Higgs Phase of Gravity in String Theory - LUTh

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 sun
observed in 1800’s… mercury
which people tried to
explain with a “dark
planet”, Vulcan,
sun
vulcan
mercury
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

Order
Parameter
Higgs Mechanism Ghost Condensation
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
Φ V(
Φ)
∂ φ
− m
Φ
2
Φ
2
( ) 2
( ∂φ)
Yukawa-type Oscillating in space
Growing in time
μ
P
φ
2 −
φ

For simplicity
L
φ
E.O.M.
= P
( 2 ) ( ∂φ)
in FRW background.
[ ]
3
∂ ′ ⋅φ
= a P
t
0
P
′φ → 0 P as a → ∞
′ φ
P
φ = 0 or ( ) 0
2 =
(unstable ghosty
background)
φ

Order
Parameter
Higgs Mechanism Ghost Condensation
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
Φ V(
Φ)
∂ φ
− m
Φ
2
Φ
2
( ) 2
( ∂φ)
Yukawa-type Oscillating in space
Growing in time
μ
P
φ
2 −
φ

Systematic construction of Low-
energy effective theory
Backgrounds characterized by
∂ φ
and timelike
μ
≠
0
Background metric is maximally
symmetric, either Minkowski or dS.

Gauge choice: φ(
t , x)
= t.
Residual symmetry:
x
→
x′
( t,
x)
(Unitary gauge)
Write down most general action invariant under
this residual symmetry.
( Action for π: undo unitary gauge!)
g = +
Start with flat background μν μν
δh = ∂ ξ + ∂
μν
μ
i
Under residual ξ
ν
0 h0i
0
i
ν
ξ
μ
ij
μν η h
δh = , δ = ∂ ξ , δh
= ∂ ξ + ∂ ξ
00
π δφ
≡ =
i
j
0
j
i

Action invariant under ξi ( ) 2
h00
( ) 2
OK
1
h0i 2
2, ij
K K K OK
ij
( )
K = ∂ h −∂ h −∂h
ij 0 ij j 0i i 0 j
( ) 2
4⎧α1 2 α2
ij ⎫
Leff = LEH+ M ⎨ h00 − K − K K
2 2 ij + ⎬
⎩ M M ⎭
Action for π
ξ 0 = π
h00 →h00 −2∂0π K → K +∂ ∂ π
ij ij i j
( ) ( ) 2
4⎧2α 1
2
Leff = LEH+ M ⎨ h00 −2 π − K +∇ π
2
⎩
M
α
2 ( ij i j
K )( K )
⎫
− +∇∇ π 2
ij +∇∇ i jπ
+ ⎬
M
⎭

( ) ( ) 2
4⎧2α 1
2
Leff = LEH+ M ⎨ h00 −2 π − K +∇ π
2
⎩
M
α
2 ( ij i j
K )( K )
⎫
− +∇∇ π 2
ij +∇∇ i jπ
+ ⎬
M
⎭
Dispersion relation
2 4
2 k
α
ω =
M
Coupling to gravity
2
2 α 4 αM
ω = k − k
2 2
M 2MPl Jeans-like Jeans like (IR) instability
ω 2 < 0 for k < k c = M 2 /2M pl
r J ~ M pl /M 2 , t J ~ M pl 2 /M 3
k 2 term is forbidden by symmetry
2
O(M 2 /M Pl 2 ) correction

Order
Parameter
Higgs Mechanism Ghost Condensation
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
Φ V(
Φ)
∂ φ
− m
Φ
2
Φ
2
( ) 2
( ∂φ)
Yukawa-type Oscillating in space
Growing in time
μ
P
φ
2 −
φ

Bounds on symmetry breaking scale M
0
Arkani-Hamed, Cheng, Luty and Mukohyama and Wiseman, hep-ph/0507120
allowed
Jeans Instability
(sun)
Twinkling from Lensing
(CMB)
Supernova time-delay
100GeV 1TeV
ruled out
ruled out
ruled out
c.f. Gauged ghost condensation allows
much higher M (M < 10 12 GeV)
Cheng, Luty, Mukohyama and Thaler, hep-th/0603010
M

