# Predicting the Scaling Relations between the Dark Matter Halo Mass and
Observables from Generalised Profiles II: Intracluster Gas Emission

A. Sullivan  International Centre for Radio Astronomy Research, The University
of Western Australia, 35 Stirling Highway, Crawley, Western Australia, 6009,
Australia  C. Power  International Centre for Radio Astronomy Research, The
University of Western Australia, 35 Stirling Highway, Crawley, Western
Australia, 6009, Australia  C. Bottrell  International Centre for Radio
Astronomy Research, The University of Western Australia, 35 Stirling Highway,
Crawley, Western Australia, 6009, Australia  A. Robotham  International Centre
for Radio Astronomy Research, The University of Western Australia, 35 Stirling
Highway, Crawley, Western Australia, 6009, Australia  S. Shabala  School of
Natural Sciences, University of Tasmania, Hobart, Tasmania, Australia  [

###### Abstract

We investigate the connection between a cluster’s structural configuration and
observable measures of its gas emission that can be obtained in X-ray and
Sunyaev-Zeldovich (SZ) surveys. We present an analytic model for the
intracluster gas density profile: parameterised by the dark matter halo’s
inner logarithmic density slope, $\alpha$, the concentration, $c$, the gas
profile’s inner logarithmic density slope, $\varepsilon$, the dilution, $d$,
and the gas fraction, $\eta$, normalised to cosmological content. We predict
four probes of the gas emission: the emission-weighted, $T_{\mathrm{X}}$, and
mean gas mass-weighted, $T_{\mathrm{m_{g}}}$, temperatures, and the
spherically, $Y_{\mathrm{sph}}$, and cylindrically, $Y_{\mathrm{cyl}}$,
integrated Compton parameters. Over a parameter space of clusters, we
constrain the X-ray temperature scaling relations, $M_{200}-T_{\mathrm{X}}$
and $M_{500}-T_{\mathrm{X}}$, within $57.3\%$ and $41.6\%$, and
$M_{200}-T_{\mathrm{m_{g}}}$ and $M_{500}-T_{\mathrm{m_{g}}}$, within $25.7\%$
and $7.0\%$, all respectively. When excising the cluster’s core, the
$M_{200}-T_{\mathrm{X}}$ and $M_{500}-T_{\mathrm{X}}$ relations are further
constrained, to within $31.3\%$ and $17.1\%$, respectively. Similarly, we
constrain the SZ scaling relations, $M_{200}-Y_{\mathrm{sph}}$ and
$M_{500}-Y_{\mathrm{sph}}$, within $31.1\%$ and $17.7\%$, and
$M_{200}-Y_{\mathrm{cyl}}$ and $M_{500}-Y_{\mathrm{cyl}}$, within $25.2\%$ and
$22.0\%$, all respectively. The temperature observable $T_{\mathrm{m_{g}}}$
places the strongest constraint on the halo mass, whilst $T_{\mathrm{X}}$ is
more sensitive to the parameter space. The SZ constraints are sensitive to the
gas fraction, whilst insensitive to the form of the gas profile itself. In all
cases, the halo mass is recovered with an uncertainty that suggests the
cluster’s structural profiles only contribute a minor uncertainty in its
scaling relations.

###### keywords:

methods: analytical - cosmology: dark matter - galaxies: clusters: general -
galaxies: clusters: intracluster medium - X-rays: galaxies: clusters

ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)
ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)
ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)
A<EMAIL_ADDRESS>
##  1 1\. Introduction

Our currently favoured standard cosmological model predicts that dark matter
halos are the building blocks of structure formation and host the galaxies,
galaxy groups, and clusters that we observe (White1978, 72, 73, e.g.). At the
mass scale of galaxy clusters, the halo’s mass can be observationally inferred
from the thermodynamic properties of its hot gaseous atmosphere, either from
its X-ray emission (Vikhlinin2006, 69, 68, 7, e.g.) or from the distortion of
the Cosmic Microwave Background (CMB) due to interactions of CMB photons with
energetic electrons (Vanderlinde2010, 67, 3, e.g.), known as the Sunyaev-
Zeldovich (SZ1970, 65, 66, SZ;) effect.

Observationally, the advent of precision X-ray telescopes such as XMM-Newton
(XMMNewton, 33, e.g.), Chandra (Chandra, 70, e.g.), and eROSITA (eROSITA, 60,
e.g.) have allowed for halo mass estimates to be routinely obtained from
statistical samples of X-ray emitting galaxy clusters. These X-ray derived
halo mass estimates typically rely on multi-parameter 3-dimensional fits to
the gas density and temperature profiles, from which the total halo mass
within a halocentric radius is estimated by requiring hydrostatic equilibrium
(Sarazin1988, 62, i.e. pressure forces balance gravitational forces; see,
e.g.). These halo masses can be correlated with mean-weighted temperature
observables, typically weighted by either the X-ray emission or the gas mass,
to establish scaling relations with the halo mass (Arnaud2005, 5, 69, 68, 7,
see, e.g.).

Similarly, SZ-selected cluster observations have been catalogued by precision
microwave telescopes such as the South Pole Telescope (SPT) (SPT, 8, e.g.),
the Atacama Cosmology Telescope (ACT) (ACT, 12, e.g.) and Planck (Planck2015,
56, e.g.). These surveys allow the integrated SZ signal, also called the
Compton parameter, to be measured for galaxy clusters, which when combined
with X-ray cluster observations permitting hydrostatic mass estimates, can
establish the scaling relation between the cluster’s SZ signal and its halo
mass (Andersson2011, 3, 16, 37, e.g.).

Generally, fits to a cluster’s X-ray emission allow tight limits to be placed
on the halo mass, up to a hydrostatic bias, which is thought to underestimate
the halo mass in relaxed galaxy clusters by $\sim 10-20\%$ (Martizzi2016, 41,
26, e.g.). However, it is notable that these hydrostatic mass estimates rely
on fitting multi-parameter models to the gas emission, which are often
empirically rather than physically motivated, producing a variety of
‘universal forms’ for the gas density profile (Vikhlinin2006, 69, 59, 40,
e.g.), the gas temperature profile (Sun2009, 64, 31, e.g.) and the gas
pressure profile (Arnaud2010, 6, e.g.). This reflects both the variance in the
forms of gas profiles devised from observational data, and the complexity of
the underlying physical processes (Power2020, 58, e.g. cosmological gas
accretion, infalling groups, outflows of feedback; see, e.g.) that shape these
profiles. Moreover, these empirical fits are often at odds with theoretically
motivated, analytic models for the gas profile (Cavaliere&Fusco-Femiano1978,
11, 34, e.g.) which struggle to capture empirical results.

Simple parametric models of galaxy clusters have been constructed to predict
their emission properties, as informed by empirical and numerical results
(Bode2009, 9, 2, e.g.). Similar work has investigated the nature of galaxy
cluster scaling relations; in particular, applying Bayesian approaches to
constraining these relations from observational data (Maughan2014, 42, e.g.),
and developing semi-analytic models that explain observed deviations of these
scaling relations from self-similarity (Ettori2015, 25, 27, e.g.). More
recently, algorithms for ‘baryon pasting’ gas profiles onto numerically
simulated dark matter-only halos have generated mock observations,
particularly for SZ observables (Osato2023, 52, e.g.), exciting the prospect
of constraining the halo mass, when informed by its reconstruction from these
mock observables.

In these parametric and statistical approaches to studying galaxy clusters and
their scaling relations, the structures of the cluster’s dark matter and
intracluster gas components are assumed, and their relation to the hydrostatic
state of the system. In this work, we investigate how these scaling relations
depend on cluster structure and composition, and how much they are expected to
vary, when the cluster is parameterised by relatively agnostic prescriptions
for its structure. This study builds on our previous work (Sullivan et al.
2024, Submitted), where we investigated the scaling relation between the
kinematic observables of tracer populations within a halo and the underlying
halo mass. In this work, we turn to the scaling of X-ray and SZ observables
with the halo mass, specifically at the mass scale of galaxy clusters, and
within the regime of self-similarity. To achieve this, one of our key goals is
to construct a physically motivated, simple analytical profile to model galaxy
clusters and their intracluster gas component and to accurately predict their
emission observables in terms of the cluster’s structural parameters.

Our analytic tool-kit is detailed in Section 2: we review theoretical
predictions and empirically motivated forms for the density profiles of the
intracluster gas and the dark matter within galaxy clusters, and postulate a
generalised profile which we call the ‘ideal baryonic cluster halo’.
Thereafter, we use the virial theorem to constrain the correspondence between
a cluster’s halo mass and its emission observables: namely, its mean-weighted
temperatures and integrated SZ signals. In Section 3 the analytic profiles for
these observables are derived, with an analysis of their bounds over the
outlined parameter space fixing constraints on the halo mass. Scaling
relations are presented in Section 4, along with a discussion on the
dependence of our results on the chosen parameters built into the cluster’s
structural model. We present our conclusions in Section 5.

##  2 2\. Theoretical background and methods

###  2.1 2.1. Constructing a generalised dark matter profile

In the first paper of our series (Sullivan et al. 2023, referred to as Paper I
hereafter), we modelled halos as dark matter-only systems, using a
generalisation of the Navarro-Frenk-White profile (NFW1995 (46, 47, 48),
hereafter NFW) which has been found to provide a good fit to the ensemble
average of dynamically relaxed halos in cosmological $N$-body simulations. We
referred to this generalisation as ‘ideal physical halos’; they describe
spherically symmetric halos with logarithmic density slope (hereafter, for
brevity, slope) of $-3$ at large radius, and inner slope $-\alpha$, such that
the density profile is parameterised by $\rho(r)\sim r^{-\alpha}$ at small
radii. This form was motivated as a reasonable approximation to relaxed and
unperturbed halos in both the observed and simulated universe.

####  2.1.1 The dark matter halo inner slope $\alpha$

The NFW profile is the $\alpha=1$ member of the ideal physical halo class,
with its divergent behaviour at small radii referred to as a ‘cusp’. If the
density profile were instead to flatten in the inner region, as $\alpha\simeq
0$, this is referred to as a ‘core’. NFW-like cusps are consistently predicted
in cosmological $N$-body simulations, while even ‘cuspier’ inner slopes of
$\alpha\simeq 1.5$ have been recovered in recent high-resolution idealised
$N$-body simulations that follow the collapse of proto-halos (OgiyaHahn2017,
50, e.g.). However, this is in tension with observational evidence of dark
matter dominated galaxies, such as dwarfs and low-surface brightness galaxies
(Moore1994, 44, 17, 18, 35, e.g.), which favour central cores; this is often
referred to as the ‘core-cusp problem’. We note that there is evidence from
cosmological hydrodynamical simulations that model galaxy formation processes
(e.g. gas cooling, stellar feedback) that these can disrupt cusps oh2011 (51,
57, 21, 20) and produce cored central densities. For these reasons, in Paper I
we chose to model the ideal physical halos within the parameter space of halo
inner slopes $\alpha\in[0,1.5]$, encompassing a plausible range of
predictions.

####  2.1.2 The concentration parameter $c$

The properties of dark matter halos are usually referenced in terms of their
so-called virial parameters. These parameters describe gravitational
structures in virial equilibrium, the state in which its gravitational
potential energy is balanced by its internal energy (approximately; cf.
colelacey1996 (13)), as expected to be established within halos, predicted
from gravitational collapse models. The virial mass, $M_{\mathrm{vir}}$,
defines the halo mass enclosed within a sphere of virial radius,
$r_{\mathrm{vir}}$, as:

$M_{\mathrm{vir}}\equiv\frac{4}{3}\pi
r_{\mathrm{vir}}^{3}\Delta\rho_{\mathrm{crit,0}},$ (1)

such that the mean enclosed mass density is given by the overdensity
parameter, $\Delta$, times the present critical density of the universe,
$\rho_{\mathrm{crit,0}}$ (white2001, 71, e.g.).

A standard convention in the literature is that $\Delta=200$, defining
$M_{200}$ and $r_{200}$ to approximate a halo’s virial mass and radius,
respectively. In X-ray observations, where temperature measurements are
usually limited to the higher density, smaller radii, regions of a halo’s gas
distribution, it is often convenient to choose $\Delta=500$, corresponding to
the halo mass $M_{500}$ and the halo radius $r_{500}$. In our investigation,
we will estimate the relationship between the observable properties of
clusters and the halo mass, in both of the conventions $M_{200}$ and
$M_{500}$.

