What is cosmology telling us about fundamental

超弦理論とインフレーション
向山信治（東大理）
第１回「超弦理論と宇宙」研究会＠尾道
Three mysteries: Inflation,
Dark Energy & Dark Matter
String Theory?
The Cosmic Uroboros by Sheldon Glashow
String theory until 2002
Bad thing
• No 4-dimensional de Sitter solution with
stabilized moduli.
• No-go theorem!
• Contradict with inflation and dark energy?
• No way to reconcile with cosmology???
Three mysteries: Inflation,
Dark Energy & Dark Matter
String Theory?
?
The Cosmic
Uroboros does
not close?
Recent Progress
• In 2003, a 4-dimensional de Sitter solution was finally found!
Kachru, Kallosh, Linde and Trivedi (KKLT)
• In the previous no-go theorem, non-perturbative effects and
branes were not taken into account.
dS  6D
4D AdS
Volume
stabilization (KKLT)
Compactification (GPK)
Fluxes
NS-NS
R-R
Warped throat （KS)
Anti-D-branes (KKLT)
3 models of stringy brane inflation
• Wrapped DBI inflation
arXiv:0708.4285 [hep-th] with
T.Kobayashi and S.Kinoshita
• Conformal rapid-roll inflation
arXiv:0709.1952 [hep-th] with L.Kofman
• Chaotic brane inflation
work in progress with L.Kofman
Conformal Rapid-roll Inflation
arXiv:0709.1952 [hep-th] with L.Kofman
The KKLMMT model
Kachru, Kallosh, Linde, Maldacena, McAllister, and Trivedi 2003
= brane inflation (Dvali&Tye 1998) in KKLT

K  3log     

W  W0  Aea
The inflaton gets the mass
D3
inflaton
anti-D3
m  2 H
2
2
However, inflation with almost scale-invariant
spectrum of perturbations requires m2 102 H 2

This can be achieved by considering -dependent W
and fine-tuning it.
KKLMMT fine-tuning
• m2 = 2H2 would stop inflation.
• This is due to the conformal coupling
-R2/12 .
• However, people have not yet looked at
modification of Einstein equation.
• We should take it into account!
[Kofman&Mukohyama 2007]
Scalar field with non-minimal
coupling to curvature
Looks like an additional mass term…
Actually, there are more terms!
Inflation without the KKLMMT fine-tuning!
Note that this is NOT a slow roll!
RAPID ROLL INFLATION!!!
Attractor behavior
de Sitter attractor !!!
Inflation with V()
Starting with:
(3)
(1)
(2)
Goal:
(1’)
(2’)
Definitions:
(3’)
Consistency condition I
Starting with:
(1)
Goal:
(1’)
(2’)
Definitions:
(1) to (1’)
(3’)
Consistency condition II
Starting with:
(2)
Goal:
(1’)
(2’)
Definitions:
(2) to (2’)
(3’)
Consistency condition III
Starting with:
Goal:
(1’)
(2’)
Definitions:
(3) to (3’)
(3’)
Condition for inflation with
conformal coupling
• Usual slow roll condition
+ additional conditions
• The 3rd condition is not satisfied by powerlaw potentials.
• The D/anti-D potential satisfies it!
Chaotic Brane Inflation
Work in progress with L.Kofman
Chaotic Inflation driven by
brane motion
in progress, with L.Kofman
• Large motion of anti-D3-brane
• In 4D, V~l4
6D space

4D universe
Anti-D3-brane
Distance to
= Inflaton
the bottom
Phase portrait for an anti-D3-brane
without non-rel. approximation
 / l M pl2 1  2 2 / V
Chaotic Inflation!
 / M pl
Length of the throat
• Can we have   M Pl ?
• For stack of anti-D3s, the answer is NO.
2
max
M
2
Pl
4 N D3
~
MK
• Better for stack of wrapped D5s and D7s.

