general...

This is contained in the Hassan-Rosen model:

S[f, g] =−M2g2

�

Md4x

�−det[g] R[g]

−M2f2

�

Md4x

�−det[f ] R[f ]

+ m2 M2g

�

Md4x

�−det[g]

4�

n=0

βn en�g−1 f

�

The (al)most general ghost-free theory of massive gravity (fully non-linear!) is the de Rham-Gabadadze-Tolley model

... why you can (almost) forget that...

0�5 �3 �1 1 3 5

0

�5

�3

�1

1

3

5

Γ

Β

.Conclusion ,

Conclusion

One definitely needs curvature terms, which are NOT included in the Hassan-Rosen theory

To me it is unclear how a non-linear theory could possibly include / dynamically generate them...

Probably nature prefers massless / undeformed gravity

Absolute stability requires proper covariantizationof the deformation matrix, at least on the linear level

.Po$er ,

22.08.11Centro de Astrofísicada Universidade do Porto, Portugal

Date Location

ONCE upon a TIME in the UNIVERSE

LMU & CP-3 proudly present a MATHEMATICA FILMdirected by FELIX BERKHAHN, DENNIS DIETRICH,

STEFAN HOFMANN, FLORIAN KÜHNELand PARVIN MOYASSARI story by and of

THE UNIVERSE

26.08.11 Florian KühnelLudwig-Maximilians-University Munich &Excellence Cluster “Universe”, Germany

AffiliationSpeaker

Generic deformations of General Relativity,

with

respect cosmological backgrounds

Any non-linear deformation mustincorporate self-stabilisation ongeneric space-times already at lowestorder in perturbation theory

Our findings include the possibility of consistent and phenomenologically viable deformations which reduce to General Relativity on Minkowski background

Mµναβ =�m20 + αR0

��g µν0 g

αβ0 − g

µβ0 g

να0

�

+ β�R µν0 g

αβ0 −R

µβ0 g

να0

+ g µν0 Rαβ0 − g

µβ0 R

να0

�

S[H] = 12

�

Md4x

�|g0|HT [E(g0,∇) +M(g0)]H

-

W a n t e di n

t h e P h y s i c al R e v i e w L e

t t e r s

.Movie ,

Stability Analysis: Time Dependence

m = 0

.On Schwarzschild ,

Degeneracy

γ!= 0

leaves a degeneracy

Hence, we study Schwarzschild, which is only sensitive to

Our result:

γ

Degeneracy is lifted!

γ = 0

Only three parameters: m, α, β at max...

On FRW we can set (vanishing Weyl tensor)