Higgs Phase of Gravity in String Theory - LUTh
Higgs Phase of Gravity in String Theory - LUTh
Higgs Phase of Gravity in String Theory - LUTh
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<strong>Higgs</strong> <strong>Phase</strong> <strong>of</strong> <strong>Gravity</strong><br />
Sh<strong>in</strong>ji Mukohyama<br />
(University <strong>of</strong> Tokyo)<br />
December 12, 2006 @ IHP<br />
Arkani-Hamed, Cheng, Luty and Mukohyama, hep-th/0312099<br />
Arkani-Hamed, Crem<strong>in</strong>elli, Mukohyama and Zaldarriaga, hep-th/0312100<br />
Arkani-Hamed, Cheng, Luty and Mukohyama and Wiseman, hep-ph/0507120<br />
Cheng, Luty, Mukohyama and Thaler, hep-th/0603010<br />
Mukohyama, hep-th/0502189, hep-th/0607181, hep-th/0610254
Motivation<br />
• <strong>Gravity</strong> at long distances<br />
Flatten<strong>in</strong>g galaxy rotation curves<br />
Dimm<strong>in</strong>g supernovae<br />
accelerat<strong>in</strong>g universe<br />
• Usual explanation: new forms <strong>of</strong> matter<br />
(DARK MATTER) and energy (DARK<br />
ENERGY).
Historical remark:<br />
Precession <strong>of</strong> perihelion sun<br />
observed <strong>in</strong> 1800’s… mercury<br />
which people tried to<br />
expla<strong>in</strong> with a “dark<br />
planet”, Vulcan,<br />
sun<br />
vulcan<br />
mercury<br />
But the right answer wasn’t “dark planet”, it was “change<br />
gravity” from Newton to GR.
Can we change gravity <strong>in</strong> IR to<br />
address these mysteries?<br />
Change theory?<br />
Macroscopic UV scale…<br />
Change state (phase)?<br />
<strong>Higgs</strong> phase <strong>of</strong> gravity<br />
The simplest: Ghost Condensation<br />
Arkani-Hamed, Cheng, Luty and Mukohyama, hep-th/0312099
Order<br />
Parameter<br />
<strong>Higgs</strong> Mechanism Ghost Condensation<br />
Instability Tachyon Ghost<br />
Condensate V’=0, V’’>0 P’=0, P’’>0<br />
Spontaneous<br />
break<strong>in</strong>g<br />
Gauge symmetry Lorents symmetry<br />
(Time translation)<br />
Modify<strong>in</strong>g Gauge force Gravitational force<br />
New<br />
potential<br />
Φ V(<br />
Φ)<br />
∂ φ<br />
− m<br />
Φ<br />
2<br />
Φ<br />
2<br />
( ) 2<br />
( ∂φ)<br />
Yukawa-type Oscillat<strong>in</strong>g <strong>in</strong> space<br />
Grow<strong>in</strong>g <strong>in</strong> time<br />
μ<br />
P<br />
φ<br />
2 −<br />
φ
For simplicity<br />
L<br />
φ<br />
E.O.M.<br />
= P<br />
( 2 ) ( ∂φ)<br />
<strong>in</strong> FRW background.<br />
[ ]<br />
3<br />
∂ ′ ⋅φ<br />
= a P<br />
t<br />
0<br />
P<br />
′φ → 0 P as a → ∞<br />
′ φ <br />
P<br />
φ = 0 or ( ) 0<br />
2 =<br />
(unstable ghosty<br />
background)<br />
φ
Order<br />
Parameter<br />
<strong>Higgs</strong> Mechanism Ghost Condensation<br />
Instability Tachyon Ghost<br />
Condensate V’=0, V’’>0 P’=0, P’’>0<br />
Spontaneous<br />
break<strong>in</strong>g<br />
Gauge symmetry Lorents symmetry<br />
(Time translation)<br />
Modify<strong>in</strong>g Gauge force Gravitational force<br />
New<br />
potential<br />
Φ V(<br />
Φ)<br />
∂ φ<br />
− m<br />
Φ<br />
2<br />
Φ<br />
2<br />
( ) 2<br />
( ∂φ)<br />
Yukawa-type Oscillat<strong>in</strong>g <strong>in</strong> space<br />
Grow<strong>in</strong>g <strong>in</strong> time<br />
μ<br />
P<br />
φ<br />
2 −<br />
φ
Systematic construction <strong>of</strong> Low-<br />
energy effective theory<br />
Backgrounds characterized by<br />
∂ φ<br />
and timelike<br />
μ<br />
≠<br />
0<br />
Background metric is maximally<br />
symmetric, either M<strong>in</strong>kowski or dS.
