Large Primordial Non-Gaussianity from early UniverseKazuya KoyamaUniversity of Portsmouth

•Proved by CMB anisotropies nearly scale invariant nearly adiabatic nearly Gaussian

•Generation mechanisms inflation curvaton collapsing universe (Ekpyrotic, cyclic)

Primordial curvature perturbations

0.960 0.013sn 0.16, /S

local 2NL

3( ) ( ) ( )5g gx x f x

1119 localNL

f

Komatsu et.al. 2008

Generation of curvature perturbations

•Delta N formalism curvature perturbations on superhorizon scales = fluctuations in local e-folding number

0in

( , ) ' ( ', )i

ti i

tN t x dt H t x ( , ) ( )i

BN t x N t

3( )P

Starobinsky ;85, Stewart&Sasaki ‘95

How to generate delta N

single field inflation multi-field inflation, new Ekpyroticisocurvature

curvaton,modulated reheating,multibrid inflation

adiabaticperturbations

entropyperturbations Sasaki. 2008

•Suppose delta N is caused by some field fluctuations at horizon crossing

• Bispectrum

2

,

1( , ) ...2

iI I J

I J I I J

N Nt x

1 2 3

, , ,1 2 , , , 1 2 3

, ,

( , , )

( ) ( ) perm. ( , , )I J IJI J K IJK

I I

B k k k

N N NP k P k N N N B k k k

N N

Local type (‘classical’)(local in real space=non-local in k-space)

Equilateral type (‘quantum’)(local in k-space)

31 2 3 1 2 3 1 2 3( ) ( ) ( ) 2 ( ) ( , , )Dk k k k k k B k k k

Lyth&Rodriguez ‘05

Observational constraints• local type maximum signal for

WMAP5

•Equilateral type maximum signal for

WMAP5

1k

2k 3k

1k

3k

2k

local 2NL

3( ) ( ) ( )5g gx x f x

3 1 2,k k k

1119 localNL

f

1 2 3k k k

NL

equil151 253f

Theoretical predictionsStandard inflation Non-standard scenario

Single field

K-inflation, DBI inflation

Features in potentialGhost inflation

Multifield

depending on the trajectory

DBI inflation

curvaton

new Ekpyrotic (simplest model)

isocurvature perturbations (axion CDM)

local, equil ( , ) 1NLf O equil 21 / 1NL sf c

local (1)NLf O

localNL (5 / 4) / curvaton decayf

equil 2 2(1 / ) / (1 ) 1NL s RSf c T

1( 1)localNL sf n

5 310localNLf

Rigopoulos, Shellard, van Tent ’06Wands and Vernizzi ’06Yokoyama, Suyama and Tanaka ‘07

Three examples for non-standard scenarios• (multi-field) K-inflation, DBI inflation

• (simplest) new ekpyrotic model

• (axion) CDM isocurvature model

Arroja, Mizuno, Koyama 0806.0619 JCAP

Koyama, Mizuno, Vernizzi, Wands 0708.4321 JCAP

Hikage, Koyama, Matsubara, Takahashi, Yamaguchi (hopefully) to appear soon

K-inflation•Non-canonical kinetic term

•Field perturbations (leading order in slow-roll)

4 1( , ),2

S d x gP X X

2 ,

, ,2X

sX XX

Pc

P XP

21 , 16T

ss pl

H PP r cc M P

sound speed

Amendariz-Picon et.al ‘99

Garriga&Mukhanov ‘99

Bispectrum

•DBI inflation

cf local-type

2

1 2 3 1 2 3 3 3 31 2 3

( , , ) ( , , ) ,P

B k k k F k k kk k k

local local 3 3 31 2 3 NL 1 2 3

3( , , ) ( )10

F k k k f k k k

1( ) 1 2 ( ) 1 ( )( )

P X f X Vf

1 2 3 2

1( , , )s

F k k kc

1k

3k

2k

1k

2k 3k

LoVerde et.al. ‘07

Aishahiha et.al. ‘04

Observational constraints• (too) large non-Gaussianity •Lyth bound

eff2

35 1108NL

s

fc

for equilateral configurations

2

2

1 6p

rM N

716 10sr c

1 4 0.04 0.013sn D3

anti-D3

inflaton

UV eff 300NLf Huston et.al ’07, Kobayashi et.al ‘08

Multi-field model

•Adiabatic and entropy decomposition adiabatic sound speed

entropy sound speed

4 1( , ),2

IJ IJ I JS d x gP X X

2( ) ( ),2

IJ J II J

bP X P X X X X X X 0 0X X

2 ,

0, ,2X

adX XX

Pc

P X P

201enc bX

s

adiabatic

entropy

Arroja, Mizuno, Koyama ’08Renaux-Petel, Steer, Langlois Tanaka‘08

•Multi-field k-inflation

•Multi-field DBI inflation

Final curvature perturbation

2 1enc

1( ) det( ( ) ) 1 ( )( )

IJ I JIJP X g f G V

f

2 2en adc c

( ) ( )IJP X P X

,H HS ss

* *RST S

Langlois&Renaux-Petel’08

Renaux-Petel, Steer, Langlois Tanaka‘08

Transfer from entropy mode•Tensor to scalar ratio

•Bispectrum k-dependence is the same as single field case!

large transfer from entropy mode eases constraints

•Trispectrum different k-dependence from single field case?

