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 2

pMS d x g R 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

aH

a

Isotropic homogeneous universe

Friedman eq.

1/ 2 188 2.4 10 GeVpM G

Action

K-G eq.

a ∝ eH tinflation

3

Origin of fluctuations2

2c

HR N H t H

pl

Hh

M

Curvature perturbations

Tensor perturbations

/ 2plM h The relation yields

The tensor to the scalar ratio2

2

816t

s p

P dr

P M dN

24

8p ij ij

ij ij

MS d x h h h h Action for GW

4 2

2 2 2

1

4 8sp

H HP

M

2

2 2

2t

p

HP

M

1( )N x 2( )N x

1( )x2( )x

1( )cR x

一定

2( )cR x

initial

inflation end 1

2 polarizations

scale invariant spectrum

4

From COBE to WMAP!

,m mm

Ta Y

T

' ' ' 'm m mma a C

CMB angular power spectrum

COBE

T

T: 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 directionGroeneboomn & 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

d4x −g gαgνβ Fν Fαβ

ds2 =−dt2 +a2(t) dx2 +dy2 +dz2⎡⎣ ⎤⎦

=a2(η) −dη2 +dx2 +dy2 +dz2⎡⎣ ⎤⎦

S −

14

d4xa4η 1a2η

1a2η

ηαηνβ Fν Fαβ

⎡

⎣⎢

⎤

⎦⎥

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 = d4x −g R+ gi j∂

i∂j −eκ

2K ()gi j DiW()DjW() −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.

+ d4x −g −

14

Re fab(i )F aνFν

b −18

Im fab(i ) νλρFν

a Fλρb⎡

⎣⎢

⎤

⎦⎥

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 constant

pρ

0pρ

ex)

ex)

a positive cosmological constant

Type IX needs a caveat.

Sketch of the proof

14

210

3

KK

t

Λ

3

3tanh 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

−13

K 2 Λ Σ ijΣ

ji

1M p

2Tν −

12

gνT⎛

⎝⎜⎞

⎠⎟ttν

3K Λ3

t Λin time scale :

21

3K Λ

+1

M p2Tνt

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 3

t Λ

in time scale :

No Shear = Isotropized(3)

0R

00 0T

Spatially Flat

No 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( )

4F F V A Aν ν L

21( )

4F 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 1

4 2 6i 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

d4x −g f 2a Fν Fν

ds2 =−dt2 +a2(t) dx2 +dy2 +dz2( )

de Sitter backgrounda(ηeHt

1−Hη

=a2(η) −dt2 +dx2 +dy2 +dz2⎡⎣ ⎤⎦

abelian gauge fields

It is possible to take the gauge

F A Aν ν ν

gauge symmetry A =A +∂ χ

A0 =0 , ∂i Ai =0

S =12

d4x −g f 2 ∂Ai

∂η⎛

⎝⎜⎞

⎠⎟

2

+ Ai ∂i2 Ai

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Ai =d3k2( )

3/2σ∑ Ak

σ (η) iσ (k)eik⋅x

ki iσ (k) =0, i

σ (−k) iσ '(k) =σσ '

12

d4x −g f 2 ∂Akσ η∂η

∂A−kσ η∂η

−k2Akσ ηA−k

σ η⎡

⎣⎢

⎤

⎦⎥

kσ =

δ S

δ ∂Akσ / ∂η( )

= f 2A−kσ

f depends on time.

Akσ , k'

σ '⎡⎣ ⎤⎦=iσσ ' (k−k')canonical commutation relation

f ((t))

Do gauge fields survive?

