1. L. Perivolaropoulos http://leandros.physics.uoi.gr Department of Physics University of Ioannina Open page S. Fay, S. Nesseris, L.P. gr-qc/0703006

2. Accelerating Cosmic Expansion Dark Energy (Quintessence, Perfect Fluid etc) Required Cosmic History: Cosmological Constant Modified Gravity ( f(R), Scalar-Tensor, BraneWorlds etc) 4 forms of f(R) (G.R. + 3) +1 form of f(R): Cosmic History H(z)Reconstruction of f(R) Dynamical System of Cosmic Evolution Modified Gravity (f(R), Scalar-Tensor, BraneWorlds etc) Carroll, Duvvuri, Trodden, Turner, astro-ph/0306438 Amendola, Polarski, Tsujikawa, gr-qc/0612180

3. Action: Generalized Friedman Equations Goal: Study the Dynamics Consistent with a LCDM Cosmic History: Flatness

4. New Variables: Constraint Equation: Dynamical System: Eliminate m assuming known x 3 Amendola et. al., gr-qc/0612180

5. Dynamical System: Eliminate m(f(R)) assuming known x 3 Reduced Dynamical System: Find x 3 assuming a ΛCDM cosmic history:

6. Effective Equation of State: Cosmic Eras : x 3 values in each era:

8. Critical Point Definition: Stationary Dynamical Evolution Stability: has only negative eigenvalues Attractor attractor saddle unstable saddle unstable saddle attractor saddle attractor saddle unstable G.R.

9. Attractor Critical Points: Numerical Evolution confirms attractor behavior of critical points

10. Ω rad Ω DE ΩmΩm

11. ΩmΩm

12. Attractor General Relativity

13. Critical Point in each era (General Relativity)

14. 1. Find F(N): 2. Find f(N): 3. Find R(N): Use numerically obtained 4. Reconstruct f(N) and f(R)

16. Required Cosmic History: 4 forms of f(R) (G.R. + 3) +1 form of f(R): Cosmic History H(z)Reconstruction of f(R) Dynamical System of Cosmic Evolution Nonlinear f(R) Theories can not mimick a ΛCDM Cosmology in a Consistent Manner