When normalising a dark matter halo’s density profile to contain
$M_{\mathrm{vir}}$ within $r_{\mathrm{vir}}$, the concentration parameter,
$c$, is typically introduced to parameterise the profile, defined as the ratio
of the outer virial radius, $r_{\mathrm{vir}}$, to the inner scale radius,
$r_{\mathrm{s}}$, as:

$c\equiv\frac{r_{\mathrm{vir}}}{r_{\mathrm{s}}}.$ (2)

The concentration parameter is found in cosmological $N$-body simulations to
be weakly dependent on halo mass, with $c=5$ corresponding to cluster-scale
halos in the standard $\Lambda\mathrm{CDM}$ cosmology (bullock2001, 10, 22,
39, 38, e.g.).

The definition of the concentration parameter $c$ depends on the choice in
overdensity, $\Delta$. As $r_{500}<r_{200}$, by definition in Equation (2),
this implies an overdensity-dependent concentration, whereby
$c_{500}<c_{200}$. In this paper, we refer to the concentration only as $c$,
and take suitable values when taking different choices of $\Delta$. In
particular, we take the typical $\Lambda\mathrm{CDM}$ cluster-scale
concentration $c=5$ when $\Delta=200$, and assume to first order
$r_{500}/r_{200}\simeq 0.5$111In general, the value of $r_{500}/r_{200}$ for a
given halo will depend on the specific form of its density profile. This exact
conversion is not of strong importance in our model, as $r_{200}$ and
$r_{500}$ are both assumed to be virial radius approximations, and so any
modification of their ratio will be quantitative rather than qualitative., so
that $c=2.5$ appropriately describes cluster-scale halos when $\Delta=500$.

####  2.1.3 The ideal physical halo profile

To model a halo’s density profile in a scale-free formalism, a dimensionless
radial scale, $s$, can be introduced, defined as:

$s\equiv\frac{r}{r_{\mathrm{vir}}},$ (3)

where $r$ is the halocentric radius. As per Paper I, this allowed us to
express the density profile of the ideal physical halos in the form:

$\frac{\rho(s,c,\alpha)}{\Delta\rho_{\mathrm{crit,0}}}=\frac{u(c,\alpha)}{3s^{\alpha}(1+cs)^{3-\alpha}},$
(4)

where $c$ is the concentration, $\alpha$ is the halo’s inner slope, and we
refer to $u(c,\alpha)$ as the generalised concentration function, defined by
the integral:

$u(c,\alpha)\equiv\left[\int_{0}^{1}\frac{s^{2-\alpha}\mathrm{d}s}{(1+cs)^{3-\alpha}}\right]^{-1}.$
(5)

###  2.2 2.2. Constructing a generalised intracluster gas profile

We begin this work by constructing a gas density profile,
$\rho_{\mathrm{gas}}(r)$, to model the distribution of hot, X-ray emitting
ionised gas within a galaxy cluster. For now, we assume that the halo’s
underlying dark matter density profile, $\rho_{\mathrm{dm}}(r)$, takes the
form given by the ideal physical halo profile in Equation (4), parameterised
by the halo’s concentration and inner slope.

To model the gas distribution within a cluster, historically the
isothermal-$\beta$ profile has been the preferred model: taken to model a
spherically symmetric isothermal gas profile with a central core
Cavaliere&Fusco-Femiano1978 (11). However, X-ray surveys have demonstrated
that the temperature distribution within clusters is far from isothermal, when
parameterised as function of halocentric radius (Vikhlinin2006, 69, 64, 40,
e.g.). These X-ray studies instead devise a cluster-averaged intracluster gas
density profile, when combining radially parameterised fits for each cluster
within a given survey; typically, this recovers a gas profile described by
dual inner and outer logarithmic density slopes, with general consensus for
the outer slope nearing $-3$ toward limiting radii (Vikhlinin2006, 69, 15, 40,
59, e.g.).

In this study, we parameterise a scale-free gas density profile in terms of a
minimal set of physically-understood parameters. To do this, we will assume an
ansatz for this profile, of the form:

$\frac{\rho_{\mathrm{gas}}(s,\varepsilon)}{\Delta\rho_{\mathrm{crit,0}}}=\frac{\delta_{\mathrm{char}}}{3s^{\varepsilon}(1+\mathcal{C}s)^{3-\varepsilon}};$
(6)

or, equivalently, that the gas density is encompassed by the set of all
spherically symmetric gas profiles with an outer slope of $-3$, and an inner
slope of $-\varepsilon$, such that $\rho_{\mathrm{gas}}(r)\sim
r^{-\varepsilon}$ models the gas distribution in the inner region of the
cluster. In this assumed form, there are two additional terms: the profile’s
characteristic density, $\delta_{\mathrm{char}}$, and the parameter
$\mathcal{C}$, which each represent undetermined normalisation factors. From
this ansatz, we will seek to constrain expressions for
$\delta_{\mathrm{char}}$ and $\mathcal{C}$ in terms of structural parameters
for the intracluster gas, such that the form of this density profile is well-
understood and physically grounded.

####  2.2.1 The dilution parameter $d$

The defining assumption built into the ansatz in Equation (6) is that the
logarithmic slope of the gas and the dark matter profile converge at some
outer radius. This must occur, as the characteristic form imposed for these
profiles imposes that each attain an outer slope of $-3$ beyond some radial
scale. This assumption is justified from modelling provided by hydrodynamic
simulations, whereby this convergence is unanimously recovered in galaxy
clusters (Frenk1999, 29, see, e.g.).

The radius in which these density slopes first converge we term the dilution
radius, $r_{\mathrm{dil}}$, describing the point where the gas slope is
‘diluted’ to the same slope as the underlying halo. To encode this structural
property, we define the dilution parameter, $d$, as the ratio of the virial
radius, $r_{\mathrm{vir}}$ to this dilution radius, $r_{\mathrm{dil}}$, as:

$d\equiv\frac{r_{\mathrm{vir}}}{r_{\mathrm{dil}}}.$ (7)

From hydrodynamic simulations of $\Lambda\mathrm{CDM}$ cluster halos, this
dilution of the gas typically occurs at or near $r_{\mathrm{dil}}\simeq
0.5r_{\mathrm{200}}$ (Yoshikawa2000, 74, e.g.). Consequently, when taking an
overdensity $\Delta=200$, we will assume a dilution of $d=2$ to model the
intracluster gas profile. Importantly, just as the halo concentration, $c$,
was overdensity-dependent, by its definition in Equation (7), the dilution,
$d$, is also overdensity-dependent. In the same way, we will assume a factor
of $0.5$ when taking an overdensity $\Delta=500$, such that the corresponding
dilution is $d=1$ in this instance.

With this definition of the dilution parameter, we can take the logarithmic
derivatives of the expected dark matter halo profile from Equation (4) and the
assumed form of the gas profile from Equation (6), and impose equality for
$r\geq r_{\mathrm{dil}}$, or in their scale-free composition, for $s\geq 1/d$.
By imposing this condition, the parameter $\mathcal{C}$ can be constrained as:

$\mathcal{C}(c,\alpha,d,\varepsilon)\equiv\frac{d(\alpha-\varepsilon)+c(3-\varepsilon)}{3-\alpha},$
(8)

as specified by the parameters $c$ and $\alpha$ that describe the dark matter
halo profile, and the dilution, $d$. We will refer to this function
$\mathcal{C}(c,\alpha,d,\varepsilon)$ as the gas’ concentration parameter, as
it appears in analogue to the halo’s concentration parameter, $c$, in the form
of the density profiles.

####  2.2.2 The fraction of cosmological baryon content $\eta$

When normalising the density profiles of a cluster comprised of both dark
matter and intracluster gas, we must ensure that the virial mass is defined by
integrating the total density ($\rho_{\mathrm{dm}}+\rho_{\mathrm{gas}}$) out
to the virial radius, rather than only the dark matter component, as was done
in Paper I. To do this, we must introduce a measure of the relative
contribution of gas to dark matter in contributing to the halo’s total mass;
in this case, we will parameterise the fraction of gas mass to total mass,
$f_{\mathrm{gas}}(r\leq r_{\mathrm{vir}})$, as measured at the cluster’s
virial radius, $r_{\mathrm{vir}}$.

The present-day cosmological value of the fraction of baryonic mass (stars +
gas) to total mass in the universe is called the cosmological baryon fraction,
$f_{\mathrm{b,cos}}$, defined in the ratio:

$f_{\mathrm{b,cos}}\equiv\frac{\Omega_{\mathrm{b,0}}}{\Omega_{\mathrm{m,0}}},$
(9)

of the present-day baryonic density parameter, $\Omega_{\mathrm{b,0}}$, to the
present-day total mass density parameter, $\Omega_{\mathrm{m,0}}$. This
cosmological parameter has been constrained from precision measurement of
anisotropies on the Cosmic Microwave Background. We take the value
$f_{\mathrm{b,cos}}=0.158$ Planck2016 (55) whilst neglecting uncertainties, as
these will be insignificant compared to the range spanned by the parameter
space of the gas profiles.

Although our cluster halo model does not consider stars, it is thought that
the hot intracluster gas component dominates the baryonic composition of X-ray
emitting clusters (Akino2022, 1, see, e.g.). This allows us to apply a
cosmological normalisation to our gas profiles, by defining the parameter
$\eta$ as the ratio of the halo cluster’s gas fraction to the cosmological
baryon fraction, as:

$\eta\equiv\frac{f_{\mathrm{gas}}(r\leq
r_{\mathrm{vir}})}{f_{\mathrm{b,cos}}},$ (10)

where $r_{\mathrm{vir}}$ is taken as either $r_{200}$ or $r_{500}$, depending
on the choice in overdensity, $\Delta$. We refer to this parameter, $\eta$, as
the halo’s fraction of cosmological baryon content.

Observational measurements for the gas fraction of clusters is routinely
devised in X-ray observations, from fits to the gas density profile, where
this fraction is consistently found to be an increasing function with cluster
halo mass (Arnaud2005, 5, 68, e.g.). Whilst early X-ray surveys measured the
mean gas fraction for cluster samples around $f_{\mathrm{gas}}(\leq
r_{500})\simeq 0.106-0.110$, corresponding to $\eta\simeq 0.67-0.70$
(Vikhlinin2006, 69, 28, e.g.), recent, more precisely calibrated surveys have
consistently measured a higher mean gas fraction of $f_{\mathrm{gas}}(\leq
r_{500})\simeq 0.130-0.132$, corresponding to $\eta\simeq 0.82-0.84$
(Eckert2013, 23, 45, 59, e.g.). At the cluster outskirts, the mean gas
fraction is found to be an increasing function of halocentric radius, and its
mean value at $r_{200}$ has been estimated around $f_{\mathrm{gas}}(\leq
r_{200})\simeq 0.150$, close to the cosmological value $\eta\simeq 1$
(Morandi2015, 45, 40, e.g.).

Encompassing this range of observational predictions — and remaining agnostic
to the choice in measuring the gas fraction within either $r_{200}$ or
$r_{500}$, so that this parameter is not overdensity-dependent — we take the
values of $\eta\in[0.6,1]$ in our parameter space of idealised halos. In terms
of this parameterisation, we can normalise the gas content in our cluster halo
model by integrating the gas density profile from Equation (6) up to
$r_{\mathrm{vir}}$, and setting the enclosed mass to contain a fraction of
$\eta$ times the virial mass, $M_{\mathrm{vir}}$. From this condition, we can
constrain the profile’s characteristic density, $\delta_{\mathrm{char}}$, in
the form:

$\delta_{\mathrm{char}}\equiv\eta
f_{\mathrm{b,cos}}\mathcal{U}(c,\alpha,d,\varepsilon),$ (11)

where we refer to $\mathcal{U}(c,\alpha,d,\varepsilon)$ as the gas’
generalised concentration function, as defined by the integral expression:

$\mathcal{U}(c,\alpha,d,\varepsilon)\equiv\left[\int_{0}^{1}\frac{s^{2-\varepsilon}\mathrm{d}s}{[1+\mathcal{C}(c,\alpha,d,\varepsilon)s]^{3-\varepsilon}}\right]^{-1},$
(12)

with this function appearing in complete analogue to the halo’s generalised
concentration function, $u(c,\alpha)$, as defined in Equation (5).