19

0 Ai (x)Ai (0) 0 =dkk PA(k)e

ik⋅x PA (k)k3 ukη

2

2a2η

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 =uk(η)ak

σ +uk* (η)a−k

σ † =A−kσ † ∂2 uk

∂η 2 + 21

f

∂ f

∂η

∂uk

∂η+ k2uk = 0

Canonical commutation relation leads to commutation relations and the normalization

uk

∂uk*

∂η−uk

* ∂uk

∂η=

if 2ak

σ , ak'σ '†⎡⎣ ⎤⎦σσ ' k−k'

akσ 0 0vacuum

Mode functions on super-horizon scales

20

sub-horizon

vk = f uk∂2 vk

∂η 2 + k2 −1

f

∂2 f

∂η 2

⎡

⎣⎢

⎤

⎦⎥vk = 0

−kη → ∞ vk 12k

e−ikη uk 1

f 2ke−ikη

−kη → 0 ∂2 uk

∂η 2 + 21

f

∂ f

∂η

∂uk

∂η= 0

∂∂η

f 2 ∂uk

∂η

⎛⎝⎜

⎞⎠⎟

= 0 uk c1 %c2

dηf 2super-horizon

f aaf

⎛

⎝⎜

⎞

⎠⎟

−2c

da d1

−Hη⎛⎝⎜

⎞⎠⎟

dηHη2 Ha2dη

uk c1 %c2

daHa2 f 2 c1 c2a

4c−1

ak H kmatching at the horizon crossing

i) c >14

ii) c <14

1

fk 2kc2ak

4c−1

1

fk 2kc1

Take a parametrization

Since we know we get

Length

time

1H

Sub-horizon

Super-horizon

a

k

Gauge fields survive!

21

i) c >14

PA (k) k3 ukη f

2

2a2η f

H2

Haf

k⎛⎝⎜

⎞⎠⎟

2c−2

ii) c <14

PA (k)k3 ukη

2

2a2η

H2

Haf

k⎛⎝⎜

⎞⎠⎟

−2c−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

fk 2k

af

ak

⎛

⎝⎜⎞

⎠⎟

4c−1

uk 1

fk 2k

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 = d4x −gM p

2

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 = d4x −gMp

2

2R−

12

∂( )2−V() −

14

f 2() FνFν

⎡

⎣⎢⎢

⎤

⎦⎥⎥

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

= −2

λlog t

Exact Anisotropic inflation

24

2 28 12 8

6 2

λ ρλ ρλ λ ρ

2 2 4

3 2

λ ρλλ λ ρ

2 2 4 0λ ρλ >

ds2 −dt2 t2 t−4dx2 t2 dy2 dz2 ⎡

⎣⎤⎦

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 =−dt2 +e2α (t) e−4σ (t)dx2 +e2σ (t) dy2 +dz2( )⎡

⎣⎤⎦

Then, the metric should be Bianchi Type-I

Watanabe, Kanno, Soda, PRL, 2009

ΣH

&σ&α13

IH

2

2

2 4

2I

λ ρλλ ρλ

H =−

&HH 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

= −2

λlog t

&Ax (t)Ctγ γ 4ρ

λ−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

2f −2()e−4α−4σ⎡

⎣⎢

⎤

⎦⎥

&&α =−3&α 2 +V() +

E2

6f −2()e−4α−4σ

&&σ =−3&α &σ +

E2

3f −2()e−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, v(t), 0 , 0 )

( )t

M p 1

&α 2

13

V ⎡⎣ ⎤⎦

22 4 4( )

2

Ef e α σ

f =e 2 2

Behavior of the vector is determined by the coupling

f (e−2α e2

V′V d

f e2 2

dαd

=&α&=−

V()′V ()

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

3 ( )Vα ′

Hamiltonian Constraint

Scalar field

21

2 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

c

1> Vector grows

1 Vector remains const.

1< Vector is negligible

α =−

V′Vd

Now we can determine the functional form of f2

2

2

mV 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−c2 −4α

&α 2

13

⎡⎣ ⎤⎦

22

2

m

cE2 e−c2 −4α

Attractor mechanism

R ≡

ρA

ρ

=E2e−c2 −4α

m22

2 3 mα

Hamiltonian Constraint

Scalar field

4σ

4σ

2σ

Define the ratio of the energy density

R ≈

1c2 ≈O(10)Typically, inflation takes place at 210

cE2e−c2 −4α ≈m2

e−2α e

2 2

4( 1)ce α

The opposite force to the mass term

Irrespective of initial conditions, we have 210R=

1c >

1

2&2

The growth should be saturated around

Inflaton dynamics in the attractor phase

&α 2 =

16

m22

3 &α &−m2 cE2e−cκ 22 −4α

ddα

−2 2cE2

m2e−c2 −4α

e−c2 −4α

m2c−1c2E2

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

1

c

2

3m

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

23 mα (1st inflationary phase)

