Anisotropic Infaltion

--- Impact of gauge fields on inflation ---

Jiro SodaKyoto University

ExDiP 2012, Hokkaido, 11 August, 2012

2

Standard isotropic inflation

24 4 1 ( )

2 2pM

S d x gR d x g V

222

1 (3 2

)1

p

H VM

3 '( ) 0H V

2 2 2 2 2 2( )ds dt a t dx dy dz

aHa

Isotropic homogeneous universe

Friedman eq.

1/ 2 188 2.4 10 GeVpM G

Action

K-G eq.

a ∝ eHtinflation

3

Origin of fluctuations2

2cHR N H t H

pl

HhM

Curvature perturbations

Tensor perturbations

/ 2plM h The relation yields

The tensor to the scalar ratio2

2

816t

s p

P drP M dN

e

24

8p ij ij

ij ij

MS d x h h h h Action for GW

4 2

2 2 2

14 8s

p

H HPM e

2

2 2

2t

p

HPM

1( )N x 2( )N x

1( )x 2( )x

1( )cR x

一定

2( )cR x

initial

inflation end 1e

2 polarizations

scale invariant spectrum

4

From COBE to WMAP!

,m mm

T a YT

' ' ' 'm m mma a C

CMB angular power spectrum

COBE

TT

: ζgravitational red shift

The predictions have been proved by cosmological observations.

We now need to look at a percent level fine structure of primordial fluctuations!

WMAP provided more precise data!

E and B Polarizations

Primordial gravitational waves via B-modes

Primordial gravitational waves

6

2

0( ) ( ) 1 ( )P k P k g k k n

n : prefered direction

Groeneboomn & Eriksen (2008)

The preferred direction may be produced by gauge fields.

Eriksen et al. 2004Hansen et al. 2009

Statistical Anisotropy?

7

quantitative improvement -> spectral tilt

qualitative improvement -> non-Gaussianity PGW

There are two directions in studying fine structures of fluctuations!

Yet other possibility is the statistical anisotropy

What should we look at?

If there exists coherent gauge fields during inflation, the expansion of the universe must be anisotropic.Thus, we may have statistical anisotropy in the primordial fluctuations.

Gauge fields and cosmic no-hair

8

The anomaly suggests gauge fields?

9

S −

14

4x −g gαgνβFνFαβ

ds2 −t2 α2t x2 y2 z2⎡⎣ ⎤⎦

α2η −η2 x2 y2 z2⎡⎣ ⎤⎦

S −

14

4xα4η1

α2η1

α2ηηαηνβFνFαβ

⎡⎣⎢

⎤⎦⎥

gauge fields

FLRW universe

In terms of a conformal time, we can explicitly see the conformal invariance

cancelled out

Thus, gauge fields are decoupled from the cosmic expansion and hence no interesting effect can be expected.In particular, gauge fields are never generated during inflation.

However, …..

The key feature of gauge fields is the conformal invariance.

Gauge fields in supergravity

10

S 4x −g R gi ∂

i∂−eκ

2Kgi D iW D W −3κ 2 W 2

⎡⎣⎢

⎤⎦⎥

While, the role of gauge kinetic function in inflation has been overlooked.

Cosmological roles of Kahler potential and super potential in inflation has been well discussed so far.

Supergravity action

The non-trivial gauge coupling breaks the conformal invariance because the scalar field has no conformal invariance.

K( : Kahler potential W ( : superpotential fab( : gauge kinetic function

Thus, there is a chance for gauge fields to make inflation anisotropic.

4x −g −

14Re αβ

iFανFνβ −

18I αβ

ie νλρFναFλρ

β⎡⎣⎢

⎤⎦⎥

Why, no one investigated this possibility?

　　　　 Black hole no-hair theorem

11

MJ

Q

Black hole has no hair other than M,J,Q

gravitational collapse

Israel 1967,Carter 1970, Hawking 1972

any initial configurations

event horizon

By analogy, we also expect the no-hair theorem for cosmological event horizons.

Let me start with analogy between black holes and cosmology.

　　 Cosmic no-hair conjecture

12

Tν ΛΛ

deSitter

Tν

Λ

Gibbons & Hawking 1977, Hawking & Moss 1982

cosmic expansion

Inhomogeneous and anisotropic universe

no-hair ？

Cosmic No-hair Theorem

Dominant energy condition: Strong energy condition:

Assumption:

Statement:

The universe of Bianchi Type I ~ VIII will be isotropized and evolves toward de Sitter space-time, provided there is a positive cosmological constant .