RST

2

1161s RS

r cT

eff2 2

35 1 1108 1NL

s RS

fc T

3eff

22,NLf

2 2

3 2

1

1

RS

RS

T

T

Mizuno, Koyama, Arroja ’09

Renaux-Petel, Steer, Langlois Tanaka‘08

New ekpyrotic models•Collapsing universe

•Ekpyrotic collapse

1H

ak

t t bounce

( ) ( ) , 1na t t n Khoury et.al. ‘01

•Old ekpyrotic model

spectrum index for is the bounce may be able to creat a scale invariant spectrum but

it depends on physics at singularity

•New ekpyrotic model

0cV V e

22/( ) ( ) ca t t

3sn

1 1 2 21 2

c cV V e V e

212/( ) ( ) ca t t

222/( ) ( ) ca t t

22/( ) ( ) ca t t

22/2 2 2

1 2

1 1 1( ) ( ) ,ca t tc c c

Khoury et.al. ‘01

Lyth ‘03

Lehners et.al , Buchbinder et.al. ‘07

B

B2

•Multi-field scaling solution is unstable entropy perturbation which has a scale invariant spectrum is converted to adiabatic perturbations

2

2 20

c Hs

1

22 2 2

1 22

T

sc H

c c

B

B

B2

B: B2:

ss

B2

Koyama, Mizuno & Wands’ 07

•Delta-N formalism

22

2

12

dN d NN s sds ds

sB

B2

s

logN a

log HN

B

B2

•Predictions

•Generalizations changing potentials conversion to adiabatic perturbations in kinetic

domination

2 21 2

local 2 1NL 1

1 11 4 0

05 5 ( 1)

12 3

s

s

nc c

r

f c n

Koyama, Mizuno, Vernizzi & Wands’ 07

Buchbinder et.al 07

Lehners &Steinhardt ‘08

Non-Gaussianity from isocurvature perturbations

• (CDM) Isocurvature perturbations are subdominant

•Non-Gaussianity can be large

0.067 ( 1)

0.008 ( 1.5) <0.001 ( 2)

SS

S

S

S

P nP P

nn

Boubekeur& Lyth ‘06, Kawasaki et.al ’08, Langlois et.al ‘08

Axion CDM•Massive scalar fields without mean

•Entropy perturbations

•Dominant nG may come from isocurvature perturbations

2 2CDM m

2 234

CDM rg g

CDM r

S

3

3/22(1)

SO

S

32

3 3eff 52

2 1/22 2

1 10NL

Sf

Linde&Mukhanov ‘97, Peebles ’98, Kawasaki et.al‘08

Bispectrum of CMB from the isocurvature perturbation

Adiabatic (fNL=10)

Equilateral triangle

SN

2

Il1l2l32 bl1l2l3

2

Cl1Cl2Cl3

l1l2l32l1l2l3

Noise: WMAP 5-year

The isocurvature perturbations can generate large non-Gaussianity (fNL~40)

adi iso

S SN N

3/2eff 41

0.067NLf

•Minkowski functional measures the topology of CMB map (=weighted some of bispectrum)

no detection of non-G from isocurvature perturbations and get constraints

comparable to constraints from power spectrum

WMAP5 constraints -Minkowski functional

0.070 ( 1)

0.042 ( 1.5)

0.0064 ( 2)

n

n

n

Hikage, Koyama, Matsubara, Takahashi, Yamaguchi ‘08

Theoretical predictionsStandard inflation Non-standard scenario

Single field

K-inflation, DBI inflation

Features in potentialGhost inflation

Multifield

depending on the trajectory

DBI inflation

curvaton

new Ekpyrotic (simplest model)

isocurvature perturbations (axion CDM)

local, equil ( , ) 1NLf O equil 21 / 1NL sf c

local (1)NLf O

localNL (5 / 4) / curvaton decayf

1( 1)localNL sf n

5 310localNLf

equil 2 2(1 / ) / (1 ) 1NL s RSf c T

Conclusions•Power spectrum from pre-WMAP to post-WMAP

•bispectrum WMAP 8year Planck

Do everything you can now!

Axion CDM

Classical mean of axion quantum fluctuations if

a aa f

inf / 2a af H

2inf / 2a H

2

inf*

*

,2

a

a

a Haa

cdm

0.82 22 *

12 11 2cdm

,

0.111.8 1010 GeV 10 GeV

a a

a

a

S r r

F arh