# Probing the circumgalactic medium with CMB polarization statistical
anisotropy

Anirban Roy  Department of Astronomy, Cornell University, Ithaca, New York
14853, USA  Alexander van Engelen  School of Earth and Space Exploration,
Arizona State University, Tempe, AZ 85287, USA  Vera Gluscevic  Department of
Physics & Astronomy, University of Southern California, Los Angeles, CA,
90007, USA  Nicholas Battaglia  Department of Astronomy, Cornell University,
Ithaca, New York 14853, USA

###### Abstract

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. We forecast the
sensitivity to measuring these observables with future experiments. The
forecasted signal-to-noise ratio is 2.6, 8.5, and 13, respectively, for a
CMB-S4-like experiment (for fractional sky coverage $f_{\rm sky}=0.5$) 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.

##  1 Introduction

Current-generation cosmic microwave background (CMB) observations by the
Planck satellite and various Stage-3 ground-based experiments have mapped the
temperature anisotropy with unprecedented precision, saturating the cosmic
variance limit on a wide range of angular scales (Planck Collaboration et al.,
2020; Aiola et al., 2020; Reichardt et al., 2021). They have, however, only
begun to tap the information encoded within the polarization and lensing
anisotropy, leaving much to be accomplished by future experiments. The next-
generation ground-based observatories such as Simons
Observatory111https://simonsobservatory.org/ (Ade et al., 2019), CMB
S4222https://cmb-s4.org/ (Abazajian et al., 2019), and
SPT-3G333https://pole.uchicago.edu/ (Sobrin et al., 2021) are envisioned to
realize a dramatic leap forward in terms of polarization and lensing
anisotropy measurements on all angular scales, optimizing the science goals
tied to the effects on the CMB-power-spectra damping tail: neutrino mass
measurements, searches for light relic particles, dark matter, etc. The
discovery potential of high-resolution polarization measurements, however,
goes beyond the damping-tail science. In particular, many physical effects
produce non-Gaussian footprints in CMB maps, which can be sought with higher-
order statistical-anisotropy estimators. In this work, we focus on non-
Gaussian signatures of secondary anisotropy imprinted by the inhomogeneous
distribution of free electrons in halos at $z\lesssim 3$.

The gas present within the virial radii of galaxies and outside their disks is
known as the circumgalactic medium (CGM). The CGM connects the interstellar
medium (ISM) to the intergalactic medium (IGM), and is affected by the physics
of the radiative process of galactic winds, active galactic nuclei (AGN) and
supernovae feedback (Werk et al., 2014; Nielsen et al., 2015; Tumlinson et
al., 2017). These processes change the pressure and density profile of
galaxies, and hence the detection of thermal Sunyaev Zel’dovich (tSZ) from CGM
is a potential probe to help understanding the thermodynamics of halos. Thus,
CMB observations can provide a new window into studying the CGM. Several
cross-correlation studies such as tSZ-galaxy clustering and tSZ-CMB lensing
are also useful to measure the pressure of gas and their evolution in host
halos (Hill & Spergel, 2014; Battaglia et al., 2015; Hurier et al., 2015;
Hojjati et al., 2017; Pandey et al., 2020; Koukoufilippas et al., 2020; Amodeo
et al., 2021; Yan et al., 2021; Pandey et al., 2021). The tSZ-galaxy
clustering cross-correlation has been extensively studied in various aspects
to measure the gas mass fraction, the gas temperature, and their evolution
(Greco et al., 2015; Singh et al., 2016; Amodeo et al., 2021; Moser et al.,
2021; Meinke et al., 2021). Furthermore, the measurements of the kinetic
Sunyaev Zel’dovich (kSZ) effect can provide constraints on the density
profiles of gas in halos, and joint analysis of kSZ and tSZ measurements can
break the degeneracies between the parameters. In addition, cross-correlations
with spectroscopic galaxy samples can help us to measure the temperature
profile of the halos (Amodeo et al., 2021).

The spatial distribution of free electrons in the Universe is inhomogeneous,
and the largest contribution to the statistical variance of these fluctuations
is generated during the epoch of reionization (EoR) since the reionization
process is highly non-Gaussian in nature. The small scale temperature and
polarization anisotropies in CMB are suppressed by a factor of
$e^{-\tau(\hat{\boldsymbol{n}})}$ due to effects of “patchy screening”, where
$\tau(\hat{\boldsymbol{n}})$ is the direction-dependent optical depth (Dvorkin
& Smith, 2009; Gluscevic et al., 2013). This can be probed by the
reconstruction of optical depth field (Dvorkin & Smith, 2009; Gluscevic et
al., 2013; Namikawa, 2018; Roy et al., 2018, 2021; Guzman & Meyers, 2021).
Similar to but smaller than the reionization process, galaxies and galaxy
clusters at low redshift contribute to these optical depth fluctuations and
provide a probe of the distribution of free electrons inside such halos.
Although the imprints of halos and the reionization process in CMB are related
to different physical phenomena, their common trait is a qualitatively similar
statistical-anisotropy signature in CMB maps that requires the same quadratic-
estimator techniques to uncover. Even though the estimators for the
reconstruction of optical depth fluctuations (Dvorkin & Smith, 2009) was first
developed to study the reionization process, we adopt it here to perform a
statistical study of the CGM in the low redshift Universe.

Quadratic estimators for reconstructing lensing potential have been widely
used to map the matter distribution in the Universe (Aghanim et al., 2018),
and an attempt to probe the electron density fluctuations was made by
implementing quadratic optical depth estimator to the Planck data (Namikawa,
2018). As secondary anisotropies due to the fluctuations in electron density
are subdominant to the lensing signal, neglecting the higher-order corrections
in lensing and optical depth can bias the reconstruction noise of an
estimator. A bispectrum approach was adopted to extract information about the
CMB lensing and optical depth simultaneously (Feng & Holder, 2019). Their
forecast shows the cross-correlation between the reconstructed optical depth
and lensing can be detected with a few $\sigma$ by upcoming CMB experiments.
Furthermore, the forecasted detectability of the cross-correlation between
optical depth reconstructed for a CMB-S4-like experiment and galaxy number-
count for a Vera C. Rubin Observatory (VRO)-like experiment is at the order of
$\sim 8\sigma$ (Feng & Holder, 2018).

