Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field...

26
Large Primordial Non- Gaussianity from early Universe Kazuya Koyama University of Portsmouth

Transcript of Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field...

Page 1: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

Large Primordial Non-Gaussianity from early Universe Kazuya Koyama University of Portsmouth

Page 2: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

• 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, /Sα ζ α<

local 2NL

3( ) ( ) ( )5g gx x f xζ ζ ζ= +

1119 localNL

<<− f

Komatsu etKomatsu et..alal. . 20082008

Page 3: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

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

0inζ =

( , ) ' ( ', )i

ti i

tN t x dt H t x= ∫ ( , ) ( )i

BN t x N tζ = −

3( )Pδρζ

ρ= Ψ −

+

Starobinsky Starobinsky ;;8585, , StewartStewart&&Sasaki Sasaki ‘‘9595

Page 4: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

How to generate delta N

single field inflation multi-field inflation, new Ekpyrotic isocurvature

curvaton, modulated reheating, multibrid inflation

adiabatic perturbations

entropy perturbations SasakiSasaki. . 20082008

Page 5: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

• 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 xζ δϕ δϕ δϕϕ ϕ ϕ

⎛ ⎞∂ ∂= + +⎜ ⎟∂ ∂ ∂⎝ ⎠

( )1 2 3

, , ,1 2 , , , 1 2 3

, ,

( , , )

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

I I

B k k kN N N

P k P k N N N B k k kN N

ζ

ζ ζ= + +

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 kζζ ζ ζ π δ= + +r r r r r r

LythLyth&&Rodriguez Rodriguez ‘‘0505

Page 6: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

Observational constraints • local type maximum signal for WMAP5 • Equilateral type maximum signal for WMAP5

1kr

2kr 3k

r

1kr

3kr

2kr

local 2NL

3( ) ( ) ( )5g gx x f xζ ζ ζ= +

3 1 2,k k k

1119 localNL

<<− f

1 2 3k k k

NL

equil151 253f− < <

Page 7: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

Theoretical predictions Standard inflation Non-standard scenario

Single field

K-inflation, DBI inflation Features in potential Ghost inflation

Multi field

depending on the trajectory

DBI inflation curvaton new Ekpyrotic (simplest model) isocurvature perturbations (axion CDM)

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

local (1)NLf O=

( )localNL (5 / 4) / curvaton decay

f ρ ρ

equil 2 2(1 / ) / (1 ) 1NL s RSf c T= + >

1( 1)localNL sf n −> −

5 310localNLf α

RigopoulosRigopoulos, , ShellardShellard, , van van Tent Tent ’’0606 Wands and Vernizzi Wands and Vernizzi ’’0606 YokoyamaYokoyama, , Suyama and Suyama and Tanaka Tanaka ‘‘0707

Page 8: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

Three examples for non-standard scenarios • (multi-field) K-inflation, DBI inflation • (simplest) new ekpyrotic model • (axion) CDM isocurvature model

ArrojaArroja, , MizunoMizuno, , Koyama Koyama 08060806..0619 0619 JCAPJCAP

KoyamaKoyama, , MizunoMizuno, , VernizziVernizzi, , Wands Wands 07080708..4321 4321 JCAP JCAP

HikageHikage, , KoyamaKoyama, , MatsubaraMatsubara, , TakahashiTakahashi, , YamaguchiYamaguchi ((hopefullyhopefully) ) to appear soonto appear soon

Page 9: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

K-inflation • Non-canonical kinetic term • Field perturbations (leading order in slow-roll)

4 1( , ),2

S d x gP X X μμφ φ φ= − = ∂ ∂∫

2 ,

, ,2X

sX XX

Pc

P XP=

+

21 , 16T

ss pl

H PP r cc M Pζ

ζ

εε

⎛ ⎞= =⎜ ⎟⎜ ⎟

⎝ ⎠

sound speed

AmendarizAmendariz--Picon etPicon et..al al ‘‘9999

GarrigaGarriga&&Mukhanov Mukhanov ‘‘9999

Page 10: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

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

ζ=

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

φ φφ

= − − − −

1 2 3 21( , , )s

F k k kc

1kr

3kr

2kr

1kr

2kr 3k

r

LoVerde etLoVerde et..alal. . ‘‘0707

Aishahiha etAishahiha et..alal. . ‘‘0404

Page 11: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

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

eff2

35 1108NL

s

fc

= − for equilateral configurations

2

21 6

p

rM N

φΔ⎛ ⎞ =⎜ ⎟Δ⎝ ⎠716 10sr c ε −= <

1 4 0.04 0.013sn ε− ±

D3

anti-D3

inflaton

UVφ φ< eff 300NLf >Huston etHuston et..al al ’’0707, , Kobayashi etKobayashi et..al al ‘‘0808

Page 12: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

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 μμφ φ φ= − = ∂ ∂∫

( )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= + δσ

adiabatic

entropy

ArrojaArroja, , MizunoMizuno, , Koyama Koyama ’’0808 RenauxRenaux--PetelPetel, , SteerSteer, , Langlois Langlois TanakaTanaka‘‘0808

