A. Malek-NejadM. M. Sheikh-Jabbari

Gauge-flation

Based onPhys.Rev.D84:043515,2011,

arXiv:1102.1513

&

JCAP01(2012)016

Cosmo-12

Inflation has many realizations…

F. R. Bouchet: “CMB anisotropies, Status & Properties”

Almost all of the inflationary models or model building ideas use one or more scalar fields with suitable potential as inflaton!

main reasons:Isotropic homogeneous FRW metric respects isotropy. turning on potential for scalar fields is easier than the

other fields.

Scalar inflation

)( t

Almost all of the inflationary models or model building ideas use one or more scalar fields with suitable potential as inflaton!

On the other hand:

non-Abelian gauge field theories are the widely

accepted framework for building particle physics models,

and beyond standard model and GUTs!

Scalar inflation

Inflation closer to particle physics

One may explore the idea of using gauge fields as

inflaton fields!!!

main obstacles:In general, gauge fields will spoil the rotational symmetry

of the background.How to satisfy the inflation condition?

( EM tensor of Yang-Mills is traceless so can not! )!0)3( P

Inflation has many realizations…

Gauge-flation

I. Gauge-flation Setup

II. Stability of the Isotropic solution

III. Perturbations and the Data

V. Summary and Outlook

Gauge-flation

I. Gauge-flation Setup

II. Stability of the Isotropic solution

III. Perturbations and the Data

V. Summary and Outlook

Gauge-flation

timespace 3,2,1,0,

3,2,1, ba

1

1hc

GevGM Pl182/1 104.2)8(

spaceji 3,2,1,

Notation:

o Throughout we will use natural unitso Reduced Planck mass

o Metric signature is

o Greek indices o Latin indices

algebra

),,,(

o Slow-roll inflation driven by non-Abelian gauge field e.g. in the SU(2) algebra a=1,2,3

o with the Field Strength

o Gauge & diff invariant Lagrangians

o Minimally coupled to Einstein gravity.

Gauge-flation setup

aA

),( gFLL a

cbabc

aaa AAgAAF

o Slow-roll inflation driven by non-Abelian gauge field e.g. in the SU(2) algebra a=1,2,3

o with the Field Strength

o Gauge & diff invariant Lagrangians

o Minimally coupled to Einstein gravity.

Restoring Rotational symmetry? Deriving Inflation from gauge fields?

Gauge-flation setup

aA

cbabc

aaa AAgAAF

non-Abelian algebra

o Gauge fields are defined up to gauge transformations

bi

ab

ai AA

aj

ji

ai ARA

oTurning on gauge (vector) fields in the background breakso rotation symmetry:

cbabc

aa AA o In the temporal gauge , we still can make time independent gauge transformations :

Rotational non-invariance is compensated by global time independent gauge transformation

(A)

(B)

1) Restoring Rotational symmetry:

o FRW metric

o Ansatz (in temporal gauge):o

o is a scalar under 3D-diffeomorphism.

o Field Strength tensor:

1) Restoring Rotational symmetry:ji

ij dxdxtadtds )(222

ai

ai HF )(0

itta

A ai

a

)()(

00

)( t

aij

aij gF 2

It is straightforward to show :

the gauge field Eq.

1) has such solution (the reduction ansatz),2) evaluated on the anstaz, it’s EOM is equivalent to the EOM of , given as

Energy-momentum tensor (for FRW metric & ansatz): has the form of perfect fluid

Cansistensy of the reduction ansatz:

0

aF

LD

redL 0))(

( 32

redred L

a

La

dta

d

),,,( PPPdiagT

an appropriate choice is:

2) Inflation from gauge fields:

24

3844

1

2aa

aa FFFF

RgxdS

2FFTr

an appropriate choice is:

energy density pressure density

where is the contribution of the Yang-Mills &

is the contribution of term to

the energy density, .

2) Inflation from gauge fields:

24

3844

1

2aa

aa FFFF

RgxdS

2FFTr

YM

YMP3

1

2FFTr

YM

an appropriate choice is:

energy density pressure density

The equation of state of term is it is perfect for driving inflationary dynamics.

2) Inflation from gauge fields:

24

3844

1

2aa

aa FFFF

RgxdS

2FFTr

YM

YMP3

1

P 2FFTr

By plugging the ansatz and the FRW metric into the action

24

3844

1

2aa

aa FFFF

RgxdS

itta

A ai

a

)()(

00

jiij dxdxtadtds )(222

Gauge-flation setup

• One can determine the reduced Lagrangain

energy density &

• pressure density

as well as the Friedman equations and the EOM.

