TOWARDS INFLATION IN STRING COSMOLOGY

小林 洸東京大学

中央大学 2010.4.27

works with Shinji Mukohyama, Shunichiro Kinoshita, Brian Powell

TODAY’S PLAN

• Why cosmic inflation?

• Why cosmic inflation in string theory?

• some stringy inflation models: D-brane inflation models and beyond

image: NASA/WMAP Science Team

THE HOT BIG BANG UNIVERSE

image: NASA/WMAP Science Team

THE HOT BIG BANG UNIVERSE

•homogeneous and isotropic•flat•without unwanted relics (gravitinos, monopoles, etc.)

COSMIC INFLATION : a period of accelerated expansion

→　makes the universe homogeneous/isotrpic and flat,

and dilutes away unwanted relics

→　makes the observable universe relatively small

figures taken from arXiv:astro-ph/9906497, Garcia-Bellido

THE “STANDARD” REALIZATION

φ

V (φ) inflation

Introduce a scalar field which acts like vacuum energy.

→　an inflaton which slowly rolls along a flat energy potential

Hubble eq.

EOMH =

a

a

φ + 3Hφ + V� = 0

3M2p H

2 =φ

2

2+ V

η ≡M2p

V ��

V� ≡

M2p

2

�V �

V

�2

slow-roll parameters

φ

V (φ)

L√−g

= −12gµν∂µφ∂νφ− V (φ)

SLOW-ROLL INFLATION

Hubble eq.

EOMH =

a

a

φ + 3Hφ + V� = 0

3M2p H

2 =φ

2

2+ V

η ≡M2p

V ��

V� ≡

M2p

2

�V �

V

�2

slow-roll parameters

φ

V (φ)

L√−g

= −12gµν∂µφ∂νφ− V (φ)

SLOW-ROLL INFLATION

slow-roll conditions �, |η|� 1

COSMIC DENSITY PERTURBATIONSAnother important role of cosmic inflation is to generate density

perturbations, which are the seeds of the structures of our universe.

φ

V (φ) ∼ Hδφ

TEMPERTURE FLUCTUATIONS OF THE COSMIC MICROWAVE BACKGROUND

The primordial density perturbations were...

nearly scale invariant,

(almost) Gaussian fluctuations

δρ

ρ∼ 10−5of amplitude ,

WMAP 7-year

UPCOMING DATE MAY TELL US ABOUT...

• non-Gaussianity

• gravitational waves (through CMB pol.)

Planck satellite

→　non “standard” inflation?

→　high energy scale inflation?

UPCOMING DATE MAY TELL US ABOUT...

• non-Gaussianity

• gravitational waves (through CMB pol.)

Planck satellite

→　non “standard” inflation?

→　high energy scale inflation?

Surely an exciting time, but are we ready for the data?

FUNDAMENTAL QUESTIONS

• What is the inflaton?

• Do we have to fine-tune slow-roll inflation?

• How did reheating occur?

• What fundamental physics governed the inflationary universe?

FUNDAMENTAL QUESTIONS

• What is the inflaton?

• Do we have to fine-tune slow-roll inflation?

• How did reheating occur?

• What fundamental physics governed the inflationary universe?

String Theory may help!

STRING COSMOLOGY

Good things for Cosmology

•allows description of the universe at high energy•provides new ideas, new ingredients, and new ways of thinking about the early universe

Good things for String Theory

•may provide tests of string theory through cosmological observables

no-go theorem for 4-dim. de Sitter spacetime

Type IIB string theory compactified on internal spaces that are compact and nonsingular, do not allow 4-dim. de Sitter spacetime in the absence of localized sources.

Gibbons 1984, de Wit et al. 1987, Maldacena & Nunez 2001

existence of many unstabilized moduli fields

DIFFICULTIES FROM STRING COMPACTIFICATION

•steepening of the inflaton potential•decompactification•cosmological moduli problem

D3ー

R-R flux

NS-NS flux

wrapped D7

Kachru, Kallosh, Linde, Trivedi ’03

THE KKLT MECHANISM

Gidding, Kachru, Polchinski ’02warped flux compactification of type IIB theoryto 4-dim. dS vacua with all moduli stabilized

Klebanov, Strassler ’00

warped throat region

non-perturbative effects

Kachru, Kallosh, Linde, Maldacena, McAllister, Trivedi ’03

THE KKLMMT INFLATION SCENARIO

brane-antibrane inflation in a warped throat region

warped throat region

the inflaton　：radial position of the D3φ

D3ー

φD3

D-BRANE INFLATION

φT

V (φ, T )