####  2.2.3 The gas inner slope $\varepsilon$

The remaining free parameter in Equation (6) is $\varepsilon$, the inner slope
of the gas profile. Recent fits for the average density profile of the
intracluster gas using XMM-Newton Pratt2023 (59) and eROSITA Lyskova2023 (40)
have observed weak cusps of $\varepsilon\simeq 0.4-0.7$ in the central regions
$r\lesssim 0.1r_{500}$ of clusters, in contrast to the central core modelled
in the isothermal-$\beta$ profile and assumed in some analytic models for the
gas density (Komatsu&Seljak2001, 34, e.g.). More comprehensive studies have
observed a large scatter in the behaviour of the gas profile in this central
region (Croston2008, 15, see, e.g.), with no clear trend between this inner
structure of the gas and the cluster’s macroscopic properties.

Numerical studies have suggested that the behaviour of gas profiles in the
central region of clusters are dominated by non-gravitational feedback
processes (McDonald2017, 43, e.g.). Analytic modelling has been employed to
attempt to relate this inner gas structure to the gas accretion properties at
or near the virial radius: in particular, by modelling the shock-wave
propagated across this boundary. This approach has established mathematical
correlations between the gas inner slope and the strength of this shock
(Patej&Loeb2015, 53, see, e.g.); however, it is not clear if this behaviour in
real halos is instead dominated by more complex, non-analytic feedback
processes.

In this study, we will model the intracluster gas density profile by inner
slopes within the range $\varepsilon\in[0,1]$: encompassing the predictions
from recent observational fits, whilst permitting a generous scatter in
values, as anticipated for real clusters undergoing complex processes in the
central regions.

Figure 1: The intracluster gas density profiles, in scale-free form $\rho_{\mathrm{gas}}/500\rho_{\mathrm{crit,0}}$ and traced over the scaled halocentric radius, $r/r_{\mathrm{500}}$, indicated by the light blue shaded region, as predicted for the ideal baryonic cluster halos. This shaded region evaluates the five parameters of our model, shown in Table 1, at values chosen to correspond to the overdensity $\Delta=500$. This prediction is compared to recent observational gas density fits: Pratt2023 (59) (the blue dotted line) and Lyskova2023 (40) (the orange dash-dotted line), with the hydrogen gas density in the latter converted to a gas density. | $\boldsymbol{c}$ | $\boldsymbol{\alpha}$ | $\boldsymbol{\eta}$ | $\boldsymbol{d}$ | $\boldsymbol{\varepsilon}$
---|---|---|---|---|---
Definition: |  Concentration |  Inner density slope of the dark matter profile |  Fraction of cosmological baryon content |  Dilution |  Inner density slope of the intracluster gas profile
Physical values: |  $c=5\quad(\Delta=200),$
$c=2.5\quad(\Delta=500)$ |  $\alpha\in[0,1.5]$ |  $\eta\in[0.6,1]$ |  $d=2\quad(\Delta=200),$
$d=1\quad(\Delta=500)$ |  $\varepsilon\in[0,1]$
Table 1: Summary of the five parameters in the ideal baryonic cluster halo
model: their symbol, definition and physical values.

####  2.2.4 The ideal baryonic cluster halo profiles

With this parameterisation of the intracluster gas profile, together with a
parameterisation for the underlying halo, we can mathematically model the
structural composition of idealised cluster halos. We refer to the structures
modelled by these idealised profiles as ‘ideal baryonic cluster halos’.

Importantly, when modelling the dark matter halo’s density profile by Equation
(4) (i.e. the ideal physical halo considered in Paper I), we must apply a
scalar factor of $(1-\eta f_{\mathrm{b,cos}})$ to take into account the gas
mass contribution. This scalar factor ensures that the total density of the
cluster halo, now considering the gas contribution, still integrates to the
normalised virial mass. This results in a slightly adjusted dark matter
density profile to model the ideal baryonic cluster halos, of the form:

$\frac{\rho_{\mathrm{dm}}(s,c,\alpha,\eta)}{\Delta\rho_{\mathrm{crit,0}}}=\frac{(1-\eta
f_{\mathrm{b,cos}})u(c,\alpha)}{3s^{\alpha}(1+cs)^{3-\alpha}},$ (13)

where the halo’s concentration function, $u(c,\alpha)$, remains defined by
Equation (5). In terms of the gas structural parameters detailed, the gas
profile to model the ideal baryonic cluster halos takes the form:

$\frac{\rho_{\mathrm{gas}}(s,c,\alpha,\eta,d,\varepsilon)}{\Delta\rho_{\mathrm{crit,0}}}=\frac{\eta
f_{\mathrm{b,cos}}\mathcal{U}(c,\alpha,d,\varepsilon)}{3s^{\varepsilon}[1+\mathcal{C}(c,\alpha,d,\varepsilon)s]^{3-\varepsilon}},$
(14)

where the gas’ concentration parameter, $\mathcal{C}(c,\alpha,d,\varepsilon)$,
and the gas’ generalised concentration function,
$\mathcal{U}(c,\alpha,d,\varepsilon)$, are defined in Equations (8) and (12),
respectively.

These idealised density profiles are constrained by five parameters — the
underlying dark matter halo’s structural parameters: its concentration, $c$,
and inner slope, $\alpha$; the gas distribution’s structural parameters: its
dilution, $d$, and inner slope, $\varepsilon$; and the relative contribution
of each component, set by $\eta$, the fraction of cosmological baryon content.
As detailed, these five parameters take on physically-motivated values: three
are bounded continuously: $\alpha\in[0,1.5]$, $\eta\in[0.6,1]$ and
$\varepsilon\in[0,1]$, and the remaining two are set at fixed values,
depending on the overdensity: $c=5$ and $d=2$ when $\Delta=200$, and $c=2.5$
and $d=1$ when $\Delta=500$. These parameters and their chosen values are
summarised in Table 1.

The resulting parameter space of intracluster gas density profiles predicted
in this model is shown in the grey shaded region of Figure 1, with the
overdensity is taken as $\Delta=500$. Here, two recent observational
constraints on the average intracluster gas density profile are shown in
comparison to our model: the best-fit from Pratt2023 (59) shown by the blue-
dotted line, from 93 clusters observed with XMM-Newton and fit to a
generalised NFW density model; and the best-fit from Lyskova2023 (40) shown by
the orange-dotted line, from 38 clusters observed with eROSITA and fit to a
ten-parameter function. Both of these independent observational best-fits are
well contained within the predicted parameter space of our model, motivating
the use of this analytic model to allow predictions for the observable X-ray
and SZ emission properties of galaxy clusters.

###  2.3 2.3. Thermodynamic profiles of cluster halos

For idealised, spherically symmetric gravitational systems consisting of gas
and dark matter components, simple analytical expressions can be used to
predict the corresponding thermodynamic emission profiles.

####  2.3.1 Hydrostatic equilibrium

To relate the gravitational mass of a cluster halo to its thermodynamic state,
it is typically assumed that the intracluster gas is in a state of hydrostatic
equilibrium: such that the thermal pressure exerted by the hot gas is balanced
by the halo’s gravitational potential. For a cluster halo consisting of dark
matter, $\rho_{\mathrm{dm}}(r)$, and hot gas, $\rho_{\mathrm{gas}}(r)$,
density contributions, the hydrostatic equilibrium state is solved by the
differential equation:

$\frac{\mathrm{d}}{\mathrm{d}r}\left[\rho_{\mathrm{gas}}(r)T(r)\right]=-\frac{G\mu
m_{\mathrm{p}}}{k_{\mathrm{B}}}\frac{\rho_{\mathrm{gas}}(r)M(r)}{r^{2}},$ (15)

where $M(r)$ is the total mass of the halo cluster within halocentric radius
$r$, given by integrating the density components of the system:

$M(r)=4\pi\int_{0}^{r}\left[\rho_{\mathrm{dm}}(r^{\prime})+\rho_{\mathrm{gas}}(r^{\prime})\right]r^{\prime}{}^{2}\mathrm{d}r^{\prime},$
(16)

and where $T(r)$ is the equilibrium temperature of the gas, to be solved for.
In Equation (15), the physical constants involved are: Newton’s gravitational
constant, $G$, the mean molecular gas weight, $\mu$, the proton mass,
$m_{\mathrm{p}}$ and the Boltzmann constant, $k_{\mathrm{B}}$. Given profiles
for the gas and the dark matter — i.e. the ideal baryonic cluster halos
posited in this investigation — and some initial or boundary condition on the
system, this differential equation can be solved for the gas’ equilibrium
temperature, $T(r)$, over the radial extent of cluster.

Analytic studies have circumvented prescribing the form of the gas density
profile or any boundary conditions in solving the cluster’s temperature
distribution, by imposing a polytropic equation of state on the gas
(Komatsu&Seljak2001, 34, see, e.g.). However, observational fits for the gas
and temperature profiles of clusters have shown that this polytropic relation
fails at large radii (DeGrandi&Molendi2002, 19, e.g.), whilst these polytropic
models fail to predict the rise in gas temperature with halo radii in the
central region as consistently found observationally (Vikhlinin2006, 69, 64,
40, e.g.). As such, in this study we will not assume a polytropic relation for
the gas, and instead solve for the hydrostatic equilibrium state of the gas by
imposing a sensible boundary condition on the gas temperature:
$\lim_{r\to\infty}T(r)=0$; or, equivalently, that the temperature goes to zero
at very large cluster radii. This allows the general solution for the gas’
equilibrium temperature, solving the hydrostatic state in Equation (15), to be
expressed as:

$T(r)=\frac{G\mu
m_{\mathrm{p}}}{k_{\mathrm{B}}}\frac{1}{\rho_{\mathrm{gas}}(r)}\int_{r}^{\infty}\frac{M(r^{\prime})\rho_{\mathrm{gas}}(r^{\prime})\mathrm{d}r^{\prime}}{r^{\prime}{}^{2}}.$
(17)

To allow this equilibrium temperature to be composed in a scale-free
formulation, we can introduce the virial temperature, $T_{\mathrm{vir}}$,
defined as the temperature of a gas distribution, at halocentric radius,
$r_{\mathrm{vir}}$, for a gravitational system in virial and hydrostatic
equilibrium, as:

$T_{\mathrm{vir}}\equiv\frac{1}{3}\frac{\mu
m_{\mathrm{p}}}{k_{\mathrm{B}}}\frac{GM_{\mathrm{vir}}}{r_{\mathrm{vir}}}.$
(18)

Of note, in this definition of $T_{\mathrm{vir}}$ in Equation (18), the
prefactor of $1/3$ is not unique: some definitions instead will use a
prefactor of $1/2$ instead, with these differences arising due to assumptions
in the form of the gas profile deriving the virial parameter. As we will only
use $T_{\mathrm{vir}}$ as a normalisation factor, and remain consistent
throughout this analysis, the choice in prefactor is entirely ambiguous and
will not impact our predictions.

This general solution for the equilibrium temperature in Equation (17) can
then be expressed as a scale-free profile, in a ratio to the virial
temperature and as a function of the dimensionless radius, $s$, taking the
form:

$\frac{T(s)}{T_{\mathrm{vir}}}=3\frac{\Delta\rho_{\mathrm{crit,0}}}{\rho_{\mathrm{gas}}(s)}\int_{s}^{\infty}\frac{M(s^{\prime})}{M_{\mathrm{vir}}}\frac{\rho_{\mathrm{gas}}(s^{\prime})}{\Delta\rho_{\mathrm{crit,0}}}\frac{\mathrm{d}s^{\prime}}{s^{\prime}{}^{2}},$
(19)

in terms of the scale-free profiles for the intracluster gas, density
$\rho_{\mathrm{gas}}(s)$, and the total mass, $M(s)$, itself encoding both
density constituents comprising the system:

$\frac{M(s)}{M_{\mathrm{vir}}}=3\int_{0}^{s}\left[\frac{\rho_{\mathrm{dm}}(s^{\prime})}{\Delta\rho_{\mathrm{crit,0}}}+\frac{\rho_{\mathrm{gas}}(s^{\prime})}{\Delta\rho_{\mathrm{crit,0}}}\right]s^{\prime}{}^{2}\mathrm{d}s^{\prime}.$
(20)

The corresponding equilibrium pressure, $p$, of the intracluster gas is then
related to the equilibrium temperature and density of the gas by the ideal gas
law:

$p=\frac{k_{\mathrm{B}}T}{\mu m_{\mathrm{p}}}\rho_{\mathrm{gas}},$ (21)

hence the scale-free pressure profile of the gas, $p(s)$, will be given by the
profile:

$\frac{p(s)}{p_{\mathrm{vir}}}=3\int_{s}^{\infty}\frac{M(s^{\prime})}{M_{\mathrm{vir}}}\frac{\rho_{\mathrm{gas}}(s^{\prime})}{f_{\mathrm{b,cos}}\Delta\rho_{\mathrm{crit,0}}}\frac{\mathrm{d}s^{\prime}}{s^{\prime}{}^{2}},$
(22)

with the total mass, $M(s)$, still defined by Equation (20), and where the
parameter $p_{\mathrm{vir}}$ is defined as the virial pressure:

$p_{\mathrm{vir}}\equiv\frac{k_{\mathrm{B}}T_{\mathrm{vir}}}{\mu
m_{\mathrm{p}}}f_{\mathrm{b,cos}}\Delta\rho_{\mathrm{crit,0}},$ (23)

as the pressure of a halo with temperature $T_{\mathrm{vir}}$ at halocentric
radius $r_{\mathrm{vir}}$, given the halo is in virial and hydrostatic
equilibrium, and with a gas fraction of exactly cosmological baryon content,
$f_{\mathrm{b,cos}}$.

