 Solution

Adopt the stipulated neutral fraction $f_{\mathrm{HI}}=10^{-6}$, independently of the equilibrium estimate in e. For pure hydrogen, $n_e=(1-f_{\mathrm{HI}})n_{\mathrm{HI}}/f_{\mathrm{HI}}\simeq 8.3{\mathrm{m}}^{-3}$. The thermal Sunyaev-Zeldovich effect measures integrated electron pressure:Using the same absorbing line of sight as in d,Thus one such filament is below the stated measurement sensitivity. If the number density inferred transversely is instead used to view the same cylinder end-on, its path is twenty times longer and $y=2.9\times 10^{-10}$, 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.