Wald (1983)

0ρ

3 0pρ

, pρ: energy density & pressure other than cosmological constantpρ

0pρ

ex)ex)

a positive cosmological constant

Type IX needs a caveat.

Sketch of the proof

14

21 03

K Kt

Λ

33

tanh 3

Kt

ΛΛ

Λ

Ricci tensor (0,0)

Einstein equation (0,0)

(3)

0R Bianchi Type I ~ VIII :

0 Strong Energy Condition

0 Dominant Energy Condition

−∂K∂t

−13K2 Λ Σ i

Σi

1M

2Tν −

12gνT

⎛⎝⎜

⎞⎠⎟t

tν

3K Λ3t Λin time scale :

213

K Λ

1M

2Tνt

tν(3)1

2R1

2i j

j iΣ Σ

2 2 ( )ijds dt h x dx dx ν 0 1 1

2 3ij ij ij ij ijh K Kt

Σ

0ijΣ 3t Λ

in time scale :

No Shear = Isotropized(3)

0R

00 0T

Spatially FlatNo matter

The cosmic no-hair conjecture kills gauge fields!

15

No gauge fields!Tν

Inflationary universe

Tν

Any preexistent gauge fields will disappear during inflation.Actually, inflation erases any initial memory other than quantum vacuum fluctuations.

Predictability is high!

Since the potential energy of a scalar field can mimic a cosmological constant, we can expect the cosmic no-hair can be applicable to inflating universe.

Initial gauge fields

Can we evade the cosmic no-hair conjecture?

All of these models breaking the energy conditionhave ghost instabilities.

・ Vector inflation with vector potential Ford (1989)

・ Lorentz violation Ackerman et al. (2007)

・ A nonminimal coupling of vector to scalar curvatureGolovnev et al. (2008), Kanno et al. (2008)

Himmetoglu et al. (2009)

1 ( )4

F F V A Aν ν L

21 ( )4

F F A A mν ν λ L

・・・ Fine tuning of the potential is necessary.

・・・ Vector field is spacelike but is necessary.0 0A

2 1 1 14 2 6

i i i iF F m R A Aν ν

L

・・・ More than 3 vectors or inflaton is required

One may expect that violation of energy conditions makes inflation anisotropic.

F A Aν ν ν

Anisotropic inflation

17

Gauge fields in inflationary background

18

S −

14

4x −g 2α FνFν

ds2 −t2 α2t x2 y2 z2

de Sitter backgrounda(ηeHt 1

−Hη α2η −t2 x2 y2 z2⎡⎣ ⎤⎦

abelian gauge fields

It is possible to take the gauge

F A Aν ν ν

gauge symmetry A A ∂ χ

A0 0 , ∂iAi 0

S 12

4x −g 2 ∂Ai

∂η⎛⎝⎜

⎞⎠⎟2

Ai ∂i2 Ai

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Ai 3κ2

3/2σ∑ Aκ

σ ηe iσ κeiκ⋅x

ki e iσ κ0, e i

σ −κe iσ 'κσσ '

12

4x −g 2 ∂Aκσ η∂η

∂A−κσ η∂η

−κ2Aκσ ηA−κ

σ η⎡⎣⎢

⎤⎦⎥

kσ = δ S

δ ∂Akσ / ∂η( )

= f 2A−kσ

f depends on time.

Akσ , κ'

σ '⎡⎣ ⎤⎦iσσ ' κ −κ'canonical commutation relation

f (t

Do gauge fields survive?

19

0 Ai (x)Ai (0) 0 κκ PAκe

iκ⋅x PA (k) κ3 uκη

2

2α2η

Thus, it is easy to obtain power spectrums

P(k)

k

blue

red

The blue spectrum means no gauge fields remain during inflation.

The red spectrum means gauge fields survive during inflation.

