Investigation of

multi-field DBI inflation

Taichi Kidani

This talk

- Motivation for the DBI inflation model

- Background dynamics

- Perturbation

- Summary

Motivation for the DBI inflation model

Inflation

Rapid expansion in the early universe solves;

-Horizon problem

-Flatness problem

-Monopole problem

FIG1. History of the UniverseNASA/WMAP Science Team(2008): http://map.gsfc.nasa.gov/media/060915/index.html

Fluctuations in CMB

CMB anisotropies

NASA(2010) :http://map.gsfc.nasa.gov/media/101080/index.html

FIG2.CMB

Almost scale invariant

Curvature perturbation

PR 2109

ns 1 0.05

Almost Gaussian

10 fNL

local 74

214 fNL

equil 266

Inflation can predict! We can distinguish different models

Komatsu et. al., 2010

Brane world scenario

In string theory, we have 10 dimensions.

Our Universe may be on a (mem)brane in the “bulk”.

Bulk:10 D

Brane:4D

y

ds2 h1/2 yK gdxdx h1/2 yK GIJ dy I dyJ

Brane Extra dimensionsFIG3.Brane in the bulk

h : warp factor

DBI action

Lagrangian for a brane in the bulk: UTP det3

(In analogy with the Nambu-Goto action)

(Coming from interaction with the bulk or other branes)

P 1

f I D 1 V I

(γμν : induced metric on the 3-brane)

(DBI kinetic term)

(T3 : brane tension)

D det I

J 2 fX I

J

X IJ 1

2

I J

&where

f h

T3,

Constraint on single field cases

Lidsey and Huston, 2007

Baumann and McAllister, 2006

ds2 d2 2dsX 5

5

Inflation is in: throatVV

max

FIG4.Calabi-Yau(courtesy of Jon Emery)

r 107

MP

r N 2

& WMAP data

UsingBoubekeur, Lyth,2005

&

1 ns 4 2 2s WMAP

Best-fit

(1-ns~0.013)

r 16cs

cs 106

fNL

equil 1

3cs

2

fNL

equil 1010

Too large!

Multi-field case

Langlois, Renaux-Petel, Steer, Tanaka , 2008

R

S

1 TRS

0 TSS

R

S

*

R: Comoving curvature perturbation

S: Entropy perturbation

*: Horizon exit

φ

R

S

χ

FIG5.Curved trajectory

Sharp curve Large TRS Small cosΘ

cos 1

1 TRS

2

We can have both “small” cs and small fNLequil.

fNL

equil 35

108

1

cs

2cos2

Multi-field potential

Chen, Gong, Koyama, Tasinato , 2010Potential:V

(Ouyang embedding case)

☆Angular mass mχ becomes:

Interaction with other branes and bulk.

FIG6.Potential

light: mχ2<1 tachyonic: mχ

2<0

The potential is naturally multi-field!!

(We have 5 angular directions + radial direction.)Complicated…

Constant sound speed model

Copeland, Mizuno, Shaeri, 2010

V , 1

2 2 0

2 2

g

2

V0

4

f f06

&

cs 3

16 f0V0 3Inflationary attractor solution with

This potential has the essential feature of the potential

derived in string theory(transition in the angular direction)!!

We can analyse this model fully numerically! FIG7.Two- field potential

Background dynamics

Specific model with

1.2106

0 0.004

V0 51012

g 3109

FIG8.x field and sound speed

FIG9.slow-roll parameters

Slow-roll background dynamics

Perturbation

Equations of motion

DecompositionAdiabatic perturbation vσ

Entropic perturbation vs

vk vsk

cs

2k2 z

z

vk

z

zvsk 0

vsk vk

cs

2k2

a2s

2

vsk

z

zvk 0

z a

d

dtHcx

3 / 2

a

cs

where &

numerically solve

vσk & vsk

PR k 3

2 2vk

2

z2

TRS PR

P*&

ξ : coupling

φ

σ

S

χ

FIG10.field decomposition

Coupling and TRS

PR 1 TRS

2 P*Power spectrum for R:

where PR*=PS*=P*(* :around horizon exit)

ξ is non-negligible only during the curve.

TRS is 0 if ξ is 0 all the time.

(ξ quantifies how much PR is amplified)

PR is sourced by PS when the trajectory curves!

(Note: PR can be observed in CMB observation)

FIG11.coupling

dR

dt

aS (on super horizon scales)

Delta-N formalism

n

n

aa

n

a

aaa

nN

nNxt ***2

21

21!

1,

This is true only if all the fields are slow-roll!

In this model, we have two fields φ

and χ, and consider the case where

N

N

2N

2

2N

2, ,…

t2,x N N

*

1

2

2N

2*

2(* : Initial hypersurface)

Sasaki, Stewart, 1995

Lyth, Malik, 1995

Bispectrum

22

2

22

3

3

3

NNN

is called “equilateral type”. This vanishes

if δχ is Gaussian.

Is called “local type” and this has some value even

with Gaussian δχ

fNL

local

22

N,

N,2

Numerical results

Full numerical result:

PR 2.3109

(all compatible with WMAP)

Results by delta-N:

TRS 103

fNL

equil 9.5

2.29109 PR 2.31109

(within 1% error)

fNL

local N,

N, 2 40

ns 0.972

FIG12.Curvature power spectrum in the long transition case

Kidani, Koyama, Mizuno, 2012

8107 r

Summary

• DBI inflation is the most promising physical model to generate equilateral type non-G.

• However, single field model is strongly constrained in string theory.

• In string theory, multi-field models are natural due to the angular directions. Multi-field effects reduce the amount of equilateral type and suppress local type. Thus, we expect a tight connection between those types of non-G.

Summary

• In toy models, we showed it is indeed possible to suppress equilateral type non-G and obtain large local type non-G so that they both satisfy the current observation.

• Measurements of both types of non-G by Plank will give us tight constraints on the form of potential in DBI inflation models.

fNL

local 5

fNL

equil 20

fNL

local 80

fNL

equil 500

WMAP PLANK

Thank you for listening.