The measurements of density profiles are particularly important to break the
degeneracy between optical depth and peculiar radial velocity in kSZ
measurements (e.g., Alonso et al., 2016; Madhavacheril et al., 2019). Other
kSZ estimators such as the projected fields estimator (Hill et al., 2016;
Ferraro et al., 2016; Kusiak et al., 2021), the tomographic kSZ estimator
(Shao & Fang, 2016), and the pairwise kSZ estimator (e.g., De Bernardis et
al., 2017; Calafut et al., 2021; Vavagiakis et al., 2021) can further provide
integral constraints optical depth of halos (the baryon abundance). Though the
statistical properties of the optical depth of halos can be inferred only from
such kSZ measurements, these cross-correlation techniques reduce various
systematic effects.

In this paper, we propose a set of independent probes such as cross-
correlation between the optical depth and galaxy clustering, as well as the
cross-correlation between the optical depth and the Compton $y$ parameter to
understand the gas physics in CGM. We thus present sensitivity forecasts for a
CMB-S4-like experiment for each of these physical effects and explore survey
strategies that maximize the quality of their measurement. We show how future
measurements of these signals can constrain the astrophysical parameters of
the CGM. Thus, we illustrate the broad range of CGM information available from
careful statistical anisotropy measurements with next-generation CMB
observations.

This paper is organized as follows. In Section 2, we describe the halo model
approach for the calculation of several auto power spectra of optical depth,
galaxy overdensity, Compton y, and cross power spectrum between them. We
compare our model to simulations in Section 3. In Section 4, we describe the
noise model for optical depth estimator, tSZ measurements, and galaxy
clustering. We present the key results of our paper in Section 5, and then we
present the constraints on astrophysical parameters in Section 6. We finally
draw our conclusions in Section 7. Throughout this work, we assume the best-
fit $\Lambda$CDM cosmological parameters derived from Planck TT, TE,
EE+lowE+lensing signals (Planck Collaboration et al., 2020).

##  2 Halo Model

In this section, we review and summarize the basic formalism of the halo-model
approach to calculate the auto and cross power spectra of different
observables such as tSZ, optical depth, and galaxy fields.

###  2.1 Optical depth

The integrated electron density along the line of sight is referred to as the
optical depth. As the underlying density field is perturbed, it introduces
spatial fluctuations in the electron distribution. This leads to the direction
dependence of the optical depth, which can be written as:

$\tau(\hat{\boldsymbol{n}})=\sigma_{\rm T}\int a\,n_{\rm
e}(\hat{\boldsymbol{n}},\chi)d\chi\,.$ (1)

Here, $n_{\rm e}(\hat{\boldsymbol{n}},\chi)$ is the electron density along the
direction $\hat{\boldsymbol{n}}$ at a comoving distance $\chi$, $\sigma_{\rm
T}$ is the Thomson scattering cross section, and $a$ is the scale factor of
the Universe. The CMB temperature and polarization anisotropy are damped as

$\Delta
T(\hat{\boldsymbol{n}})=\Delta\tilde{T}(\hat{\boldsymbol{n}})e^{-\tau(\hat{\boldsymbol{n}})}$
(2)

$\Delta(Q\pm iU)(\hat{\boldsymbol{n}})=\Delta(\tilde{Q}\pm
i\tilde{U})(\hat{\boldsymbol{n}})e^{-\tau(\hat{\boldsymbol{n}}})\,,$ (3)

where $\tilde{T}$ is the temperature, $\tilde{Q}$ and $\tilde{U}$ are the
Stokes parameters of CMB polarization at the last scattering surface.

The perturbation on the free electron density $\Delta n_{\rm
e}(\hat{\boldsymbol{n}},\chi)$ introduces the perturbation
$\Delta\tau(\hat{\boldsymbol{n}})$ on the sky averaged optical depth
$\bar{\tau}$ as

$\tau(\hat{\boldsymbol{n}})=\bar{\tau}+\Delta\tau(\hat{\boldsymbol{n}})\,.$
(4)

We now proceed to relate the optical depth to the distribution of electrons
within halos. To model the electron density within halos, we adopt the fitting
formula derived from cosmological simulations for the gas density from
Battaglia (2016) as

$\rho_{\rm
fit}(x)=\rho_{0}(x/x_{c})^{\gamma}\left[1+(x/x_{c})^{\alpha}\right]^{-\left(\frac{\beta_{\rho}+\gamma}{\alpha}\right)}\,,$
(5)

where $x=r/r_{s}$, $\rho_{0}$ is the amplitude of density profile, $\alpha$ is
the intermediate slope, and $\beta_{\rho}$ is the power law index. Then, the
average stacked profile of the gas density becomes $\rho_{3D}=f_{\rm
b}\rho_{\rm fit}\rho_{\rm crit}(z)$, where $f_{b}$ is the baryon fraction of
the Universe, $\Omega_{\rm b}/\Omega_{\rm M}$, and $\rho_{\rm crit}$ is the
critical density of the Universe. Thoughout this work, we keep fixed
$\gamma=-0.2$ (Battaglia et al., 2012). The dependence of the parameters
$\rho_{0}$, $\alpha$, and $\beta_{\rho}$ on the redshift and mass of the halo
is fitted with a generic form of equation as:

$X=A_{X}\left(\frac{M_{200}}{10^{14}\,M_{\odot}}\right)^{\alpha_{m,X}}(1+z)^{\alpha_{z,X}}\,.$
(6)

In the above equation, index $X$ refers to the parameters of our choice
related to density profiles, $\rho_{0}$, $\alpha$, and $\beta_{\rho}$. $A_{X}$
is the amplitude of the parameter, $\alpha_{m,X}$, and $\alpha_{z,X}$ are the
power-law indices for the mass and redshift scaling, respectively. The values
of such fitted parameters are given in the Table 1

Parameter | $A$ | $\alpha_{m}$ | $\alpha_{z}$
---|---|---|---
$\rho_{0}$ | $4\times 10^{3}$ | 0.29 | -0.66
$\alpha$ | 0.88 | -0.03 | 0.19
$\beta_{\rho}$ | 3.83 | 0.04 | -0.025
Table 1: The parameters fitted for the density profile of halos with AGN
feedback (Battaglia, 2016).

