Also at ]Kavli Institute for the Physics and Mathematics of the Universe
(Kavli IPMU, WPI), Todai Institutes for Advanced Study, The University of
Tokyo, Kashiwa 277-8583, Japan.

# Testing the SZ-based tomographic approach to the thermal history of the
universe with pressure-density cross-correlations: Insights from the
_Magneticum_ simulation

Sam Young<EMAIL_ADDRESS>Eiichiro Komatsu  [  Max Planck
Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany
Klaus Dolag  University Observatory Munich, Scheinerstr. 1, 81679 Munich,
Germany  Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1,
85748 Garching, Germany

###### Abstract

The thermal Sunyaev-Zeldovich effect contains information about the thermal
history of the universe, observable in maps of the Compton $y$ parameter;
however, it does not contain information about the redshift of the sources.
Recent papers have utilized a tomographic approach, cross-correlating the
Compton $y$ map with the locations of galaxies with known redshift, in order
to deproject the signal along the line of sight. In this paper, we test the
validity and accuracy of this tomographic approach to probe the thermal
history of the universe. We use the state-of-the-art cosmological
hydrodynamical simulation, _Magneticum_ , for which the thermal history of the
universe is a known quantity. The key ingredient is the Compton-$y$-weighted
halo bias, $b_{y}$, computed from the halo model. We find that, at redshifts
currently available, the method reproduces the correct mean thermal pressure
(or the density-weighted mean temperature) to high accuracy, validating and
confirming the results of previous papers. At higher redshifts ($z\gtrsim
2.5$), there is significant disagreement between $b_{y}$ from the halo model
and the simulation.

##  I Introduction

As cosmological structures form, the gravitational potential wells seeded by
primordial density fluctuations become deeper [1]. As this process occurs,
gravitational potential energy is converted into kinetic energy in an expended
universe, following the Layzer-Irvine equation [2, 3, 4]. A part of the
kinetic energy is converted into thermal energy via the process of shock
heating in the large-scale structure of the universe [5, 6, 7], while the rest
remains in the form of bulk/turbulent motion until it decays and thermalizes
[8, 9]. Therefore, measurements of the mean thermal energy of the cosmic gas
content at different redshifts can be used to probe the growth of structure,
as recently demonstrated in Refs. [10, 11].

To this end, the thermal Sunyaev-Zeldovch (SZ) effect [12, 13] can be used to
probe the baryons in the universe [14, 15, 16]. As photons in the cosmic
microwave background (CMB) are inverse-Compton scattered off of the free
electrons in the ionized gas, they leave an imprint in the form of a spectral
distortion in the CMB. The amplitude of the SZ effect depends on the electron
pressure integrated along the line of sight, which is parametrized by the
Compton $y$ parameter [12, 13],

$y(\hat{\phi})=\frac{\sigma_{\mathrm{T}}}{m_{\mathrm{e}}c^{2}}\int\frac{\mathrm{d}\chi}{1+z}P_{\mathrm{e}}(\chi\hat{\phi}),$
(1)

where $\sigma_{\mathrm{T}}$ is the Thomson scattering cross section,
$m_{\mathrm{e}}$ is the electron mass, $c$ is the speed of light,
$\chi=\chi(z)$ is the comoving radial distance out to a given redshift $z$,
and $P_{\mathrm{e}}$ is the electron pressure. The integral runs from zero up
to the surface of last scattering, $z\simeq 1090$. The electron pressure
$P_{\mathrm{e}}$ is related to the electron temperature $T_{\mathrm{e}}$ as
$P_{\mathrm{e}}=n_{\mathrm{e}}k_{\mathrm{B}}T_{\mathrm{e}}$, where
$n_{\mathrm{e}}$ and $k_{\mathrm{B}}$ are the (proper) electron number density
and the Boltzmann constant, respectively.

