L. Perivolaropoulos Department of Physics University of Ioannina Open page S. Fay, S. Nesseris, L.P. gr-qc/0703006.

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