  Following our previous work in [JCAP 1206, 041 (2012)], in this paper, we
continue our study of reconstructing $f(R)$ modified gravity models that can be
connected to a single scalar field in general relativity via conformal
transformation, which lead to scale-invariant power spectrum in the early
universe. With $f(R)$ modified gravity, one does not need to introduce extra
scalar, the nature of which are to be explained. Different from general
nonminimal coupling theory, the behavior of the $f(R)$ theory has been fixed by
its counterpart in Einstein frame, and thus have one to one correspondence.
Numerical plots of the functional form of $f(R)$ as well as the evolution of
$R$ in terms of cosmic time $t$ are also presented.