D3D3ー

φ

Coulomb interaction D3-D3 annihilationー

a stringy realization of “hybrid inflation”

INFLATON KINETIC TERM

ds2 = h(ρ)2g(4)µν dxµdxν + h(ρ)−2(dρ2 + ρ2dΣ2

X5)

throat geometryD3 ρ

−T3

�d4x

�−det(Gµν)

−T3

�d4x

�−g(4)h4

�1 + h−4g(4)αβ∂αρ∂βρ

−T3

�d4x

�−g(4)

�h4 +

12g(4)αβ∂αρ∂βρ

�

SDBI =

=

�

dφ =�

T3 dρ

inflaton

DBI action

=�

d4x�−g(4)

�−T3h

4 − 12g(4)αβ∂αφ∂βφ

�

INFLATON POTENTIAL

H2 � V

3M2p

h0 : warp factor at the throat tip

H2φ

2 + ∆V (φ)2h40T3

�1− 3h4

0T3

8π2φ4

�+V (φ) =

η = M2p

V ��

V� 2

3

Coulomb interactionfrom moduli stabilizationD3-tensionー

−→

φ

V (φ)

2h40T3

η-problem

needs to cancel the Hubble-size mass for slow-roll inflation∆V

TOWARDS SLOW-ROLL INFLATION

H2φ

2 + ∆V (φ)2h40T3

�1− 3h4

0T3

8π2φ4

�+V (φ) =

Calculate ∆V Baumann, Dymarsky, Klebanov, McAllister ’07!

bulk CY

warped throat

D7

r

D3

D3

r0 !2.18

2.16

2.14

2.12 0.2 0.4 0.6 0.8 1.0

!/!"

V(!) x 1012

Delicate fine-tuning required

figures taken from arXiv:0706.0360

TOWARDS SLOW-ROLL INFLATION

H2φ

2 + ∆V (φ)2h40T3

�1− 3h4

0T3

8π2φ4

�+V (φ) =

Calculate ∆V

•bulk fluxes•distant branes•nonperturbative effects•further moduli stabilization effects•・・・

Baumann, Dymarsky, Kachru, Klebanov, McAllister ’08

Baumann, Dymarsky, Kachru, Klebanov, McAllister ’10

may work out, may not work out...

!

ALTERNATIVE APPROACH: DBI INFLATION

Inflation driven by D-branes moving with relativistic velocities.

ds2 = h(ρ)2g(4)µν dxµdxν + h(ρ)−2(dρ2 + ρ2dΣ2

X5)

ρ

−T3

�d4x

�−det(Gµν)SDBI =

The warping of the throat enforces the D-brane to slow down, regardless of the potential.

Silverstein, Tong ’04

= −T3

�d4x

�−g(4)h(ρ)4

�

1− ρ2

h(ρ)4

ρ2

h(ρ)4< 1

DBI INFLATION

L�−g(4)

= −T (φ)

�

1− φ2

T (φ)+ T (φ)− V (φ)

T (φ) ≡ T3h(φ)4

•inflation can occur for some　　　,

•a remedy to the η-problem (no need for slow-roll)

•produces large non-Gaussianity

T (φ) V (φ)

→　a new stringy inflation mechanism

→　can be tested in future experiments

NEW DIFFICULTIES: GEOMETRICAL CONSTRAINTS ON FIELD RANGE

• DBI inflation generates (too) large non-Gaussianity. In order to suppress non-Gaussianity down to a level consistent with WMAP data, the inflaton need to travel at least some field range　　.

• On the other hand, the inflaton field range is geometrically restricted by the throat length, which is restricted by the Planck mass.

Baumann, McAllister ’06 Lidsey, Huston ’07

M2p =

2V6

(2π)7g2sα�4 >

2Vol(X5)(∆φ)6

(2π)7g2sα�4h4

∗T33

∆φ

NEW DIFFICULTIES: GEOMETRICAL CONSTRAINTS ON FIELD RANGE

• DBI inflation generates (too) large non-Gaussianity. In order to suppress non-Gaussianity down to a level consistent with WMAP data, the inflaton need to travel at least some field range　　.

• On the other hand, the inflaton field range is geometrically restricted by the throat length, which is restricted by the Planck mass.

