https://arxiv.org/pdf/0804.3865.pdf

for Eq.(B3) on page 11, there is this equation of number of free electron per hydrogen atom.

But for this equation, if [tex]z \rightarrow[/tex] large values, then y=(1+z)

^{3}/2 will also become large values, while y(z

_{re}) and [tex]\Delta_y[/tex] are fixed. Since "[tex]\tanh(x \rightarrow \infty) \rightarrow 1[/tex]", then for large redshift, [tex]x_{\rm e} \rightarrow 1[/tex].

If [tex]z \rightarrow 0[/tex], then [tex]\tanh[/tex] function will becomes a negative value but greater than −1, then [tex]x_{\rm e} \rightarrow 0[/tex].

So this is completely opposite to the trend of Fig.6 on the same page. Can anyone explain what is going on here?

Perhaps I made some stupid mistake, please point it out. Thank you.