Higgs Phase of Gravity Shinji Mukohyama (University of Tokyo) - - PowerPoint PPT Presentation

Higgs Phase of Gravity

Shinji Mukohyama (University of Tokyo) December 12, 2006 @ IHP

Arkani-Hamed, Cheng, Luty and Mukohyama, hep-th/0312099 Arkani-Hamed, Creminelli, Mukohyama and Zaldarriaga, hep-th/0312100 Arkani-Hamed, Cheng, Luty and Mukohyama and Wiseman, hep-ph/0507120 Cheng, Luty, Mukohyama and Thaler, hep-th/0603010 Mukohyama, hep-th/0502189, hep-th/0607181, hep-th/0610254

Motivation

• Gravity at long distances

Flattening galaxy rotation curves Dimming supernovae accelerating universe

• Usual explanation: new forms of matter

(DARK MATTER) and energy (DARK ENERGY).

Historical remark:

Precession of perihelion

• bserved in 1800’s…

mercury

sun

which people tried to explain with a “dark planet”, Vulcan,

mercury

vulcan sun But the right answer wasn’t “dark planet”, it was “change gravity” from Newton to GR.

Can we change gravity in IR to address these mysteries?

Change theory? Macroscopic UV scale… Change state (phase)? Higgs phase of gravity The simplest: Ghost Condensation

Arkani-Hamed, Cheng, Luty and Mukohyama, hep-th/0312099

Higgs Mechanism Ghost Condensation Order Parameter Instability Tachyon Ghost Condensate V’=0, V’’>0 P’=0, P’’>0 Spontaneous breaking Gauge symmetry Lorents symmetry (Time translation) Modifying Gauge force Gravitational force New potential Yukawa-type Oscillating in space Growing in time

φ

• (

)

2

) ( φ ∂ P Φ ( ) Φ V

Φ φ

μ

∂

2 2Φ

− m

2

φ

• −

For simplicity

( )

2

) ( φ

φ

∂ = P L

P in FRW background. E.O.M.

] [

3

= ⋅ ′ ∂ φ

• P

a

t

→ ′φ

• P

as

∞ → a = φ

• r

) (

2 =

′ φ

• P

(unstable ghosty background)

φ

Higgs Mechanism Ghost Condensation Order Parameter Instability Tachyon Ghost Condensate V’=0, V’’>0 P’=0, P’’>0 Spontaneous breaking Gauge symmetry Lorents symmetry (Time translation) Modifying Gauge force Gravitational force New potential Yukawa-type Oscillating in space Growing in time

φ

• (

)

2

) ( φ ∂ P Φ ( ) Φ V

Φ φ

μ

∂

2 2Φ

− m

2

φ

• −

• and timelike

Background metric is maximally symmetric, either Minkowski or dS. Systematic construction of Low- energy effective theory Backgrounds characterized by

≠ ∂ φ

μ

Gauge choice:

. ) , ( t x t =

• φ

(Unitary gauge)

) , ( x t x x

Residual symmetry:

• ′

→

Write down most general action invariant under this residual symmetry. ( Action for π: undo unitary gauge!) Start with flat background

μν μν μν

η h g + =

μ ν ν μ μν

ξ ξ δ ∂ + ∂ = h

i

Under residual ξ

i j j i ij i i

h h h ξ ξ δ ξ δ δ ∂ + ∂ = ∂ = = , ,

00

π δφ ≡ =

Action invariant under Action invariant under ξ ξi

i

( )

2 00

h

( )

2 0i

h

2, ij ij

K K K

( )

1 2

ij ij j i i j

K h h h = ∂ − ∂ − ∂

OK OK

( )

2 4 2 1 2 00 2 2 ij eff EH ij

L L M h K K K M M α α ⎧ ⎫ = + − − + ⎨ ⎬ ⎩ ⎭

• Action for

Action for π π

ij ij i j

K K π → + ∂ ∂

00 00

2 h h π → − ∂

ξ0 = π

( )

( )

2 2 4 2 1 00 2

2

eff EH

L L M h K M α π π ⎧ = + − − + ∇ ⎨ ⎩

• (

)( )

2 2 ij i j ij i j

K K M α π π ⎫ − + ∇ ∇ + ∇ ∇ + ⎬ ⎭

Dispersion relation Dispersion relation

2 4 2 k

M α ω = ( )

( )

2 2 4 2 1 00 2

2

eff EH

L L M h K M α π π ⎧ = + − − + ∇ ⎨ ⎩

• (

)( )

2 2 ij i j ij i j

K K M α π π ⎫ − + ∇ ∇ + ∇ ∇ + ⎬ ⎭

• 2

2 4 2 2

2

Pl

M k k M M α α ω = −

2

Coupling to gravity k2 term is forbidden by symmetry O(M2/MPl

2) correction

Jeans Jeans-

• like (IR) instability

like (IR) instability

ω2 < 0 for k < kc = M2/2Mpl

rJ ~ Mpl/M2, tJ ~ Mpl

2/M3

Higgs Mechanism Ghost Condensation Order Parameter Instability Tachyon Ghost Condensate V’=0, V’’>0 P’=0, P’’>0 Spontaneous breaking Gauge symmetry Lorents symmetry (Time translation) Modifying Gauge force Gravitational force New potential Yukawa-type Oscillating in space Growing in time

φ

• (

)

2

) ( φ ∂ P Φ ( ) Φ V

Φ φ

μ

∂

2 2Φ

− m

2

φ

• −

Bounds on symmetry breaking scale M

M

Jeans Instability (sun) 100GeV 1TeV Supernova time-delay ruled out ruled out ruled out

allowed

Twinkling from Lensing (CMB) Arkani-Hamed, Cheng, Luty and Mukohyama and Wiseman, hep-ph/0507120

c.f. Gauged ghost condensation allows much higher M (M < 1012 GeV)