Now, the Fourier transform of $\tau$ under the Limber approximation becomes
(Komatsu & Kitayama, 1999; Hill & Pajer, 2013),

$K^{\tau}_{\ell}(M,z)=\frac{4\pi r_{s}\sigma_{T}}{\ell^{2}_{s}}\int
dx\frac{\sin(\ell x/\ell_{s})}{\ell x/\ell_{s}}\rho_{\rm 3D}(x,M,z)x^{2}\,,$
(7)

where $K^{\tau}_{\ell}(M,z)$ is the kernel of the anisotropies of $\tau$
related to the density profiles of a halo and $\ell_{s}$ is the multipole
moment corresponding to the characteristic scale radius $r_{s}$, of the
$\rho_{\rm 3D}$ profile. The effective bias of a field is related to the bias
of a dark matter halo. The effective bias of the optical depth profile of such
a halo can be written as:

$b_{\ell}^{\rm\tau}(M,z)=\int_{M_{\rm min}}^{M_{\rm
max}}dM\frac{dn(M,z)}{dM}K^{\tau}_{\ell}(M,z)b(M,z)\,,$ (8)

where $b(M,z)$ is the halo bias and $dn(M,z)/dM$ is the halo mass function. We
use the Tinker halo mass function throughout this work (Tinker et al., 2008).

###  2.2 Compton y parameter

The CMB photons get scattered off the hot electron gas present in the halos,
which causes additional anisotropies in CMB temperature (Sunyaev & Zeldovich,
1970, 1972). The temperature fluctuation due to the tSZ effect can be
expressed as

$\frac{\Delta T_{\rm tSZ}(\nu,\hat{\boldsymbol{n}})}{T_{\rm
CMB}}=g(\nu)y(\hat{\boldsymbol{n}})\,.$ (9)

In the above equation $T_{\rm CMB}$ is the sky-averaged temperature and
$\Delta T_{\rm tSZ}(\nu)$ is change of temperature with respect to $T_{\rm
CMB}$ at frequency $\nu$. The term $g(\nu)$, describes how the temperature
fluctuation due to tSZ effect changes with frequency. In the non-relativistic
limit, $g(\nu)=x\coth(x/2)-4$, where $x={h\nu}/k_{\rm B}T_{\rm e}$, where
$T_{\rm e}$ is the temperature of the electron gas, $h$ is the Planck constant
and $K_{\rm B}$ is the Boltzman constant. We neglect the relativistic
corrections to the spectral shape of the tSZ signal (Nozawa et al., 2006;
Chluba et al., 2012).

The Compton $y$ parameter in a direction $\hat{\boldsymbol{n}}$ is given by

$y(\hat{\boldsymbol{n}})=\frac{\sigma_{\rm T}}{m_{\rm e}c^{2}}\int P_{\rm
e}(\hat{\boldsymbol{n}},\chi)d\chi\,.$ (10)

Here, $c$ is the speed of light in free space and $m_{\rm e}$ is the mass of
an electron. $P_{\rm e}(\hat{\boldsymbol{n}},\chi)$ is the pressure along the
direction $\hat{\boldsymbol{n}}$ at a distance $\chi$. For an isothermal
medium, the Compton $y$ parameter is proportional to the optical depth of that
medium.

To calculate the $y$ parameter, we use the analytic pressure profiles
described in Battaglia et al. (2012) that use a GNFW profile (Zhao, 1996) to
model the pressure. If $r$ is the radius of a halo and $r_{s}$ is the
characteristic scale of such a halo, then the pressure fitted to the
generalized NFW profile is

${P}_{\rm
e}(x)=P_{0}(x/x_{c})^{\gamma}\left[1+(x/x_{c})^{\alpha}\right]^{-\beta_{\rm
P}}\,.$ (11)

Here, $P_{0}$ is the amplitude of pressure, and $x_{c}$ is the fractional core
scale radius, and $\beta_{P}$ is the power-law index. $\alpha$ and $\gamma$
are two free parameters of the model. In equation Eq. 6, index $X$ refers to
the parameters of our choice related to pressure profiles, $P_{0}$, $x_{c}$,
and $\beta_{P}$. The best-fit values of these parameters are given in the
Table 2.

Parameter | $A$ | $\alpha_{m}$ | $\alpha_{z}$
---|---|---|---
$P_{0}$ | 18.1 | 0.154 | -0.758
$x_{c}$ | 0.497 | -0.00865 | 0.731
$\beta_{P}$ | 4.35 | 0.0393 | 0.415
Table 2: The parameters fitted for the pressure profile of halos (Battaglia et
al., 2012).

Now, the Fourier transform of the $y$ parameter under the Limber approximation
becomes (Komatsu & Kitayama, 1999; Hill & Pajer, 2013),:

$K^{y}_{\ell}(M,z)=\frac{4\pi r_{s}}{\ell^{2}_{s}}\int dx\frac{\sin(\ell
x/\ell_{s})}{\ell x/\ell_{s}}y(x,M,z)x^{2}\,.$ (12)

The effective bias of the $y$ parameter is given by (Pandey et al., 2020):

$b_{\ell}^{y}(M,z)=\int_{M_{\rm min}}^{M_{\rm
max}}dM\frac{dn(M,z)}{dM}K^{y}_{\ell}(M,z)b(M,z)\,.$ (13)

###  2.3 Galaxy overdensity

We model the distributions of galaxies inside dark matter halos using the halo
occupation distribution (HOD) model. The mean occupation function of the
central and satellite galaxies in a halo of mass $M_{h}$ are given by (Zheng
et al., 2005)

$\langle N_{\rm
cen}(M_{h})\rangle=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{\log M_{\rm
h}-\log M_{\rm th}}{\sigma_{\rm logM}}\right)\right]\,,$ (14)

and

$\langle N_{\rm sat}(M_{h})\rangle=\left(\frac{M_{\rm h}-M_{\rm
cut}}{M_{1}}\right)^{\alpha_{\rm g}}\,.$ (15)

Here, $\langle N_{\rm cen}(M_{h})\rangle$ and $\langle N_{\rm
sat}(M_{h})\rangle$ are the average number density of central and satellite
galaxies. $M_{\rm th}$ is the threshold halo mass to host a central galaxy,
$M_{\rm cut}$ is the minimum mass for hosting satellite galaxies, $\sigma_{\rm
logM}$ is the width of the transition of step-like error function,
$\alpha_{\rm g}$ is the power law exponent, and $M_{1}$ is the mass
normalization factor. The value of HOD-model parameters are given as $\log
M_{\rm th}=12.712$, $\sigma_{\rm logM}=0.287$, $\log M_{\rm cut}=12.95$, $\log
M_{1}=13.62$, and $\alpha=0.98$ (Zheng et al., 2005).

