5 1. Before Recombination 2. After Recombination Before the recombination of electrons, photons and matters had been coupled and had the same tempera- ture. The number density of helium during this epoch is proportional to the cube of the temperature and written as Matters are assumed to completely decouple from ra- diation after the recombination. Thus, n$_{He}$ and R after the recombination are denoted as functions of the gas temperature T , Here T$_{0}$ = 2 . 73 K is the present temperature of CMBR and n$_{γ}$$_{0}$ is the present number density of photons and expressed as n$_{γ}$$_{0}$ = 2 ζ (3) /π $^{2}$( k$_{B}$T$_{0}$/ ℏ c ) 3 = 416 cm − 3 where ℏ is the reduced Planck constant. For the case of η = 10 − 4 and Y$_{p}$ = 0 . 4, Thus, the size R is expressed as a function of the tem- perature Here the temperature T$_{re}$ when free electrons recombine with hydrogen ions is about 5,500 K due to the higher density. We have assumed that all of helium and hy- drogen atoms are neutral in this stage. Therefore the cross section for helium atoms becomes of the order of ∼ 10 − 16 cm $^{2}$. Accordingly, D becomes and Next, we estimate D before the recombination. As- suming that the universe has no net charge, the number density of electrons n$_{e}$ is written as The diffusion timescale is obtained as a constant value. This diffusion time is also too large. The last value is, again, for the case of η = 10 − 4 and Y$_{p}$ = 0 . 4. From [32], Since the region with enhanced helium content has a much higher density than the surrounding region, the region accretes matters from the outside, which might dilute the helium fraction. For simplicity, the timescale of this process is estimated assuming the Bondi accretion [33]. The time scale of the Bondi accretion is inferred from the relation where Thus, From R and D derived above, t$_{diff}$ is estimated to be This diffusion time scale is so large that the effect of diffusion is negligible. where M is the total mass of the region (10 $^{6}$M ⊙ in this case), ρ $^{(}$∞) and c$_{s}$ $^{(}$∞) are the density and the sound speed at infinity. The eigen value of the steady state so- lution λ ( γ ) is a function of the adiabatic index γ written as The values of ρ $^{(}$∞) and c$_{s}$ $^{(}$∞) are taken from those in the surrounding region with Y$_{p}$ = 0 . 25 and η $^{= 6}$×10 − $^{10}$. n$_{He}$ = Y$_{p}$ 4 n$_{b}$ = Y$_{p}$ 4 ( T T$_{0}$ ) 3 ηn$_{γ}$$_{0}$. (8) n$_{He}$ = Y$_{p}$ 4 ηn$_{γ}$$_{0}$ ( T$_{re}$T ) 3 / 2 T 3 0 = 80 T 3 / 2 cm − $^{3}$, (18) R = $^{7}$× 10 19 T 1 / 2 cm . (19) n$_{He}$ $^{= 2}$× 10 − $^{4}$T 3 cm − $^{3}$. (9) R = ( 3 Y$_{p}$ 10 $^{6}$M ⊙ 16 πm$_{H}$n$_{He}$ ) 1 / 3 = $^{5}$× 10 21 T cm . (10) D = $^{3}$× 10 17 T cm $^{2}$sec − $^{1}$. (20) t$_{diff}$ $^{= 2}$× 10 22 sec . (21) B. Accretion n$_{e}$ = n$_{H}$ + 2 n$_{He}$ (11) = ( $^{1}$− Y$_{p}$ 2 ) ( T T$_{0}$ ) 3 ηn$_{γ}$$_{0}$ (12) $^{= 2}$× 10 − $^{3}$T 3 cm − $^{3}$. (13) λ = 9 k 2 $_{B}$T 2 4 n$_{e}$e $^{4}$ln Λ = $^{8}$× 10 $^{5}$T 2 n$_{e}$ ln Λ cm , (14) Λ = 3 4 e 2 ( k 3 $_{B}$T 3 πn$_{e}$ ) 1 / 2 $^{= 7}$× 10 3 ( T 3 n$_{e}$ ) 1 / 2 . (15) D = λv 3 = $^{8}$× 10 11 T 1 / 2 cm $^{2}$sec − $^{1}$. (16) t$_{diff}$ ∼ $^{3}$× 10 31 T 3 / 2 sec ∼ $^{9}$× 10 20 ( T 10 $^{7}$K ) 3 / 2 sec . (17) λ ( γ ) =     ( 2 $^{5}$−3 γ ) 5 − 3 γ 2( γ − 1) ( γ = 5 $_{3}$) e 5 / 3 ( γ = 5 $_{3}$) . (23) ˙ M M = 4 πG $^{2}$M ρ $^{(}$∞) c 3 $^{$_{s}$(}$∞ ) λ ( γ ) , (22)