The SZ effect, observable in maps of the Compton $y$ parameter, contains
information about $P_{\mathrm{e}}$, but does not contain information about $z$
of the sources. To probe the growth history of structure using the SZ effect,
we therefore require an external source of information about $z$ [17, 18]. As
the SZ signal is dominated by massive structure, the signal can be deprojected
along the line of sight using clustering-based redshift inference [19, 20,
21]. Following this method, an external sample of reference sources with known
$z$ is taken, and cross-correlated with the Compton $y$ parameter as a
function of $z$. This allows correlated intensities to be extracted
tomographically [22, 23, 24], as has been studied in recent papers for the SZ-
galaxy cross-correlation with spectroscopic redshifts [25, 26, 10]. See Refs.
[27, 28, 29] for the cross-correlation with photometric redshifts.

The halo bias-weighted mean electron pressure, $\langle
bP_{\mathrm{e}}\rangle$, is the direct observable of the SZ-galaxy cross-
correlation function on large scales [25]. Here, $\langle...\rangle$ denotes
an ensemble average. To infer the mean electron pressure, $\langle
P_{\mathrm{e}}\rangle$, we need to know the Compton $y$-weighted halo bias,
$b_{y}\equiv\langle bP_{\mathrm{e}}\rangle/\langle P_{\mathrm{e}}\rangle$. In
Refs. [10, 11], the halo model developed in Refs. [30, 31, 32, 26, 33] was
used to calculate $b_{y}$ and infer $\langle P_{\mathrm{e}}\rangle$ from the
measured $\langle bP_{\mathrm{e}}\rangle$.

How accurate is this approach? It is the aim of this paper to test the
validity of the tomographic approach to the mean thermal history of the
universe in Refs. [10, 11]. Specifically, we test this approach against the
_Magneticum_ simulation [34]. We use cross-correlations of the density and
pressure to calculate the density-weighted mean temperature of baryonic gas,
which is a known quantity in the simulation.

The rest of this paper is organized as follows. In Sec. II, we describe the
_Magneticum_ simulation. We then present the comparison of the simulations and
observations for $\langle bP_{\mathrm{e}}\rangle$ in Sec. III and the density-
weighted mean electron temperature $\bar{T}_{\mathrm{e}}$ in Sec. V, while we
compare the halo model calculation and the simulation result for $b_{y}$ in
Sec. IV. We conclude in Sec. VI.

##  II The _Magneticum_ simulation

The _Magneticum_ simulation is a set of state-of-the-art, cosmological
hydrodynamical simulations of different cosmological volumes with different
resolutions performed with an improved version of the smoothed-particle
hydrodynamics (SPH) code GADGET3 [35, 36]. They follow a standard $\Lambda$
Cold Dark Matter (CDM) cosmology with parameters close to the best-fitting
values of the WMAP 7-year results [37] for a flat $\Lambda$CDM cosmology with
the total matter density $\Omega_{\mathrm{m}}=0.272$ (16.8% baryons), the
cosmological constant $\Omega_{\Lambda}=0.728$, the Hubble constant
$H_{0}=70.4~{}\mathrm{km~{}s^{-1}~{}Mpc^{-1}}$ ($h=0.704$), the index of the
primordial power spectrum $n_{\mathrm{s}}=0.963$, and the overall
normalisation of the power spectrum $\sigma_{8}=0.809$.

Here we describe the simulations briefly; for a more detailed description we
refer to the previous work using these simulations [34, 38, 39, 40, 41, 42].
The simulations follow a wide range of physical processes (see Refs. [43, 44]
for details) which are important for studying the formation of active galactic
nuclei (AGN), galaxies, and galaxy groups and clusters. The simulation set
covers a huge dynamical range following the same underlying treatment of the
physical processes controlling galaxy formation, thereby allowing to reproduce
the properties of the large-scale, inter-galactic and intra-cluster medium
[34, 39, 45], as well as the detailed properties of galaxies including
morphological classifications and internal properties [44, 46, 47]. This also
includes the distribution of different metal species within galaxies and
galaxy clusters [48] and the properties of the AGN population [43, 49].
Especially the simulations well reproduce the observed pressure profiles of
galaxy clusters [50, 39] and X-ray scaling relations [51].

