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DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Modified theories of gravity and mechanisms ofmass screening
Radouane GANNOUJI
Tokyo University of Science
@ Nagoya Daigaku
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Dark Energy in a nutshell
Chameleon
Symmetron
Vainshtein
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Dark Energy in a nutshell
Chameleon
Symmetron
Vainshtein
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Dark Energy in a nutshell• Friedmann equation
3H2 = ρ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Dark Energy in a nutshell• Friedmann equation
3H2 = ρ+ Λ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
ΩDE = 0.72± 0.015, −1.11< wDE = P/ρ < −0.86
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
ΩDE = 0.72± 0.015, −1.11< wDE = P/ρ < −0.86
⇒ The Universe is accelerating
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Dark Energy in a nutshell• Friedmann equation
3H2 = ρ+ Λ
• Cosmological constant problem
ρvac/ρΛ ≈ 1055
• How to modify this equation• Add exotic matter (Dark Energy)
3H2 = ρ+ ρDE
• Modify the spacetime reaction (Modified gravity)
3H2 + H = ρ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Dark Energy in a nutshell• Friedmann equation
3H2 = ρ+ Λ
• Cosmological constant problem
ρvac/ρΛ ≈ 1055
• How to modify this equation• Add exotic matter (Dark Energy)
3H2 = ρ+ ρDE
• Modify the spacetime reaction (Modified gravity)
3H2 + H = ρ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Dark Energy in a nutshell• Friedmann equation
3H2 = ρ
• Cosmological constant problem
ρvac/ρΛ ≈ 1055
• How to modify this equation• Add exotic matter (Dark Energy)
3H2 = ρ+ ρDE
• Modify the spacetime reaction (Modified gravity)
3H2 + H = ρ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Dark Energy in a nutshell• Friedmann equation
3H2 = ρ
• Cosmological constant problem
ρvac/ρΛ ≈ 1055
• How to modify this equation• Add exotic matter (Dark Energy)
3H2 = ρ+ ρDE
• Modify the spacetime reaction (Modified gravity)
3H2 + H = ρ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Dark Energy in a nutshell• Friedmann equation
3H2 = ρ
• Cosmological constant problem
ρvac/ρΛ ≈ 1055
• How to modify this equation• Add exotic matter (Dark Energy)
3H2 = ρ+ ρDE
• Modify the spacetime reaction (Modified gravity)
3H2 + H = ρ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Dark Energy in a nutshell• Friedmann equation
3H2 = ρ
• Cosmological constant problem
ρvac/ρΛ ≈ 1055
• How to modify this equation• Add exotic matter (Dark Energy)
3H2 = ρ+ ρDE
• Modify the spacetime reaction (Modified gravity)
3H2 + H = ρ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Local bounds• Laboratory Bounds
V(r) = −Gm1m2
r
[
1+ α exp−r/λ]
g00 = −1+ 2GMrc2
− 2βPPN(GM
rc2
)2+ · · ·
gij = δij
[
1+ 2γPPNGMrc2
+ · · ·]
• Solar System Bounds
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
• Problem?Solar System and lab experiments rule out any long rangegravitationally coupled fifth force.So the challenge is to modify gravity only at large (cosmic) distances,while keeping it unaltered at short (Solar System) distances.
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
• Problem?Solar System and lab experiments rule out any long rangegravitationally coupled fifth force.So the challenge is to modify gravity only at large (cosmic) distances,while keeping it unaltered at short (Solar System) distances.
• How to modify the theory of gravity?Weinberg’s theorem: A Lorentz-invariant theory of a massless spin-twofield must be equivalent to GR in the low-energy limit.Low-energy modifications of gravity therefore necessarilyinvolveintroducing new degrees of freedom, such as a scalar.Giving the graviton a small mass width turns out to introducea scalar.Theories that invoke a Lorentz violating massive graviton also involve ascalar.
