Celebrating “DBI in the Sky” April 2012, IPM,...
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Celebrating “DBI in the Sky” April 2012, IPM, Tehran
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Celebrating “DBI in the Sky” April 2012, IPM, Tehran
Our Universe today, looks Homogeneous and Isotropic at cosmological scales.
Cosmic evolution may start froma generic initial condition
Is there any Dynamical explanationIs the Isotropic & Homogeneous Universe an attractor of the cosmic evolution?
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Our Universe today, looks Homogeneous and Isotropic at cosmological scales.
Cosmic evolution may start froma generic initial condition
Is there any Dynamical explanationIs the Isotropic & Homogeneous Universe an attractor of the cosmic evolution?
Celebrating “DBI in the Sky” April 2012, IPM, Tehran
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Our Universe today, looks Homogeneous and Isotropic at cosmological scales.
Cosmic evolution may start froma generic initial condition
Is there any Dynamical explanation ?Is the Isotropic & Homogeneous Universe an attractor of the cosmic evolution?
Celebrating “DBI in the Sky” April 2012, IPM, Tehran
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Our Universe today, looks Homogeneous and Isotropic at cosmological scales.
Cosmic evolution may start froma generic initial condition
Is there any Dynamical explanation ?Is the Isotropic & Homogeneous Universe an attractor of the cosmic evolution?
?
Celebrating “DBI in the Sky” April 2012, IPM, Tehran
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( by Gibbons & Hawking 1977, Hawking & Moss 1982)
all expanding-universe models with a positivecosmological constant (Λ>0) asymptotically approach the de-Sitter solution.
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( by Gibbons & Hawking 1977, Hawking & Moss 1982)
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( by Gibbons & Hawking 1977, Hawking & Moss 1982)
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The 1st attempt to formulate that made by Wald 1983.
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( by Gibbons & Hawking 1977, Hawking & Moss 1982)
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( by R. Wald 1983)
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( by R. Wald 1983)
Theorem: In the General Relativity, consider Initially expanding Bianchi-type ( homogeneous but anisotropic)models, with the total energy-momentum tensor as
, whereis a positive cosmological constant &
satisfies Strong & Dominant energy conditions.
This system will approach de-Sitter space exponentially fast,
with the time-scale .
µνµνµν T~g -=T 0 +Λ00 >Λ
µνT~
0/3 Λ
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( Homogenous But Anisotropic space)jit eeetadtds ij ⊗+−= )(222 )( β
0tr =ijβ
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FRW (Homogeneous and Isotropic Space)ji
ij eetadtds ⊗+−= δ)(222
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FRW (Homogeneous and Isotropic Space)ji
ij eetadtds ⊗+−= δ)(222
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FRW (Homogeneous and Isotropic Space)ji
ij eetadtds ⊗+−= δ)(222
Bianchi (Homogeneous But Anisotropic Space)jit eeetadtds ij ⊗+−= )(222 )( β
0tr =ijβ
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FRW (Homogeneous and Isotropic Space)ji
ij eetadtds ⊗+−= δ)(222
Bianchi (Homogeneous But Anisotropic Space)
0tr =ijβ
jit eeetadtds ij ⊗+−= )(222 )( β
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Strong Energy Condition (SEC):
for all time-like 0)21( ≥− υµ
µνµν ttTgT µt
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Strong Energy Condition (SEC):
for all time-like
Dominant Energy Conditions (DEC):
for all time-like and
0)21( ≥− υµ
µνµν ttTgT µt
0' ≥υµµν ttT µt µ't
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Strong Energy Condition (SEC):
for all time-like
Dominant Energy Conditions (DEC):
for all time-like and
Weak Energy Condition (WEC):
for all time-like
0)21( ≥− υµ
µνµν ttTgT µt
0' ≥υµµν ttT µt µ't
0≥υµµν ttT µt
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( by R. Wald 1983)
Theorem: In the General Relativity, consider Initially expanding Bianchi-type ( homogeneous but anisotropic)models, with the total energy-momentum tensor as
, whereis a positive cosmological constant &
satisfies Strong & Dominant energy conditions.
This system will approach de-Sitter space exponentially fast,
with the time-scale .
µνµνµν T~g -=T 0 +Λ00 >Λ
µνT~
0/3 Λ
Celebrating “DBI in the Sky” April 2012, IPM, Tehran
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( by Gibbons & Hawking 1977, Hawking & Moss 1982)
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( by R. Wald 1983)If the conditions of the Wald’s theorem hold:
Regardless of the enlargement of the initial anisotropy, system exponentially approaches the isotropic solution.
Inflation never ends in these systemsInflationary models do not satisfy in Wald’s
theorem!