1c >Remember

ρA

E2

2e−c2 −4α

Energy Density

Solvable

m2(c −1)2c2

Phase flow: Inflaton

23 m

cα

23 mα

2c Numerically solution at

Scalar field

1

2

(2nd inflationary phase)

(1st inflationary phase)

c0 17 m 10−5

ΣH

≡&σ&α=

E2e−c2 −4α

9 &α 2

The degree of Anisotropy

3 &α &σ

E2

3e−c2 −4α

ΣH

23R t

23

c−1c22

2

αα

1 1

3

c

H cΣ

σ

Anisotropy

4σ

2( )

3t R

The degree of anisotropy is determined by

The slow-roll parameter is given by

Attractor point

2c2

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

Attractor point

e−c2 −4α

m2c−1c2E2

Attractor Point

3 &α 2

12

m22:Hamiltonian Constraint

R ≡

ρA

ρ

E2e−c2 −4α

m22

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 =−dt2 + e2Ht e−4Σ tdx2 + e2Σ t dy2 + dz2( )⎡

⎣⎤⎦

1

3 HIH

Σ 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

ρem∝ IH

ff≈

VV

≈1

H

vector-tensor

vector-scalar

ρem ≈ IH

ff

ρem ≈1

H

IH ≈ I

−ggαgνβ f 2()Fν

ρemn

1 24 34Fαβ

−ggαgνβ f 2()ff

{ Fν

1f 2

ρemn

{Fαβ

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

k1 k2 k1 k2Psk1 |k1 |

h(k

1)h(k

2) k1 k2Ptk1 |k1 |

The power spectrum is isotropic

However, in anisotropic inflation, we have the following couplingslength

t

1H

a

k

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 k

224 ( )c H

Gr I N k

P

s(k) =Ps(k) 1−gs n⋅k̂( )

2⎡⎣⎢

⎤⎦⎥

P

t(k) =Pt(k) 1−gt n⋅k̂( )

2⎡⎣⎢

⎤⎦⎥

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

These results give consistency relations between observables.

64gt=r gs 4c tr g

Watanabe, Kanno, Soda, PTP, 2010

Dulaney, Gresham, PRD, 2010

Gumrukcuoglu,, Himmetoglu., Peloso PRD, 2010

k̂

preferred direction

n

37

WMAP constraint Pullen & Kamionkowski 2007

Now, suppose we detected

Then we could expect

0.02H

• 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

G

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′ ′ ′ ′ 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′ ′ ′ 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 = d4x −g

12

R−12

∂( )2−V()−1

4f 2() Fν F

ν⎡

⎣⎢

⎤

⎦⎥

f (ec22

e−2cα aη−2c

PA (k)=k3 uk(η)

2

2a2 (η)=

H2

Haf

k⎛⎝⎜

⎞⎠⎟

2c−2

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

a(η)= 1−Hη

deSitter

P =P 1+192 P NCMB

2 (Ntot −NCMB)⎡⎣ ⎤⎦ : P

f

NL

local =1280PNCMB3 (Ntot −NCMB) : 10

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

k13k3

3 k1 k2

k3∝1+cos2(k̂1⋅k̂3) ∝ cosY0

0 +sinY20

; 0.22anisotropy

Non-gaussianity in anisotropic inflation

&α 2

16

m22

3 &α &−m2cE2e−c2−4α

ddα

=−2+

2cm2

E2e−c2−4αec2

=c2E2

m2 (c−1)e−4α

f ()=ec22

=e−2α =a(η)−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 N2(k) < 0.3

I =

λ2 + 2ρλ −4λ2 + 2ρλ

<0.3

24 N2(k)

　 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.