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Three ways to hide
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
• The scalar field is weakly coupled to matter
• Quintessence
• The scalar field is massive in vicinity of matter
• Chameleon (Khoury and Weltman: 2004)• Symmetron (Olive and Pospelov: 2008)
• The scalar field is weakly coupled in vicinity of matter• Vainshtein (1972)
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Three ways to hide
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
• The scalar field is weakly coupled to matter
• Quintessence
• The scalar field is massive in vicinity of matter
• Chameleon (Khoury and Weltman: 2004)• Symmetron (Olive and Pospelov: 2008)
• The scalar field is weakly coupled in vicinity of matter• Vainshtein (1972)
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Three ways to hide
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
• The scalar field is weakly coupled to matter
• QuintessenceL = −1
2gµν∂µφ∂νφ− V(φ) + T
• The scalar field is massive in vicinity of matter
• Chameleon (Khoury and Weltman: 2004)• Symmetron (Olive and Pospelov: 2008)
• The scalar field is weakly coupled in vicinity of matter• Vainshtein (1972)
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Three ways to hide
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
• The scalar field is weakly coupled to matter
• QuintessenceL = −1
2gµν∂µφ∂νφ− V(φ) + T
• The scalar field is massive in vicinity of matter
• Chameleon (Khoury and Weltman: 2004)• Symmetron (Olive and Pospelov: 2008)
• The scalar field is weakly coupled in vicinity of matter• Vainshtein (1972)
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Three ways to hide
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
• The scalar field is weakly coupled to matter
• QuintessenceL = −1
2gµν∂µφ∂νφ− V(φ) + T
• The scalar field is massive in vicinity of matter
• Chameleon (Khoury and Weltman: 2004)• Symmetron (Olive and Pospelov: 2008)
L = −12
gµν∂µφ∂νφ− V(φ) + A(φ)T
• The scalar field is weakly coupled in vicinity of matter• Vainshtein (1972)
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Three ways to hide
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
• The scalar field is weakly coupled to matter
• QuintessenceL = −1
2gµν∂µφ∂νφ− V(φ) + T
• The scalar field is massive in vicinity of matter
• Chameleon (Khoury and Weltman: 2004)• Symmetron (Olive and Pospelov: 2008)
L = −12
gµν∂µφ∂νφ− V(φ) + A(φ)T
• The scalar field is weakly coupled in vicinity of matter• Vainshtein (1972)
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Three ways to hide
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
• The scalar field is weakly coupled to matter
• QuintessenceL = −1
2gµν∂µφ∂νφ− V(φ) + T
• The scalar field is massive in vicinity of matter
• Chameleon (Khoury and Weltman: 2004)• Symmetron (Olive and Pospelov: 2008)
L = −12
gµν∂µφ∂νφ− V(φ) + A(φ)T
• The scalar field is weakly coupled in vicinity of matter• Vainshtein (1972)
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Three ways to hide
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
• The scalar field is weakly coupled to matter
• QuintessenceL = −1
2gµν∂µφ∂νφ− V(φ) + T
• The scalar field is massive in vicinity of matter
• Chameleon (Khoury and Weltman: 2004)• Symmetron (Olive and Pospelov: 2008)
L = −12
gµν∂µφ∂νφ− V(φ) + A(φ)T
• The scalar field is weakly coupled in vicinity of matter• Vainshtein (1972)
L = −12
Zµν(φ)∂µφ∂νφ+ A(φ)T
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Dark Energy in a nutshell
Chameleon
Symmetron
Vainshtein
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Chameleon
S=
∫
d4x√−g
(
R−12(∂φ)2 − V(φ)
)
+ Sm[ψm;A2(φ)gµν ]
• Equation of motion
• Dynamics governed by effective potential:
