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![Page 1: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/1.jpg)
A. Malek-NejadM. M. Sheikh-Jabbari
Gauge-flation
Based onPhys.Rev.D84:043515,2011,
arXiv:1102.1513
&
JCAP01(2012)016
Cosmo-12
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Inflation has many realizations…
F. R. Bouchet: “CMB anisotropies, Status & Properties”
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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
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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
![Page 5: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/5.jpg)
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
![Page 6: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/6.jpg)
Inflation has many realizations…
Gauge-flation
![Page 7: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/7.jpg)
I. Gauge-flation Setup
II. Stability of the Isotropic solution
III. Perturbations and the Data
V. Summary and Outlook
Gauge-flation
![Page 8: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/8.jpg)
I. Gauge-flation Setup
II. Stability of the Isotropic solution
III. Perturbations and the Data
V. Summary and Outlook
Gauge-flation
![Page 9: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/9.jpg)
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
),,,(
![Page 10: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/10.jpg)
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
![Page 11: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/11.jpg)
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
![Page 12: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/12.jpg)
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:
![Page 13: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/13.jpg)
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
![Page 14: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/14.jpg)
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
![Page 15: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/15.jpg)
an appropriate choice is:
2) Inflation from gauge fields:
24
3844
1
2aa
aa FFFF
RgxdS
2FFTr
![Page 16: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/16.jpg)
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
![Page 17: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/17.jpg)
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
![Page 18: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/18.jpg)
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
![Page 19: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/19.jpg)
• 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
![Page 20: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/20.jpg)
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
![Page 21: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/21.jpg)
143102 10733.1,105.2,10,105.3 gii
Numerical Results
A. Maleknejad, M. M. Sheikh-Jabbari Phys.Rev.D 84:043515, 2011
![Page 22: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/22.jpg)
I. Gauge-flation Setup
II. Stability of the Isotropic solution
III. Perturbations and the Data
V. Summary and Outlook
Gauge-flation
![Page 23: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/23.jpg)
• 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
![Page 24: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/24.jpg)
• 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
![Page 25: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/25.jpg)
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.
![Page 26: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/26.jpg)
I. Gauge-flation Setup
II. Stability of the Isotropic solution
III. Perturbations and the Data
V. Summary and Outlook
Gauge-flation
![Page 27: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/27.jpg)
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
![Page 28: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/28.jpg)
• 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
![Page 29: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/29.jpg)
• 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(
![Page 30: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/30.jpg)
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
![Page 31: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/31.jpg)
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]
![Page 32: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/32.jpg)
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(
![Page 33: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/33.jpg)
I. Gauge-flation Setup
II. Stability of the Isotropic solution
III. Perturbations and the Data
V. Summary and Outlook
Gauge-flation
![Page 34: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/34.jpg)
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
![Page 35: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/35.jpg)
-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
![Page 36: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/36.jpg)
Thank you 謝謝
![Page 37: Based on Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016 Phys.Rev.D84:043515,2011,arXiv:1102.1513&JCAP01(2012)016.](https://reader036.fdocuments.in/reader036/viewer/2022062717/56649e365503460f94b2639f/html5/thumbnails/37.jpg)
Thank you 謝謝