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( by Gibbons & Hawking 1977, Hawking & Moss 1982)
Celebrating “DBI in the Sky” April 2012, IPM, Tehran
( by R. Wald 1983)
We need to extend Wald’s theorem for inflationary settings.
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Celebrating “DBI in the Sky” April 2012, IPM, Tehran
( by R. Wald 1983)
( by A. Malek-Nejad & M. M. Sheikh-Jabbari 2012)
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( by A. Malek-Nejad & M. M. Sheikh-Jabbari 2012)
Theorem: In the GR, assume Inflation in Bianchi-type modelswith the total energy-momentum tensor as
where &
satisfies Strong & Weak energy conditions.
I) In principle, anisotropies can grow ( in contrast to the cosmic no-hair conjecture)
2) however, there is an upper-bound on the growth of anisotropies of the order of the slow-roll parameter.
µνµνµν Tg )(-=T +Λ t 0)(t ≥Λ
µνT
!!!
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0)(t ≤Λ
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It is always possible to: describe the energy-momentum tensor of any inflationary system as
where &
satisfies Strong & Weak Energy Conditions.A. M-N and M. M. S-J arXiv:1203.0219v2
µνµνµν Tg )(-=T +Λ t
0)(t ≥Λ
µνT
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General form for the energy-momentum tensor:
anisotropic stress tensor
)()())()((T tgtPuutPt µνµννµµν ρ Π+++=
Celebrating “DBI in the Sky” April 2012, IPM, Tehran
)( tµνΠ jitTij ≠)(
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General form for the energy-momentum tensor:
anisotropic stress tensor
is the source of anisotropy dynamics!
A. M-N and M. M. S-J arXiv:1203.0219v2
)()())()((T tgtPuutPt µνµννµµν ρ Π+++=
Celebrating “DBI in the Sky” April 2012, IPM, Tehran
)( tµνΠ jitTij ≠)(
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Scalar driven inflationary models
Models of inflation involving vector gauge fields
Celebrating “DBI in the Sky” April 2012, IPM, Tehran
Ordinary multi-scalar filed models K-inflation DBI inflation
Gauge-flation Inflationary universe with anisotropic hair
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Scalar driven inflationary models
In all of the above cases , is identically zero. So,anisotropy damps out exponentially fast in few Hubble times.
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Ordinary multi-scalar filed models K-inflation DBI inflation
ijT ji ≠
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Models of inflation involving vector gauge fields Gauge-flation:
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Models of inflation involving vector gauge fields Gauge-flation:
f
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(non-Abelian gauge field inflation) is a novel inflationary scenario in which inflation is driven by su(2) non-Abelian gauge field minimallycoupled to Einstein gravity.
A. Maleknejad & M.M. Sheikh-Jabbari Phys. Rev. D 84 (2011)
A. Maleknejad & M.M. Sheikh-JabbariarXiv:1102.1513
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Models of inflation involving vector gauge fields Gauge-flation:
f
Celebrating “DBI in the Sky” April 2012, IPM, Tehran
(non-Abelian gauge field inflation) is a novel inflationary scenario in which inflation is driven by su(2) non-Abelian gauge field minimallycoupled to Einstein gravity.
A. Maleknejad & M.M. Sheikh-Jabbari Phys. Rev. D 84 (2011)
A. Maleknejad & M.M. Sheikh-JabbariarXiv:1102.1513
Due to its vector nature
Stability of Isotropic background ?
A. Maleknejad , M.M. Sheikh-Jabbari and Jiro Soda, JCAP 1201,016 (2012)
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Celebrating “DBI in the Sky” April 2012, IPM, Tehran
A. Maleknejad , M.M. Sheikh-Jabbari and Jiro Soda, JCAP 1201,016 (2012)
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Models of inflation involving vector gauge fields Inflationary universe with anisotropic hair
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fM. Watanabe, S. kanno and J. Soda
has introduced an inflationary model with anisotropic hair. Their model includes a scalar field as inflation coupled to a mass-less U(1) gauge-field.
M. Watanabe, S. kanno and J. Soda, Phys. Rev. Lett. 102, 191302 (2009)
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B. Himmetoglu, JCAP 1003, 023 (2010)
M. Watanabe, S. kanno and J. Soda, Phys. Rev. Lett. 102, 191302 (2009)
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we Extended cosmic no-hair theorem for general inflationary setups.
We find, the behavior of anisotropies are governed by the anisotropic stress tensor (anisotropic part of )
It is shown that Anisotropies can grow during inflation, but
There is an upper-bound value on their enlargement (assuming slow-roll, its equal to )
Celebrating “DBI in the Sky” April 2012, IPM, Tehran
)(38
00 ηε −
µνT
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