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Chameleon
S=
∫
d4x√−g
(
R−12(∂φ)2 − V(φ)
)
+ Sm[ψm;A2(φ)gµν ]
• Equation of motion
φ = V,φ + A,φρ
• Dynamics governed by effective potential:
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Chameleon
S=
∫
d4x√−g
(
R−12(∂φ)2 − V(φ)
)
+ Sm[ψm;A2(φ)gµν ]
• Equation of motion
φ = V,φ + A,φρ
• Dynamics governed by effective potential:
φ = Veff,φ
Veff(φ) ≡ V(φ) + A(φ)ρ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Dynamics governed by effective potential
φ = Veff,φ, with Veff(φ) ≡ V(φ) + A(φ)ρ
Largeρ Smallρ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Solution for a compact object
Rc
Mc,Ρc
Ρ¥
d2φ
dr2+
2r
dφdr
= V,φ + A(φ)ρ(r)
The boundary conditions specify that the solution be nonsingular at the
origin:dφdr
= 0 atr = 0
And that the force on a test particle vanishes at infinity:φ→ φ∞ asr → ∞
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
RcΡc
Φ=Φc
VeffHΦL
Φ
VeffHΦL
Φ
Ρ¥
Φ=Φ¥
0 20 40 60 80 10099.90
99.91
99.92
99.93
99.94
99.95
99.96
99.97
r
Φ
φ(r) ≈ −
(
β
4πMpl
)
Mce−m∞(r−Rc)
r+φ∞
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
RcΡc
Φ=Φc
VeffHΦL
Φ
DRcVeffHΦL
Φ
Ρ¥
Φ=Φ¥
0 20 40 60 80 1000
10
20
30
40
50
60
70
r
Φ
φ(r) ≈ −
(
β
4πMpl
)(
3∆Rc
Rc
)
Mce−m∞(r−Rc)
r+φ∞
• Field reaches effective minimum almost everywhere in the body• All field variations confined to small region near body surface
• This region is called a thin-shell∆Rc
Rc≪ 1
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Fifth Force on Body with Thin-Shell
A test massM1 in a static chameleon fieldφ experiences a force−→F φ
−→F φ
M1= − β
Mpl
−→∇φ
⇒ φ is the potential for the chameleon force.
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Fifth Force on Body with Thin-Shell
A test massM1 in a static chameleon fieldφ experiences a force−→F φ
−→F φ
M1= − β
Mpl
−→∇φ
⇒ φ is the potential for the chameleon force.
Fφ
M1=
GMc
r2(6β2∆Rc
Rce−m∞r)
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Fifth Force on Body with Thin-Shell
A test massM1 in a static chameleon fieldφ experiences a force−→F φ
−→F φ
M1= − β
Mpl
−→∇φ
⇒ φ is the potential for the chameleon force.
Fφ
M1=
GMc
r2(6β2∆Rc
Rce−m∞r)
⇒ F12 =GNm1m2
r2 (1+ αe−mr)
• deviations from Newton’s law are given byα = 6β2∆Rc
Rc
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
• Thin Shell
deviations from Newton’s law are given byα = 6β2∆Rc
Rcwith
∆Rc
Rc≈
1Newton’s potential
for large objects (sun, earth, moon) , Newton’s potential islarge and athin shell is always present
⇒ Planetary motion unaffected by the chameleon field
• Example (Earth)Modeled as a sphere of radiusRc = 6000Km and densityρc = 10g/cm3.Far away the matter density is approximately that of baryonic gas anddark matter:ρ∞ = 10−24g/cm3
∆Rc
Rc≈ 10−8 ≪ 1
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
• Thin Shell
deviations from Newton’s law are given byα = 6β2∆Rc
Rcwith
∆Rc
Rc≈
1Newton’s potential
for large objects (sun, earth, moon) , Newton’s potential islarge and athin shell is always present
⇒ Planetary motion unaffected by the chameleon field
• Example (Earth)Modeled as a sphere of radiusRc = 6000Km and densityρc = 10g/cm3.Far away the matter density is approximately that of baryonic gas anddark matter:ρ∞ = 10−24g/cm3
∆Rc
Rc≈ 10−8 ≪ 1
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Gravity Test
F12 =GNm1m2
r2(1+ αe−mr)
α = 10−8
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Dark Energy in a nutshell
Chameleon
Symmetron
Vainshtein
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Symmetron
φ = Veff,φ, with Veff(φ) ≡ V(φ) + A(φ)ρ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Symmetron
φ = Veff,φ, with Veff(φ) ≡ V(φ) + A(φ)ρ
V(φ) = −12µ2φ2 +
14λφ4
A(φ) = 1+φ2
2M2
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Symmetron