Cheng, Luty, Mukohyama and Thaler, hep-th/0603010

Applications to Cosmology (I)

! ≠ φ

• φ

NOT SLOW ROLL Scale-invariant perturbations

ρ δρ φ δπ

• H

~

δπ

4 / 1

) / ( M H M ⋅

~

4 / 5

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ M H φ ~

2

~

M

scaling dim of π [compare ]

ε

Pl

M H

Ghost Inflation Ghost Inflation

• eg. hybrid type

Arkani-Hamed, Creminelli, Mukohyama and Zaldarriaga hep-th/0312100

• cf. This is the reason why higher terms such as

are irrelevant at low E.

∫

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ∇ −

• 2

2 2 2 3

) ( 2 1 M x dtd π α π

π π

4 / 1 2 / 1 1

r dx r dx dt r dt rE E → → → →

− −

Make invariant

Scaling dim of π is 1/4! not the same as the mass dim 1!

2 4

) )( ( π ∇ ′ − M P

∫

∇

2 2 3

~ ) ( M x dtd π π

Prediction of Large (visible) non-Gauss. Leading non-linear interaction

2 2

( ) M π π ∇

• non-G of ~

H M ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ 1/4

scaling dim of op.

1/5

δρ ρ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠

~

[Really “0.1” ~ 10-2. VISIBLE. Compare with usual inflation where non-G ~ ~ 10-5 too small.]

( )

1/5

/ δρ ρ ×

( )

/ δρ ρ

3 2 1 2 3 6 1 1 1

1 ( , , ) , k k F k k k F k k k ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠

3-point function for ghost inflation 3-point function for “local” non-G

3 1

/ k k

2 1

/ k k

2 1

/ k k

3 1

/ k k

( )

2 2

3 5

G NL G G

f ς ς ς ς = − ⋅ −

2 1

/ k k 1

3 1

/ k k

Cosmological Application (II)

Alternative to DE/DM Alternative to DE/DM

• For FRW universe, it behaves like c.c. + CDM

behaves like c.c. + CDM.

• Clustering properties remain unexplored and

may be different from c.c. + CDM.

dS dS Λ=0

Ghost condensation

Φ ( ) Φ V

Usual Higgs mechanism

eff

Λ <

eff

Λ <

Λ=0

( )

2

( ) P φ ∂

Λeff>0 CDM

Cosmic Cosmic Uroboros Uroboros

DE/DM DE/DM

String/M theory? String/M theory?

Higgs phase Higgs phase

• f gravity
• f gravity

KKLT setup KKLT setup

Anti-D3-branes Warped Throat Warped Throat 10D = 4D universe x 6D internal space CY CY Shape & Volume Shape & Volume stabilized stabilized Non-SUSY NS5-brane

Kachru, Pearson & Verlinde (2002)

Correspondence principle Correspondence principle Stringy Stringy Object Object Black Black-

• Brane

Brane

Size > Rgrav Size < Rgrav

Horowitz & Polchinski (1997)

Black Black-

• Brane

Brane Non Non-

• SUSY

SUSY NS5 NS5-

• brane

brane

( )

2 3 3

/ 1

RR s

M N g N >

• ∼

MRR: # of R-R flux : # of ‘s gs : string coupling

3

N 3 D

Mukohyama, hep-th/0610254

Black Black brane brane at the tip at the tip

Non Non-

• extremal

extremal black 3 black 3-

• brane

brane

4D Universe ⊗

Spontaneous Lorentz breaking Spontaneous Lorentz breaking

• The (3+1)-dim spacetime is spanned by

( t , xi ). Non-extremal black 3-brane

Projection onto t = const. surfaces Projection onto xi = const. surfaces

r

r = const.

xi t

xi t

Warp factors for the tt-component and the ij-components are different.

Spontaneous Lorentz breaking!

Gauged Ghost Condensation

GL instability

• Non-extremal Black branes are gravitationally
• unstable. [Gregory-Laflamme, PRL70, 2837

(1993); NPB428, 399 (1994)]

• The dispersion relation is similar to that for

the NG boson in our setup with gGCC

2 < gc 2.

[Gregory-Laflamme, NPB428, 399 (1994)]

2

ω − =

2

k =

• Charged black string

with r+=2

• In our geometrical setup there is a black

brane at the bottom of the warped throat.

• The world-volume of the black brane is

parallel to our world.

Geometrical: GL instability Low-E EFT: Jeans-like instability

DUAL Conjecture Conjecture

Mukohyama, hep-th/0610254

Summary Summary

• Ghost condensation is the simplest Higgs

phase of gravity, including only one Nambu- Goldstone boson. No ghost included.

• Can drive inflation.
• Can be alternative to DE/DM.
• The KKLT setup in the regime of parameters

is a UV completion (string theory version) of the gauged ghost condensation.

( )

2 3 3

/ 1

RR s

M N g N >

• ∼

Higgs Mechanism Ghost Condensation Order Parameter Instability Tachyon Ghost Condensate V’=0, V’’>0 P’=0, P’’>0 Spontaneous breaking Gauge symmetry Lorents symmetry (Time translation) Modifying Gauge force Gravitational force New potential Yukawa-type Oscillating in space Growing in time

φ

• (

)

2

) ( φ ∂ P Φ ( ) Φ V

Φ φ

μ

∂

2 2Φ

− m

2

φ

• −