Accelerating Multidimensional Cosmologies

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Accelerating MultidimensionalCosmologies

• Alexander Zhuk
• Department of Theoretical Physics
• and
• Astronomical Observatory,
• Odessa National University
• zhuk_at_paco.net

A.Z.,V.Baukh, hep-th/0601205 A.Z.,T.Saidov,
hep-th/0612217
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Energy-density components of the Universe
?!
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Evidence for dark matter

• Galactic rotation curves
• Gravitational lensing
• Large-scale structure formation

Evidence for dark energy
?CDM- model

• Observations of type Ia supernovae
• Cosmic microwave background anisotropy
• Universe is flat
• Total amount of matter
• Large-scale structure formation
• Total amount of matter

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Observations
Late-time accelerating expansion of the Universe
Origin of acceleration ?
Modified gravity
Extra-dimensions

• - term,
• Quintessence,
• K-essence

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- term
Einstein 1917
(the biggest blinder of my life )
Present-day acceleration
Quintessence
K-essence
Modified scalar field kinetic term (negative)
Modified gravity
Non-linear models
E.g.
Conventional cosmology
Late-time acceleration
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Acceleration from extra dimensions
Stabilized internal spaces (geometrical moduli)
Dynamical internal spaces (Sp-brane models)
Varying effective 4-D fundamental constants
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Topology
. . .
M
M
M
M
0
1
n
Metric
Einstein spaces
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Gravitational excitons
Stable compactification
time
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Conformal excitations of the internal spaces
MR
R
S
Gravitational excitons
D3
a
Equilibrium position
U.Günther, A.Zhuk, PRD56 (1997)
D
2
eff
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General method of stabilization
U.Günther, A.Zhuk, PRD56 (1997) 6391
Action
D-dimensional gravitational constant
Dimensional reduction (Einstein frame)
Effective potential
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stably compactified background
gravexcitons
Minima of define
positions of stabilization of the internal spaces
4-D gravitational constant
internal space volume
Einstein Brans-Dicke frames
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The model
- dimensional non-linear action
-form-field strength with block-orthogonal
structure
Equivalency
A.Z,. U.Günther, V.Bezerra, C.Romero, CQG
22(2005)3135
Linear model scalar field
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Conformal transformation
Nonlinearity scalar field
Potential
Freund-Rubin ansatz for form-fields
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Dimensional reduction
where
and
is defined by
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The form of nonlinearity
for
for

Potential
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Freezing stabilization conditions
Extremum condition
extremum position
Hessian positivity
4-D effective cosmological constant
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I. Zero effective cosmological constant
Quadratic equation
,
Solutions
,
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Zero-minimum conditions
Zero minimum of exists for the
following combinations of the parameter
1.
2.
Zero minimum
absence of acceleration
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II. Positive effective cosmological constant
Particular case
Cosmic acceleration
Decoupling of excitations
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Extremum conditions
1. Quadratic equation
Solutions
2. Relations between parameters
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Positive minimum conditions

Minimum exists for the following combination of
the parameters
Profile of the
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Positive minimum conditions



positive curvature internal space
2




1TeV
Extreme fine tuning
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Saddle point of the effective potential
In the case all possible extrema
of may take place only for
which correspond to the solutions of
the above quadratic equation.
Local minimum
Saddle
Contour plot of
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Domain walls
Profile of the
Domain walls separates regions with different
vacua in the Universe.
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Non-inflating domain wall
K.Maeda et al PRD53(1996) 655 Necessary
condition for inflating domain wall in
double-well potential
In our case
Domain walls do not inflate.
Slow roll parameters (in the saddle point)
but
Our potential is too steep!
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Conclusions
1. It is shown the double role of the effective
potential a) it provides the freezing
stabilization of the internal spaces
b) it ensures the cosmic acceleration
(fine-tuning). 2. The effective potential has
the saddle point it results in non-inflating
domain walls which separates regions
with different vacua in the Universe. 3.
The effective potential has branchpoints (q-1).
There is a possibility of transitions from
one branch to another (on the analogy of
the catastrophe theory or phase transitions)
(problem of future investigation).
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Part II Dynamical internal spaces
Sp-brane cosmological models
M.Gutperle, A.Strominger hep-th/0202210
Time dependent solutions with (p1)-dimensional
Euclidean world-volume (D-p-2)-dimensional
hyperbolic, flat or spherical spaces as
transverse/additional dimensions
gauge of time
Space-like brane. If p2 ? our 3-D space
Dynamical internal space
11-D M-theory
SM2-branes
10-D Dirichlet strings
SD2-branes
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D-dimensional Einstein-Yang-Mills theory (Abelian
part)
D-dimensional electric charge
Dimensional reduction
A.Z., U.Günther, A.Starobinsky, hep-ph/0306191
- Internal space volume
4-D effective electric charge
4-D effective fine structure constant
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4-D effective fine structure constant
Dynamical behavior of the internal spaces
results in variation with time of the 4-D
effective fine structure constant
Observational constraints
QSO ( Webb et all )
Oklo ( Olive et al )
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The model
Action
Metric
Synchronous time in Einstein frame

Topology
E-frame
BD-frame
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Solution for the model
(in harmonic time gauge)
A.Z., U.Bleyer, gr-qc/9405199405028
External space scale-factor
Internal space scale-factor
Scalar field
- momenta in minisuperspace
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Hubble parameters
synchronous time
harmonic time
Deceleration parameter (external space)
A.Riess et al (astro-ph/0402512) transition
readshift between deceleration and acceleration
stages
4-D fine structure constant
Variations
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SM2-brane case
Scalar field is absent
Acceleration does not end at present time, z0
Deceleration parameter
QSO
Oklo
Fine structure constant variations does
not satisfy these constraints
Fine structure constant variation
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SD2-Brane
- redshift of the completion of accelerating
expansion
Scale factor momentum
Scalar field is present
Condition for acceleration
Scalar field momentum
Large scalar field kinetic energy suppress
acceleration
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Variations of the fine structure constant
Such variations contradict observations (too
large!)
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Conclusions

• Sp-brane models can provide stages of the
  late-time
• accelerating expansion.

2. For considered models, acceleration lasts for
limiting period of time.
3. All considered models conflict with
observational data on variations of the fine
structure constant.
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Thank you