The Fourier transform of the distribution of satellite galaxies is given by
(e.g. Scoccimarro et al., 2001; Pandey et al., 2020):

$K^{g}_{\ell}=\frac{W_{\rm g}(z)}{\chi^{2}\bar{n}_{\rm
g}(z)}\times\sqrt{2N_{\rm cen}N_{\rm sat}u_{\rm sat}+N_{\rm cen}^{2}N_{\rm
sat}^{2}u_{\rm sat}^{2}}\,,$ (16)

where, $\bar{n}_{\rm g}(z)$ is the mean galaxy number density and $W_{\rm
g}(z)$ is the normalized redshift distribution of galaxies that can be
expressed as $\frac{c}{H(z)}\frac{dn_{\rm g}}{dz}$. We assume that the
distribution of satellite galaxies follows the underlying dark matter
overdensity field. The term $u_{\rm sat}$, for the truncated NFW profile is
given by (Navarro et al., 1996)

$u_{\rm sat}=\frac{\delta_{c}\rho(z)}{x(1+x)^{2}}\,,$ (17)

where $\rho(z)$ is the mean density of the Universe, and $\delta_{c}$ is the
characteristic over density of halos.

Putting everything together, the effective bias of the galaxies becomes

$b_{\ell}^{\rm g}(M,z)=\frac{W_{\rm g}(z)}{\chi^{2}\bar{n}_{\rm
g}(z)}\int_{M_{\rm min}}^{M_{\rm max}}dM\frac{dn(M,z)}{dM}\\\ N_{\rm
cen}(M)(1+N_{\rm sat}(M))u_{\rm sat}b(M,z)\,.$ (18)

###  2.4 Power spectrum

The 1-halo contribution to the power spectra can be expressed within the mass
range, $M_{\rm min}$ to $M_{\rm max}$, and the redshift range between $z_{\rm
min}$ and $z_{\rm max}$ as (Komatsu & Kitayama, 1999; Cooray & Sheth, 2002)

$C_{\ell}^{{\rm XY}\,{\rm(1-halo)}}=\int_{z_{\rm min}}^{z_{\rm
max}}dz\frac{d^{2}V}{d\Omega dz}\int_{M_{\rm min}}^{M_{\rm
max}}dM\frac{dn(M,z)}{dM}K_{\ell}^{X}K_{\ell}^{Y}\,.$ (19)

Here, $K_{\ell}^{X}$ and $K_{\ell}^{Y}$ are the Fourier transform of kernel of
the X and Y observables, respectively. $dV$ is the comoving volume element at
$\chi(z)$.

The 2-halo term can be written as (Komatsu & Kitayama, 1999; Cooray & Sheth,
2002)

$C_{\ell}^{{\rm XY}\,{\rm(2-halo)}}=\int_{z_{\rm min}}^{z_{\rm
max}}dz\frac{d^{2}V}{d\Omega dz}b_{\ell}^{X}(M,z)b_{\ell}^{Y}(M,z)\\\ \,P_{\rm
lin}\left(k=\ell/\chi,k\right)\,,$ (20)

where $P_{\rm lin}$ is the matter power spectrum that we calculated using
CAMB444https://camb.info in the linear regime (Lewis et al., 2000). The cross
power spectrum is the sum of 1-halo and 2-halo terms:

$C_{\ell}^{\rm XY}=C_{\ell}^{{\rm XY}\,{\rm(1-halo)}}+C_{\ell}^{{\rm
XY}\,{\rm(2-halo)}}\,.$ (21)

Using Eq. 19, 20 and 21, we calculate the auto power spectrum by setting
$X=Y$.

##  3 Comparison with Simulations

Figure 1: The agreement between the power spectra of $\tau g$, $\tau y$, and
$\tau\tau$ calculated from the hydrodynamic simulations and analytic halo
model. We fix $M_{\rm min}=3\times 10^{13}\,M_{\odot}$, $z_{\rm min}=0$ and
$z_{\rm max}=5$. This validity check ensures us using halo model calculations
reliably for constraining astrophysical parameters of gas thermodynamics in
halos.

We validate our results from the analytic halo model against existing
cosmological hydrodynamic simulations. These simulations were performed using
a modified version of the smoothed particle hydrodynamics (SPH) code, GADGET-2
(Springel, 2005). Included in these simulations were sub-grid models for
active galactic nuclei (AGN) feedback (for more details see Battaglia et al.,
2010), radiative cooling, star formation, galactic winds, supernova feedback
(for more details see Springel & Hernquist, 2003), and cosmic ray physics (for
more details see Pfrommer et al., 2006; Enßlin et al., 2007; Jubelgas et al.,
2008). The box sizes for these simulations were 165 Mpc$/h$, with a resolution
of 2563 gas and dark matter (DM) particles. This yields mass resolutions of
$M_{\mathrm{gas}}=3.2\times 10^{9}\mathrm{M}_{\odot}/h$ and
$M_{\mathrm{DM}}=1.54\times 10^{10}\mathrm{M}_{\odot}/h$. Thus, these
simulations can resolve more than 1000 particles in halos with masses of
$M>3\times 10^{13}\,M_{\odot}$. The cosmological parameters used for these
simulations were
$\Omega_{\mathrm{M}}=\Omega_{\mathrm{DM}}+\Omega_{\mathrm{b}}=0.25$,
$\Omega_{\mathrm{b}}=0.043$, $\Omega_{\Lambda}=0.75$,
$H_{0}=100\,h\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, $h=0.72$,
$n_{\mathrm{s}}=0.96$ and $\sigma_{8}=0.8$. These parameters differ from those
used in the halo model but not at the level to cause substantial differences
larger than what is currently shown in Figure 1.

We compare the amplitude of $C_{\ell}^{\tau g}$, $C_{\ell}^{\tau y}$, and
$C_{\ell}^{\tau\tau}$ estimated from the halo-model against the hydrodynamic
simulations. The halo-model approach is computationally efficient, and hence,
if our results from the halo-model are consistent with the results from
simulations, we can calibrate our model to study CGM. In addition, we have
more freedom to select the different combinations of astrophysical parameters
that we can constrain with future datasets.

In Figure 1 we show a comparison between the power spectrum estimated from the
halo-model approach and hydrodynamic simulations. In doing so, we fixed the
minimum halo mass to $3\times 10^{13}\,M_{\odot}$ so that it matches with the
mass resolution of the simulations. At small scales, $\ell>1000$, all power
spectra agree very well. This consistency check allows us to study the
detectability of this signal by changing the different values of $M_{\rm
min}$.

##  4 Noise Model

In this section, we review the reconstruction method for optical depth
fluctuations and sources of noise for the measurement of the galaxy and tSZ
power spectrum.