Akσ uκηακ

σ uκ*ηα−κ

σ † A−κσ † ∂2 uk

∂η 2 + 21f

∂ f∂η

∂uk

∂η+ k 2uk = 0

Canonical commutation relation leads to commutation relations and the normalization

uk∂uκ

*

∂η−uκ

* ∂uκ

∂η

i2ak

σ , ακ'σ '†⎡⎣ ⎤⎦σσ ' κ−κ'

akσ 0 0vacuum

Mode functions on super-horizon scales

20

sub-horizon

vk uκ∂2 vk

∂η 2 + k2 − 1f

∂2 f∂η 2

⎡⎣⎢

⎤⎦⎥vk = 0

−kη → ∞ vk 12κ

e−iκη uk 1

2κe−iκη

−kη → 0 ∂2 uk

∂η 2 + 2 1f

∂ f∂η

∂uk

∂η= 0 ∂

∂ηf 2 ∂uk

∂η⎛⎝⎜

⎞⎠⎟

= 0 uk χ1 %χ2

η2super-horizon

f αα

⎛⎝⎜

⎞⎠⎟−2χ

da 1−Hη

⎛⎝⎜

⎞⎠⎟

ηHη2 Hα2η

uk χ1 %χ2

αHα2 2

χ1 χ2α4χ−1

ak H κmatching at the horizon crossing

i) c >14

ii) c <14

1fk 2k

χ2ακ4χ−1

1fk 2k

χ1

Take a parametrization

Since we know we get

Length

time

1H

Sub-horizon

Super-horizon

ak

Gauge fields survive!

21

i) c >14

PA (k) κ3 uκη

2

2α2η

H2

Hα

κ⎛⎝⎜

⎞⎠⎟2χ−2

ii) c < 14

PA (k) κ3 uκη

2

2α2η

H2

Hα

κ⎛⎝⎜

⎞⎠⎟−2χ−1

c >1 red spectrum

c <−12

red spectrum

For a large parameter region, we have a red spectrum, which means that there exists coherent long wavelength gauge fields!

Finally, at the end of inflation, we obtain the power spectrum of gauge fields on super-horizon scales

uk 1

κ 2κα

ακ

⎛⎝⎜

⎞⎠⎟4χ−1

uk 1

κ 2κ

We should overcome prejudice!

22

According to the cosmic no-hair conjecture, the inflation should be isotropic and no gauge fields survive during inflation.

However, we have shown that gauge fields can survive during inflation.

It implies that the cosmic no-hair conjecture does not necessarily hold in inflation.

Hence, there may exist anisotropic inflation.

23

S 4x −g

M2

2R −

12

∂ 2−V

⎡

⎣⎢⎢

⎤

⎦⎥⎥ 0

pMV V eλ

22 2 4/ 2 2 2ds dt t dx dy dzλ

In this case, it is well known that there exists an isotropic power law inflation

Gauge fields and backreaction

S 4x −g

M2

2R −

12

∂ 2−V −

142 FνF

ν⎡

⎣⎢⎢

⎤

⎦⎥⎥

0pMf f e

ρ

In this background, one can consider generation of gauge fields

gauge kinetic function

power-law inflation

our universe

H −1

There appear coherent gauge fields in each Hubble volume.Thus, we need to consider backreaction of gauge fields.

M p

= − 2λ

log t

Exact Anisotropic inflation

24

2 28 12 8

6 2λ ρλ ρ

λ λ ρ

2 2 43 2λ ρλzλ λ ρ

2 2 4 0λ ρλ >

ds2 −t2 t2 t−4zx2 t2z y2 z2 ⎡⎣ ⎤⎦

For the parameter region , we found the following new solution

Kanno, Watanabe, Soda, JCAP, 2010

For homogeneous background, the time component can be eliminated by gauge transformation.Let the direction of the vector to be x – axis.

ds2 −t2 e2α t e−4σ tx2 e2σ t y2 z2 ⎡⎣ ⎤⎦

Then, the metric should be Bianchi Type-I

Watanabe, Kanno, Soda, PRL, 2009

ΣH

&σ&α13IeH

2

2

2 42

I λ ρλλ ρλ

eH −&H

H 2

6λ λ 2ρ λ2 8ρλ 12ρ2 8

0 1I <

Apparently, the expansion is anisotropic and its degree of anisotropy is given by

slow roll parameter

A 0, Axt, 0, 0

M p

= −2λ

log t &Ax (t)Ctg g 4 ρλ

−ω − 4ζ

The phase space structure

25

Isotropic inflation

Anisotropic inflation

2 2 4 0λ ρλ >

After a transient isotropic inflationary phase, the universe enter into an anisotropic inflationary phase.

Kanno, Watanabe, Soda, JCAP, 2010

The result universally holds for other set of potential and gauge kinetic functions.