))()((2

3 242422 HggHLred

YM

))((2

3 242 Hg

Gauge-flation setup

YMP3

1

))((2

3 422 gHYM

The slow-roll parameter, is :

Slow-roll inflationEnd of inflation

Then, the slow-roll parameters are approximately

where or equivalently

YM

YM

H

H 22

1 YM

1 YM

2

2)1(

2

22

H

g

Gauge-flation setup

)1(

22

g

H

143102 10733.1,105.2,10,105.3 gii

Numerical Results

A. Maleknejad, M. M. Sheikh-Jabbari Phys.Rev.D 84:043515, 2011

I. Gauge-flation Setup

II. Stability of the Isotropic solution

III. Perturbations and the Data

V. Summary and Outlook

Gauge-flation

• Due to the gauge-vector nature of our inflaton

Question: stability of the classical inflationary trajectory

against (classical) initial anisotropies.

• Bianchi type-I metric

• Anisotropic ansatz

where &

Stability of the Isotropic Gauge-flation

))(( 22)(22)(4)(222 dzdyedxeedtds ttt

(Homogeneous But Anisotropic Space)

,aii

ai eA

,21

2

• Due to the gauge-vector nature of our inflaton

Question: stability of the classical inflationary trajectory

against (classical) initial conditions.

• Bianchi type-I metric

• Anisotropic ansatz parametrizing

where & anisotropy

Stability of the Isotropic Gauge-flation

))(( 22)(22)(4)(222 dzdyedxeedtds ttt

(Homogeneous But Anisotropic Space)

,aii

ai eA

,21

2

Stability of the Isotropic Gauge-flation

Result: Isotropic FLRW cosmology is an attractor of the gauge-flation dynamics.

A. Maleknejad, M. M. Sheikh-Jabbari, Jiro Soda JCAP01(2012)016

The phase-diagram in plane. The vertical axis is and the horizontal is . This Fig. shows that the isotropic FRW metric is an attractor of the system.

I. Gauge-flation Setup

II. Stability of the Isotropic solution

III. Perturbations and the Data

V. Summary and Outlook

Gauge-flation

Perturbed metric

Perturbations of the gauge field

Coordinate transformations

Gauge transformations

Perturbations during inflation

jiijjiijij

iii

dxdxtxhtxWtxEtxCta

dtdxtxStxBtadttxAds

)),(),(2),(2)),(21)(((

)),(),()((2)),(21(

)(2

22

ijaj

jajiji

jak

akiik

akai

ai

jj

akka

a

twvPgMQA

uYA

0

iV

aii

aia

iVi

ijii xxxx

ttt

• FIVE physical (gauge and diff invariant) scalar variables:

• Three physical vectors, are exponentially damped, as in the usual

inflationary models.

• TWO physical tensor variables: is the same as usual tensor perturbations, while shows exponential damping at super-horizon scales.

Perturbations during inflation

ijij th ,

))((

)(

2

2

a

BEa

dt

dA

a

BEHaC

.~

,~

),(~ 2

EPYY

EPMM

a

BEaQQ

ijh

ijt

• There are four constraints and one dynamical equation for the five physical scalar perturbations:

• Three constraints and the dynamical equation are coming from perturbed Einstein equations:

• one constraint comes from equations of motion of in the second order action:

TG

y

Perturbations during inflation

totS)2(

Our resultsGauge-flation’s prediction for the values of

power spectrum of R ,

scalar spectral tilt,

tensor to scalar ratio,

tensor spectral tilt:

Also, # e-folds is

2

,)12(

)2)(1(4

,)1

13(21

,)(8

1

)2)(1(

)12(4

2

2

222

2

t

R

t

R

plR

n

P

Pr

n

M

HP

Gauge-flation

)1

ln(2

1

eN

CMB Observations

Current observations of CMB provide values for power spectrum of R and spectral tilt and impose an upper bound on tensor to scalar ratio:

.24.0

,012.0968.0

,105.2 9

r

n

P

R

R

Komatsu et al. 2010, Astrophys. J. Suppl. 192:18 [astro-ph/1001.4538]

Contact with the observation from our results and the CMB data, we obtain:

41372

10)176.4(,10)0.513.0(4

PlMg

22 102.1109.01.0~ r

plMH 5105.3

plM210)0.85.3(

I. Gauge-flation Setup

II. Stability of the Isotropic solution

III. Perturbations and the Data

V. Summary and Outlook

Gauge-flation

Successful slow-roll inflation can be driven by non-Abelian gauge fields with gauge invariant actions

minimally coupled to Einstein gravity.

Our specific gauge-flation model has two parameters Yang-Mills coupling and the dimensionful

coefficient of term .

Cosmic data implies

g2)( FF

3105.2 g414 )106( GeV

Summery and Outlook

-term may be obtained by integrating out massive axions of the non-Abelian gauge theory, if the axion mass is

above Hubble scale H. In this model, the axion coupling scale is

a reasonable value from particle physics viewpoint.

Although a small field model, , we have sizeable gravity-wave power spectrum

M. M. Sheikh-Jabbari , arXiv:1203.2265

GeVH 151010~

plM210~

1.0~r

Summery and Outlook

Thank you 謝謝

Thank you 謝謝