2
max
2
Pl
M
4 N D5
~
M

2
max
2
Pl
M
N D7 K
~
M
String theorists &
Cosmologists must
talk to each other!
The Cosmic Uroboros by Sheldon Glashow
質問
1. Warped conifold内のコンパクトな4-cycle
で、角度方向に対して安定なものは？
2. D5とD3のbound state（橋本さん＆今村さ
ん＠昨日の飲み会）で、D5の枚数をどこ
まで増やせるか？
3. Conformal coupling (m2=2H2)を、対称性
（AdS?)でprotectすることは可能か？
Wrapped DBI inflation
arXiv:0708.4285 [hep-th] with
T.Kobayashi and S.Kinoshita
The KKLMMT model
Kachru, Kallosh, Linde, Maldacena, McAllister, and Trivedi 2003
= brane inflation (Dvali&Tye 1998) in KKLT

K  3log     

W  W0  Aea
The inflaton gets the mass
D3
inflaton
anti-D3
m  2 H
2
2
However, inflation with almost scale-invariant
spectrum of perturbations requires m2 102 H 2

This can be achieved by considering -dependent W
and fine-tuning it.
KKLMMT fine-tuning
• m2 = 2H2 would stop inflation.
• This is based on dynamics of a scalar
with canonical kinetic term   / 2 .
• However, the brane position is described by
nonlinear DBI kinetic action.
• We should take it into account!
[Silberstein&Tong 2003]
DBI inflation: model description
• Mobile D3-brane with relativistic speed
• Action
S   d 4  g (4) T ( ) 1     / T ( )  T ( )  V ( ) 


d  T d 
T ( )  T h
1/ 2
3
1/ 2 4
3
DBI part
 : radial position of the brane
• Energy density & Pressure
  T ( )(  1)  V ( )
p  T ( )(1   1 )  V ( )

1
1   2 / T ( )
DBI inflation: good things
• A kind of k-inflation with general sound speed:
a new model of stringy inflation!
• No need for slow-roll:
a remedy to the η problem (KKLMMT finetuning)?
1 2
• Large non-Gaussianity:
f NL  (  1)
3
signature of stringy inflation?
DBI inflation: bad thing
Baumann&Mcllister 2006; Lidsey&Huston 2007
• (UV) DBI inflation with large non-Gaussianity
seems inconsistent with WMAP data.
• Can be consistent only in the limit when it goes
back to the slow-roll KKLMMT inflation.
• The reason:
large but not too large |fNL| (say, 20<|fNL|<300)
large r
large D/MPl
i) large D
long throat
large V6
ii) not large MPl
not large V6
confliction
Useful equations to derive
constraints
Our attempt: wrapped DBI
Kobayashi, Mukohyama and Kinoshita 2007
• The essential reason for the inconsistency of
DBI inflation with WMAP data:
large D
long throat
• For D3, this is inevitable: d  T31/ 2d 
• For a wrapped D5 or D7, we can get
larger D from the same throat!
d  T
1/ 2
32 n
 d
 det(Gkl  Bkl )
2n

1/ 2
d
large volume of 2- or 4-cycle
• This significantly ameliorates the confliction!
More stringent bound on
wrapped DBI inflation
• Wrapped DBI inflation with large nonGaussianity still requires a long throat.
(Not as long as for D3 but still long.)
• Known Calabi-Yau manifold cannot sustain
such a long throat, (let alone for D3).
long throat
large background charge
large Euler number of CY,
exceeding the known maximal value
c = 1820448 [Klemm,Lian,Roan&Yau 1997]
Summary of wrapped
DBI inflation
• Wrapping D5 or D7 over a cycle changes the
relation between the brane position and the
inflaton field.
• This significantly ameliorates the confliction
between (UV) DBI inflation and WMAP data.
• However, successful wrapped (UV) DBI
inflation requires Euler number larger than
the known maximal value.
e-foldings & mass hierarchy
• e-foldings:
a  ~ const.
N ~ ln ( i / e )
• Mass hierarchy a la Randall-Sundrum :
M / Mpl ~ e-N
• Enough inflation vs TeV gravity:
N ~ 62 + ln ( M / 1016GeV )
M ~ TeV
These conditions are equivalent!
Summary of conformal
rapid-roll inflation
• Conformal coupling does NOT necessarily
spoil inflation.
• Brane / anti-brane inflation may work
without severe fine-tuning.
• E-foldings & mass hierarchy are related.
• Modulated reheating can generate scaleinvariant density perturbation.
Summary of this talk
• It seems that we can really enjoy cosmology in
the framework of string theory.
• Model I: Wrapping D5 or D7 over a cycle
ameliorates the confliction between the DBI
inflation and WMAP data. However, a
successful DBI inflation requires large Euler
number of CY manifold.
• Model II: Conformal rapid-roll inflation is
possible without the KKLLMT fine-tuning.
The mass hierarchy and e-foldings are
related!
• Model III: Chaotic brane inflation may be
possible with wrapped D7.

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