Big Bang Nucleosynthesis with Gaussian Inhomogeneous Neutrino Degeneracy

We consider the effect of inhomogeneous neutrino degeneracy on Big Bang nucleosynthesis for the case where the distribution of neutrino chemical potentials is given by a Gaussian. The chemical potential fluctuations are taken to be isocurvature, so that only inhomogeneities in the electron chemical potential are relevant. Then the final element abundances are a function only of the baryon-photon ratio $\eta$, the effective number of additional neutrinos $\mathrm{\Delta }{N}_{\nu }$, the mean electron neutrino degeneracy parameter $\bar{\xi }$, and the rms fluctuation of the degeneracy parameter, ${\sigma }_{\xi }$. We find that for fixed $\eta$, $\mathrm{\Delta }{N}_{\nu }$, and $\bar{\xi }$, the abundances of helium-4, deuterium, and lithium-7 are, in general, increasing functions of ${\sigma }_{\xi }$. Hence, the effect of adding a Gaussian distribution for the electron neutrino degeneracy parameter is to decrease the allowed range for $\eta$. We show that this result can be generalized to a wide variety of distributions for $\xi$.