####  2.3.2 The emission-weighted temperature

In X-ray observations, the temperature of a cluster is typically weighted by
the properties of the emitting gas for an average measure of the cluster’s
temperature. Most commonly, this is the emission-weighted temperature, or
simply the X-ray temperature, $T_{\mathrm{X}}(<r_{\mathrm{det}})$, as measured
when the cluster’s temperature is weighted by the gas’ emission out to a
halocentric radius, called the detection radius, $r_{\mathrm{det}}$. For a
spherically symmetric galaxy cluster, the emission-weighted temperature can be
calculated by the integral:

$T_{\mathrm{X}}(<r_{\mathrm{det}})=\frac{\int_{0}^{r_{\mathrm{det}}}\rho_{\mathrm{gas}}^{2}(r)\Lambda(T)T(r)r^{2}\mathrm{d}r}{\int_{0}^{r_{\mathrm{det}}}\rho_{\mathrm{gas}}^{2}(r)\Lambda(T)r^{2}\mathrm{d}r},$
(24)

where the emission is taken to be proportional to
$\rho_{\mathrm{gas}}^{2}(r)\Lambda(T)$, in terms of the cooling function,
$\Lambda(T)$, which depends on the emission mechanism that dominates at the
cluster’s physical temperature. For X-ray emitting hot gas, Bremsstrahlung or
‘free-free’ emission dominates, with X-ray emission proportional to
$\rho_{\mathrm{gas}}^{2}(r)T^{1/2}(r)$. In this regime, the emission-weighted
temperature is given by:

$T_{\mathrm{X}}(<r_{\mathrm{det}})=\frac{\int_{0}^{r_{\mathrm{det}}}\rho_{\mathrm{gas}}^{2}(r)T^{3/2}(r)r^{2}\mathrm{d}r}{\int_{0}^{r_{\mathrm{det}}}\rho_{\mathrm{gas}}^{2}(r)T^{1/2}(r)r^{2}\mathrm{d}r},$
(25)

which is accessible in X-ray cluster observations, given fits for a cluster’s
radial gas density and temperature profiles out to some detection radius,
usually limited to $r_{500}$. Formulating this expression in terms of the
dimensionless radius, $s$, and a corresponding dimensionless detection radius,
$s_{\mathrm{det}}$, the scale-free formulation of this X-ray temperature is:

$\frac{T_{\mathrm{X}}(<s_{\mathrm{det}})}{T_{\mathrm{vir}}}=\frac{\int_{0}^{s_{\mathrm{det}}}\left[\frac{\rho_{\mathrm{gas}}(s)}{\Delta\rho_{\mathrm{crit,0}}}\right]^{2}\left[\frac{T(s)}{T_{\mathrm{vir}}}\right]^{3/2}s^{2}\mathrm{d}s}{\int_{0}^{s_{\mathrm{det}}}\left[\frac{\rho_{\mathrm{gas}}(s)}{\Delta\rho_{\mathrm{crit,0}}}\right]^{2}\left[\frac{T(s)}{T_{\mathrm{vir}}}\right]^{1/2}s^{2}\mathrm{d}s},$
(26)

as weighted by the scale-free profiles for the cluster’s gas density and
temperature.

####  2.3.3 The mean gas mass-weighted temperature

Often, rather than measuring the emission-weighted temperature, observational
surveys can instead measure a mean gas mass-weighted temperature,
$T_{\mathrm{m_{g}}}(<r_{\mathrm{det}})$, similarly measured out to a
halocentric radius of $r_{\mathrm{det}}$. For a spherically symmetric mass
distribution, this observable is recovered by weighting the temperature by the
gas density profile, such that:

$T_{\mathrm{m_{g}}}(<r_{\mathrm{det}})=\frac{\int_{0}^{r_{\mathrm{det}}}\rho_{\mathrm{gas}}(r)T(r)r^{2}\mathrm{d}r}{\int_{0}^{r_{\mathrm{det}}}\rho_{\mathrm{gas}}(r)r^{2}\mathrm{d}r}.$
(27)

This weighted temperature is similarly accessible in X-ray observations, given
fits to the cluster’s gas density and temperature. Due to its weighting by the
mean gas mass, which should be proportional to the total halo mass, this
observable is often preferred in X-ray scaling relations, as a tight
proportionality to the halo mass is expected. In terms of our scale-free
framework, this temperature measure is:

$\frac{T_{\mathrm{m_{g}}}(<s_{\mathrm{det}})}{T_{\mathrm{vir}}}=\frac{\int_{0}^{s_{\mathrm{det}}}\frac{\rho_{\mathrm{gas}}(s)}{\Delta\rho_{\mathrm{crit,0}}}\frac{T(s)}{T_{\mathrm{vir}}}s^{2}\mathrm{d}s}{\int_{0}^{s_{\mathrm{det}}}\frac{\rho_{\mathrm{gas}}(s)}{\Delta\rho_{\mathrm{crit,0}}}s^{2}\mathrm{d}s},$
(28)

as weighted by the scale-free profiles within the dimensionless detection
radius, $s_{\mathrm{det}}$.

####  2.3.4 The Sunyaev-Zeldovich signal

The SZ effect is known to induce a frequency-dependent temperature shift,
$\Delta T_{\mathrm{SZE}}$, on the CMB temperature, $T_{\mathrm{CMB}}$, of the
form SZ1970 (65, 66):

$\frac{\Delta T_{\mathrm{SZE}}}{T_{\mathrm{CMB}}}=f(\nu)\cdot y,$ (29)

where $f(\nu)$ encodes the frequency dependence and $y$ is the Compton
parameter. The SZ signal for a galaxy cluster is encoded in this Compton
parameter, which measures the cluster’s electron pressure, $p_{\mathrm{e}}$,
integrated inside some volume, $\mathcal{V}$, as:

$y\equiv\frac{\sigma_{\mathrm{T}}}{m_{\mathrm{e}}c_{\gamma}^{2}}\int_{\mathcal{V}}p_{\mathrm{e}}\mathrm{d}\mathcal{V},$
(30)

usually integrated along the line of sight. In Equation (30), the physical
constants are: the Thompson cross-section, $\sigma_{\mathrm{T}}$, the speed of
light, $c_{\gamma}$ 222The speed of light is denoted with a subscript, as
$c_{\gamma}$, to avoid confusion with the halo’s concentration, denoted $c$.,
and the electron mass, $m_{\mathrm{e}}$. The electron pressure is related to
the intracluster gas pressure, $p$, by the ratio of the mean molecular weight,
$\mu$, to the mean molecular weight of electrons, $\mu_{\mathrm{e}}$, and with
this gas pressure related to the gas’ density and temperature by the ideal gas
law, Equation (21), such that the electron pressure can be expressed as:

$p_{\mathrm{e}}=\frac{\mu}{\mu_{\mathrm{e}}}p=\frac{k_{\mathrm{B}}T}{\mu_{\mathrm{e}}m_{\mathrm{p}}}\rho_{\mathrm{gas}}.$
(31)

As such, the Compton parameter can be equivalently defined by the volume
integral:

$y\equiv\frac{k_{\mathrm{B}}\sigma_{\mathrm{T}}}{m_{\mathrm{e}}c_{\gamma}^{2}\mu_{\mathrm{e}}m_{\mathrm{p}}}\int_{\mathcal{V}}\rho_{\mathrm{gas}}T\mathrm{d}\mathcal{V}.$
(32)

In observational surveys, the SZ signal for a galaxy cluster can be measured
by calculating the Compton parameter as in Equation (32) when integrated over
some well-defined volume: typically, this is either a spherical volume, by
integrating fits for the cluster’s 3-dimensional gas and temperature profiles,
or in a cylindrical volume, by integrating these fits along the line of sight
direction within some 2-dimensional projected radius, $R_{\mathrm{ap}}$, known
as the aperture radius. In these integral expressions, we will consistently
notate 3-dimensional halo radii with a lower-case $r$, and 2-dimensional
projected radii with an upper-case $R$, inclusive of all subscripts.

For a spherically-integrated, $Y_{\mathrm{sph}}(<r_{\mathrm{det}})$, Compton
parameter, measured within the sphere of halocentric radius of
$r_{\mathrm{det}}$, the associated SZ signal is given by:

$Y_{\mathrm{sph}}(<r_{\mathrm{det}})=\frac{4\pi
k_{\mathrm{B}}\sigma_{\mathrm{T}}}{m_{\mathrm{e}}c_{\gamma}^{2}\mu_{\mathrm{e}}m_{\mathrm{p}}}\int_{0}^{r_{\mathrm{det}}}\rho_{\mathrm{gas}}(r)T(r)r^{2}\mathrm{d}r.$
(33)

Similarly, for a cylindrically-integrated,
$Y_{\mathrm{cyl}}(<R_{\mathrm{ap}})$, Compton parameter, measured along the
line of sight direction within an aperture radius, $R_{\mathrm{ap}}$, the
associated SZ signal will be:

$Y_{\mathrm{cyl}}(<R_{\mathrm{ap}},r_{\mathrm{b}})=\frac{4\pi
k_{\mathrm{B}}\sigma_{\mathrm{T}}}{m_{\mathrm{e}}c_{\gamma}^{2}\mu_{\mathrm{e}}m_{\mathrm{p}}}\int_{0}^{R_{\mathrm{ap}}}R\mathrm{d}R\Biggl{[}\int_{R}^{r_{\mathrm{b}}}\frac{\rho_{\mathrm{gas}}(r)T(r)r\mathrm{d}r}{\sqrt{r^{2}-R^{2}}}\Biggr{]};$
(34)

or, equivalently, by subtracting from the total spherically-integrated
parameter at the halocentric cluster boundary, $r_{\mathrm{b}}$, as Arnaud2010
(6):

$\displaystyle Y_{\mathrm{cyl}}(<R_{\mathrm{ap}},r_{\mathrm{b}})$
$\displaystyle=Y_{\mathrm{sph}}(<r_{\mathrm{b}})$ (35)
$\displaystyle-\frac{4\pi
k_{\mathrm{B}}\sigma_{\mathrm{T}}}{m_{\mathrm{e}}c_{\gamma}^{2}\mu_{\mathrm{e}}m_{\mathrm{p}}}\int_{R_{\mathrm{ap}}}^{r_{\mathrm{b}}}\rho_{\mathrm{gas}}(r)T(r)r\sqrt{r^{2}-R_{\mathrm{ap}}^{2}}\mathrm{d}r.$

In Equations (34) and (35), this halocentric cluster boundary,
$r_{\mathrm{b}}$, is taken to occur at the cluster’s accretion shock, beyond
which the electron pressure is expected to fall to the ambient pressure of the
intergalactic medium, thus truncating the cluster’s SZ signal. This cluster
boundary is usually taken as $r_{\mathrm{b}}\simeq 5r_{500}$, as predicted in
hydrodynamic simulations (Arnaud2010, 6, see, e.g.).