Here we focus on the largest box (‘Box0’) which follows the evolution of
$2\times 4536^{3}$ particles in a large box of the comoving volume
$\left(2688~{}h^{-1}~{}\mathrm{Mpc}\right)^{3}$ [38, 52, 40], making it the
largest cosmological hydrodynamical simulation performed to date.

##  III The halo bias-weighted mean electron pressure $\langle
bP_{\mathrm{e}}\rangle$

The tomographic technique provides direct constraints on the halo bias-
weighted mean electron pressure, $\langle bP_{\mathrm{e}}\rangle$ [25]. This
can be rewritten using the the large-scale, $y$-weighted halo clustering bias
$b_{y}$ and the mean electron pressure $\langle P_{\mathrm{e}}\rangle$ as
$\langle bP_{\mathrm{e}}\rangle=b_{y}\langle P_{\mathrm{e}}\rangle$ [10].
Making use of the _Magneticum_ simulation, knowledge of the density and
pressure is readily available for different $z$, and instead of using the
clustering redshift technique, we can perform an equivalent measurement using
a cross-correlation of the density $\rho$ and thermal gas pressure
$P_{\mathrm{th}}$:

$\frac{\langle\delta_{\rho}P_{\mathrm{th}}\rangle(k)}{\langle\delta_{\rho}\delta_{\rho}\rangle(k)}\to\langle
bP_{\mathrm{th}}\rangle=b_{y}\langle P_{\mathrm{th}}\rangle,$ (2)

where $\delta_{\rho}\equiv(\rho-\bar{\rho})/\bar{\rho}$ and the bar denotes
the mean value in the simulation. Assuming the gas is fully ionized, the
electron pressure $P_{\mathrm{e}}$ is related to the total thermal gas
pressure $P_{\mathrm{th}}$ as $P_{\mathrm{th}}=(8-5Y)/(4-2Y)P_{\mathrm{e}}$,
where $Y=0.24$ is the primordial helium mass fraction (we assume a primordial
mixture of elements and full ionization). Here,
$\langle\delta_{\rho}P_{\mathrm{th}}\rangle(k)$ and
$\langle\delta_{\rho}\delta_{\rho}\rangle(k)$ denote the density-pressure
cross power spectrum and the density power spectrum, respectively, and $k$ is
the wavenumber. The arrow indicates the large-scale limit, $k\to 0$.

Why do we call $\langle bP_{\mathrm{th}}\rangle$ the “halo-bias weighted”
quantity, when we cross-correlate the pressure and density fields with no
explicit reference to collapsed structures such as halos? The reason is that
the thermal pressure is dominated by halos [6, 31]; specifically, the density-
pressure cross correlation is dominated by galaxy groups and clusters [10,
11]. Therefore, our estimator using the density fields yields the halo-bias
weighted mean pressure.

In order to obtain $\langle\delta_{\rho}P_{\mathrm{th}}\rangle(k)$ and
$\langle\delta_{\rho}\delta_{\rho}\rangle(k)$, particles in the _Magneticum_
simulation are assigned to the nearest point in a $100^{3}$ grid spanning the
simulation box. The coarse grain of the grid is acceptable since we are only
interested in the large-scale limit of the cross-correlation. The nbodykit
package [53] is used to calculate the power spectra and cross-correlation of
the resulting maps.

Figure 1: The ratio of the density-pressure cross power spectrum and the
density power spectrum,
$\langle\delta_{\rho}\delta_{P}\rangle(k)/\langle\delta_{\rho}\delta_{\rho}\rangle(k)$
with
$\delta_{P}\equiv(P_{\mathrm{th}}-\bar{P}_{\mathrm{th}})/\bar{P}_{\mathrm{th}}$,
as measured in the _Magneticum_ simulation. We show the results for 3
different redshifts: $z=1.98$, $0.90$ and $0.00$ (from top to bottom). At
large scales, the value approaches a constant value, $b_{y}$, to within
scatter due to the finite box size.