Quantum fluctuations generate seeds of coherent vector fields.

anisotropy

scalar

vector

More general cases

&α 2 &σ 2

13

12&2 V

E2

2−2e−4α−4σ⎡

⎣⎢

⎤⎦⎥

&&α −3&α 2 V

E2

6−2e−4α−4σ

&&σ −3&α &σ

E2

3−2e−4α−4σ

2 3 4 43 ( ) ( ) ( )V E f f e α σ α

Hamiltonian Constraint

Scale factor

Anisotropy

Scalar field

t

'

2 2 2 ( ) 4 ( ) 2 2 ( ) 2 2t t tds dt e e dx e dy dzα σ σ

const. of integration

2 4 ( ) Ev f e α σ

A 0, vt, 0 , 0

( )t

M p 1

&α 2

13

V ⎡⎣ ⎤⎦2

2 4 4( )2

E f e α σ

f e 2 2

Behavior of the vector is determined by the coupling

f (e−2α e2

VV

f e2 2

dα

&α&−

V V

Conventional slow-roll equations(The 1st Inflationary Phase)

3 ( )Vα

Hamiltonian Constraint

Scalar field

212 2σ

2 3 4 4( ) ( )E f f e α σ

c

To go beyond the critical case, we generalize the function by introducing a parameter

c1> Vector grows1 Vector remains const.1< Vector is negligible

α −

VV

Now we can determine the functional form of f2

2

2mV 2( )f e α

Critical Case

The vector field should grow in the 1st inflationary phase.

Can we expect that the vector field would keep growing forever?

E2

2e−χ2 −4α

&α 2

13

⎡⎣ ⎤⎦2

2

2m

χE2 e−χ

2 −4α

Attractor mechanism

R ≡

ρA

ρ

E2e−χ

2 −4α

22

2 3 mα

Hamiltonian Constraint

Scalar field

4σ

4σ

2σ

Define the ratio of the energy density

R ≈

1χ2 ≈O10Typically, inflation takes place at 210

cE2e−χ2 −4α ≈ 2

e−2α e2 2

4( 1)ce α

The opposite force to the mass term

Irrespective of initial conditions, we have 210R=

1c >

12

&2

The growth should be saturated around

Inflaton dynamics in the attractor phase

&α 2

16 22

3 &α &− 2 χE2e−χκ22 −4α

α

−2 2χE2

2e−χ

2 −4α

e−χ2 −4α

2χ−1χ2E2

Hamiltonian Constraint

Scalar field

α

const. of integration

4( 1)cDe α 11

We find becomes constant during the second inflationary phase.Aρ

The modified slow-roll equations: The second inflationary phase

1c

2

3 mc

α E.O.M. for Φ 　 : (2nd inflationary phase)

23 mα (1st inflationary phase)

1c >Remember

ρA

E2

2e−χ

2 −4α

Energy Density

Solvable

m2(c −12χ2

Phase flow: Inflaton

23 m

cα

23 mα

2c Numerically solution at

Scalar field

12

(2nd inflationary phase)(1st inflationary phase)

c0 17 m 10−5

ΣH

≡&σ&αE2e−χ

2 −4α

9 &α 2

The degree of Anisotropy

3 &α &σ

E2

3e−χ

2 −4α

ΣH

23R t

23χ−1χ22

2

αeα

1 13

cH c

eΣ

σ

Anisotropy4σ

2 ( )3

t R

The degree of anisotropy is determined by

The slow-roll parameter is given by

Attractor point

e

2χ2

We find that the degree of anisotropy is written by the slow-roll parameter.

Attractor point

e−χ2 −4α

2χ−1χ2E2

Attractor Point

3 &α 2

12 22:Hamiltonian Constraint

R ≡

ρA

ρ

E2e−χ

2 −4α

22

Ratio

Compare

: A universal relation

Evolutions of the degree of anisotropy

32

c0 17

( ) AtH

ρρ

Σ =R

Numerically solution at

Initially negligible

grows fast

becomes constant disappears

0.3%HΣ

increase

x

Anisotropic Inflation is an attractor

33

ds2 −t2 e2Ht e−4Σ tx2 e2Σ t y2 z2 ⎡⎣ ⎤⎦

13 HI

HeΣ

0 1I <

Statistical Symmetry Breaking in the CMB

It is true that exponential expansion erases any initial memory.In this sense, we have still the predictability. However, the gauge kinetic function generates a slight anisotropy in spacetime.

34

Phenomenology of anisotropic inflation

　　　　 What can we expect for CMB observables?

