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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