In Fig. 1, we show the ratio
$\langle\delta_{\rho}\delta_{P}\rangle(k)/\langle\delta_{\rho}\delta_{\rho}\rangle(k)$
as a function of $k$, where
$\delta_{P}\equiv(P_{\mathrm{th}}-\bar{P}_{\mathrm{th}})/\bar{P}_{\mathrm{th}}$.
This quantity is equal to $b_{y}$ in the large-scale limit. We obtain the
large-scale value by averaging the correlation functions over
$k<0.03~{}\mathrm{Mpc}^{-1}$. This reduces the scatter in the correlation
functions at small $k$ due to the finite box size. We have checked for
convergence with respect to our choice of $k<0.03~{}\mathrm{Mpc}^{-1}$. The
standard deviation in the correlation functions is used to derive error bars
on the quantities derived using this method. We also checked that fitting the
correlation functions to various forms has only a small effect on the
calculated values for $b_{y}$ of approximately 2%, which is subdominant to the
uncertainty due to cosmic variance.

Figure 2: The bias-weighted mean electron pressure $\langle
bP_{\mathrm{e}}\rangle$ is shown as a function of $z$. The points are taken
from the observed data given in Ref. [10], and the blue dashed line shows the
value from the _Magneticum_ simulation. The (small) shaded blue region shows
the $1\sigma$ uncertainty in the _Magneticum_ simulation due to the scatter in
the correlation functions at small $k$, whilst the shaded green region shows
the uncertainty due to cosmic variance.

We estimate the uncertainty due to cosmic variance by dividing the simulation
box into 64 sub-regions and calculating the variance of $\langle
bP_{\mathrm{e}}\rangle$ within those regions. In Fig. 2, we compare the values
of $\langle bP_{\mathrm{e}}\rangle$ from the _Magneticum_ simulation and the
observations [10]. They are in excellent agreement. To quantify how well the
_Magneticum_ values fit the data, we consider an overall rescaling of the
electron pressure $\langle bP_{\mathrm{e}}\rangle\rightarrow C\langle
bP_{\mathrm{e}}\rangle$, and consider if this can give a better fit to the
data as presented in [10]. A $\chi^{2}$ analysis, comparing the _Magneticum_
values with the data up to $z=0.98$ (above this value only upper bounds are
available), gives a value $C=0.936\pm 0.095$. This confirms that the
measurements from the _Magneticum_ simulation are in agreement with
observations - with an uncertainty at the $10\%$ level.

##  IV The large-scale, $y$-weighted halo clustering bias $b_{y}$

While $\langle bP_{\mathrm{e}}\rangle$ is the direct observable of the SZ-
based tomography, we need the knowledge of $b_{y}$ to obtain $\langle
P_{\mathrm{e}}\rangle$ from the observed $\langle bP_{\mathrm{e}}\rangle$. In
Refs. [10, 11], the halo model developed in Refs. [30, 31, 32, 26, 33], which
resulted in the pysz code 111 https://github.com/ryumakiya/pysz, was used to
calculate $b_{y}$. How accurate is this approach?

The halo model calculation yields (see Appendix B of [10] for details)

$b_{y}(z)=\frac{\int dM\frac{dn}{dM}M^{5/3+\alpha_{p}}b_{h}(M,z)}{\int
dM\frac{dn}{dM}M^{5/3+\alpha_{p}}},$ (3)

where $dn/dM$ and $b_{h}$ are the mass function and linear clustering bias of
dark matter halos, respectively [55, 56]. The physics is simple: the total
pressure of gas in a halo $\int dVP_{\mathrm{th}}(M)$ is proportional to the
halo mass $M$ times the virial temperature, the latter of which is
proportional to $M^{2/3}$; thus, the virial relation gives $\int
dVP_{\mathrm{th}}\propto M^{5/3}$. The X-ray observation of galaxy clusters
shows a small empirical correction to this relation, $M^{5/3+\alpha_{p}}$ with
$\alpha_{p}=0.12$ [57]. We use the pysz code to calculate $b_{y}$ with the
following parameters: $h=0.704$, $\Omega_{\mathrm{b}}h^{2}=0.02265$,
$\Omega_{\mathrm{c}}h^{2}=0.11216$, $A_{\mathrm{s}}=2.42\times 10^{-9}$, and
$n_{\mathrm{s}}=0.963$. The neutrino mass $m_{\nu}$ is set to zero because the
_Magneticum_ simulation does not include massive neutrinos.