Gauge choice: φ(<br />
t , x)<br />
= t.<br />
Residual symmetry:<br />
<br />
x<br />
→<br />
<br />
x′<br />
( t,<br />
x)<br />
(Unitary gauge)<br />
Write down most general action <strong>in</strong>variant under<br />
this residual symmetry.<br />
( Action for π: undo unitary gauge!)<br />
g = +<br />
Start with flat background μν μν<br />
δh = ∂ ξ + ∂<br />
μν<br />
μ<br />
i<br />
Under residual ξ<br />
ν<br />
0 h0i<br />
0<br />
i<br />
ν<br />
ξ<br />
μ<br />
ij<br />
μν η h<br />
δh = , δ = ∂ ξ , δh<br />
= ∂ ξ + ∂ ξ<br />
00<br />
π δφ<br />
≡ =<br />
i<br />
j<br />
0<br />
j<br />
i
Action <strong>in</strong>variant under ξi ( ) 2<br />
h00<br />
( ) 2<br />
OK<br />
1<br />
h0i 2<br />
2, ij<br />
K K K OK<br />
ij<br />
( )<br />
K = ∂ h −∂ h −∂h<br />
ij 0 ij j 0i i 0 j<br />
( ) 2<br />
4⎧α1 2 α2<br />
ij ⎫<br />
Leff = LEH+ M ⎨ h00 − K − K K<br />
2 2 ij + ⎬<br />
⎩ M M ⎭<br />
<br />
Action for π<br />
ξ 0 = π<br />
h00 →h00 −2∂0π K → K +∂ ∂ π<br />
ij ij i j<br />
( ) ( ) 2<br />
4⎧2α 1<br />
2<br />
Leff = LEH+ M ⎨ h00 −2 π − K +∇ π<br />
2<br />
⎩<br />
M<br />
α <br />
2 ( ij i j<br />
K )( K )<br />
⎫<br />
− +∇∇ π 2<br />
ij +∇∇ i jπ<br />
+ ⎬<br />
M<br />
⎭
( ) ( ) 2<br />
4⎧2α 1<br />
2<br />
Leff = LEH+ M ⎨ h00 −2 π − K +∇ π<br />
2<br />
⎩<br />
M<br />
α <br />
2 ( ij i j<br />
K )( K )<br />
⎫<br />
− +∇∇ π 2<br />
ij +∇∇ i jπ<br />
+ ⎬<br />
M<br />
⎭<br />
Dispersion relation<br />
2 4<br />
2 k<br />
α<br />
ω =<br />
M<br />
Coupl<strong>in</strong>g to gravity<br />
2<br />
2 α 4 αM<br />
ω = k − k<br />
2 2<br />
M 2MPl Jeans-like Jeans like (IR) <strong>in</strong>stability<br />
ω 2 < 0 for k < k c = M 2 /2M pl<br />
r J ~ M pl /M 2 , t J ~ M pl 2 /M 3<br />
k 2 term is forbidden by symmetry<br />
2<br />
O(M 2 /M Pl 2 ) correction
Order<br />
Parameter<br />
<strong>Higgs</strong> Mechanism Ghost Condensation<br />
Instability Tachyon Ghost<br />
Condensate V’=0, V’’>0 P’=0, P’’>0<br />
Spontaneous<br />
break<strong>in</strong>g<br />
Gauge symmetry Lorents symmetry<br />
(Time translation)<br />
Modify<strong>in</strong>g Gauge force Gravitational force<br />
New<br />
potential<br />
Φ V(<br />
Φ)<br />
∂ φ<br />
− m<br />
Φ<br />
2<br />
Φ<br />
2<br />
( ) 2<br />
( ∂φ)<br />
Yukawa-type Oscillat<strong>in</strong>g <strong>in</strong> space<br />
Grow<strong>in</strong>g <strong>in</strong> time<br />
μ<br />
P<br />
φ<br />
2 −<br />
φ
Bounds on symmetry break<strong>in</strong>g scale M<br />
0<br />
Arkani-Hamed, Cheng, Luty and Mukohyama and Wiseman, hep-ph/0507120<br />
allowed<br />
Jeans Instability<br />
(sun)<br />
Tw<strong>in</strong>kl<strong>in</strong>g from Lens<strong>in</strong>g<br />
(CMB)<br />
Supernova time-delay<br />
100GeV 1TeV<br />
ruled out<br />
ruled out<br />
ruled out<br />
c.f. Gauged ghost condensation allows<br />
much higher M (M < 10 12 GeV)<br />
Cheng, Luty, Mukohyama and Thaler, hep-th/0603010<br />
M
Applications to Cosmology (I)<br />
Ghost Inflation<br />
φ <br />
≠<br />