35

ρe ∝ IeH

f≈VV

≈1

eH

vector-tensor

vector-scalar

ρe ≈ IeH

f

ρe ≈1

eH

IeH ≈ I

−ggαgνβ 2Fν

ρe ν

1 24 34Fαβ

−ggαgνβ 2

{ Fν

1

2ρe ν

{Fαβ

In the isotropic inflation, scalar, vector, tensor perturbations are decoupled.

z κ1z κ 2 κ1 κ 2Pσκ1 |κ1 |

h(k1)h(k 2 ) κ1 κ 2Ptκ1 |κ1 |

The power spectrum is isotropic

However, in anisotropic inflation, we have the following couplingslength

t

1H

ak

N(k)

Preferred direction

n ∝A

36

Predictions of anisotropic inflation

statistical anisotropy in curvature perturbations

cross correlation between curvature perturbations and primordial GWs

statistical anisotropy in primordial GWs

TB correlation in CMB

224 ( )sg I N k

26 ( )t Hg I N ke

224 ( )c H

Gr I N k

ze

zz

Ps(k) Pσκ 1−gσ ν⋅κ̂

2⎡⎣⎢

⎤⎦⎥

Pt (k) Ptκ 1−gt ν⋅κ̂

2⎡⎣⎢

⎤⎦⎥

Thus, we found the following nature of primodordial fluctuations in anisotropic inflation.

These results give consistency relations between observables.

64gt ρgσ 4c tr g

Watanabe, Kanno, Soda, PTP, 2010Dulaney, Gresham, PRD, 2010Gumrukcuoglu,, Himmetoglu., Peloso PRD, 2010

k̂

preferred direction

n

37

WMAP constraint Pullen & Kamionkowski 2007

Now, suppose we detected

Then we could expect

0.02He

• statistical anisotropy in GWs

• cross correlation between curvature perturbations and GWs

224 ( ) 0.3sg I N k

224 ( ) 0.3sg I N k

31.5 10tg 36 10

Gzzz

If these predictions are proved, it must be an evidence of anisotropic inflation!

How to test the anisotropic inflation?

The current observational constraint is given by

How does the anisotropy appear in the CMB spectrum?

38

* ˆ ˆXYk XY s m s mC d P Y Y W k k k

P P kk XYC

The off-diagonal part of the angular power spectrum tells us if the gauge kinetic function plays a role in inflation.

For isotropic spectrum, , we have

Angular power spectrum of X and Y reads

For anisotropic spectrum, there are off-diagonal components.

For example, *

2ˆ ˆTB

k TB m mC d P Y Y W k k k

, 1

We should look for the following signals in PLANCK data!

39

0.3r 0.3sg

When we assume the tensor to the scalar ratio

and scalar anisotropy

The off-diagonal spectrum becomes

The anisotropic inflation can be tested through the CMB observation!

Watanabe, Kanno, Soda, MNRAS Letters, 2011

Non-gaussianity in isotropic inflation

S 4x −g

12R −

12

∂ 2−V −

142 FνF

ν⎡⎣⎢

⎤⎦⎥

f (eχ22

e−2χα αη−2χ

PA(k)κ3 uκη

2

2α2η

H2

Hα

κ⎛⎝⎜

⎞⎠⎟2χ−2

For c=1, we have the scale invariant spectrum.

a(η1

−HηdeSitter

Pz P 1192 PNCMB2 Ntot −NCMB⎡⎣ ⎤⎦ : P

fNL

local 1280PNCMB3 Ntot −NCMB : 10

−2Ntot −NCMB< 600 Barnaby, Namba, Peloso 2012

k13k3

3 z κ1z κ2

z κ3∝1χoσ2κ̂1⋅κ̂3∝ χoσY0

0 σiνY20 ; 0.22

anisotropy

Non-gaussianity in anisotropic inflation

&α 2

16 22

3 &α &− 2χE2e−χ2−4α

dα

−2

2χ 2

E2e−χ2−4α ec2

χ2E2

2χ−1e−4α

f (eχ22

e−2α αη−2

c >1

ceff 1Scale invariant Is an attractor!

Summary

42

　 We have shown that anisotropic inflation with a gauge kinetic function induces the statistical symmetry breaking in the CMB.

　 Off-diagonal angular power spectrum can be used to prove or disprove our scenario.

More precisely, we have given the predictions:

　 We have already given a first cosmological constraint on gauge kinetic functions.

the statistical anisotropy in scalar and tensor fluctuations the cross correlation between scalar and tensor the sizable non-gaussianity

gs 24 I N2κ < 0.3 I

λ2 2ρλ −4λ2 2ρλ

<0.3

24 N 2κ

　 Anisotropic inflation can be realized in the context of supergravity.

　 As a by-product, we found a counter example to the cosmic no-hair conjecture.