Applications to Cosmology (I)
Ghost Inflation
φ
≠
0!
NOT SLOW ROLL
Scale-invariant perturbations
δρ
ρ
Hδπ
φ
⎛ H ⎞
⎜ ⎟
⎝ M ⎠
~ δπ
~
5/
4
Arkani-Hamed, Creminelli, Mukohyama and Zaldarriaga
hep-th/0312100
φ
~ M ⋅(
H / M
scaling dim of π
1/
4
)
[compare ]
M
H
Pl
eg. hybrid type
ε
φ ~ 2
M

E
dt
dx
→
→
→
π →
rE
r
r
r
−1
−1/
2
1/
4
dt
π
dx
Make
invariant
Scaling dim of π is 1/4!
not the same as the mass dim 1!
2 2 ⎤
∫
3 ⎡1
2 α(
∇ π )
dtd x⎢
π − +
2 ⎥
⎣2
M ⎦
− P′
M
cf. This is the reason why higher terms such as
2
∇ are irrelevant at low E.
∫
3
~ 2
) ( π
π
dtd x
M
4 2
( )( ∇π
)

Prediction of Large (visible) non-Gauss.
Leading non-linear interaction
2
π ( ∇π
)
2
M
non-G of ~
1/4
⎛ H ⎞ scaling dim of op.
~
⎜ ⎟
⎝M⎠ ⎛δρ ⎞
⎜ ⎟
⎝ ρ ⎠
1/5
[Really “0.1” ~ 10-2 × δρ / ρ . VISIBLE.
( ) 1/5
Compare with usual inflation where
δρ/ ρ
( )
non-G ~ ~ 10 -5 too small.]

3-point function for ghost inflation
k3/ k1
k2 / k1
3-point function for “local” non-G
k3/ k1
k2 / k1
1 2 3
( 1, 2, 3) , 6
1 1 1
k k ⎛ ⎞
Fk k k = F⎜
⎟
k ⎝ k k ⎠
k3/ k1
1
k2 / k1
3
ς = ς − f ⋅ ς − ς
5
( 2 2 )
G NL G G

Cosmological Application (II)
Alternative to DE/DM
• For FRW universe, it behaves like c.c. + CDM. CDM
Ghost condensation
dS
( 2
( ∂ ) )
P φ
Λ=0
Λeff >0 CDM
dS
( Φ)
• Clustering properties remain unexplored and
may be different from c.c. + CDM.
V
Usual Higgs mechanism
Λ < 0 Λ < 0
eff
Λ=0
eff
Φ

Cosmic Uroboros
String/M theory?
Higgs phase
DE/DM
of gravity

Warped Throat
KKLT setup
10D = 4D universe x 6D internal space
CY
Shape & Volume
stabilized
Anti-D3-branes
Non-SUSY NS5-brane
Kachru, Pearson & Verlinde (2002)

Correspondence principle
Stringy
Object
Size > R grav
Non-SUSY
Non SUSY
NS5-brane
NS5 brane
( ) 2
M / N 1
RR 3 ∼
> gN s 3
Mukohyama, hep-th/0610254
Horowitz & Polchinski (1997)
Black-Brane
Black Brane
Size < R grav
Black-Brane
Black Brane
MRR : # of R-R flux
N3 : # of D3
‘s
gs : string coupling

4D
Universe ⊗
Black brane at the tip
Non-extremal
Non extremal
black 3-brane 3 brane

x i
Spontaneous Lorentz breaking
• The (3+1)-dim spacetime is spanned by
( t , xi ).
Non-extremal
black 3-brane
t
r = const.
r
x i
t
Projection onto
t = const. surfaces
Projection onto
x i = const. surfaces
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 g GCC 2 < gc 2 .
2
− ω =
2
k =
Charged black string
with r + =2
[Gregory-Laflamme, NPB428, 399 (1994)]

• 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.
Conjecture Mukohyama, hep-th/0610254
Low-E EFT: Jeans-like instability
DUAL
Geometrical: GL instability

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
( ) 2
M RR / N 3 > gsN 3 1
∼
is a UV completion (string theory version) of
the gauged ghost condensation.

Order
Parameter
Higgs Mechanism Ghost Condensation
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
Φ V(
Φ)
∂ φ
− m
Φ
2
Φ
2
( ) 2
( ∂φ)
Yukawa-type Oscillating in space
Growing in time
μ
P
φ
2 −
φ