= Solution
Adopt the stipulated neutral fraction $f_{\rm HI}=10^{-6}$, independently of the equilibrium estimate in <1/e>. For pure <hydrogen>, $n_e=(1-f_{\rm HI})n_{\rm HI}/f_{\rm HI}\simeq8.3\ \mathrm{m^{-3}}$. The <thermal Sunyaev-Zeldovich effect> measures integrated electron <pressure>:
$$
y=\frac{\sigma_T}{m_ec^2}\int n_ek_BT_e\,ds
\simeq\frac{\sigma_Tk_BT_e}{m_ec^2}\frac{N_{\rm HI}}{f_{\rm HI}}.
$$
Using the same absorbing <line of sight> as in <1/d>,
$$
\boxed{y=1.44\times10^{-11},\qquad y/(10^{-9})=0.0144.}
$$
Thus one such filament is below the stated measurement sensitivity. If the <number density> inferred transversely is instead used to view the \i[same cylinder] end-on, its path is twenty times longer and $y=2.9\times10^{-10}$, still below the stated sensitivity. Changing the assumed path of the original absorption measurement while keeping its <column density> fixed does not change its predicted $y$. The quoted neutral-fraction floor is a premise of this estimate, not a universal bound derivable from <thermal equilibrium>.
Back to article page
