Constraining the ionized gas evolution with CMB-spectroscopic survey
  cross-correlation
  We forecast the prospective constraints on the ionized gas model $f_{\rm
gas}(z)$ at different evolutionary epochs via the tomographic cross-correlation
between kinetic Sunyaev-Zeldovich (kSZ) effect and the reconstructed momentum
field at different redshifts. The experiments we consider are the Planck and
CMB Stage-4 survey for CMB and the SDSS-III for the galaxy spectroscopic
survey. We calculate the tomographic cross-correlation power spectrum, and use
the Fisher matrix to forecast the detectability of different $f_{\rm gas}(z)$
models. We find that for constant $f_{\rm gas}$ model, Planck can constrain the
error of $f_{\rm gas}$ ($\sigma_{f_{\rm gas}}$) at each redshift bin to $\sim
0.2$, whereas four cases of CMB-S4 can achieve $\sigma_{f_{\rm gas}} \sim
10^{-3}$. For $f_{\rm gas}(z)=f_{\rm gas,0}/(1+z)$ model the error budget will
be slightly broadened. We also investigate the model $f_{\rm gas}(z)=f_{\rm
gas,0}/(1+z)^{\alpha}$. Planck is unable to constrain the index of redshift
evolution, but the CMB-S4 experiments can constrain the index $\alpha$ to the
level of $\sigma_{\alpha} \sim 0.01$--$0.1$. The tomographic cross-correlation
method will provide an accurate measurement of the ionized gas evolution at
different epochs of the Universe.
