  In models featuring extra spatial dimensions, particle collisions in the
early universe can produce Kaluza-Klein gravitons. Such particles will later
decay, potentially impacting the process of Big Bang nucleosynthesis. In this
paper, we consider scenarios in which gravity is free to propagate throughout
$n$ flat, compactified extra dimensions, while the fields of the Standard Model
are confined to a 3+1 dimensional brane. We calculate the production and decay
rates of the states that make up the Kaluza-Klein graviton tower and determine
the evolution of their abundances in the early universe. We then go on to
evaluate the impact of these decays on the resulting light element abundances.
We identify significant regions of previously unexplored parameter space that
are inconsistent with measurements of the primordial helium and deuterium
abundances. In particular, we find that for the case of one extra dimension
(two extra dimensions), the fundamental scale of gravity must be $M_{\star} > 2
\times 10^{13} \, {\rm GeV}$ ($M_{\star} > 10^{10} \, {\rm GeV}$) unless the
temperature of the early universe was never greater than $T \sim 2 \, {\rm
TeV}$ ($T \sim 1 \, {\rm GeV}$). For larger values of $n$, these constraints
are less stringent. For the case of $n=6$, for example, our analysis excludes
all values of $M_{\star}$ less than $\sim 10^{6} \, {\rm GeV}$, unless the
temperature of the universe was never greater than $T \sim 3 \, {\rm TeV}$. The
results presented here severely limit the possibility that black holes were
efficiently produced through particle collisions in the early universe's
thermal bath.