In Fig. 3, we show the comparison of $b_{y}$ calculated with the halo model
and from _Magneticum_. We find that for low $z$, there is excellent agreement
between the two calculations - well within the uncertainties. However, at
large $z\gtrsim 2.5$, there is significant disagreement, with $b_{y}$ measured
in _Magneticum_ being significantly lower than predicted by the halo model.
This suggests that the measurements made in Ref. [10] of the mean electron
temperature (see Sec. V) are reliable up to a redshift $z\lesssim 2.5$, but
implies uncertainty in the upper bounds which were derived at $z\gtrsim 2.5$.

The reason for this disagreement is not clear, but it is plausible that the
assumption (made for the halo model) that pressure is dominated by the
virialized structures and the contribution from supernova and AGN feedback are
subdominant compared to the thermal pressure of virialized gas may be violated
at such a high $z$. There may be an additional term arising from a response of
internal properties of galaxy groups and clusters to the large-scale
overdensity [58, 59], which we do not include in Eq. (3). As Eq. (3) agrees
with $b_{y}$ from the _Magneticum_ simulation, which should include all the
relevant effects, precisely at $z\lesssim 2.5$, these contributions are small
in this redshift range. Further analysis of this disagreement at $z\gtrsim
2.5$ is left for future work.

Figure 3: The $y$-weighted halo bias $b_{y}$ is shown as a function of $z$.
The red, dotted line is the prediction from the halo model, whilst the blue-
line is that derived from the _Magneticum_ simulation. The shaded blue region
represents the $1\sigma$ uncertainty in $b_{y}$ due to scatter in the large-
scale correlation functions. We find excellent agreement between the two until
$z\gtrsim 2.5$.

To quantify how well the _Magneticum_ values fit the halo model prediction, we
again consider an overall rescaling factor to the halo bias calculated with
the halo model, $b_{y}\rightarrow Ab_{y}$, and consider whether this can
provide a better fit to the the simulation. Performing a $\chi^{2}$ analysis
(and excluding values from $z>2.5$) gives a value $A=1.0027\pm 0.0043$,
indicating that the halo model prediction is able to predict the values from
_Magneticum_ to better than $1\%$ accuracy.

##  V The density-weighted mean electron temperature $\bar{T}_{\mathrm{e}}$

The mean electron pressure $\langle P_{\mathrm{e}}\rangle$ is related to the
density-weighted mean temperature of electrons in the universe, defined as
$\bar{T}_{\mathrm{e}}\equiv{\langle
n_{\mathrm{e}}T_{\mathrm{e}}\rangle}/{\langle n_{\mathrm{e}}\rangle}$ [5, 6].
Specifically, using $\langle bP_{\mathrm{e}}\rangle$ and $b_{y}$, we find

$\bar{T}_{\mathrm{e}}=\frac{2m_{\mathrm{H}}}{\rho_{\mathrm{c}}\Omega_{\mathrm{b}}k_{\mathrm{B}}(2-Y)(1+z)^{3}}\frac{1}{b_{y}}\langle
bP_{\mathrm{e}}\rangle,$ (4)

where $\rho_{\mathrm{c}}=3H_{0}^{2}/(8\pi G)$ is the critical density of the
universe at $z=0$ and $m_{\mathrm{H}}$ is the hydrogen mass.

