  We study the chameleon field dark matter, dubbed \textit{scalaron}, in $F(R)$
gravity in the Big Bang Nucleosynthesis (BBN) epoch. With an $R^{2}$-correction
term required to solve the singularity problem for $F(R)$ gravity, we first
find that the scalaron dynamics is governed by the $R^{2}$ term and the
chameleon mechanism in the early universe, which makes the scalaron physics
model-independent regarding the low-energy scale modification. In viable $F(R)$
dark energy models including the $R^{2}$ correction, our analysis suggests the
scalaron universally evolves in a way with a bouncing oscillation irrespective
of the low-energy modification for the late-time cosmic acceleration.
Consequently, we find a universal bound on the scalaron mass in the BBN epoch,
to be reflected on the constraint for the coupling strength of the $R^2$ term,
which turns out to be more stringent than the one coming from the fifth force
experiments. It is then shown that the scalaron naturally develops a small
enough fluctuation in the BBN epoch, hence can avoid the current BBN constraint
placed by the latest Planck 2018 data, and can also have a large enough
sensitivity to be hunted by the BBN, with more accurate measurements for light
element abundances as well as the baryon number density fraction.
