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 .