When the SZ effect is integrated over some volume, it becomes an important
observational probe, as the integrated Compton parameter will be proportional
to the number of electrons inside the region, and so proportional to the
cluster’s mass. To capture these SZ observables in a scale-free formalism, we
introduce the virial Compton parameter, $Y_{\mathrm{vir}}$, defined as the
spherically-integrated value enclosed by the halocentric radius
$r_{\mathrm{vir}}$, as:

$Y_{\mathrm{vir}}\equiv\frac{4}{3}\pi
r_{\mathrm{vir}}^{3}\frac{\sigma_{\mathrm{T}}}{m_{\mathrm{e}}c_{\gamma}^{2}}\frac{\mu}{\mu_{\mathrm{e}}}p_{\mathrm{vir}},$
(36)

where $p_{\mathrm{vir}}$ is the virial pressure, defined in Equation (23).
Taking this definition of $Y_{\mathrm{vir}}$, the scale-free spherically-
integrated Compton parameter, measured within a dimensionless detection
radius, $s_{\mathrm{det}}$, is given by the profile:

$\frac{Y_{\mathrm{sph}}(<s_{\mathrm{det}})}{Y_{\mathrm{vir}}}=3\int_{0}^{s_{\mathrm{det}}}\frac{\rho_{\mathrm{gas}}(s)}{f_{\mathrm{b,cos}}\Delta\rho_{\mathrm{crit,0}}}\frac{T(s)}{T_{\mathrm{vir}}}s^{2}\mathrm{d}s.$
(37)

To devise an expression for the scale-free cylindrically-integrated Compton
parameter, we must define a dimensionless projected radius scale, denoted by
$S$, as the ratio of the projected radius to the virial radius, as:

$S\equiv\frac{R}{r_{\mathrm{vir}}}.$ (38)

As such, in terms of a dimensionless aperture radius, $S_{\mathrm{ap}}$, the
scale-free cylindrically-integrated Compton parameter is given by the profile:

$\displaystyle\frac{Y_{\mathrm{cyl}}(<S_{\mathrm{ap}},s_{\mathrm{b}})}{Y_{\mathrm{vir}}}$
$\displaystyle=\frac{Y_{\mathrm{sph}}(<s_{\mathrm{b}})}{Y_{\mathrm{vir}}}$
(39)
$\displaystyle-3\int_{S_{\mathrm{ap}}}^{s_{\mathrm{b}}}\frac{\rho_{\mathrm{gas}}(s)}{f_{\mathrm{b,cos}}\Delta\rho_{\mathrm{crit,0}}}\frac{T(s)}{T_{\mathrm{vir}}}s\sqrt{s^{2}-S_{\mathrm{ap}}^{2}}\mathrm{d}s,$

as subtracted from the total, scale-free spherically-integrated signal from
Equation (37), and where all three-dimensional, dimensionless halo radii are
denoted by a lower-case $s$, and all two-dimensional, dimensionless projected
radii are denoted by an upper-case $S$, inclusive of all subscripts.

In this scale-free composition, $s_{\mathrm{b}}\equiv
r_{\mathrm{b}}/r_{\mathrm{vir}}$ is now the overdensity-dependent cluster
boundary, requiring its specification in units of $r_{200}$ or $r_{500}$, when
$\Delta$ is fixed. Taking the predicted value $r_{\mathrm{b}}=5r_{500}$ when
$\Delta=500$, we will assume this is equivalently approximated to
$r_{\mathrm{b}}=2.5r_{200}$ when $\Delta=200$, approximately consistent with
numerical predictions (Lau2015, 36, e.g.).

###  2.4 2.4. Predictions for the X-ray and SZ scaling relations

With this scale-free framework for X-ray and SZ observables, we now seek to
construct an analytical correspondence between these observables and the halo
mass, in the form of scaling relations.

####  2.4.1 Dimensional analysis

For this derivation, we use the method of dimensional analysis. This technique
allows us to link the halo’s mass, $M_{\mathrm{halo}}$ (measured in
$\mathrm{M}_{\odot}$), to the temperature observables, $T_{\mathrm{X}}$,
$T_{\mathrm{m_{g}}}$ (measured in $\mathrm{K}$), and the SZ observables,
$Y_{\mathrm{sph}}$, $Y_{\mathrm{cyl}}$ (measured in $\mathrm{Mpc}^{2}$), by
some combination of physical constants that ensure dimensionality in each
scaling relation, and by the introduction of some dimensionless prefactor,
denoted $\mathcal{A}$.

For the temperature observables, we can reasonably assume that the
correspondence contains the physical constants: $G$ (measured in
$\mathrm{Mpc\,km^{2}\,s^{-2}\,\mathrm{M}_{\odot}^{-1}}$),
$\rho_{\mathrm{crit,0}}$ (measured in $\mathrm{M_{\odot}\,Mpc^{-3}}$),
$k_{\mathrm{B}}$ (measured in $\mathrm{M_{\odot}\,km^{2}\,s^{-2}\,K^{-1}}$),
and $m_{\mathrm{p}}$ (measured in $\mathrm{M}_{\odot}$), such that, by
dimensionality, this relation will take the form:

$M_{\mathrm{halo}}=\mathcal{A}\cdot\sqrt{\frac{1}{\rho_{\mathrm{crit,0}}}\left[\frac{k_{\mathrm{B}}}{m_{\mathrm{p}}G}\right]^{3}}\cdot
T_{\mathrm{X}}^{3/2},$ (40)

where the temperature observable is either $T_{\mathrm{X}}$, as shown, or
$T_{\mathrm{m_{g}}}$; this correspondence is interchangeable. For constraints
to be placed on this relationship, the dimensionless prefactor, $\mathcal{A}$,
must be predicted and constrained. For this, we can use the virial theorem, to
relate the halo’s virial mass to its virial temperature, by utilising the
definitions in Equations (1) and (18), whereby:

$M_{\mathrm{vir}}=\frac{9}{2}\sqrt{\frac{1}{\pi\Delta\rho_{\mathrm{crit,0}}}\left[\frac{k_{\mathrm{B}}}{G\mu
m_{\mathrm{p}}}\right]^{3}}\cdot T_{\mathrm{vir}}^{3/2},$ (41)

with this above relationship encoding the same dimensional form as Equation
(40). By taking the virial mass, $M_{\mathrm{vir}}$, as an approximation to
the halo’s mass, in either convention $M_{200}$ or $M_{500}$, we can use the
form of this correspondence to recover the desired scaling relation, such
that:

$M_{\mathrm{vir}}=\frac{9}{2}\sqrt{\frac{1}{\pi\Delta\rho_{\mathrm{crit,0}}}\left[\frac{k_{\mathrm{B}}}{G\mu
m_{\mathrm{p}}}\right]^{3}}\cdot\left[\frac{T_{\mathrm{X}}}{\tau_{1}}\right]^{3/2},$
(42)

in terms of the emission-weighted temperature, $T_{\mathrm{X}}$, by the
introduction of a dimensionless parameter, denoted $\tau_{1}$. Equivalently,
this correspondence can be constructed in terms of the mean gas mass-weighted
temperature, $T_{\mathrm{m_{g}}}$, by the introduction of a similar
dimensionless parameter, $\tau_{2}$, such that:

$M_{\mathrm{vir}}=\frac{9}{2}\sqrt{\frac{1}{\pi\Delta\rho_{\mathrm{crit,0}}}\left[\frac{k_{\mathrm{B}}}{G\mu
m_{\mathrm{p}}}\right]^{3}}\cdot\left[\frac{T_{\mathrm{m_{g}}}}{\tau_{2}}\right]^{3/2}.$
(43)

In these scaling relations above, the dimensionless parameters $\tau_{1}$ and
$\tau_{2}$ are each defined as:

$\tau_{1}\equiv\frac{T_{\mathrm{X}}(<r_{\mathrm{det}})}{T_{\mathrm{vir}}},\quad\tau_{2}\equiv\frac{T_{\mathrm{m_{g}}}(<r_{\mathrm{det}})}{T_{\mathrm{vir}}},$
(44)

as the scale-free form of the weighted temperatures, $T_{\mathrm{X}}$ and
$T_{\mathrm{m_{g}}}$, respectively, when measured within some detection
radius, $r_{\mathrm{det}}$. Each of these parameters can then be analytically
predicted, by Equations (26) and (28).

Using this same technique, we can predict the scaling relations between the
halo mass and the SZ observables. When considering the correspondence between
the virial mass and the virial Compton parameter, defined in Equation (36),
these scaling relations can be recovered by the introduction of similarly
defined dimensionless parameters. For the spherically-integrated Compton
parameter, the scaling takes the form:

$M_{\mathrm{vir}}=\left[\frac{81}{4\pi\Delta\rho_{\mathrm{crit,0}}}\left(\frac{\mu_{\mathrm{e}}m_{\mathrm{e}}c_{\gamma}^{2}}{\mu\sigma_{\mathrm{T}}f_{\mathrm{b,cos}}G}\right)^{3}\right]^{1/5}\cdot\left[\frac{Y_{\mathrm{sph}}}{\zeta_{1}}\right]^{3/5},$
(45)

and for the cylindrically-integrated Compton parameter, the form:

$M_{\mathrm{vir}}=\left[\frac{81}{4\pi\Delta\rho_{\mathrm{crit,0}}}\left(\frac{\mu_{\mathrm{e}}m_{\mathrm{e}}c_{\gamma}^{2}}{\mu\sigma_{\mathrm{T}}f_{\mathrm{b,cos}}G}\right)^{3}\right]^{1/5}\cdot\left[\frac{Y_{\mathrm{cyl}}}{\zeta_{2}}\right]^{3/5}.$
(46)

In these relations, the dimensionless parameters $\zeta_{1}$ and $\zeta_{2}$
are each defined as:

$\zeta_{1}\equiv\frac{Y_{\mathrm{sph}}(<r_{\mathrm{det}})}{Y_{\mathrm{vir}}},\quad\zeta_{2}\equiv\frac{Y_{\mathrm{cyl}}(<R_{\mathrm{ap}})}{Y_{\mathrm{vir}}},$
(47)

as the scale-free form of the SZ observables, each which can be analytically
predicted in Equations (37) and (39), respectively.

####  2.4.2 Constraining the scaling relations

When modelling each of these dimensionless parameters, $\tau_{1}$, $\tau_{2}$,
$\zeta_{1}$ and $\zeta_{2}$, continuously over the parameter space of ideal
baryonic cluster halos, as detailed in Table 1, each of these dimensionless
parameters will be bounded within some interval, when measured within a
detection radius, $r_{\mathrm{det}}$, or in the case of $Y_{\mathrm{cyl}}$,
within an aperture radius, $R_{\mathrm{ap}}$. From these bounds, the scaling
relations in Equations (42), (43), (45) and (46) will each be constrained,
within a corresponding minimum and maximum proportionality.

Evaluating the physical constants in Equation (42) and assuming a mean
molecular weight $\mu=0.60$, the halo’s X-ray temperature scaling relation
$M_{\mathrm{vir}}-T_{\mathrm{X}}$ reduces to the form:

$\frac{M_{\mathrm{vir}}}{\mathrm{M}_{\odot}}=\frac{2.757\times
10^{4}}{\tau_{\mathrm{1}}^{3/2}\Delta^{1/2}h}\left[\frac{T_{\mathrm{X}}}{\mathrm{K}}\right]^{3/2},$
(48)

which will have the same numerical form as the
$M_{\mathrm{vir}}-T_{\mathrm{m_{g}}}$ scaling relation, Equation (43), when
replacing $\tau_{1}$ with $\tau_{2}$. Here, $h$ is the Hubble parameter,
$h\equiv H_{0}/100\mathrm{km\,s^{-1}Mpc^{-1}}$, for $H_{0}$ the Hubble
constant. In this analysis, we take the value $h=0.6751$ Planck2016 (55),
neglecting its uncertainty, as this will be tiny compared to the anticipated
bounds in the scaling relations. In Equation (48), the values for the
overdensity, $\Delta$, must be chosen to specify the virial mass
approximation. Taking the convention $\Delta=200$ in Equation (48), the
$M_{200}-T_{\mathrm{X}}$ scaling relation takes the form:

$\frac{M_{200}}{\mathrm{M}_{\odot}}=\frac{2.888\times
10^{3}}{\tau_{\mathrm{1}}^{3/2}}\left[\frac{T_{\mathrm{X}}}{\mathrm{K}}\right]^{3/2},$
(49)

and similarly, in the convention $\Delta=500$, the $M_{500}-T_{\mathrm{X}}$
scaling relation takes the form:

$\frac{M_{500}}{\mathrm{M}_{\odot}}=\frac{1.826\times
10^{3}}{\tau_{\mathrm{1}}^{3/2}}\left[\frac{T_{\mathrm{X}}}{\mathrm{K}}\right]^{3/2},$
(50)

each dependent on constraints derived for the parameter $\tau_{1}$. The
analogous scaling relations $M_{200}-T_{\mathrm{m_{g}}}$ and
$M_{500}-T_{\mathrm{m_{g}}}$ will each take on the same numeric form, but
again will instead depend on the constraints derived for $\tau_{2}$, rather
than $\tau_{1}$.