0!<br />
NOT SLOW ROLL<br />
Scale-<strong>in</strong>variant perturbations<br />
δρ<br />
ρ<br />
Hδπ<br />
φ<br />
⎛ H ⎞<br />
⎜ ⎟<br />
⎝ M ⎠<br />
~ δπ<br />
~<br />
5/<br />
4<br />
Arkani-Hamed, Crem<strong>in</strong>elli, Mukohyama and Zaldarriaga<br />
hep-th/0312100<br />
φ<br />
~ M ⋅(<br />
H / M<br />
scal<strong>in</strong>g dim <strong>of</strong> π<br />
1/<br />
4<br />
)<br />
[compare ]<br />
M<br />
H<br />
Pl<br />
eg. hybrid type<br />
ε<br />
φ ~ 2<br />
M
E<br />
dt<br />
dx<br />
→<br />
→<br />
→<br />
π →<br />
rE<br />
r<br />
r<br />
r<br />
−1<br />
−1/<br />
2<br />
1/<br />
4<br />
dt<br />
π<br />
dx<br />
Make<br />
<strong>in</strong>variant<br />
Scal<strong>in</strong>g dim <strong>of</strong> π is 1/4!<br />
not the same as the mass dim 1!<br />
2 2 ⎤<br />
∫<br />
3 ⎡1<br />
2 α(<br />
∇ π )<br />
dtd x⎢<br />
π − + <br />
2 ⎥<br />
⎣2<br />
M ⎦<br />
− P′<br />
M<br />
cf. This is the reason why higher terms such as<br />
2<br />
∇ are irrelevant at low E.<br />
∫<br />
3<br />
~ 2<br />
) ( π<br />
π<br />
dtd x<br />
M<br />
4 2<br />
( )( ∇π<br />
)
Prediction <strong>of</strong> Large (visible) non-Gauss.<br />
Lead<strong>in</strong>g non-l<strong>in</strong>ear <strong>in</strong>teraction<br />
2<br />
π ( ∇π<br />
)<br />
2<br />
M<br />
non-G <strong>of</strong> ~<br />
1/4<br />
⎛ H ⎞ scal<strong>in</strong>g dim <strong>of</strong> op.<br />
~<br />
⎜ ⎟<br />
⎝M⎠ ⎛δρ ⎞<br />
⎜ ⎟<br />
⎝ ρ ⎠<br />
1/5<br />
[Really “0.1” ~ 10-2 × δρ / ρ . VISIBLE.<br />
( ) 1/5<br />
Compare with usual <strong>in</strong>flation where<br />
δρ/ ρ<br />
( )<br />
non-G ~ ~ 10 -5 too small.]
3-po<strong>in</strong>t function for ghost <strong>in</strong>flation<br />
k3/ k1<br />
k2 / k1<br />
3-po<strong>in</strong>t function for “local” non-G<br />
k3/ k1<br />
k2 / k1<br />
1 2 3<br />
( 1, 2, 3) , 6<br />
1 1 1<br />
k k ⎛ ⎞<br />
Fk k k = F⎜<br />
⎟<br />
k ⎝ k k ⎠<br />
k3/ k1<br />
1<br />
k2 / k1<br />
3<br />
ς = ς − f ⋅ ς − ς<br />
5<br />
( 2 2 )<br />
G NL G G
Cosmological Application (II)<br />
Alternative to DE/DM<br />
• For FRW universe, it behaves like c.c. + CDM. CDM<br />
Ghost condensation<br />
dS<br />
( 2<br />
( ∂ ) )<br />
P φ<br />
Λ=0<br />
Λeff >0 CDM<br />
dS<br />
( Φ)<br />
• Cluster<strong>in</strong>g properties rema<strong>in</strong> unexplored and<br />
may be different from c.c. + CDM.<br />
V<br />
Usual <strong>Higgs</strong> mechanism<br />
Λ < 0 Λ < 0<br />
eff<br />
Λ=0<br />
eff<br />
Φ
Cosmic Uroboros<br />
Str<strong>in</strong>g/M theory?<br />
<strong>Higgs</strong> phase<br />
DE/DM<br />
<strong>of</strong> gravity
Warped Throat<br />
KKLT setup<br />
10D = 4D universe x 6D <strong>in</strong>ternal space<br />
CY<br />
Shape & Volume<br />
stabilized<br />
Anti-D3-branes<br />
Non-SUSY NS5-brane<br />
Kachru, Pearson & Verl<strong>in</strong>de (2002)
Correspondence pr<strong>in</strong>ciple<br />
Str<strong>in</strong>gy<br />
Object<br />
Size > R grav<br />
Non-SUSY<br />
Non SUSY<br />
