Modified theories of gravity and mechanisms of …...2012/10/02 · ⇒ The Universe is accelerating DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN Dark Energy in a nutshell

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



Chameleon

Symmetron

Vainshtein


Dark Energy in a nutshell• Friedmann equation

3H2 = ρ



3H2 = ρ+ Λ


ΩDE = 0.72± 0.015, −1.11< wDE = P/ρ < −0.86


ΩDE = 0.72± 0.015, −1.11< wDE = P/ρ < −0.86

⇒ The Universe is accelerating



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



3H2 = ρ+ Λ


ρvac/ρΛ ≈ 1055


3H2 = ρ+ ρDE


3H2 + H = ρ



3H2 = ρ


ρvac/ρΛ ≈ 1055


3H2 = ρ+ ρDE


3H2 + H = ρ



3H2 = ρ


ρvac/ρΛ ≈ 1055


3H2 = ρ+ ρDE


3H2 + H = ρ



3H2 = ρ


ρvac/ρΛ ≈ 1055


3H2 = ρ+ ρDE


3H2 + H = ρ



3H2 = ρ


ρvac/ρΛ ≈ 1055


3H2 = ρ+ ρDE


3H2 + H = ρ


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


• 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.


• 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.


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)


Three ways to hide

L = −12



• Quintessence





Three ways to hide

L = −12



• QuintessenceL = −1

2gµν∂µφ∂νφ− V(φ) + T





Three ways to hide

L = −12









Three ways to hide

L = −12







L = −12

gµν∂µφ∂νφ− V(φ) + A(φ)T



Three ways to hide

L = −12







L = −12




Three ways to hide

L = −12







L = −12




Three ways to hide

L = −12







L = −12



L = −12

Zµν(φ)∂µφ∂νφ+ A(φ)T



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:


Chameleon

S=

∫

d4x√−g

(

R−12(∂φ)2 − V(φ)

)



φ = V,φ + A,φρ



Chameleon

S=

∫

d4x√−g

(

R−12(∂φ)2 − V(φ)

)



φ = V,φ + A,φρ


φ = Veff,φ

Veff(φ) ≡ V(φ) + A(φ)ρ


Dynamics governed by effective potential

φ = Veff,φ, with Veff(φ) ≡ V(φ) + A(φ)ρ

Largeρ Smallρ


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 → ∞


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∞


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


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.




−→F φ

M1= − β

Mpl

−→∇φ


Fφ

M1=

GMc

r2(6β2∆Rc

Rce−m∞r)




−→F φ

M1= − β

Mpl

−→∇φ


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


• 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


• 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


Gravity Test

F12 =GNm1m2

r2(1+ αe−mr)

α = 10−8



Chameleon

Symmetron

Vainshtein


Symmetron



Symmetron


V(φ) = −12µ2φ2 +

14λφ4

A(φ) = 1+φ2

2M2


Symmetron


V(φ) = −12µ2φ2 +

14λφ4

A(φ) = 1+φ2

2M2

Veff(φ) =12

( ρ

M2− µ2

)

φ2 +14λφ4


Symmetron

Veff(φ) =12

( ρ

M2− µ2

)

φ2 +14λφ4

Large density Low density

Coupling to matter≃φ2

M2ρ


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


Conclusion

• Strengthens:• Useful for many models:f (R,G)• Fifth force is suppressed locally• Interesting cosmological consequences


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



Chameleon

Symmetron

Vainshtein


Vainshtein

L = −12


• At low energyZµν ≈ gµν• At high energyZµν ≫ 1

• The field is strongly coupled to itself and becomes weakly coupled toexternal sources


Vainshtein

L = −12


= −12

Zµν(φ)∂µφ∂νφ+φ

MplT




Vainshtein

L = −12


= −12


MplT

Zµν ≈ gµν +1Λ3∂µ∂νφ+

1Λ6

(

∂µ∂νφ)2

+ · · ·




Vainshtein

L = −12


= −12


MplT


1Λ6

(

∂µ∂νφ)2

+ · · ·




Vainshtein

L = −12


= −12


MplT


1Λ6

(

∂µ∂νφ)2

+ · · ·




Vainshtein

L = −12


= −12


MplT


1Λ6

(

∂µ∂νφ)2

+ · · ·


−12

Zµν(φ)∂µφ∂νφ = −12

(

∂φ)2, φ =

φ√Z



Vainshtein

L = −12


= −12


MplT


1Λ6

(

∂µ∂νφ)2

+ · · ·


−12


(

∂φ)2, φ =

φ√Z

φ

MplT =

φ√ZMpl

T ≪ 1



Vainshtein

L = −12


= −12


MplT


1Λ6

(

∂µ∂νφ)2

+ · · ·


−12


(

∂φ)2, φ =

φ√Z

φ

MplT =

φ√ZMpl

T ≪ 1



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)




S= M3(5)

∫

d5X√−GR+ M2

pl

∫

d4x√−gR+ Sm






S= M3(5)

∫

d5X√−GR+ M2

pl

∫

d4x√−gR+ Sm






S= M3(5)

∫

d5X√−GR+ M2

pl

∫

d4x√−gR+ Sm






S= M3(5)

∫

d5X√−GR+ M2

pl

∫

d4x√−gR+ Sm






S= M3(5)

∫

d5X√−GR+ M2

pl

∫

d4x√−gR+ Sm






S= M3(5)

∫

d5X√−GR+ M2

pl

∫

d4x√−gR+ Sm



S=

∫

d4x[

−3(∂φ)2 − 1Λ3

φ(∂φ)2 + 2φ

MplT]

, Λ =M2

(5)

Mpl




S= M3(5)

∫

d5X√−GR+ M2

pl

∫

d4x√−gR+ Sm



S=

∫

d4x[

−3(∂φ)2 − 1Λ3

φ(∂φ)2 + 2φ

MplT]

, Λ =M2

(5)

Mpl

Zµν = gµν[

6+2Λ3

φ]


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


Decoupling limit

Lφ = −3(∂φ)2 − 1Λ3

φ(∂φ)2 + 2φ

MplT





Decoupling limit

Lφ = −3(∂φ)2 − 1Λ3

φ(∂φ)2 + 2φ

MplT



3φ+1Λ3

(

(φ)2 − (∂µ∂νφ)2)

= −T

Mpl



Decoupling limit

Lφ = −3(∂φ)2 − 1Λ3

φ(∂φ)2 + 2φ

MplT



3φ+1Λ3

(

(φ)2 − (∂µ∂νφ)2)

= −T

Mpl



Decoupling limit

Lφ = −3(∂φ)2 − 1Λ3

φ(∂φ)2 + 2φ

MplT



3φ+1Λ3

(

(φ)2 − (∂µ∂νφ)2)

= −T

Mpl


⇒ Galileon models


Fifth force

3φ+1Λ3

(

(φ)2 − (∂µ∂νφ)2)

= −T

Mpl


Fifth force

3φ+1Λ3

(

(φ)2 − (∂µ∂νφ)2)

= −T

Mpl

Fφ =34

rΛ3(

−1+

√

1+r3

v

r3

)

,

r3v =

8M9MplΛ3


Fifth force

3φ+1Λ3

(

(φ)2 − (∂µ∂νφ)2)

= −T

Mpl

Fφ =34

rΛ3(

−1+

√

1+r3

v

r3

)

,

r3v =

8M9MplΛ3


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


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


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

Documents

Modified theories of gravity and mechanisms of …...2012/10/02 · ⇒ The Universe is accelerating DARK ENERGY IN A NUTSHELL CHAMELEON SYMMETRON VAINSHTEIN Dark Energy in a nutshell