Does inhomogeneous big bang nucleosynthesis produce an inhomogeneous element distribution today?

Inhomogeneous big bang nucleosynthesis (BBN) produces a spatially inhomogeneous distribution of element abundances at $T\sim {10}^9\mathrm{K}$, but subsequent element diffusion will tend to erase these inhomogeneities. We calculate the cosmological comoving diffusion length for the BBN elements. This diffusion length is limited by atomic scattering and is therefore dominated by diffusion when the atoms are neutral, between the redshifts of recombination and reionization. We find that the comoving diffusion length today is $d_{\mathrm{com}}\approx 70\mathrm{pc}$ for all of the elements of interest except ${}^7\mathrm{Li}$, for which $d_{\mathrm{com}}$ is an order of magnitude smaller because ${}^7\mathrm{Li}$ remains ionized throughout the relevant epoch. This comoving diffusion length corresponds to a substellar baryonic mass scale and is roughly equal to the horizon scale at BBN. These results lend support to the possibility that inhomogeneities on scales larger than the horizon at BBN could lead to a spatially inhomogeneous distribution of elements today, while purely subhorizon fluctuations at BBN can result only in a homogeneous element distribution at present.