###  4.1 $\tau$ reconstruction noise

Future CMB experiments promise to reach high-resolution and low-noise
measurements of polarization anisotropies, and it increases our ability to
measure the $C_{\ell}^{\tau\tau}$ by applying a reconstruction method (Roy et
al., 2018; Namikawa et al., 2021). The fluctuations in optical depth introduce
correlations between different Fourier modes. We can generalize the mode
coupling of CMB temperature and polarization as (Dvorkin & Smith, 2009):

$\langle a^{X1}_{\ell_{1}m_{1}}a^{X2}_{\ell_{2}m_{2}}\rangle=\sum_{\ell
m}\Gamma^{X_{1}X_{2}}_{\ell_{1}\ell_{2}\ell}\begin{pmatrix}\ell_{1}&\ell_{2}&\ell\\\
m_{1}&m_{2}&m\end{pmatrix}(\Delta\tau)^{*}_{\ell m}\,,$ (22)

where $X_{1}$ and $X_{2}$ are any combinations of $T$, $E$, and $B$,
$\ell=\ell_{1}+\ell_{2}$, and $\Gamma^{X_{1}X_{2}}_{\ell_{1}\ell_{2}\ell}$ is
the coupling factor that correlates $\ell_{1}$ and $\ell_{2}$. To minimize the
noise of the estimator, we only consider the $EB$ quadratic estimator (QE) as
it provides the highest signal-to-noise ratio (Gluscevic et al., 2013). The
coupling factor for the $EB$ estimator can be written in terms of Wigner-3j
symbols as

$\Gamma^{EB}_{\ell_{1}\ell_{2}\ell}=\frac{C_{\ell_{1}}^{EE}}{2i}\sqrt{\frac{(2\ell_{1}+1)(2\ell_{2}+1)(2\ell+1)}{4\pi}}\\\
\left[\begin{pmatrix}\ell_{1}&\ell_{2}&\ell\\\
-2&2&0\end{pmatrix}-\begin{pmatrix}\ell_{1}&\ell_{2}&\ell\\\
2&-2&0\end{pmatrix}\right]\,,$ (23)

where $C_{\ell}^{EE}$ is the power spectrum of the $E$-mode polarization
field. The noise for $\tau$ reconstruction depends on the sensitivity and
resolution of a CMB experiment. The reconstructed $\tau$ field is given by
(Dvorkin & Smith, 2009)

$\widetilde{\tau}^{*}_{\ell
m}=N_{\ell}^{\tau\tau}\sum_{\ell_{1}\ell_{2}}\sum_{m_{1}m_{2}}\Gamma^{EB^{*}}_{\ell_{1}\ell_{2}\ell}\begin{pmatrix}\ell_{1}&\ell_{2}&\ell\\\
m_{1}&m_{2}&m\end{pmatrix}\\\
\times\frac{a^{E^{*}}_{\ell_{1}m_{1}}a^{B^{*}}_{\ell_{2}m_{2}}}{(C^{EE}_{\ell_{1}}+N^{EE}_{\ell_{1}})(C^{BB}_{\ell_{2}}+N^{BB}_{\ell_{2}})}\,,$
(24)

where $C_{\ell}^{EE}$ and $C_{\ell}^{BB}$ are the power spectra of $E$ and $B$
modes. We note that here we only consider the ”screening” contribution arising
from an inhomogeneous optical depth, and do not include the extra information
from new polarization generated from scattering from incident quadrupoles
(e.g., Dvorkin et al., 2009). For $C_{\ell}^{BB}$ our forecast includes
iterated delensing (Smith et al., 2012) which we assume does not get modulated
by the $B$ modes induced by $\tau$ fluctuations. $N_{\ell}^{EE}$ and
$N_{\ell}^{BB}$ are instrumental noise power spectra for $E$ and $B$-mode
signals, respectively. The instrumental noise power spectra are given by
(Knox, 1995)

$N_{\ell}^{EE}=N_{\ell}^{BB}=\Delta_{\rm
P}^{2}\textrm{exp}\left[\frac{\ell(\ell+1)\theta_{\rm f}^{2}}{8\ln
2}\right]\,.$ (25)

In the above equation, $\Delta_{\rm P}$ is the polarization sensitivity of a
CMB experiment, which is $\sqrt{2}$ times larger than the sensitivity in
temperature, $\Delta_{\rm T}$, and $\theta_{\rm f}$ is the full width at half
maximum (FWHM) of the beam size.

Finally, the reconstruction noise, $N_{\ell}^{\tau\tau}$ is given by (Dvorkin
& Smith, 2009)

$N_{\ell}^{\tau\tau}=\left[\frac{1}{2\ell+1}\sum_{\ell_{1}\ell_{1}}\frac{\left|\Gamma^{EB^{*}}_{\ell_{1}\ell_{2}\ell}\right|^{2}}{(C^{EE}_{\ell_{1}}+N^{EE}_{\ell_{1}})(C^{BB}_{\ell_{2}}+N^{BB}_{\ell_{2}})}\right]^{-1}\,.$
(26)

We calculate the $\tau$ reconstruction noise for CMB S4 like experiments with
1$\mu K$ arcmin sensitivity in temperature and 1 arcmin beam size (Abazajian
et al., 2019). Unless mentioned, we use $l_{min}=30$ and $l_{max}=5000$
throught this work because a CMB-S4-like experiment will make observations at
these scales.

Figure 2: Reconstruction noise of optical depth power spectrum corresponding
to different sensitivities in temperature of a CMB experiment while the FWHM
of the beam is fixed at 1 arcmin. $C_{\ell}^{\tau\tau}$ from reionization and
halos ($M>3\times 10^{13}\,M_{\odot}$) are calculated from hydrodynamic
simulations (Battaglia et al., 2012).

This process is analogous to the reconstruction method of the lensing
potential from the CMB. In practice, both lensing and patchy reionization
signals are present in the CMB data, and if we apply only the $\tau$
estimator, there will be leakage from the lensing field to the $\tau$ field
(Dvorkin & Smith, 2009; Su et al., 2011). This problem can be dealt with by
applying a “bias-hardened” $\tau$ estimator that is insensitive to lensing at
leading order fields (Namikawa et al., 2013, 2021). In order to determine the
impact of bias-hardening the $\tau$ estimator from lensing, we computed the
$\tau$ reconstruction noise $N_{\ell}^{\tau\tau}$ for bias-hardened
estimators, for both $TT$ and $EB$. We found that for the $TT$ estimator, the
bias hardening can increase the lensing reconstruction noise
$N_{\ell}^{\tau\tau}$ by factors of $2$–$3$ across all relevant $\ell$ values
for the S4-like experiment. However, for the $EB$ estimator, bias hardening
for lensing has a negligible effect on $N_{\ell}^{\tau\tau}$ for all but the
very highest $\ell$ values, namely $\ell\gtrsim 4500$. Thus, we do not use the
bias-hardened $EB$ noise curves in this work.