Baumann, McAllister ’06 Lidsey, Huston ’07

M2p =

2V6

(2π)7g2sα�4 >

2Vol(X5)(∆φ)6

(2π)7g2sα�4h4

∗T33

∆φ

CONTRADICTION!

Throat too short for a relativistically moving D3-brane to drive sufficient inflation.

WRAPPED BRANESTK, Mukohyama, Kinoshita ’07

Becker, Leblond, Shandera ’07

D3 D5,D7

ρ

��d2nξ

�det(Gkl −Bkl)

�1/2

dφ ≡ T 1/23+2n

dρ

～volume of the wrapped cycle

dφ ≡ T 1/23 dρ

D3-brane D(3+2n)-brane

effective field range increased for wrapped branes

→　Geometrical constraint for DBI inflation can be solved!

BACKGROUND CHARGE

A long throat is sufficient in any case.

→　Large flux number required.

tadpole-cancellation condition

: Euler number of a Calabi-Yau fourfoldχ

N =χ

24

The large flux number required exceeds the largest known Euler number for a CY 4-fold χ = 1820448

Klemm, Lian, Roan, Yau 1998

SUMMARY SO FAR FOR D-BRANE INFLATION

Slow-Roll Inflation

DBI Inflation (relativistic branes)

•suffers from geometrical constraints

•wrapped branes can significantly improve the situtation, but still may not be enough

•suffers from the η-problem (i.e. Hubble size mass)

•additional contributions to the potential　　may help,

but delicate fine-tuning required?

∆V

RAPID-ROLLING D-BRANES

• An inflationary attractor solution DOES exist even for 　　　　.

• This corresponds to D-branes moving rapidly, but non-relativistically.

• The inflaton is quickly decelerating / accelerating.(　　　slow-roll inf.)

Kofman, Mukohyama ’07TK, Mukohyama ’08TK, Mukohyama, Powell ’09

η ∼ O(1)

RAPID-ROLL INFLATION

L√−g

= −12gµν∂µφ∂νφ− V (φ)

η ≡M2p

V ��

V� ≡

M2p

2

�V �

V

�2

Hubble eq.

EOM

3M2p H

2 =φ

2

2+ V

φ + 3Hφ = −V�

RAPID-ROLL INFLATION

L√−g

= −12gµν∂µφ∂νφ− V (φ)

η ≡M2p

V ��

V� ≡

M2p

2

�V �

V

�2

Hubble eq.

EOM

3M2p H

2 =φ

2

2+ V

φ + 3Hφ = −V�

conditions

����η +c2

3− c

����� 1�� 1

cHφ � −V�

3M2p H

2 � V

RAPID-ROLL INFLATION

L√−g

= −12gµν∂µφ∂νφ− V (φ)

η ≡M2p

V ��

V� ≡

M2p

2

�V �

V

�2

Hubble eq.

EOM

3M2p H

2 =φ

2

2+ V

φ + 3Hφ = −V�

conditions

����η +c2

3− c

����� 1�� 1

cHφ � −V�

3M2p H

2 � V

one recovers slow-roll

upon η → 0

INFLATION FROM RAPID-ROLLING D3-BRANE

L√−g

= −12gµν∂µφ∂νφ− V (φ)

V (φ) = V0

�1 +O(1)

�φ

Mp

�2

−∆4

φ4

�

φ

H� − 6η

3 +√

9− 12ηφ

inflationary attractor

φ

V (φ)

φ

stronger warping gives longer inflationary period

INFLATION FROM RAPID-ROLLING D3-BRANE

• D3-D3 system in a warped geometry can give rise to rapid-roll inflation

• no need to cancel the Hubble size mass

• sufficient inflationary expansion can be obtained in a single throat

• However, the inflaton itself cannot produce scale-invariant density perturbations.

→　remedy to the η-problem

ー

→　geometrical constraints circumvented

→　in need of a perturbation-generating mechanism

GENERATING DENSITY PERTURBATIONS

• Various phenomenological mechanisms are known to be able to generate density perturbations by introducing additional fields.

• Since many fields show up from string compactification, why don’t we seek such mechanisms (instead of asking the inflaton to do all the work)?