φ = Veff,φ, with Veff(φ) ≡ V(φ) + A(φ)ρ
V(φ) = −12µ2φ2 +
14λφ4
A(φ) = 1+φ2
2M2
Veff(φ) =12
( ρ
M2− µ2
)
φ2 +14λφ4
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Symmetron
Veff(φ) =12
( ρ
M2− µ2
)
φ2 +14λφ4
Large density Low density
Coupling to matter≃φ2
M2ρ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Symmetron
Veff(φ) =12
( ρ
M2− µ2
)
φ2 +14λφ4
Large density Low density
Coupling to matter≃φ2
M2ρ
Fluctuations(δφ) around the local background(φ0) ≃φ0
M2δφρ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Conclusion
• Strengthens:• Useful for many models:f (R,G)• Fifth force is suppressed locally• Interesting cosmological consequences
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Conclusion
• Strengthens:• Useful for many models:f (R,G)• Fifth force is suppressed locally• Interesting cosmological consequences
• Weaknesses• Singularity problem ?• We do not solve the cosmological constant problem:
mφ < 10−33eV
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Dark Energy in a nutshell
Chameleon
Symmetron
Vainshtein
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Vainshtein
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
• At low energyZµν ≈ gµν• At high energyZµν ≫ 1
• The field is strongly coupled to itself and becomes weakly coupled toexternal sources
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Vainshtein
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
= −12
Zµν(φ)∂µφ∂νφ+φ
MplT
• At low energyZµν ≈ gµν• At high energyZµν ≫ 1
• The field is strongly coupled to itself and becomes weakly coupled toexternal sources
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Vainshtein
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
= −12
Zµν(φ)∂µφ∂νφ+φ
MplT
Zµν ≈ gµν +1Λ3∂µ∂νφ+
1Λ6
(
∂µ∂νφ)2
+ · · ·
• At low energyZµν ≈ gµν• At high energyZµν ≫ 1
• The field is strongly coupled to itself and becomes weakly coupled toexternal sources
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Vainshtein
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
= −12
Zµν(φ)∂µφ∂νφ+φ
MplT
Zµν ≈ gµν +1Λ3∂µ∂νφ+
1Λ6
(
∂µ∂νφ)2
+ · · ·
• At low energyZµν ≈ gµν• At high energyZµν ≫ 1
• The field is strongly coupled to itself and becomes weakly coupled toexternal sources
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Vainshtein
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
= −12
Zµν(φ)∂µφ∂νφ+φ
MplT
Zµν ≈ gµν +1Λ3∂µ∂νφ+
1Λ6
(
∂µ∂νφ)2
+ · · ·
• At low energyZµν ≈ gµν• At high energyZµν ≫ 1
• The field is strongly coupled to itself and becomes weakly coupled toexternal sources
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Vainshtein
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
= −12
Zµν(φ)∂µφ∂νφ+φ
MplT
Zµν ≈ gµν +1Λ3∂µ∂νφ+
1Λ6
(
∂µ∂νφ)2
+ · · ·
• At low energyZµν ≈ gµν• At high energyZµν ≫ 1
−12
Zµν(φ)∂µφ∂νφ = −12
(
∂φ)2, φ =
φ√Z
• The field is strongly coupled to itself and becomes weakly coupled toexternal sources
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Vainshtein
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
= −12
Zµν(φ)∂µφ∂νφ+φ
MplT
Zµν ≈ gµν +1Λ3∂µ∂νφ+
1Λ6
(
∂µ∂νφ)2
+ · · ·
• At low energyZµν ≈ gµν• At high energyZµν ≫ 1
−12
Zµν(φ)∂µφ∂νφ = −12
(
∂φ)2, φ =
φ√Z
φ
MplT =
φ√ZMpl
T ≪ 1
• The field is strongly coupled to itself and becomes weakly coupled toexternal sources
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Vainshtein
L = −12
Zµν(φ)∂µφ∂νφ− V(φ) + A(φ)T
= −12
Zµν(φ)∂µφ∂νφ+φ
MplT
Zµν ≈ gµν +1Λ3∂µ∂νφ+
1Λ6
(
∂µ∂νφ)2
+ · · ·
• At low energyZµν ≈ gµν• At high energyZµν ≫ 1
−12
Zµν(φ)∂µφ∂νφ = −12
(
∂φ)2, φ =
φ√Z
φ
MplT =
φ√ZMpl
T ≪ 1
• The field is strongly coupled to itself and becomes weakly coupled toexternal sources
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
DGP• Dvali, Gabadadze, Porrati (2000).
