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Waseda University
Kei-ichi Maeda
On Acceleration of the Universe
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Present Acceleration
cosmological constant
Acceleration of cosmic expansion
Inflation: early stage of the Universe
Dark Energy
Inflaton ?
Modified gravity
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Two Comments
[I] Matter couplings
[2] Negative cosmological constant
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[I] Matter couplings
When we discuss acceleration by some unknown field
(or modification of gravity), we ignore matter fields.
Its energy density will drop when the Universe expands
(2) Perfect Fluid with
Matter: (1) Particles in Standard Model
However, if there exists some couplings between matter and field
(or gravity), dynamics may change.
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(1) Coupling with gauge field
What is an inflaton f ?
top-down superstring (or 10D supergravity)
In compactification,
we naturally expect a dilaton or moduli coupling.
This coupling may spoil the inflationary models
unless the moduli is fixed.
Townsend (2003)
Flux compactification
Hyperbolic compactification
No accelerated expansion
Note: power-law inflationary solution if
scale factor with
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There exists another natural ingredient in the unified theories : gauge fields
Abelian [U(1)] or non-Abelian [e.g. SU(2)] gauge fields
Heterotic string theory
Flux compactification U(1) multiplet
In effective 4D action,
moduli coupling may appear:
If VEVs of gauge fields exist, it will change the dynamics of a scalar field.
Hull-Townsend (1995) :
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U(1) field
U(1) multiplet with the same gauge-kinetic coupling
Non-Abelian gauge field
Anisotropic Inflation
Kanno, Soda, Watanabe (2009), Watanabe, Kanno, Soda (2010)
Kanno. Soda, Watabnabe (2010)
The isotropic inflationary universe is an attractor.
Anisotropic inflation can be possible as a transient state
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Inflation with Gauge Fields
:SU(2) Yang-Mills field
FLRW metric
YM potential
Isotropic and homogeneous universe
Scalar field
KM & K. Yamamoto (2013)
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Basic equations:
Einstein equations
YM equation
electric component magnetic component
YM energy density
scalar field equation
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U(1) triplet no non-linear coupling
power-law solution with larger power exponent
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power-law solutions
The case with electric field (E )
The case with magnetic field (B )
The scalar field dominance (S )
U1
U1
U1
Inflation
E-B duality E U1 B U1
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phase diagram each phase is an attractor
a
l
B U1
E U1 S U1
B - I U1
S - I U1
E - I U1
conventional power-law inflation Inflation is possible even for
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YM field
Both electric and magnetic components exist
Non-linear coupling complicated
The case with dominant electric component (E -I)
The case with dominant magnetic component (B -I)
The scalar field dominance (S -I)
YM
YM
YM
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phase diagram
a
l
1/3<p<1/2 p>1
p>1
1/2<p<
1
p=1/2
p=1
B - I YM
E - I YM
NA +
NA -
S - I YM
O +
O -
S - I YM
+YM oscillation
Stable fixed points
Asymptotic spacetime +YM oscillations Inflationary spacetime
+YM oscillations
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The power-law inflation with the gauge field is
possible even for a steep potential such as
which is expected in the higher-dimensional
unified theories.
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(2) Coupling with perfect fluid KM, Y. Fujii (09)
Modified gravity (e.g. scalar tensor theory)
MODEL
conformal transformation
g → g exp(2zks)
Einstein gravity (g) + scalar field s U=V exp (-4zks)
z = x/(e+6x)
But, coupling with matter is important
Dynamics without matter is well-known
U=V exp (-4zks)
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g
8z 2
FP1
No attractor
FP2
Two fixed points
FP1
FP2
Scalar field dominant
Scaling solution
Minkowski in Jordan frame
power exponent
of attractor sol.
const
FP1
FP2
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power-law potential
a
z
g=1(dust) g=4/3(radiation) g=2 (stiff)
power exponent
of attractor sol.
Inflation with a steep potential
p +
g = 1 z
a = 5
FP1 FP2
New type
Power-law inflation
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[2] Negative cosmological constant
Supergravity (Superstring)
cosmological constant
effective cosmological constant
Accelerating universe
(1) Quantum corrections
(2) KKLT compactification
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20
Heterotic superstring theory
Quantum corrections R.R. Metsaev A.A. Tseytlin,
(’87)
B. Zwiebach (’85)
Ambiguity in the effective action due to field redefinition
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Negative cosmological constant + higher-curvature terms KM & Ohta (14)
Accelerating universe
EH action
unstable
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EH+GB
q-sphere
q-hyperbolic space
two branches:
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L: negative
stable
Branch (2)
L: positive
unstable
Branch (1)
de Sitter solution with GB term is
stable if L is negative.
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(2) KKLT compactification
CY compactification:
fixing moduli
AdS dS
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Two types of strings
g-string & f-string bigravity theory in 10-dim
similar interactions to ghost-free bigravity
two metrics & twin matter fluid
Interactions ? g, f
: coupling constants
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Not need to introduce anti-branes
dS
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Interactions between three forms
CY VEV of three forms
the same as the previous interaction term
Does this explain smallness of the “graviton mass” ?
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Matter coupling may change the dynamics
[I] Matter couplings
[2] Negative cosmological constant
Two examples to find de Sitter solution
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Thank you for your attention