  As cosmic microwave background (CMB) photons traverse the Universe,
anisotropies can be induced via Thomson scattering (proportional to the
integrated electron density; optical depth) and inverse Compton scattering
(proportional to the integrated electron pressure; thermal Sunyaev-Zel'dovich
effect). Measurements of anisotropy in optical depth $\tau$ and Compton $y$
parameter are imprinted by the galaxies and galaxy clusters and are thus
sensitive to the thermodynamic properties of circumgalactic medium and
intergalactic medium. We use an analytic halo model to predict the power
spectrum of the optical depth ($\tau\tau$), the cross-correlation between the
optical depth and the Compton $y$ parameter ($\tau y$), as well as the
cross-correlation between the optical depth and galaxy clustering ($\tau g$),
and compare this model to cosmological simulations. We constrain the optical
depths of halos at $z\lesssim 3$ using a technique originally devised to
constrain patchy reionization at a much higher redshift range. The forecasted
signal-to-noise ratio is 2.6, 8.5, and 13, respectively, for a CMB-S4-like
experiment and a VRO-like optical survey. We show that a joint analysis of
these probes can constrain the amplitude of the density profiles of halos to
6.5% and the pressure profile to 13%, marginalizing over the outer slope of the
pressure profile. These constraints translate to astrophysical parameters
related to the physics of galaxy evolution, such as the gas mass fraction,
$f_{\rm g}$, which can be constrained to 5.3% uncertainty at $z\sim 0$,
assuming an underlying model for the shape of the density profile. The
cross-correlations presented here are complementary to other CMB and galaxy
cross-correlations since they do not require spectroscopic galaxy redshifts and
are another example of how such correlations are a powerful probe of the
astrophysics of galaxy evolution.
