A linear programming approach to inhomogeneous primordial nucleosynthesis

We examine inhomogeneous primordial nucleosynthesis for {\it arbitrary} distributions $f$ of the baryon-to-photon ratio $\eta$, in the limit where neither particle diffusion nor gravitational collapse is important. By discretizing $f(\eta )$ and using linear programming, we show that for a set of $m$ constraints on the primordial element abundances, the maximum and minimum possible values of $\bar{\eta }$ (the final mean value of $\eta$) are given when $f(\eta )$ is a sum of at most $m+1$ delta functions. Our linear programming results indicate that when $f$ is taken to be an arbitrary function, there is no lower bound on $\bar{\eta }$, while the upper bound is essentially the homogeneous upper bound.