CURVATON MECHANISM

σ

12m2σ2

δρσ

Hlog ρ

log a

∝ a−3

∝ a−4

ρσ

ρφ

curvaton : a light field (i.e. ) which generates density perturbations after inflation

m2 � H

2

Linde, Mukhanov ’96Enqvist, Sloth ’01Lyth, Wands ’01Moroi, Takahashi ’01

CURVATON MECHANISM

σ

12m2σ2

δρσ

Hlog ρ

log a

∝ a−3

∝ a−4

ρσ

ρφ

δρσ

curvaton : a light field (i.e. ) which generates density perturbations after inflation

m2 � H

2

Linde, Mukhanov ’96Enqvist, Sloth ’01Lyth, Wands ’01Moroi, Takahashi ’01

CURVATON MECHANISM

σ

12m2σ2

δρσ

Hlog ρ

log a

∝ a−3

∝ a−4

ρσ

ρφ

δρσ

curvaton : a light field (i.e. ) which generates density perturbations after inflation

m2 � H

2

δρσ

ρ

Linde, Mukhanov ’96Enqvist, Sloth ’01Lyth, Wands ’01Moroi, Takahashi ’01

CURVATONS IN WARPED THROATSTK, Mukohyama ’09

a curvaton model from D-branes oscillating in a warped throat with approximate isometries

WARPED DEFORMED CONIFOLD

warp

6D Bulk

Conifold

S3 × S2

× S2S3angular isometry

Giddings, Kachru, Polchinski ’02Klebanov, Strassler ’00

WARPED DEFORMED CONIFOLD

warp

6D Bulk

Conifold

S3 × S2

× S2S3angular isometry

Giddings, Kachru, Polchinski ’02Klebanov, Strassler ’00

Standard Model brane (anti-D3)

ANGULAR POTENTIAL

• isometry breaking bulk effects

• moduli stabilizing non-perturbative effects

+ warping

small mass to theangular degrees of freedom of the SM brane

CURVATON σ

=

Standard Model brane (anti-D3)

→ eventually decays into other open string modes (reheating)

ACTION

−(m2bulk + m2

np)σ2 +gsM1/2α�3/2

h30

m2bulkm

2np

m2bulk + m2

np

σψiD/ψ + · · ·�

∼�

d4x�−g(4)

�−(∂σ)2 + ψiD/ψ

S = −T3

�d4ξ

�−det Gµν

�1− ΨiD/Ψ

�− T3

�C4

h0 : warp factor at the tip ds2 = h2g(4)µν dxµdxν + h−2g(6)

mndxmdxn��

m2bulk =

h∆−20

gsMα� : bulk effects

m2np =

hλ−20

gsMα� : nonperturbative effects

masslessness

4.5 5.0 5.5 6.0

4.5

5.0

5.5

6.0

BBN

fNL < 100

σ dom.sub-dom.

mσ < Hinf

PARAMETER CONSTRAINTS: nonperturbative effectsλ

: bulk effects∆

gs = 0.1

h0 = 10−5

N = 50000

Mplα�1/2 = 300

hence� �

Hinf ∼ 10−11fNLMpl

masslessness

4.5 5.0 5.5 6.0

4.5

5.0

5.5

6.0

BBN

fNL < 100

σ dom.sub-dom.

mσ < Hinf

PARAMETER CONSTRAINTS: nonperturbative effectsλ

: bulk effects∆KS throat

D7-brane(Kuperstein embedding)

gs = 0.1

h0 = 10−5

N = 50000

Mplα�1/2 = 300

hence� �

Hinf ∼ 10−11fNLMpl

EARLY UNIVERSE COSMOLOGY WITH D-BRANES

• slow-rolling / rapid-rolling / relativistic D-branes can drive inflation

• due to the many physical degrees of freedom of D-branes, functions of inflationary cosmology can be shared among multiple fields

• D-branes allow geometric descriptions of inflaton/curvaton fields

OTHER STRINGY INFLATION MODELS

• moduli inflation (inflaton : closed string modes)

• large field inflation by “recycling” finite field ranges (e.g. monodromy models)

Today, we focused on D-brane inflation (inflaton: open string modes) with small field ranges.

Silverstein, Westphal ’08McAllister, Silverstein, Westphal ’08

SUMMARY & FUTURE PROSPECTS

• Phenomenological inflation models are starting to be realized in the framework of string theory.

• String cosmology thus far has provided new mechanisms, new views on naturalness, and new constraints.

• For a unified description of the early universe, we need better understanding of string compactifications, Standard Model realization, and an integration of both.