Page 13: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

• 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 μν μ νφ φ φ φφ

= − − − ∂ ∂ − −

2 2en adc c=

( ) ( )IJP X P X=

,H HS ss

ζ δσ δσ

= =& &* *RST Sζ ζ= +

LangloisLanglois&&RenauxRenaux--PetelPetel’’0808

RenauxRenaux--PetelPetel, , SteerSteer, , Langlois TanakaLanglois Tanaka‘‘0808

Page 14: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

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

2116

1sRS

r cT

ε=+

eff2 2

35 1 1108 1NL

s RS

fc T

= −+

3eff

22,NLf

ζ

ζ

2 2

3 2

1

1

RS

RS

T

T

ζ

ζ

∝ +

∝ +

MizunoMizuno, , KoyamaKoyama, , Arroja Arroja ’’0909

RenauxRenaux--PetelPetel, , SteerSteer, , Langlois TanakaLanglois Tanaka‘‘0808

Page 15: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

New ekpyrotic models • Collapsing universe • Ekpyrotic collapse

1H −

ak

t t− ↔bounce

( ) ( ) , 1na t t n= −Khoury etKhoury et..alal. . ‘‘0101

Page 16: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

• 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 ϕ−= −

22/( ) ( ) ca t t= −

ζ 3sn =

1 1 2 21 2

c cV V e V eϕ ϕ− −= − −

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 etKhoury et..alal. . ‘‘0101

Lyth Lyth ‘‘0303

Lehners etLehners et..al al , , Buchbinder etBuchbinder et..alal. . ‘‘0707

B

B2

Page 17: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

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

2

2 20

c Hsδπ

ζ

= 1

22 2 2

1 22

T

sc H

c c

δ δϕ

ζ δϕπ

∝ =+

B

B

B2

B: B2:

sδsδ

B2

KoyamaKoyama, , Mizuno Mizuno & & WandsWands’’ 0707

Page 18: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

• Delta-N formalism

22

212

dN d NN s sds ds

δ δ δ= +

sδB

B2

logN a=

log H

B

B2

Page 19: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

• 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 −

⎛ ⎞− = + >⎜ ⎟

⎝ ⎠=

= > −

KoyamaKoyama, , MizunoMizuno, , Vernizzi Vernizzi & & WandsWands’’ 0707

Buchbinder etBuchbinder et..al al 0707

Lehners Lehners &&Steinhardt Steinhardt ‘‘0808

Page 20: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

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

ζ

α = < =+

< ==

BoubekeurBoubekeur& & Lyth Lyth ‘‘0606, , Kawasaki etKawasaki et..al al ’’0808, , Langlois etLanglois et..al al ‘‘0808

Page 21: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

Axion CDM • Massive scalar fields without mean • Entropy perturbations • Dominant nG may come from isocurvature perturbations

2 2CDM mδρ σ

2 234

CDM rg g

CDM r

S δρ δρ η ηρ ρ

= − ≡ −3

3/22(1)

SO

S

32

3 3eff 52

2 1/22 2

1 10NL

Sf α α

ζ ζ

LindeLinde&&Mukhanov Mukhanov ‘‘9797, , Peebles Peebles ’’9898, , Kawasaki etKawasaki et..alal‘‘0808

Page 22: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

Bispectrum of CMB from the isocurvature perturbation

Adiabatic (fNL=10)

Equilateral triangle

SN

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2

= Il1l2l3

2 bl1l2l3

2

Cl1Cl2

Cl3Δ l1l2l32≤ l1 ≤ l2 ≤ l3

Noise: WMAP 5-year

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

adi iso

S SN N

⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

3/2eff 41

0.067NLf α⎛ ⎞= ⎜ ⎟⎝ ⎠

Page 23: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

• 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

η

η

η

α

α

α

< =

< =

< =

HikageHikage, , KoyamaKoyama, , MatsubaraMatsubara, , TakahashiTakahashi, , Yamaguchi Yamaguchi ‘‘0808

Page 24: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

Theoretical predictions Standard inflation Non-standard scenario

Single field

K-inflation, DBI inflation Features in potential Ghost inflation

Multi field

depending on the trajectory

DBI inflation curvaton new Ekpyrotic (simplest model) isocurvature perturbations (axion CDM)

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

local (1)NLf O=

( )localNL (5 / 4) / curvaton decay

f ρ ρ

1( 1)localNL sf n −> −

5 310localNLf α

equil 2 2(1 / ) / (1 ) 1NL s RSf c T= + >

Page 25: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

Conclusions • Power spectrum from pre-WMAP to post-WMAP • bispectrum WMAP 8year Planck Do everything you can now!

Page 26: Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field inflation, new Ekpyrotic isocurvature curvaton, modulated reheating, multibrid inflation

Axion CDM

Classical mean of axion quantum fluctuations if

a aa f θ=

inf / 2a af Hθ π<

2inf / 2a Hδ π=

2

inf*

*

,2

a

a

a Haa

δρ δρ π

⎛ ⎞= =⎜ ⎟

⎝ ⎠

cdm0.82 2

2 *12 11 2

cdm

,

0.111.8 1010 GeV 10 GeV

a a

a

a

S r r

F arh

δρρ

−−

Ω= =

Ω

⎛ ⎞⎛ ⎞ ⎛ ⎞= × ⎜ ⎟⎜ ⎟ ⎜ ⎟ Ω⎝ ⎠ ⎝ ⎠ ⎝ ⎠