In complete analogy, the SZ scaling relations are determined by evaluating the
physical constants in Equation (45), assuming a mean molecular weight
$\mu=0.60$, and a mean molecular weight of electrons $\mu_{\mathrm{e}}=1.148$,
and taking the integrated Compton parameters in units of $\mathrm{Mpc}^{2}$.
As such, the $M_{\mathrm{vir}}-Y_{\mathrm{sph}}$ scaling relation takes the
form:

$\frac{M_{\mathrm{vir}}}{\mathrm{M}_{\odot}}=\frac{6.388\times
10^{17}}{\zeta_{1}^{3/5}\Delta^{1/5}h^{2/5}}\left[\frac{Y_{\mathrm{sph}}}{\mathrm{Mpc}^{2}}\right]^{3/5},$
(51)

which will have the same numerical form as the
$M_{\mathrm{vir}}-Y_{\mathrm{cyl}}$ scaling relation, Equation (46), when
replacing $\zeta_{1}$ with $\zeta_{2}$. Evaluating this scaling relation for
chosen values of $\Delta$, the $M_{200}-Y_{\mathrm{sph}}$ scaling relation
takes the form:

$\frac{M_{200}}{\mathrm{M}_{\odot}}=\frac{2.591\times
10^{17}}{\zeta_{1}^{3/5}}\left[\frac{Y_{\mathrm{sph}}}{\mathrm{Mpc}^{2}}\right]^{3/5},$
(52)

and the $M_{500}-Y_{\mathrm{sph}}$ scaling relation, the form:

$\frac{M_{500}}{\mathrm{M}_{\odot}}=\frac{2.157\times
10^{17}}{\zeta_{1}^{3/5}}\left[\frac{Y_{\mathrm{sph}}}{\mathrm{Mpc}^{2}}\right]^{3/5},$
(53)

as dependent on constraints for $\zeta_{1}$; and analogously, the scaling
relations $M_{200}-Y_{\mathrm{cyl}}$ and $M_{500}-Y_{\mathrm{cyl}}$ will take
the same form, but instead depend on constraints for $\zeta_{2}$.

####  2.4.3 Regimes to bound the X-ray and SZ scaling relations

When deriving these anticipated bounds in the dimensionless parameters,
$\tau_{1}$, $\tau_{2}$, $\zeta_{1}$ and $\zeta_{2}$, the overdensity-dependent
parameters — the concentration, $c$, the dilution, $d$, and the SZ cluster
boundary, in the scale-free form $s_{\mathrm{b}}\equiv
r_{\mathrm{b}}/r_{\mathrm{vir}}$ — must each be specified at fixed values. The
scale-free detection radius, $s_{\mathrm{det}}\equiv
r_{\mathrm{det}}/r_{\mathrm{vir}}$, and aperture radius,
$S_{\mathrm{ap}}\equiv R_{\mathrm{ap}}/r_{\mathrm{vir}}$, in real surveys
depend on observational contingencies: as they are intrinsically dependent on
the available data within a given cluster. Generally, fits for the radial gas
profiles are limited to within $r_{500}$, and so $r_{\mathrm{det}}=r_{500}$
and $R_{\mathrm{ap}}=r_{500}$ are the most common choice in the literature. As
such, we will adopt these values when $\Delta=500$, and when $\Delta=200$ we
will adopt the values $r_{\mathrm{det}}=0.5r_{200}$ and
$R_{\mathrm{ap}}=0.5r_{200}$ in rough likeness.

Given these choices in fixed parameters, the four scaling relations will be
evaluated in two distinct regimes, at each overdensity, as summarised below:

  1. 1.

The $M_{200}-T_{\mathrm{X}}$, $M_{200}-T_{\mathrm{m_{g}}}$,
$M_{200}-Y_{\mathrm{sph}}$ scaling relations:
with parameters: $r_{\mathrm{det}}=0.5r_{200}$, $c=5$, $d=2$.

  2. 2.

The $M_{200}-Y_{\mathrm{cyl}}$ scaling relation:
with parameters: $R_{\mathrm{ap}}=0.5r_{200}$, $r_{\mathrm{b}}=2.5r_{200}$,
$c=5$, $d=2$.

  3. 3.

The $M_{500}-T_{\mathrm{X}}$, $M_{500}-T_{\mathrm{m_{g}}}$,
$M_{500}-Y_{\mathrm{sph}}$ scaling relations:
with parameters: $r_{\mathrm{det}}=r_{500}$, $c=2.5$, $d=1$.

  4. 4.

The $M_{500}-Y_{\mathrm{cyl}}$ scaling relation:
with parameters: $R_{\mathrm{ap}}=r_{500}$, $r_{\mathrm{b}}=5r_{500}$,
$c=2.5$, $d=1$.

For further clarity, these regimes and their associated parameters are
detailed in Tables 2 and 3 in the following section, which show the bounds
devised for the dimensionless parameters corresponding to each of these
scaling relations, in each of these regimes.

##  3 3\. Analysis

###  3.1 3.1. Analytical profiles for the ideal baryonic cluster halos

To derive and constrain the dimensionless parameters $\tau_{1}$, $\tau_{2}$,
$\zeta_{1}$ and $\zeta_{2}$, in terms of the outlined parameter space within
each of the desired regimes, we will model the dark matter and intracluster
gas density profiles with the ideal baryonic cluster halo model, as per
Equations (13) and (14), respectively. With this structural model, we can
analytically predict and numerically trace the associated scale-free X-ray and
SZ emission profiles, and hence the associated dimensionless parameters, by
substituting these idealised density forms into the equations for each
emission observable, as detailed in Section 2.3.

####  3.1.1 The temperature profile of the ideal baryonic cluster halos

Figure 2: The temperature profiles for the ideal baryonic cluster halos, in
scale-free form $T/T_{\mathrm{vir}}$, traced over the scaled halocentric
radius, $r/r_{\mathrm{vir}}$. Each row varies the halo concentration, $c$, and
the dilution, $d$, and each column varies the gas inner slope, $\varepsilon$.
Within each box, each colour varies the halo inner slope, $\alpha$, with the
solid coloured lines tracing a fraction of cosmological baryon content of
$\eta=0.8$, and the shaded colour region around each solid line (not visible
for all curves) tracing this value continuously between $\eta=0.6$ and
$\eta=1$. Figure 3:  The equilibrium temperature and pressure profiles, in
scale-free form $T/T_{500}$ and $p/p_{500}$, shown in the top and bottom
panels, respectively, each traced over the scaled halocentric radius,
$r/r_{\mathrm{500}}$, indicated by the light blue shaded regions, as predicted
for the ideal baryonic cluster halos. These shaded regions evaluate the five
parameters of our model, from Table 1, at values chosen to correspond to the
overdensity $\Delta=500$. These predictions are compared to recent
observational fits for the temperature profile of galaxy clusters, from
Ghirardini2019 (31), for samples of cool core clusters (the blue dotted line,
in the top panel) and non-cool core clusters (the orange dash-dotted line, in
the top panel), as well as to the universal gas pressure profile from
Arnaud2010 (6) (the purple dotted line, in the bottom panel).

Each of the cluster emission observables depend on the equilibrium temperature
profile of the intracluster gas, which is derived from the general solution to
the hydrostatic equilibrium state, Equation (19). For the ideal baryonic
cluster halos, substituting the corresponding density profiles into this
solution, the associated scale-free temperature profile takes the analytic
form:

$\frac{T(s,c,\alpha,\eta,d,\varepsilon)}{T_{\mathrm{vir}}}=3s^{\varepsilon}\left[1+\mathcal{C}(c,\alpha,d,\varepsilon)s\right]^{3-\varepsilon}\cdot\mathcal{I}(s,c,\alpha,\eta,d,\varepsilon),$
(54)

where, for notation simplicity, we define
$\mathcal{I}(s,c,\alpha,\eta,d,\varepsilon)$ as the integral function:

$\displaystyle\mathcal{I}(s,c,\alpha,\eta,d,\varepsilon)$
$\displaystyle\equiv\int_{s}^{\infty}\frac{\mathrm{d}s^{\prime}\Biggl{\\{}(1-\eta
f_{\mathrm{b,cos}})u(c,\alpha)\cdot\int_{0}^{s^{\prime}}\frac{s^{\prime}{}^{\prime}{}^{2-\alpha}\mathrm{d}s^{\prime}{}^{\prime}}{(1+cs^{\prime}{}^{\prime})^{3-\alpha}}}{s^{\prime}{}^{2+\varepsilon}\left[1+\mathcal{C}(c,\alpha,d,\varepsilon)s^{\prime}\right]^{3-\varepsilon}}$
(55) $\displaystyle+\quad\eta
f_{\mathrm{b,cos}}\mathcal{U}(c,\alpha,d,\varepsilon)\cdot\int_{0}^{s^{\prime}}\frac{s^{\prime}{}^{\prime}{}^{2-\varepsilon}\mathrm{d}s^{\prime}{}^{\prime}}{\left[1+\mathcal{C}(c,\alpha,d,\varepsilon)s^{\prime}{}^{\prime}\right]^{3-\varepsilon}}\Biggr{\\}}.$

These scale-free temperature profiles, $T/T_{\mathrm{vir}}$, are traced in
Figure 2, as a function of the dimensionless halocentric radius, $s\equiv
r/r_{\mathrm{vir}}$. These panels encompass the variation of the halo’s
temperature within the desired parameter space, for $\alpha\in[0,1.5]$,
$\eta\in[0.6,1]$ and $\varepsilon\in[0,1]$, and at fixed concentration, $c$,
and dilution, $d$, values corresponding to the two desired overdensity
choices. Except for some curves in the left panels of Figure 2, those with gas
cores, $\varepsilon=0$, inside cuspy, $\alpha=1,\alpha=1.5$, dark matter
halos, all other temperature profiles in Figure 2 exhibit a characteristic
temperature peak within $0.03r_{\mathrm{vir}}-0.1r_{\mathrm{vir}}$. This
general shape is consistent with cluster temperature fits devised in X-ray
observations (Vikhlinin2006, 69, 64, 31, 40, e.g.).

####  3.1.2  The pressure profile of the ideal baryonic cluster halos

Similarly, by taking the density profiles of the ideal baryonic cluster halos
into Equation (22), the scale-free, equilibrium gas pressure profiles of these
halos take the form:

$\frac{p(s,c,\alpha,\eta,d,\varepsilon)}{p_{\mathrm{vir}}}=\eta\mathcal{U}(c,\alpha,d,\varepsilon)\cdot\mathcal{I}(s,c,\alpha,\eta,d,\varepsilon).$
(56)

To illustrate the predictive power of these simple analytic profiles, in
Figure 3 we trace the total variation of these scale-free temperature and gas
pressure profiles, from Equation (54) and (56), respectively, over the
physical parameter space corresponding to $\Delta=500$. For observational
comparison, in the top panel of Figure 3 we compare our temperature profiles
to the best-fit profile of observed X-COP galaxy clusters derived in
Ghirardini2019 (31):— split into cool core clusters, the blue-dotted line, and
non-cool core clusters, the orange dash-dotted line, and with their profiles
re-scaled to meet our definition of $T_{500}$. Similarly, in the bottom panel
of Figure 3, we compare our gas pressure profiles to the universal profile
from Arnaud2010 (6), shown by the purple dotted line, again re-scaled to meet
out definition of $p_{500}$.

