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L. Perivolaropouloshttp://leandros.physics.uoi.gr
Department of Physics
University of Ioannina
Open page
S. Fay, S. Nesseris, L.P. gr-qc/0703006
Accelerating
Cosmic ExpansionDark Energy (Quintessence, Perfect Fluid etc)
Required Cosmic History:
Cosmological Constant
Modified Gravity (f(R), Scalar-Tensor, BraneWorlds etc)
1 0 1
3eff eff effw rad w mat w
1r 1m
4 forms of f(R) (G.R. + 3)
+ 1 form of f(R): 2f R R
Cosmic History H(z) Reconstruction of f(R)
Dynamical System of Cosmic Evolution
Modified Gravity (f(R), Scalar-Tensor, BraneWorlds etc)
2f R R
Carroll, Duvvuri, Trodden, Turner, astro-ph/0306438Amendola, Polarski, Tsujikawa, gr-qc/0612180
Action:
Generalized Friedman Equations
Goal: Study the Dynamics Consistent with a LCDM Cosmic History:
Flatness
New Variables:
Constraint Equation:
Dynamical System:
Eliminate m assuming known x3
3H N x N
Amendola et. al., gr-qc/0612180
Dynamical System:
Eliminate m(f(R)) assuming known x3
3H N x N
Reduced Dynamical System:
Find x3 assuming a ΛCDM cosmic history:
Effective Equation of State:
Cosmic Eras :
x3 values in each era:
Critical Point Definition: 1 2 4 0x x x 1 2 4, ,x x x
Stationary Dynamical Evolution
Stability: l l
iij x x
j
xM
x
has only negative eigenvalues Attractor
attractor
saddle
unstable
saddle
unstable
saddle
saddle
attractor
saddle
attractor
saddle
unstable
G.R.
G.R.
G.R.
Attractor Critical Points:
Numerical Evolutionconfirms attractor behavior
of critical points
ΩradΩDE
Ωm
1 2 3 4
1 2 3
4
1m
DE
rad
x x x x
x x x
x
1 2 3 4
1 2 3
4
1m
DE
rad
x x x x
x x x
x
ΩDE
Ωm
1 2 314,5,0,0 3,386,3,886, ,0 1,0,2,02A A A
1 2 31 10,0,0,1 0, , ,0 0, 1,2,02 2B B B
Attractor
General Relativity
1 0x
Critical Point 1 2 3 4, , ,x x x x f R
20 1H
1 0x
in each era
(General Relativity)
20 1H
1. Find F(N):
2. Find f(N):
3. Find R(N):
Use numerically obtained 1 2 3 4, , ,x N x N x N x N
4. Reconstruct f(N) and f(R)
Required Cosmic History:
1 0 1
3eff eff effw rad w mat w
1r 1m
4 forms of f(R) (G.R. + 3)
+ 1 form of f(R): 2f R R
Cosmic History H(z) Reconstruction of f(R)
Dynamical System of Cosmic Evolution
2f R R
Nonlinear f(R) Theories can not mimick a ΛCDM Cosmology in a Consistent Manner