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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. 

## 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. 

<img src='content_image/393907.jpg'>

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 

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 

where R is the size of the region. 

## A. Diffusion 

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<img src='content_image/393917.jpg'>

<img src='content_image/393922.jpg'>

