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### Is the x_e equation in CAMB correct or not?

Posted: July 31 2017
I am looking at the Antony Lewis' paper

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 $$z \rightarrow$$ large values, then y=(1+z)^3/2 will also become large values, while y(z_re) and $$\Delta_y$$ are fixed. Since "$$\tanh(x \rightarrow \infty) \rightarrow 1$$", then for large redshift, $$x_{\rm e} \rightarrow 1$$.

If $$z \rightarrow 0$$, then $$\tanh$$ function will becomes a negative value but greater than -1, then $$x_{\rm e} \rightarrow 0$$.

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.

### Re: Is the x_e equation in CAMB correct or not?

Posted: July 31 2017
I think it's just a typo (code has y(zre)-y in the tanh):

https://github.com/cmbant/CAMB/blob/mas ... on.f90#L80

### Is the x_e equation in CAMB correct or not?

Posted: July 31 2017
Oh, then this typo affects the Planck reionization paper, the published version of Planck intermediate results XLVII. Planck constraints on reionization history,

page 5, equation 2

also takes this typo.