introduction to the vainshtein mechanism · 2013-09-26 · for the vainshtein mechanism it is...
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LPT, OrsayEugeny Babichev
Introduction to the Vainshtein mechanism
School Paros23-28 September 2013
based on arXiv:1107.1569with C.Deffayet
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OUTLINE
✦ Introduction and motivation
✦ k-mouflage
✦ Galileons
✦ Non-linear massive gravity
✦ de Rham-Gabadadze-Tolley massive gravity
✦ Other examples
✦ Problems
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Introduction and motivations: modifying graivity
- Modifying gravity - explain Dark energy, cosmological constant problem, to cure non-renormalizability problem, theoretical curiosity etc.
- There are many ways to modify gravity: f(R), scalar-tensor theories, Galileons, Horndeski theory, KGB, Fab-four, higher-dimensions, DGP, massive gravity...
We want to recover General Relativity at short distances
When modifying gravity, extra degrees of freedom appear, which alter gravitational interaction between bodies
A trick to comply with both requirements is needed !
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Mechanisms to recover General Relativity:
➡ Chameleon (non-linear potential for a canonical extra propagating scalar) - scalar-tensor theories, f(R)
➡ Symmetron (coupling to matter depends of the environment)
➡ Vainshtein mechanism (nonlinear kinetic term effectively hides extra degree(s) of freedom) - k-essence, DGP, Galileon, Horndeski theory, massive gravity
Introduction and motivations: how to recover GR at small distances
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Fierz-Pauli action (Fierz&Pauli’39):
Equations of motion:
Introduction and motivations: Linearized massive graviton
Linearized Einstein-Hilbert term
Minimal coupling to mattermass term
To be compared with linearized GR:
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Solution for a point-like sourcein GR:
Introduction and motivations: vDVZ discontinuity
van Dam-Veltman-Zakharov discontinuity ! (vanDam&Veltman’70&Zakharov’70)How to fit observation in MG ?
Tested with high precision !
Solution for a point-like sourcein massive gravity:
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Introduction and motivations: Vainshtein’s idea
Vainshtein’72: In non-linear massive gravity GR can be restored!
The linear approximation breaks down at the Vainshtein radius
Inside the Vainshtein radius GR is restored, outside -- linear MG
Linear massive gravity
Non GR
Non-perturbative regimeGR??
Source
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k-mouflage (EB,Deffayet,Ziour’09)
For the Vainshtein mechanism it is (generically) sufficient to have a non-linear (non-canonical) kinetic term.
k-mouflage simple way to understand the Vainshtein mechanism
- There are two scales in the action: m and the Planck mass - Composite scales- Nonlinear regimes happen for different scales
EH term modificationof gravity
(“Brans-Dicke term”)
Non-linear kinetic term
Do not expand
Expansion:
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k-mouflage simple way to understand the Vainshtein mechanism
⇥2� = M�2P T
h �= hGR
⇥2� � E� = M�2P T
h � hGR
Variation wrt
Variation wrt
Expanded action
Two regimes:
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k-mouflage simple explanation 2
Demix kinetic terms for
Variation wrt
Variation wrt
⇥2� = M�2P T
h �= hGR
⇥2� � E� = M�2P T
h � hGR
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k-mouflage action
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k-mouflage expanded action
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k-mouflage expanded action
- the spin-0 and spin-2 modes decouple- a non-minimal scalar-matter coupling appears
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k-mouflage two regimes
EOMs:
where
Linear regime
non-GR
h � hGR
Non-linear regime
GR is restored
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k-mouflage scalings
n>k
r0
�
rV
- Schematic form of kinetic self-interacting
- strong coupling scale
Spherical symmetry:
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k-mouflage k-essence
1.000.50 5.000.10 10.000.05R !RV ,H0.01
0.1
1
10
100
1000
104
Linear regime
Vainshtein regime:~ GR
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Galileons cubic galileon
current:
EOM:
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Galileons other galileons
Galileon Lagrangians (Horndeski’74,Fairlie et al’92, Nicolis’09, Deffayet et al’09+many others)
The Vainshtein mechanism works for a generic galileon:
➡ non-covariant galileons (Nicolis’09)➡ ...➡ Horndeski model (Koyama et al’13, Kase&Tsujikawa’13)
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Galileons induced coupling
The Vainshtein mechanism with time dependent boundary conditions (EB&Esposito-Fasere’12)
- because of cosmological evolution (e.g. in KGB, deffayet et al’10)
Solution for a spherically symmetric source
Naturally the induced coupling is of the order of 1 !