In the panels in Figure 3, the predicted variation within our model does a
reasonable job at encompassing the observed fits, particularly within
halocentric radii of $r\lesssim 0.5r_{500}$. Toward larger halocentric radii,
as seen in the top panel of Figure 3, the temperature predictions in our model
begin to deviate from the observed fits; in particular, our model
systematically over-predicts the gas’ temperature. To a lesser extent this
trend is visible in the pressure comparison in the bottom panel of Figure 3,
where our model loses centring of the universal pressure profile. This over-
prediction of the gas’ temperature and pressure toward large halocentric radii
is theoretically expected, due to non-thermal pressure contributors, such as
turbulence and shocks, increasing toward the outer radii of real clusters, as
studied in non-radiative hydrodynamic simulations (Nelson2014, 49, 4, e.g.)
and inferred observationally from measuring the hydrostatic bias of cluster
masses (Eckert2019, 24, 63, e.g.). As these non-thermal pressure contributions
were neglected in our model, our solutions for the gas’ temperature and
pressure will continue to satisfy the balance of hydrostatic equilibrium
without their consideration, and so will expectedly overestimate the trends in
real systems wherever non-thermal pressure becomes significant, as at these
large halocentric radii. In future work, we will endeavour to improve our
model by considering the non-thermal pressure profile of galaxy clusters.

####  3.1.3 Convectional instabilities in the parameter space

The minority of temperature profiles in Figure 2 that do not exhibit a
temperature peak, as in the left panels, corresponding to $\varepsilon=0$, gas
cores, instead show a flattening or increase of the temperature with
decreasing halocentric radii. In particular, for cuspy, $\alpha=1.5$ halos,
shown in magenta, the temperature in the central region becomes divergent. To
discuss the physical nature of these halos, we can quantify the effective
polytropic index, $\Gamma$, for each halo over the parameter space, with this
index defined by the logarithmic derivative:

$\Gamma\equiv 1+\frac{\mathrm{d}\ln T}{\mathrm{d}\ln\rho_{\mathrm{gas}}}.$
(57)

Physically, the index $\Gamma$ must not exceed the gas’ adiabatic gas
constant, $\gamma=5/3$ for a monatomic gas, or the system will be unstable
against convection. This allows us to categorise whether a particular
structural configuration of a cluster halo represents a stable configuration
in its gas and dark matter components, or whether the halo will be unable to
maintain such a configuration over time.

When calculating the index $\Gamma$ over the parameter space of ideal baryonic
cluster halos, we find that $\Gamma>5/3$ for all clusters with a halo inner
slope $\alpha=1.5$ and a gas core shallower than $\varepsilon\lesssim 0.05$;
thus, the divergent temperature curves in Figure 2 are unstable halo
configurations, as expected. When measuring $\Gamma$ down to very small
halocentric radii, we find that $\Gamma>5/3$ for all $\varepsilon=0$ gas cores
inside halos cuspier than $\alpha\gtrsim 1.04$. This implies that modelling a
gas core inside a cuspier-than-NFW halo is an unstable parameter choice,
particularly in the inner region of a cluster. Interestingly, this instability
is corrected whenever the gas’ inner slope becomes just slightly positive,
i.e. $\varepsilon=0^{+}$, for most of these halos, up to a maximum requirement
of $\varepsilon\gtrsim 0.05$ for $\alpha=1.5$ halos, as already discussed. As
this unstable parameter region essentially comprises a boundary of the
parameter space chosen in Table 1, it is not mathematically necessary to
remove this region for our analysis. As such, we will maintain the parameter
choices in Table 1, but we note that this unstable region of halo
configurations should be kept in mind for application of our model to real
clusters.

Aside from this unstable region, the effective polytropic index of all other
halos in our model is a monotonically increasing function of halocentric radii
in the region $0.01r_{\mathrm{vir}}\leq r\leq r_{\mathrm{vir}}$, with all
halos converging toward a value of $\Gamma\simeq 1.1-1.2$ near the virial
radius, in both overdensity conventions. This general trend is broadly
consistent with observational constraints on the effective polytropic index of
the intracluster gas in X-ray emitting clusters (Ghirardini2019b, 32, see,
e.g.).

####  3.1.4 The emission-weighted temperature of the
ideal baryonic cluster halos

Figure 4: The emission-weighted temperature profiles for the ideal baryonic
cluster halos, in scale-free form $\tau_{1}\equiv
T_{\mathrm{X}}(<r_{\mathrm{det}})/T_{\mathrm{vir}}$, traced over the scaled
detection radius, $r_{\mathrm{det}}/r_{\mathrm{vir}}$. Each row varies the
halo concentration, $c$, and the dilution, $d$, and each column varies the gas
inner slope, $\varepsilon$. Within each box, each colour varies the halo inner
slope, $\alpha$, with the solid coloured lines tracing a fraction of
cosmological baryon content of $\eta=0.8$, and the shaded colour region around
each solid line (not visible for all curves) tracing this value continuously
between $\eta=0.6$ and $\eta=1$.

Taking the scale-free temperature profile for the ideal baryonic cluster
halos, as in Equation (54), we can model the mean-weighted temperature
observables for these halos over the desired parameter space. From Equation
(26), we predict the emission-weighted temperature profiles for these halos,
in our scale-free framework, as:

$\frac{T_{\mathrm{X}}(<s_{\mathrm{det}},c,\alpha,\eta,d,\varepsilon)}{T_{\mathrm{vir}}}=3\cdot\frac{\int_{0}^{s_{\mathrm{det}}}\frac{\Bigl{[}\mathcal{I}(s,c,\alpha,\eta,d,\varepsilon)\Bigr{]}^{3/2}s^{2}\mathrm{d}s}{\left\\{s^{\varepsilon}\left[1+\mathcal{C}(c,\alpha,d,\varepsilon)s\right]^{3-\varepsilon}\right\\}^{1/2}}}{\int_{0}^{s_{\mathrm{det}}}\frac{\Bigl{[}\mathcal{I}(s,c,\alpha,\eta,d,\varepsilon)\Bigr{]}^{1/2}s^{2}\mathrm{d}s}{\left\\{s^{\varepsilon}\left[1+\mathcal{C}(c,\alpha,d,\varepsilon)s\right]^{3-\varepsilon}\right\\}^{3/2}}}.$
(58)

These scale-free temperature profiles,
$T_{\mathrm{X}}(<r_{\mathrm{det}})/T_{\mathrm{vir}}$, are shown in Figure 4,
as a function of the dimensionless detection radius, $s_{\mathrm{det}}\equiv
r_{\mathrm{det}}/r_{\mathrm{vir}}$.

####  3.1.5 The mean gas mass-weighted temperature of the
ideal baryonic cluster halos

Figure 5: The mean gas mass-weighted temperature profiles for the ideal
baryonic cluster halos, in scale-free form $\tau_{2}\equiv
T_{\mathrm{m_{g}}}(<r_{\mathrm{det}})/T_{\mathrm{vir}}$, traced over the
scaled detection radius, $r_{\mathrm{det}}/r_{\mathrm{vir}}$. Each row varies
the halo concentration, $c$, and the dilution, $d$, and each column varies the
gas inner slope, $\varepsilon$. Within each box, each colour varies the halo
inner slope, $\alpha$, with the solid coloured lines tracing a fraction of
cosmological baryon content of $\eta=0.8$, and the shaded colour region around
each solid line (not visible for all curves) tracing this value continuously
between $\eta=0.6$ and $\eta=1$.

From Equation (27), we predict the mean gas mass-weighted temperature profiles
for the ideal baryonic cluster halos, in our scale-free framework, as:

$\frac{T_{\mathrm{m_{g}}}(<s_{\mathrm{det}},c,\alpha,\eta,d,\varepsilon)}{T_{\mathrm{vir}}}=3\cdot\frac{\int_{0}^{s_{\mathrm{det}}}\mathcal{I}(s,c,\alpha,\eta,d,\varepsilon)\,s^{2}\mathrm{d}s}{\int_{0}^{s_{\mathrm{det}}}\frac{s^{2-\varepsilon}\mathrm{d}s}{\left[1+\mathcal{C}(c,\alpha,d,\varepsilon)s\right]^{3-\varepsilon}}}.$
(59)

These scale-free temperature profiles,
$T_{\mathrm{m_{g}}}(<r_{\mathrm{det}})/T_{\mathrm{vir}}$, are shown in Figure
5, as a function of the dimensionless detection radius,
$s_{\mathrm{det}}\equiv r_{\mathrm{det}}/r_{\mathrm{vir}}$.

Figure 6: The spherically-integrated Sunyaev-Zeldovich signal for the ideal
baryonic cluster halos, in scale-free form $\zeta_{1}\equiv
Y_{\mathrm{sph}}(<r_{\mathrm{det}})/Y_{\mathrm{vir}}$, traced over the scaled
detection radius, $r_{\mathrm{det}}/r_{\mathrm{vir}}$. Each row varies the
halo concentration, $c$, and the dilution, $d$, and each column varies the gas
inner slope, $\varepsilon$. Within each box, each colour varies the halo inner
slope, $\alpha$, with the solid coloured lines tracing a fraction of
cosmological baryon content of $\eta=0.8$, and the shaded colour region around
each solid line tracing this value continuously between $\eta=0.6$ and
$\eta=1$. Figure 7: The cylindrically-integrated Sunyaev-Zeldovich signal for
the ideal baryonic cluster halos, in scale-free form $\zeta_{2}\equiv
Y_{\mathrm{cyl}}(<R_{\mathrm{ap}})/Y_{\mathrm{vir}}$, traced over the scaled
aperture radius, $R_{\mathrm{ap}}/r_{\mathrm{vir}}$. Each row varies the
cluster boundary, in scale-free form $r_{\mathrm{b}}/r_{\mathrm{vir}}$, the
halo concentration, $c$, and the dilution, $d$, and each column varies the gas
inner slope, $\varepsilon$. Within each box, each colour varies the halo inner
slope, $\alpha$, with the solid coloured lines tracing a fraction of
cosmological baryon content of $\eta=0.8$, and the shaded colour region around
each solid line tracing this value continuously between $\eta=0.6$ and
$\eta=1$. Figure 8: The emission-weighted and mean gas mass-weighted
temperature profiles, in scale-free form $\tau_{1}\equiv
T_{\mathrm{X}}(<r_{\mathrm{det}})/T_{\mathrm{vir}}$ and $\tau_{2}\equiv
T_{\mathrm{m_{g}}}(<r_{\mathrm{det}})/T_{\mathrm{vir}}$, shown in the top and
bottom panels, respectively, for the ideal baryonic cluster halos, evaluated
at fixed detection radii, and traced over halo inner slopes, $\alpha$. Each
column fixes the values of the detection radius, in scale-free form
$r_{\mathrm{det}}/r_{\mathrm{vir}}$, the halo concentration, $c$, and the
dilution, $d$, with these choices corresponding to a particular choice in the
overdensity: the left panels for $\Delta=500$, and the right panels for
$\Delta=200$. Within each box, each colour varies the gas inner slope,
$\varepsilon$, with the solid coloured lines tracing a fraction of
cosmological baryon content of $\eta=0.8$, and the shaded colour region around
each solid line (not visible for all curves) tracing this value continuously
between $\eta=0.6$ and $\eta=1$.

####  3.1.6 The spherically-integrated Compton parameter of the
ideal baryonic cluster halos

By integrating the product of the scale-free gas density and temperature
profiles for the ideal baryonic cluster halos, as in Equations (14) and (54),
the integrated Compton parameters for these halos can be devised. From
Equation (37), the spherically-integrated Compton parameter profiles for these
halos are given by:

$\displaystyle\frac{Y_{\mathrm{sph}}(<s_{\mathrm{det}},c,\alpha,\eta,d,\varepsilon)}{Y_{\mathrm{vir}}}$
$\displaystyle=3\eta\,\mathcal{U}(c,\alpha,d,\varepsilon)$ (60)
$\displaystyle\times\int_{0}^{s_{\mathrm{det}}}\mathcal{I}(s,c,\alpha,\eta,d,\varepsilon)s^{2}\mathrm{d}s.$

These scale-free SZ profiles, $\zeta_{1}\equiv
Y_{\mathrm{sph}}(<r_{\mathrm{det}})/Y_{\mathrm{vir}}$, are shown in Figure 6,
as a function of the dimensionless detection radius, $s_{\mathrm{det}}\equiv
r_{\mathrm{det}}/r_{\mathrm{vir}}$.

