Modified theories of gravity and mechanisms of …...2012/10/02 · ⇒ The Universe is...

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

⇒ 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

1+ α exp−r/λ]

g00 = −1+ 2GMrc2

− 2βPPN(GM

)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.

• 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

gµν∂µφ∂νφ− V(φ) + A(φ)T

Three ways to hide

L = −12

Three ways to hide

L = −12

Three ways to hide

L = −12

Zµν(φ)∂µφ∂νφ+ A(φ)T

Chameleon

Symmetron

Vainshtein

Chameleon

d4x√−g

R−12(∂φ)2 − V(φ)

+ Sm[ψm;A2(φ)gµν ]

• Equation of motion

• Dynamics governed by effective potential:

Chameleon

d4x√−g

R−12(∂φ)2 − V(φ)

φ = V,φ + A,φρ

Chameleon

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

Mc,Ρc

= 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 → ∞

Φ=Φc

VeffHΦL

Φ=Φ¥

0 20 40 60 80 10099.90

φ(r) ≈ −

4πMpl

Mce−m∞(r−Rc)

r+φ∞

Φ=Φc

VeffHΦL

DRcVeffHΦL

Φ=Φ¥

0 20 40 60 80 1000

φ(r) ≈ −

4πMpl

3∆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

−→∇φ

⇒ φ is the potential for the chameleon force.

−→F φ

M1= − β

−→∇φ

r2(6β2∆Rc

Rce−m∞r)

−→F φ

M1= − β

−→∇φ

r2(6β2∆Rc

Rce−m∞r)

⇒ F12 =GNm1m2

r2 (1+ αe−mr)

• deviations from Newton’s law are given byα = 6β2∆Rc

• Thin Shell

deviations from Newton’s law are given byα = 6β2∆Rc

Rcwith

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≈ 10−8 ≪ 1

• Thin Shell

deviations from Newton’s law are given byα = 6β2∆Rc

Rcwith

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≈ 10−8 ≪ 1

Gravity Test

F12 =GNm1m2

r2(1+ αe−mr)

α = 10−8

Chameleon

Symmetron

Vainshtein

Symmetron

V(φ) = −12µ2φ2 +

14λφ4

A(φ) = 1+φ2

Symmetron

V(φ) = −12µ2φ2 +

14λφ4

A(φ) = 1+φ2

Veff(φ) =12

M2− µ2

φ2 +14λφ4

Symmetron

Veff(φ) =12

M2− µ2

φ2 +14λφ4

Large density Low density

Coupling to matter≃φ2

Symmetron

Veff(φ) =12

M2− µ2

φ2 +14λφ4

Large density Low density

Coupling to matter≃φ2

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

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

Vainshtein

L = −12

= −12

Zµν ≈ gµν +1Λ3∂µ∂νφ+

∂µ∂νφ)2

+ · · ·

Vainshtein

L = −12

= −12

∂µ∂νφ)2

+ · · ·

Vainshtein

L = −12

= −12

∂µ∂νφ)2

+ · · ·

Vainshtein

L = −12

= −12

∂µ∂νφ)2

+ · · ·

Zµν(φ)∂µφ∂νφ = −12

∂φ)2, φ =

φ√Z

Vainshtein

L = −12

= −12

∂µ∂νφ)2

+ · · ·

∂φ)2, φ =

φ√Z

MplT =

φ√ZMpl

T ≪ 1

Vainshtein

L = −12

= −12

∂µ∂νφ)2

+ · · ·

∂φ)2, φ =

φ√Z

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

d4x√−gR+ Sm

S= M3(5)

d5X√−GR+ M2

d4x√−gR+ Sm

S= M3(5)

d5X√−GR+ M2

d4x√−gR+ Sm

S= M3(5)

d5X√−GR+ M2

d4x√−gR+ Sm

S= M3(5)

d5X√−GR+ M2

d4x√−gR+ Sm

S= M3(5)

d5X√−GR+ M2

d4x√−gR+ Sm

−3(∂φ)2 − 1Λ3

φ(∂φ)2 + 2φ

, Λ =M2

S= M3(5)

d5X√−GR+ M2

d4x√−gR+ Sm

−3(∂φ)2 − 1Λ3

φ(∂φ)2 + 2φ

, Λ =M2

Zµν = gµν[

6+2Λ3

Decoupling limit

Lφ = −3(∂φ)2 − 1Λ3

φ(∂φ)2 + 2φ

• 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

Decoupling limit

Lφ = −3(∂φ)2 − 1Λ3

φ(∂φ)2 + 2φ

3φ+1Λ3

(φ)2 − (∂µ∂νφ)2)

= −T

Decoupling limit

Lφ = −3(∂φ)2 − 1Λ3

φ(∂φ)2 + 2φ

3φ+1Λ3

(φ)2 − (∂µ∂νφ)2)

= −T

Decoupling limit

Lφ = −3(∂φ)2 − 1Λ3

φ(∂φ)2 + 2φ

3φ+1Λ3

(φ)2 − (∂µ∂νφ)2)

= −T

⇒ Galileon models

Fifth force

3φ+1Λ3

(φ)2 − (∂µ∂νφ)2)

= −T

Fifth force

3φ+1Λ3

(φ)2 − (∂µ∂νφ)2)

= −T

Fφ =34

8M9MplΛ3

Fifth force

3φ+1Λ3

(φ)2 − (∂µ∂νφ)2)

= −T

Fφ =34

8M9MplΛ3

Fifth force

3φ+1Λ3

(φ)2 − (∂µ∂νφ)2)

= −T

Fφ =34

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

Modified theories of gravity and mechanisms of …...2012/10/02 · ⇒ The Universe is...

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