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Non-linear massive gravity potential for metric
Need to construct a mass term -> introduce an extra metric
Construct a potential, following the rules:- general covariance under diffeomorphisms (common to the two metrics)- has flat spacetime as solution for physical metric- when expanding around flat metric the potential takes a specific form, the Pauli-Fierz form
(Boulware & Deser‘72)
(Arkani-Hamed et al’03)
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Non-linear massive gravity Boulware-Deser ghost
There are two propagating scalars: one is a ghost ! (Boulware & Deser‘72)
The presence of ghosts is not connected to the Vainshtein mechanism.
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Non-linear massive gravity equations of motion
EOMs
ansatz
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Non-linear massive gravity equations of motion
EOMs
ansatz
NUMERICS ?SIMPLIFY ?
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Non-linear massive gravity Stuckelberg approach
Stuckelberg approach (Arkani-Hamed et al’03)
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Non-linear massive gravity Stuckelberg approach
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Non-linear massive gravity action and EOM in decoupling limit
Decoupling limit
k-mouflage:
Vainshtein mechanism in decoupling limit of massive gravity (EB,Deffayet,Ziour’09)
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Non-linear massive gravity EOM - spherical symmetry
Q is a non-linear function
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Non-linear massive gravity solutions in decoupling limit
Newtonian gauge:
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Non-linear massive gravity solutions in decoupling limit
0.01 0.1 1
0.1
1
10
100
1000
Vainshtein scaling
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dRGT the model
Massive gravity without Boulware-Deser ghost (de Rham-Gabadabze-Tolley’10)
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dRGT decoupling limit
Decoupling limit :
The same procedure as in case of NLMG
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dRGT decoupling limit
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dRGT decoupling limit
Nonlinear field redefinition:
Additional constraint because of stability (Berezhiani et al’13)
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dRGT decoupling limit - special case
Vainshtein mechanism for (Koyama et al’11)
Solution inside the Vainshtein radius
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dRGT decoupling limit - general case
Vainshtein mechanism for (Chkareuli et al’11)
EOMs:
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dRGT decoupling limit - general case
Vainshtein mechanism for (Chkareuli et al’11)
EOMs:
Solution inside the Vainshtein radius
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Vainshtein mechanism in bi-gravity the model
Action for bi-gravity (Hassan&Rosen’11)
Decoupling limit ?
Weak-field approximation (EB,Deffayet,Ziour’10)Vainshtein mechanism in bi-gravity (EB,Crisostomi’13)
see talk by Marco Crisostomi
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Vainshtein mechanism in full NLMG
10!10
10!9
10!8
10!7
10!6
0.001 0.01 0.1 1 10 100
10!12
10!10
10 100
!
"#
"#
!
, DL
, DL
Numerical solution for the full system of equations (EB,Deffayet,Ziour’09)
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Vainshtein mechanism in full dRGT
Numerical solution for the full system of equations (Volkov’12,Gruzinov& Mirbabayi’12)
Also in bi-gravity (Volkov’12)
The Vainshtein mechanism does work !
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Super-Vainshtein mechanism and MOND
Improving MOND (EB,Deffayet,Esposito-Farese’11)
r0
�
rV
linear solutionVainshtein scaling
Super-Vainshtein scaling
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CONCLUSIONS
✦ The Vainshtein mechanism as k-mouflage
‣ k-essence
‣ galileons
✦ The Vainshtein mechanism in the decoupling limit of the non-linear massive gravity
✦ The Vainshtein mechanism in the decoupling limit of the dRGT theory
✦ Other examples: bi-gravity, full NLMG and full dRGT
✦ Stability of solutions (ghosts, Laplace instabilities?)
✦ Strong coupling and problems on quantum level ?
✦ Superluminality
✦ Very compact stars (no solutions for far) ?