4 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4  0  20  40  60  80  100  120  140  160 Number Fraction Ratio Mass Number =9 10 -6 =1 10 -5 =2 10 -5 =3 10 -5 Fig. 2: The number fractions of the produced elements rel- ative to the solar abundances. All the calculations are done with ξ e = 0. Fig. 1: Results of Y$_{p}$ for some sets of η and ξ e . The horizontal line at the middle represents Y$_{p}$ = 0 . 35. Helium ions in the region with a higher Y$_{p}$ might dif- fuse outward the surrounding region with lower helium contents. This effect takes place after the horizon scale exceeds the size of the region, that is, after the tem- perature becomes lower than 10 8 − 10 7 K. The diffusion equation for helium is written as there is a considerable jump in the amount of heavy el- ements with A ∼ 100. Fig. 7 of Matsuura et al. [23] shows that η = 10 − 4 yields a significant amount of heavy elements with A ∼ 150, comparable to the solar abun- dances. With higher η , the abundances of heavy ele- ments continue to increase. Observations show that the amounts of heavy elements are at least a factor of a few less than the solar abundances in the relevant GC [31]. Therefore the upper limit of η is somewhere between $^{3}$× 10 − 5 and 10 − $^{4}$. In Fig. 2, some elements A ∼ 100 are very close to the expected abundances of the GC stars. Thus, it is a good touchstone of our model to check the abundances of the elements A ∼ 100 in the dwarfs in bMS. where n$_{He}$ is the number density of helium and D is the diffusion coefficient of helium with electrons and its mag- nitude is estimated as where λ is the mean free path of helium ions and v = √ 3 k$_{B}$T / 4 m$_{H}$ is the thermal speed of helium ions. Here the constant m$_{H}$ denotes the mass of hydrogen. The time scale t$_{diff}$ of the diffusion is approximated as ∂n$_{He}$ ∂t = $^{D}$∇ $^{2}$n$_{He}$ , (5) D = λv 3 , (6) A. Diffusion IV. EVOLUTION OF HELIUM ENHANCED REGION Even if a primordial gas with high Y$_{p}$ was once formed at the time of BBN, it might be mixed with other gases in the outer regions and end up with a lower fraction of helium. We briefly show that the effect of such mix- ing would not affect the region with high helium abun- dance. Here, we do not only consider effects of diffusion and accretion, but also take into account self-gravity as a competing process. where R is the size of the region. t$_{diff}$ = ∣ ∣$^{∂}$$^{ln}$$^{n$_{He}$}$ ∂t ∣ ∣$^{−}$$^{1}$ = ∣ D ∇ $^{2}$n$_{He}$ n$_{He}$ ∣ ∣$^{−}$$^{1}$ ∼ R 2 D , (7)