We consider f ( R ) modified gravity theories in the metric variation formalism and attempt to reconstruct the function f ( R ) by demanding a background \Lambda CDM cosmology .
In particular we impose the following requirements : a .
A background cosmic history H ( z ) provided by the usual flat \Lambda CDM parametrization though the radiation ( w _ { eff } = 1 / 3 ) , matter ( w _ { eff } = 0 ) and deSitter ( w _ { eff } = -1 ) eras .
b .
Matter and radiation dominate during the ‘ matter ’ and ‘ radiation ’ eras respectively i.e . \Omega _ { m } = 1 when w _ { eff } = 0 and \Omega _ { r } = 1 when w _ { eff } = 1 / 3 .
We have found that the cosmological dynamical system constrained to obey the \Lambda CDM cosmic history has four critical points in each era which correspondingly lead to four forms of f ( R ) .
One of them is the usual general relativistic form f ( R ) = R - 2 \Lambda .
The other three forms in each era , reproduce the \Lambda CDM cosmic history but they do not satisfy requirement b. stated above .