• The 3-brane is embedded in a Minkowski bulk spacetime with infinitelylarge 5th extra dimensions:
• Boundary effective action (Luty,Porrati,Rattazzi: 2003)• ADM decomposition(gµν ,N,Nµ)• GMN = ηMN + hMN
• We focus on the scalar mode (φ)
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
DGP• Dvali, Gabadadze, Porrati (2000).
• The 3-brane is embedded in a Minkowski bulk spacetime with infinitelylarge 5th extra dimensions:
S= M3(5)
∫
d5X√−GR+ M2
pl
∫
d4x√−gR+ Sm
• Boundary effective action (Luty,Porrati,Rattazzi: 2003)• ADM decomposition(gµν ,N,Nµ)• GMN = ηMN + hMN
• We focus on the scalar mode (φ)
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
DGP• Dvali, Gabadadze, Porrati (2000).
• The 3-brane is embedded in a Minkowski bulk spacetime with infinitelylarge 5th extra dimensions:
S= M3(5)
∫
d5X√−GR+ M2
pl
∫
d4x√−gR+ Sm
• Boundary effective action (Luty,Porrati,Rattazzi: 2003)• ADM decomposition(gµν ,N,Nµ)• GMN = ηMN + hMN
• We focus on the scalar mode (φ)
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
DGP• Dvali, Gabadadze, Porrati (2000).
• The 3-brane is embedded in a Minkowski bulk spacetime with infinitelylarge 5th extra dimensions:
S= M3(5)
∫
d5X√−GR+ M2
pl
∫
d4x√−gR+ Sm
• Boundary effective action (Luty,Porrati,Rattazzi: 2003)• ADM decomposition(gµν ,N,Nµ)• GMN = ηMN + hMN
• We focus on the scalar mode (φ)
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
DGP• Dvali, Gabadadze, Porrati (2000).
• The 3-brane is embedded in a Minkowski bulk spacetime with infinitelylarge 5th extra dimensions:
S= M3(5)
∫
d5X√−GR+ M2
pl
∫
d4x√−gR+ Sm
• Boundary effective action (Luty,Porrati,Rattazzi: 2003)• ADM decomposition(gµν ,N,Nµ)• GMN = ηMN + hMN
• We focus on the scalar mode (φ)
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
DGP• Dvali, Gabadadze, Porrati (2000).
• The 3-brane is embedded in a Minkowski bulk spacetime with infinitelylarge 5th extra dimensions:
S= M3(5)
∫
d5X√−GR+ M2
pl
∫
d4x√−gR+ Sm
• Boundary effective action (Luty,Porrati,Rattazzi: 2003)• ADM decomposition(gµν ,N,Nµ)• GMN = ηMN + hMN
• We focus on the scalar mode (φ)
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
DGP• Dvali, Gabadadze, Porrati (2000).
• The 3-brane is embedded in a Minkowski bulk spacetime with infinitelylarge 5th extra dimensions:
S= M3(5)
∫
d5X√−GR+ M2
pl
∫
d4x√−gR+ Sm
• Boundary effective action (Luty,Porrati,Rattazzi: 2003)• ADM decomposition(gµν ,N,Nµ)• GMN = ηMN + hMN
• We focus on the scalar mode (φ)
S=
∫
d4x[
−3(∂φ)2 − 1Λ3
φ(∂φ)2 + 2φ
MplT]
, Λ =M2
(5)
Mpl
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
DGP• Dvali, Gabadadze, Porrati (2000).