The temperature data can also be read directly from the output files of the
_Magneticum_ simulation, and the uncertainty in the mean temperature (and
pressure) due to cosmic variance is estimated by dividing the simulation box
into 64 sub-regions and calculating the variance of the mean temperature
within those regions. We assume throughout that the electron temperature and
the total temperature are equal, although there is likely to be some
difference between the two, as discussed in more detail in Refs. [60, 61].

For a direct comparison to the measurements of Ref. [10], we here use the
value of $b_{y}$ calculated using the halo model, rather than the measured
value from the _Magneticum_ simulation. In Fig. 4, we compare
$\bar{T}_{\mathrm{e}}$ from the data and the simulation as a function of $z$.
As expected from the excellent agreement for $\langle bP_{\mathrm{e}}\rangle$
(Fig. 2) and $b_{y}$ (Fig. 3), we find excellent agreement for
$\bar{T}_{\mathrm{e}}$. This completes the validation of the methodology for
obtaining $\langle T_{\mathrm{e}}\rangle$, i.e., the thermal history of the
universe, developed in Refs. [10, 11].

Figure 4: The density-weighted mean electron temperature
$\bar{T}_{\mathrm{e}}$ (in units of million K) is shown as a function of $z$.
The points are taken from the observed data given in Ref. [10]. The solid
green line shows the temperature read directly from the _Magneticum_
simulation, while the blue dashed line shows that calculated from the pressure
data in the _Magneticum_ simulation. The blue and green regions show the
$1\sigma$ confidence intervals for their respective data.

##  VI Conclusion

This study aimed to answer two principle questions:

  1. 1.

The bias-weighted mean electron pressure $\langle bP_{\mathrm{e}}\rangle$ is
observable from cosmological surveys. Does this quantity measured from the
_Magneticum_ simulation agree with the data given in Ref. [10]?

  2. 2.

In Ref. [10], the density-weighted mean electron temperature
$\bar{T}_{\mathrm{e}}$ is derived by dividing $\langle bP_{\mathrm{e}}\rangle$
by $b_{y}$ calculated from the halo model. Is this method accurate?

The results presented here show that the answer to both questions is: yes. The
values of $\langle bP_{\mathrm{e}}\rangle$ (as well as $\bar{T}_{\mathrm{e}}$)
from _Magneticum_ agree well with the data - with the best-fitting value to
rescale the overall pressure given by $C=0.936\pm 0.095$, consistent with
unity to within the uncertainty. Likewise, the halo model prediction for
$b_{y}$ matches very well to that measured in _Magneticum_ , with the best-
fitting rescaling value given by $A=1.0027\pm 0.0043$.

The results presented here confirm the validity of the tomographic method in
order to determine the thermal history of gas in the universe, although
highlight a need for further study of the halo model calculation at high
redshifts, $z\gtrsim 2.5$, which are yet to be probed by observations.

We also suggest that the measured mean temperature can be used as a simple
“thermometer test” of the baryonic physics of simulations to confirm that they
are capable of accurately reproducing observed values, which can be performed
for a host of state-of-the-art cosmological hydrodynamical simulations [62,
63, 64, 65, 66, 67]. The _Magneticum_ simulation has passed this test. How
about yours?

###### Acknowledgements.

EK thanks Y.-K. Chiang, R. Makiya and B. Ménard for the collaboration on Refs.
[10, 11], which motivated this work. SY thanks R. Makiya for his help with the
pysz code [26, 33] and was supported during the research by a Humboldt
Research Fellowship. This work was supported in part by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy - EXC-2094 - 390783311. KD acknowledges support for the
COMPLEX project from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation program grant agreement
ERC-2019-AdG 860744. The Kavli IPMU is supported by World Premier
International Research Center Initiative (WPI), MEXT, Japan. The calculations
were carried out at the Leibniz Supercomputer Center (LRZ) under the project
pr83li. We are especially grateful for the support by M. Petkova through the
Computational Center for Particle and Astrophysics (C2PAP) and the support by
N. Hammer at LRZ when carrying out the Box0 simulation with the Extreme Scale-
Out Phase on the new SuperMUC Haswell extension system.

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