  Modified theories of gravity have recently been studied by several authors as
possibly viable alternatives to the cosmological concordance model. Such
theories attempt to explain the accelerating expansion of the universe by
changing the theory of gravity, instead of introducing dark energy. In
particular, a class of models based on higher order curvature invariants,
so-called $f(R)$ gravity models, has drawn attention. In this letter we show
that within this framework, the expansion history of the universe does not
uniquely determine the form of the gravitational action and it can be radically
different from the standard Einstein-Hilbert action. We demonstrate that for
any barotropic fluid, there always exists a class of $f(R)$ models that will
have exactly the same expansion history as that arising from the
Einstein-Hilbert action. We explicitly show how one can extend the
Einstein-Hilbert action by constructing a $f(R)$ theory that is equivalent on
the classical level. Due to the classical equivalence between $f(R)$ theories
and Einstein-Hilbert gravity with an extra scalar field, one can also hence
construct equivalent scalar-tensor theories with standard expansion.