In Figure 2, we show the amplitude and shape of $N_{\ell}^{\tau\tau}$ that
depends on the experiment. We vary the sensitivity of an experiment while
keeping the beam size fixed to 1 arcmin. The reconstruction noise power
spectrum decreases by a factor of $56$ if the sensitivity of a CMB experiment
is decreased from 9 $\mu K$-arcmin to 1 $\mu K$-arcmin at $\ell=1000$. The
power spectra of optical depth from reionization and halos at $\ell=1000$ are
roughly 3 and 4 orders of magnitude smaller than the noise with
$\Delta_{T}=1\,\mu K$-arcmin.

###  4.2 Noise for tSZ measurements

We calculate the noise contribution in the $yy$ auto power spectrum,
$N_{\ell}^{yy}$, by applying a constrained internal linear combination (cILC)
algorithm. The cILC method depends on the assumption that the signal of our
interest, tSZ, does not depend on the other signals and noise components and
nulls a given contamination signal (Eriksen et al., 2004; Remazeilles et al.,
2011). In the non-relativistic limit, the tSZ spectral dependence is very well
defined, and it helps to minimize the variance to reconstruct the signal from
the total observed signal. Since we are interested in measuring the $yy$ auto
power spectrum in the absence of the cosmic infrared background (CIB,
generally the largest contaminant to tSZ observations), we can write down a
model for an angular scale ($d$) per frequency ($\nu$) as (Planck
Collaboration et al., 2016)

$d_{\ell,\nu}=a_{\ell,\nu}f^{y}_{\nu}C_{\ell}^{y}+b_{\ell,\nu}f^{\mathrm{CIB}}_{\nu}C_{\ell}^{\mathrm{CIB}}+n_{\ell,\nu},$
(27)

where $a$ and $b$ are the scale and frequency-dependent coefficients. We will
refer to $n$ as noise, but it also contains primordial CMB fluctuations, radio
sources, instrumental noise, etc. The cILC algorithm constructs a weight
vector $W$ such that we get a unit response to $a$ (the tSZ coefficients) and
a null response to $b$ (the CIB coefficients), $W^{T}\vec{a}=1\quad
W^{T}\vec{b}=0$. The general solution for $W$ was shown in Remazeilles et al.
(2011),

$W_{a=1,b=0}=\frac{b^{T}n^{-1}ba^{T}n^{-1}-a^{T}n^{-1}bb^{T}n^{-1}}{a^{T}n^{-1}ab^{T}n^{-1}b-(a^{T}n^{-1}b)^{2}}.$
(28)

Note that we dropped the frequency dependence for clarity. From the weight we
can calculate $N_{\ell}^{yy}$ via

$N_{\ell}^{yy}=\sum_{\nu}(W_{\nu}n_{\nu}W_{\nu}^{T})_{\ell}.$ (29)

We use the SILC555https://github.com/nbatta/SILC package to estimate
$N_{\ell}^{yy}$ in presence of primordial CMB, cosmic infrared background,
foreground due to the dust and synchrotron radiation. In the near future, low
noise experiments will be able to produce signal dominated measurements of the
tSZ effect (Hanany et al., 2019).

###  4.3 Shot noise of galaxy power spectrum

For the case of the noise power spectrum of galaxies, we assume that the
process of local galaxy formation in halos is linear and that the dominant
source of noise is the shot noise term for the measurement of galaxy power
spectra. In this regime, the shot noise becomes equal to $1/\bar{n}_{\rm g}$,
where $\bar{n}_{\rm g}$ is the average number density of galaxies. The average
number density of galaxies at $z$ can be written as

$\bar{n}_{g}(z)=\int_{M_{min}}^{M_{max}}dM\frac{dn(M,z)}{dM}N_{g}(M,z)\,.$
(30)

We consider the specifications of VRO666https://www.lsst.org/ to determine the
shot noise term. We fix $M_{\rm max}=10^{13}\,M_{\odot}$ and consider a
redshift distribution of galaxies in the redshift raange $z_{\rm min}=0$ and
$z_{\rm max}=5$ as described in Abell et al. (2009). The total number of
galaxies that will be probed by a VRO-like-experiment is 50 per square arc-
minute. Using Eq. 30, we find $M_{\min}=2\times 10^{11}\,M_{\odot}$ that
satisfies this condition and we use this value of $M_{\rm min}$ for making
forecasts for a VRO-like experiment.

##  5 Detectability of signals

In this section, we discuss the detectability of the cross-correlation between
different observational probes that are described in Section 2. For the
forecasts, we used the sensitivity of CMB-S4 like experiment with the
sensitivity, $\Delta_{T}=1\,\mu K$-arcmin, and the FWHM of the beam,
$\theta_{\rm f}=1$ arcmin.

As discussed in the previous section, we assume that the estimator described
in Dvorkin & Smith (2009) to reconstruct a fluctuating $\tau$ map, and this
map is then cross-correlated with a given galaxy sample or with a Compton-$y$
map, which produces cross spectra ($C^{\tau g}_{\ell}$ and $C^{\tau
y}_{\ell}$, respectively). A similar approach was proposed to measure the
fluctuations in $\tau$ from reionization by cross-correlating with
measurements from 21cm experiments (Meerburg et al., 2013; Roy et al., 2020).

The total $\mathrm{S}/\mathrm{N}$ is calculated by summing the signal-to-noise
per $\ell$ mode,

$\left(\mathrm{S}/\mathrm{N}\right)^{2}=\sum_{\ell}{\left(\frac{C_{\ell}^{XY}}{\Delta
C_{\ell}^{XY}}\right)^{2}},$ (31)

where $\Delta C_{\ell}^{XY}$ is theoretical error on $C_{\ell}^{XY}$ given
$N_{\ell}^{\tau\tau}$. The exact form of $\Delta C_{\ell}^{XY}$ we use is

$\left(\Delta
C_{\ell}^{XY}\right)^{2}=\frac{1}{(2\ell+1)f^{XY}_{\mathrm{sky}}}[(C_{\ell}^{XY})^{2}+\\\
\left(C_{\ell}^{XX}+N_{\ell}^{XX}\right)\left(C_{\ell}^{YY}+N_{\ell}^{YY}\right)]\,,$
(32)

where $f^{XY}_{\mathrm{sky}}$ is the fraction of sky covered, $C_{\ell}^{XX}$
and $N_{\ell}^{YY}$ are the auto spectrum and noise term of a given tracer,
respectively. Here we assume that $N_{\ell}^{\tau\tau}\gg C_{\ell}^{\tau\tau}$
to simplify Eq. 32.

