Constraining Density Fluctuations with Big Bang Nucleosynthesis in the
  Era of Precision Cosmology
  We reexamine big bang nucleosynthesis with large-scale baryon density
inhomogeneities when the length scale of the density fluctuations exceeds the
neutron diffusion length ($\sim 10^7-10^8$ cm at BBN), and the amplitude of the
fluctuations is sufficiently small to prevent gravitational collapse. In this
limit, the final light element abundances can be determined by simply mixing
the abundances from regions with different baryon/photon ratios without
interactions. We examine gaussian, lognormal, and gamma distributions for the
baryon/photon ratio, $\eta $. We find that the deuterium and lithium-7
abundances increase with the RMS fluctuation in $\eta $, while the effect on
helium-4 is much smaller. We show that these increases in the deuterium and
lithium-7 abundances are a consequence of Jensen's inequality, and we derive
analytic approximations for these abundances in the limit of small RMS
fluctuations. Observational upper limits on the primordial deuterium abundance
constrain the RMS fluctuation in $\eta $ to be less than $17\%$ of the mean
value of $\eta $. This provides us with a new limit on the graininess of the
early universe.