• The 3-brane is embedded in a Minkowski bulk spacetime with infinitelylarge 5th extra dimensions:
S= M3(5)
∫
d5X√−GR+ M2
pl
∫
d4x√−gR+ Sm
• Boundary effective action (Luty,Porrati,Rattazzi: 2003)• ADM decomposition(gµν ,N,Nµ)• GMN = ηMN + hMN
• We focus on the scalar mode (φ)
S=
∫
d4x[
−3(∂φ)2 − 1Λ3
φ(∂φ)2 + 2φ
MplT]
, Λ =M2
(5)
Mpl
Zµν = gµν[
6+2Λ3
φ]
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Decoupling limit
Lφ = −3(∂φ)2 − 1Λ3
φ(∂φ)2 + 2φ
MplT
• The effective action is local
• The equation of motion remain second order in derivative
• The equations of motion are invariant under shift and Galilean globaltransformationsφ→ φ+ c+ vµxµ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Decoupling limit
Lφ = −3(∂φ)2 − 1Λ3
φ(∂φ)2 + 2φ
MplT
• The effective action is local
• The equation of motion remain second order in derivative
• The equations of motion are invariant under shift and Galilean globaltransformationsφ→ φ+ c+ vµxµ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Decoupling limit
Lφ = −3(∂φ)2 − 1Λ3
φ(∂φ)2 + 2φ
MplT
• The effective action is local
• The equation of motion remain second order in derivative
3φ+1Λ3
(
(φ)2 − (∂µ∂νφ)2)
= −T
Mpl
• The equations of motion are invariant under shift and Galilean globaltransformationsφ→ φ+ c+ vµxµ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Decoupling limit
Lφ = −3(∂φ)2 − 1Λ3
φ(∂φ)2 + 2φ
MplT
• The effective action is local
• The equation of motion remain second order in derivative
3φ+1Λ3
(
(φ)2 − (∂µ∂νφ)2)
= −T
Mpl
• The equations of motion are invariant under shift and Galilean globaltransformationsφ→ φ+ c+ vµxµ
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Decoupling limit
Lφ = −3(∂φ)2 − 1Λ3
φ(∂φ)2 + 2φ
MplT
• The effective action is local
• The equation of motion remain second order in derivative
3φ+1Λ3
(
(φ)2 − (∂µ∂νφ)2)
= −T
Mpl
• The equations of motion are invariant under shift and Galilean globaltransformationsφ→ φ+ c+ vµxµ
⇒ Galileon models
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Fifth force
3φ+1Λ3
(
(φ)2 − (∂µ∂νφ)2)
= −T
Mpl
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Fifth force
3φ+1Λ3
(
(φ)2 − (∂µ∂νφ)2)
= −T
Mpl
Fφ =34
rΛ3(
−1+
√
1+r3
v
r3
)
,
r3v =
8M9MplΛ3
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Fifth force
3φ+1Λ3
(
(φ)2 − (∂µ∂νφ)2)
= −T
Mpl
Fφ =34
rΛ3(
−1+
√
1+r3
v
r3
)
,
r3v =
8M9MplΛ3
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Fifth force
3φ+1Λ3
(
(φ)2 − (∂µ∂νφ)2)
= −T
Mpl
Fφ =34
rΛ3(
−1+
√
1+r3
v
r3
)
,
r3v =
8M9MplΛ3
⇒ The fifth force is suppressed locally
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Conclusion
• Three ways to hide the additional degrees of freedom• Chameleon• Symmetron• Vainshtein
• Nontrivial interactions between the scalar field and matterallowed
• Candidate for dark energy
• Bounds from laboratories, solar system and astrophysical experimentsare exponentially relaxed
• The best model is the cosmological constant
ご清聴ありがとうございました
DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN
Conclusion
• Three ways to hide the additional degrees of freedom• Chameleon• Symmetron• Vainshtein
• Nontrivial interactions between the scalar field and matterallowed
• Candidate for dark energy
• Bounds from laboratories, solar system and astrophysical experimentsare exponentially relaxed
• The best model is the cosmological constant
ご清聴ありがとうございました
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