Figure 3:  Forecast of signal-to-noise ratio of $C_{\ell}^{\tau g}$,
$C_{\ell}^{\tau y}$, and $C_{\ell}^{\tau\tau}$ signals with FWHM of beam for
CMB-S4 like experiment for $M_{\rm min}=2\times 10^{11}\,M_{\odot}$ and
$M_{\rm max}=10^{13}\,M_{\odot}$. The cross-correlations are calculated using
analytic halo model and the instrumental sensitivity in temperature is set to
1 $\mu K$-arcmin. The signal-to-noise ratio increases with the smaller beam
size because the number of accessible modes increases with resolutions of an
experiment. In addition to that, delensing of $B$ modes help to reduce the
noise in polarization that lower the reconstruction noise of $\tau$
fluctuations. Figure 4: Top panel: $\tau\tau$ (left panel), $\tau g$ (middle
panel), and $\tau y$ (right panel) power spectra shown as a function of the
minimum halo mass, $M_{\rm min}$. Bottom panel: cumulative SNR calculated for
the $\tau\tau$ (left panel), $\tau y$ (middle panel), and $\tau g$ (right
panel) with $1\,\mu K$arcmin sensitivities in temperature and 1 arcmin beam
size. This figure shows that signal-to-noise ratio are dominated by the low
mass halos ($M_{\rm halo}\lesssim 10^{12}\,M_{\odot}$) rather than the high
mass halos ($M_{\rm halo}\gtrsim 10^{13}\,M_{\odot}$). Figure 5: Forecasted
constraints on the density, pressure and HOD parameters from the different
combinations of $\tau g$, $\tau y$, and $yg$. We set $M_{\rm min}=2\times
10^{11}\,M_{\odot}$ and $M_{\rm max}=10^{13}M_{\odot}$ to study the properties
of circumgalactic medium. Tight constraints are placed on HOD parameters from
the measuremnts of $C_{\ell}^{gg}$.

Figure 6: left: solid lines show the constraints on the amplitude of the
density profile, $A_{\rho}$, from $\tau g$ cross-correlations and dashed lines
represent the constraints inferred from $\tau y$ cross-correlation. right: The
forecasted 1$\sigma$ uncertainty of gas mass fraction, $f_{\rm g}$, calculated
from a joint analysis of different probes of cross-correlation. Dashed white
line is the best fit curve for the $\tau y$ \+ $\tau g$ \+ $yg$ \+ $gg$.

We use the simulations described in Section 3 to calculate the cross power
spectra $C^{\tau g}_{\ell}$ and $C^{\tau y}_{\ell}$. Here the field $g$ refers
to a number-count map where a non-zero value is assigned at the location of
each halo above a given minimum mass, and $y$ is a Compton-$y$ map with the
same mass threshold. We cross-correlate the number-count and Compton-$y$ maps
with the simulated $\tau$ maps. Then, for each redshift, we average over all
simulation realizations and add these differential cross-spectra to get the
signal.

Parameters | meaning | fiducial values |  $\tau g$ \+ $gg$ |  $\tau y$ \+ $yg$\+ $gg$ |  $yg$ \+ $gg$ | joint analysis
---|---|---|---|---|---|---
$A_{\rho_{0}}$ | amplitude of density profile | $4000$ | $7.27$ | $13.2$ | - | $6.38$
$A_{P_{0}}$ | amplitude of pressure profile | 18.1 | - | $13.64$ | $14.32$ | $13.65$
$A_{\rm xc}$ | amplitude of characteristics scale of halo | 0.497 | - | $12.9$ | $13.54$ | $12.86$
$A_{\beta p}$ | power law index of the fall of pressure profile | 4.35 | - | $6.0$ | $6.35$ | $6.0$
Table 3: The forecasted $1\sigma$ uncertainties on the parameters of density
and pressure profiles.

Note that all the cross-correlations have contributions both from the high
redshift ($z\gtrsim 5$) and low redshift ($z\lesssim 5$). The $yy$ from
reionization is subdominant as the power spectrum is at least 3-4 orders of
magnitude smaller than the galaxy and cluster contributions (Hill et al.,
2015), but the amplitude of the $\tau y$ power spectrum from reionization is
comparable with the contributions from halos at low redshift (though they may
have different shapes as the characteristic scale of ionized bubbles and halos
are different; Hill et al. (2015); Namikawa et al. (2021)). We calculate the
$\tau y$ signal for a fiducial model of reionization with the characteristic
bubble size of ionized bubbles $\bar{R}_{b}=5\,$ Mpc following a semi-analytic
model described in Namikawa et al. (2021). Hence, we consider the $\tau y$
signal from reionization as a source of noise in our forecasts. $yg$ and $\tau
g$ from high and low redshift do not correlate as we choose the galaxy
distribution function at the low redshift only, so they have different
kernels.

In Figure 3, we show the total $\mathrm{S}/\mathrm{N}$ as a function of beam
size for CMB-S4 like survey that has a noise level of 1$\mu$K acrmin. The
$\ell$ range we consider is from 100-5000 in polarization. For a halo mass
threshold of $M_{\rm min}=2\times 10^{11}M_{\odot}$ and $\theta_{\rm f}=1$
arcmin, we find the total $\mathrm{S}/\mathrm{N}f_{\mathrm{sky}}^{-1/2}\approx
15$ and 10 for $C^{\tau g}_{\ell}$ and $C^{\tau y}_{\ell}$, respectively.
After delensing the lensed $B$-mode power spectrum at level of 90% (Smith et
al., 2012; Abazajian et al., 2019), we find that the signal-to-noise ratio
increases by 14% and 12.5% for $C^{\tau g}_{\ell}$ and $C^{\tau y}_{\ell}$,
respectively.

In Figure 4, we show the variation of $\tau y$, $\tau g$, and $\tau y$ signals
for different minimum halo mass, $M_{\rm min}$. The cumulative SNRs for $\tau
g$, $\tau y$ and $\tau\tau$ are 20, 16 and 5.5 corresponding to $M_{\rm
min}=10^{11}\,M_{\odot}$. The SNRs change to 11, 11.5 and 4 if $M_{\rm min}$
is set to $10^{13}\,M_{\odot}$. We find SNR for $yg$ is 1710 and 950 for
$M_{\rm min}=10^{11}\,M_{\odot}$ and $10^{14}\,M_{\odot}$ respectively.