NS5-brane<br />
NS5 brane<br />
( ) 2<br />
M / N 1<br />
RR 3 ∼<br />
> gN s 3<br />
Mukohyama, hep-th/0610254<br />
Horowitz & Polch<strong>in</strong>ski (1997)<br />
Black-Brane<br />
Black Brane<br />
Size < R grav<br />
Black-Brane<br />
Black Brane<br />
MRR : # <strong>of</strong> R-R flux<br />
N3 : # <strong>of</strong> D3<br />
‘s<br />
gs : str<strong>in</strong>g coupl<strong>in</strong>g
4D<br />
Universe ⊗<br />
Black brane at the tip<br />
Non-extremal<br />
Non extremal<br />
black 3-brane 3 brane
x i<br />
Spontaneous Lorentz break<strong>in</strong>g<br />
• The (3+1)-dim spacetime is spanned by<br />
( t , xi ).<br />
Non-extremal<br />
black 3-brane<br />
t<br />
r = const.<br />
r<br />
x i<br />
t<br />
Projection onto<br />
t = const. surfaces<br />
Projection onto<br />
x i = const. surfaces<br />
Warp factors for the<br />
tt-component and<br />
the ij-components<br />
are different.<br />
Spontaneous<br />
Lorentz break<strong>in</strong>g!<br />
Gauged Ghost<br />
Condensation
GL <strong>in</strong>stability<br />
• Non-extremal Black branes are gravitationally<br />
unstable. [Gregory-Laflamme, PRL70, 2837<br />
(1993); NPB428, 399 (1994)]
• The dispersion relation is similar to that for<br />
the NG boson <strong>in</strong> our setup with g GCC 2 < gc 2 .<br />
2<br />
− ω =<br />
<br />
2<br />
k =<br />
Charged black str<strong>in</strong>g<br />
with r + =2<br />
[Gregory-Laflamme, NPB428, 399 (1994)]
• In our geometrical setup there is a black<br />
brane at the bottom <strong>of</strong> the warped throat.<br />
• The world-volume <strong>of</strong> the black brane is<br />
parallel to our world.<br />
Conjecture Mukohyama, hep-th/0610254<br />
Low-E EFT: Jeans-like <strong>in</strong>stability<br />
DUAL<br />
Geometrical: GL <strong>in</strong>stability
Summary<br />
• Ghost condensation is the simplest <strong>Higgs</strong><br />
phase <strong>of</strong> gravity, <strong>in</strong>clud<strong>in</strong>g only one Nambu-<br />
Goldstone boson. No ghost <strong>in</strong>cluded.<br />
• Can drive <strong>in</strong>flation.<br />
• Can be alternative to DE/DM.<br />
• The KKLT setup <strong>in</strong> the regime <strong>of</strong> parameters<br />
( ) 2<br />
M RR / N 3 > gsN 3 1<br />
∼<br />
is a UV completion (str<strong>in</strong>g theory version) <strong>of</strong><br />
the gauged ghost condensation.
Order<br />
Parameter<br />
<strong>Higgs</strong> Mechanism Ghost Condensation<br />
Instability Tachyon Ghost<br />
Condensate V’=0, V’’>0 P’=0, P’’>0<br />
Spontaneous<br />
break<strong>in</strong>g<br />
Gauge symmetry Lorents symmetry<br />
(Time translation)<br />
Modify<strong>in</strong>g Gauge force Gravitational force<br />
New<br />
potential<br />
Φ V(<br />
Φ)<br />
∂ φ<br />
− m<br />
Φ<br />
2<br />
Φ<br />
2<br />
( ) 2<br />
( ∂φ)<br />
Yukawa-type Oscillat<strong>in</strong>g <strong>in</strong> space<br />
Grow<strong>in</strong>g <strong>in</strong> time<br />
μ<br />
P<br />
φ<br />
2 −<br />
φ