##  6 Fisher Forecast

In this section, we discuss how the future measurements of these cross-
correlations can be used to constrain the astrophysical parameters related to
the pressure and density of CGM. For the rest of our analysis, we use the
upper limit on the halo mass of $M_{\rm max}=10^{13}\,M_{\odot}$ so that we
can study the properties of CGM. We assume that the pressure and density
profile is self-similar to the redshift and mass of halos.

We use Fisher matrix analysis to forecast the measurement uncertainties of the
parameters of our interest. The Fisher matrix is given by

$F_{ij}=\sum_{\ell}\frac{1}{(\Delta C^{XY}_{\ell})^{2}}\frac{\partial
C_{\ell}^{XY}}{\partial P_{i}}\frac{\partial C_{\ell}^{XY}}{\partial
P_{j}}\,,$ (33)

where $P_{i}$ and $P_{j}$ are the parameters which we aim to constrain from
the measurements of cross-correlations.

We select parameters of pressure and density as described in Table 3 and show
the constraints on parameters in Figure 5 for the different combinations of
the $\tau g$, $yg$, $\tau y$, and $gg$ signal. Note that the $yg$ signal can
probe the parameters related to the pressure profile, whereas the $\tau g$
signal is sensitive to the density profile of halos. The $\tau y$ signal is
sensitive to both the pressure and density profile, and for this reason, we
use it to break the degeneracy between the amplitude of density and pressure
profiles. We use $M_{\rm min}=2\times 10^{11}\,M_{\odot}$, $z_{max}=5$, and
$f_{\rm sky}=0.5$ for this analysis. The Galaxy power spectrum puts tighter
constraints on the HOD model than the parameters that are inferred from the
$yg$ signal, and it helps to break the degeneracy between parameters of
pressure and the HOD model. We present our forecasts of the parameters in
Table 3.

The detection of these cross-correlations leads us to understand some other
astrophysical parameters that are directly related to the pressure and density
profiles. We specifically focus on the gas mass fraction, $f_{\rm g}={M_{\rm
gas}}/M_{\rm halo}$. It should be noted that this quantity depends on the
definition of radius and mass of halos. For example, the value of $f_{\rm g}$
will be different for radius $R_{200}$, $R_{500}$, and $R_{2500}$ as they
correspond to different halo masses $M_{200}$, $M_{500}$, and $M_{2500}$.

The $M_{\rm gas}$ inside $R_{500}$ can be written as

$M_{\rm gas,500}(z)=\int_{0}^{R_{500}}\,\rho_{\rm gas}(r,z)\;4\pi r^{2}dr\,.$
(34)

In Figure 6, we show the forecasted constraints on the amplitude of the
density profile $A_{\rho}$ from the $\tau g$ and $\tau y$ signals. As
expected, $\tau g$ put tighter limits on the $A_{\rho}$ as the SNR is larger
than the $\tau y$ for the same $M_{\rm min}$. We find that the gas mass
fraction, $f_{g\rm}$, can be measured with more than $7\sigma$ confidence for
$\tau y$ \+ $gg$ and $17\sigma$ for $\tau g$ \+ $gg$.

##  7 Discussion

In the near future, there will be signal dominated high-resolution tSZ
measurements that will bring a unique opportunity to break the degeneracy
among astrophysical parameters related to the thermodynamic properties of CGM
by performing cross-correlation with other probes. We explored such cross-
correlations to measure the density and pressure profiles of halos by
independent observational probes. In this paper, we presented the
detectability of cross-correlations between the optical depth anisotropy
(reconstructed from the CMB polarization anisotropy), Compton-y parameter
(derived from the CMB temperature measurements), and galaxy overdensity field.
The optical depth anisotropy is a tracer of the gas density distribution in
halos, the Compton $y$ parameter probes the pressure profile, and the galaxy
number-count carries information of the distribution of galaxies in dark
matter halos.

We used a semi-analytic halo-model formalism (compared against hydrodynamic
simulations) to estimate the power spectra of these cross-correlated signals.
The $yg$ signal is the most promising probe for constraining the pressure
parameters, which can be performed using VRO galaxies and Compton $y$ map from
the CMB-S4 experiment. We find the signal-to-noise ratio can reach as high as
500 to 2000 depending on the $M_{\rm min}$. Such a high signal-to-noise ratio
can be used to understand the sub-grid physics in halos, such as AGN feedback,
turbulence, and depletion efficiency of gas in CGM.

On the other hand, the $\tau g$ signal is a useful probe for measuring the
density profile along with the priors on HOD model parameters that can be well
achieved by the precise measurement of the $gg$ signal alone. The measurements
of pressure and density profiles by different probes like $\tau g$, $yg$, and
$yg$ will also lead us to determine the temperature profile of halos. A joint
analysis will not only improve our understanding of the low redshift Universe
but will also help us to investigate the reionization process. The $y\tau$
signal at low redshift contaminates the $y\tau$ signal from reionization.
Hence a proper estimation of $y\tau$ from halos will help to remove it from
the total observed signal that can put tighter constraints on reionization
parameters such as size and temperature of ionized bubbles (Namikawa et al.,
2021).

The multiple probes presented in this paper are complementary to tSZ and kSZ
measurements and their cross-correlations with galaxy surveys. Previous kSZ
measurements have probed the average density profile of halos (Schaan et al.,
2016, 2021; Amodeo et al., 2021). However, spectroscopic galaxy redshifts are
required to reconstruct the velocity field to achieve these constraints on the
density profile. The cross-correlations we propose do not require
spectroscopic information of galaxies. Instead, we forecast complementary and
comparable constraints by utilizing a larger sample of galaxies selected from
future imaging surveys. Our forecasts show that the cross-correlation between
$\tau$ and $g$ is detectable with more than 15 $\sigma$ for upcoming
experiments, whereas the auto power spectrum of $\tau$ is detectable at $\sim
4\sigma$.

Baryons that are outside the halos are hard to see in optical and X-ray bands,
but their presence is encoded in the optical depth anisotropies as these
baryons interact with CMB photons via Thomson scattering. The cross-
correlations presented here are sensitive to the global properties of gas
inside and outside halos. Thus, probing the baryon abundances inside and
outside halos with such cross-correlations can be used to resolve the question
of are we missing baryons? Furthermore, we demonstrated that our proposed
joint cross-correlations are robust. We found that “bias hardening” the
estimator to reduce leakage from CMB lensing did not have an appreciable
impact on the results.

The authors would like to thank Stefania Amodeo, Joel Meyers, Connor Sheere,
Kendrick Smith, and David Spergel. VG is supported by the National Science
Foundation under Grant No. PHY-2013951. NB acknowledges support from NSF grant
AST-1910021 and NASA grants 21-ADAP21-0114 and 21-ATP21-0129.

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