####  3.1.7 The cylindrically-integrated Compton parameter of the
ideal baryonic cluster halos

Finally, taking the profiles for the ideal baryonic cluster halos into
Equation (39), the cylindrically-integrated Compton parameter profiles are
given as:

$\displaystyle\frac{Y_{\mathrm{cyl}}(<S_{\mathrm{ap}},s_{\mathrm{b}},c,\alpha,\eta,d,\varepsilon)}{Y_{\mathrm{vir}}}$
$\displaystyle=3\eta\,\mathcal{U}(c,\alpha,d,\varepsilon)\,\,\,\times$ (61)
$\displaystyle\Biggl{[}\int_{0}^{s_{\mathrm{b}}}\mathcal{I}(s,c,\alpha,\eta,d,\varepsilon)s^{2}\mathrm{d}s-\int_{S_{\mathrm{ap}}}^{s_{\mathrm{b}}}\mathcal{I}(s,c,\alpha,\eta,d,\varepsilon)\sqrt{s^{2}-S^{2}_{\mathrm{ap}}}s\mathrm{d}s\Biggr{]}.$

These scale-free SZ profiles, $\zeta_{2}\equiv
Y_{\mathrm{cyl}}(<R_{\mathrm{ap}})/Y_{\mathrm{vir}}$, are shown in Figure 7,
as a function of the dimensionless aperture radius, $S_{\mathrm{ap}}\equiv
R_{\mathrm{ap}}/r_{\mathrm{vir}}$.

###  3.2 3.2. Constraints on $\tau_{1}$, $\tau_{2}$, $\zeta_{1}$ and
$\zeta_{2}$

Overdensity parameter |  Detection radius |  Concentration |  Dilution | Bounds in $\tau_{1}\equiv T_{\mathrm{X}}(<r_{\mathrm{det}})/T_{\mathrm{vir}}$
---|---|---|---|---
$\Delta=200$ |  $r_{\mathrm{det}}=0.5r_{200}$ |  $c=5$ |  $d=2$ |  $\tau_{\mathrm{1,min}}=0.899$ |  $\tau_{\mathrm{1,max}}=2.145$
$\Delta=500$ |  $r_{\mathrm{det}}=r_{500}$ |  $c=2.5$ |  $d=1$ |  $\tau_{\mathrm{1,min}}=0.956$ |  $\tau_{\mathrm{1,max}}=1.725$

Overdensity parameter |  Detection radius |  Concentration |  Dilution | Bounds in $\tau_{2}\equiv T_{\mathrm{m_{g}}}(<r_{\mathrm{det}})/T_{\mathrm{vir}}$
---|---|---|---|---
$\Delta=200$ |  $r_{\mathrm{det}}=0.5r_{\mathrm{200}}$ |  $c=5$ |  $d=2$ |  $\tau_{\mathrm{2,min}}=1.162$ |  $\tau_{\mathrm{2,max}}=1.650$
$\Delta=500$ |  $r_{\mathrm{det}}=r_{\mathrm{500}}$ |  $c=2.5$ |  $d=1$ |  $\tau_{\mathrm{2,min}}=1.210$ |  $\tau_{\mathrm{2,max}}=1.329$

Table 2: The constraints placed on $\tau_{1}\equiv
T_{\mathrm{X}}(<r_{\mathrm{det}})/T_{\mathrm{vir}}$ and $\tau_{2}\equiv
T_{\mathrm{m_{g}}}(<r_{\mathrm{det}})/T_{\mathrm{vir}}$ over the parameter
space of the ideal baryonic cluster halos, each evaluated at two conventions
in the overdensity, $\Delta=200$ and $\Delta=500$.

Overdensity parameter |  Detection radius |  Concentration |  Dilution | Bounds in $\zeta_{1}\equiv Y_{\mathrm{sph}}(<r_{\mathrm{det}})/Y_{\mathrm{vir}}$
---|---|---|---|---
$\Delta=200$ |  $r_{\mathrm{det}}=0.5r_{200}$ |  $c=5$ |  $d=2$ |  $\zeta_{\mathrm{1,min}}=0.336$ |  $\zeta_{\mathrm{1,max}}=0.983$
$\Delta=500$ |  $r_{\mathrm{det}}=r_{500}$ |  $c=2.5$ |  $d=1$ |  $\zeta_{\mathrm{1,min}}=0.726$ |  $\zeta_{\mathrm{1,max}}=1.319$

Overdensity parameter |  Aperture radius |  Cluster boundary |  Concentration |  Dilution | Bounds in $\zeta_{2}\equiv Y_{\mathrm{cyl}}(<R_{\mathrm{ap}})/Y_{\mathrm{vir}}$
---|---|---|---|---|---
$\Delta=200$ |  $R_{\mathrm{ap}}=0.5r_{\mathrm{200}}$ |  $r_{\mathrm{b}}=2.5r_{200}$ |  $c=5$ |  $d=2$ |  $\zeta_{\mathrm{2,min}}=0.492$ |  $\zeta_{\mathrm{2,max}}=1.161$
$\Delta=500$ |  $R_{\mathrm{ap}}=r_{\mathrm{500}}$ |  $r_{\mathrm{b}}=5r_{500}$ |  $c=2.5$ |  $d=1$ |  $\zeta_{\mathrm{2,min}}=0.916$ |  $\zeta_{\mathrm{2,max}}=1.928$

Table 3: The constraints placed on $\zeta_{1}\equiv
Y_{\mathrm{sph}}(<r_{\mathrm{det}})/Y_{\mathrm{vir}}$ and $\zeta_{2}\equiv
Y_{\mathrm{cyl}}(<R_{\mathrm{ap}})/Y_{\mathrm{vir}}$ over the parameter space
of the ideal baryonic cluster halos, each evaluated at two conventions in the
overdensity, $\Delta=200$ and $\Delta=500$.

With these scale-free predictions for the X-ray and SZ observables, the
dimensionless parameters $\tau_{1}$, $\tau_{2}$, $\zeta_{1}$ and $\zeta_{2}$
are constrained inside minimum and maximum values, when evaluating these
profiles in each of the desired regimes.

####  3.2.1 Constraints on $\tau_{1}$ and $\tau_{2}$ within each regime

The two regimes in which the dimensionless weighted temperature parameters,
$\tau_{1}$ and $\tau_{2}$, are to be bounded can be traced within Figures 4
and 5, respectively. Each row of panels in each of these figures fixes the
halo concentration, $c$, and dilution, $d$, to the set of values required in
each regime, such that when these profiles are evaluated at a chosen detection
radius, in scale-free form $r_{\mathrm{det}}/r_{\mathrm{vir}}$, the bounds in
$\tau_{1}$ and $\tau_{2}$ will be determined. Figure 8 evaluates these scale-
free weighted temperatures from Figures 4 and 5 as a continuous function of
the halo inner slope, $\alpha$, at detection radii corresponding to the two
regimes — $r_{\mathrm{det}}/r_{\mathrm{vir}}=0.5$ for $\Delta=200$ and
$r_{\mathrm{det}}/r_{\mathrm{vir}}=1$ for $\Delta=500$ — producing four
distinct windows, two for each observable. In the panels of Figure 8, within
each box, the scale-free weighted temperatures are always bounded between the
minimum, $\varepsilon=1$, in purple, and the maximum, $\varepsilon=0$, in
teal, fixed gas slope curves. As such, the values $\tau_{\mathrm{1,min}}$ and
$\tau_{\mathrm{1,max}}$, or $\tau_{\mathrm{2,min}}$ and
$\tau_{\mathrm{2,max}}$, are simply the minimum and maximum values of the
parameter space traced between these curves, in each box, corresponding to
each of the two regimes, for each parameter. These results produce the
constraints in $\tau_{1}$ and $\tau_{2}$ as summarised in Table 2.

####  3.2.2 Constraints on $\zeta_{1}$ and $\zeta_{2}$ within each regime

Similarly, the two regimes in which the dimensionless SZ parameters
$\zeta_{1}$ and $\zeta_{2}$ are to be bounded can be traced within Figures 6
and 7, respectively. As before, each row of panels fixes the halo
concentration, $c$, and dilution, $d$, to the set of values required in each
of the required regimes. At the desired detection and aperture radii —
$r_{\mathrm{det}}/r_{\mathrm{vir}}=0.5$ and
$R_{\mathrm{ap}}/r_{\mathrm{vir}}=0.5$ when $\Delta=200$ and
$r_{\mathrm{det}}/r_{\mathrm{vir}}=1$ and $R_{\mathrm{ap}}/r_{\mathrm{vir}}=1$
when $\Delta=500$ — the integrated Compton parameter profiles in Figures 6 and
7 are considerably degenerate for the parameters shown across these panels. By
taking the minimum and maximum values of $\zeta_{1}\equiv
Y_{\mathrm{sph}}(<r_{\mathrm{det}})/Y_{\mathrm{vir}}$ and
$Y_{\mathrm{cyl}}(<R_{\mathrm{ap}})/Y_{\mathrm{vir}}$ at each desired
detection and aperture radius, respectively, the bounds summarised in Table 3
are calculated.

##  4 4\. Results

###  4.1 4.1. The $M_{\mathrm{vir}}-T_{\mathrm{X}}$ and
$M_{\mathrm{vir}}-T_{\mathrm{m_{g}}}$ scaling relations

Figure 9: Our predictions for the halo mass - temperature scaling relations,
in terms the emission-weighted temperature, $M_{200}-T_{\mathrm{X}}$ and
$M_{500}-T_{\mathrm{X}}$, in the left panels, and the mean gas mass-weighted
temperature, $M_{200}-T_{\mathrm{m_{g}}}$ and $M_{500}-T_{\mathrm{m_{g}}}$, in
the right panels. The light blue intervals trace the uncertainties in each of
these scaling relations, as quantified by constraints in the dimensionless
parameters, $\tau_{1}$ and $\tau_{2}$, given in Table 2, derived over
specified halo parameters. In the left panels, the additional pink interval
traces the scaling relations formed by excising the cluster’s central core,
given by the constraints in Table 4. For each of these coloured intervals, the
solid dotted lines enclosing each interval correspond to the minimum and
maximum bounds in the scaling relation, with the solid central line tracing
its mid-range value. These predictions are shown in comparison to
observational fits to these scaling relations, from Arnaud2005 (5) (the blue
dotted lines), Vikhlinin2006 (69) (the orange dash-dotted line) and
Vikhlinin2009 (68) (the purple dotted line).

From our analysis, we are now able to present the predicted scaling relations
of the halo mass with its weighted temperatures observables. In particular,
from the minimum and maximum values of $\tau_{1}$ and $\tau_{2}$ detailed in
Table 2, the scaling relations of the halo mass with the weighted
temperatures, $T_{\mathrm{X}}$ and $T_{\mathrm{m_{g}}}$, as devised in Section
2.4 — Equations (49) and (50) — will be constrained. These predictions are
illustrated in Figure 9, with the left panels showing the
$M_{200}-T_{\mathrm{X}}$ and $M_{500}-T_{\mathrm{X}}$ scaling relations, and
the right panels showing the $M_{200}-T_{\mathrm{m_{g}}}$ and
$M_{500}-T_{\mathrm{m_{g}}}$ scaling relations. Each of these scaling
relations and their predicted uncertainties are presented analytically below.

####  4.1.1 The $M_{200}-T_{\mathrm{X}}$ and $M_{500}-T_{\mathrm{X}}$ scaling
relations

In terms of the mid-range value and the associated uncertainty, the
$M_{200}-T_{\mathrm{X}}$ scaling relation is predicted as:

$\displaystyle M_{200}$ $\displaystyle=\left(6.81\pm
3.90\right)\cdot\left[\frac{T_{\mathrm{X}}}{10^{7}\,\mathrm{K}}\right]^{3/2}\cdot
10^{13}\,\mathrm{M}_{\odot},$ (62)

and similarly, the $M_{500}-T_{\mathrm{X}}$ scaling relation, as:

$\displaystyle M_{500}$ $\displaystyle=\left(4.36\pm
1.81\right)\cdot\left[\frac{T_{\mathrm{X}}}{10^{7}\,\mathrm{K}}\right]^{3/2}\cdot
10^{13}\,\mathrm{M}_{\odot},$ (63)

with each of these relations attaining an uncertainty of $57.3\%$ and
$41.6\%$, respectively.

####  4.1.2 The $M_{200}-T_{\mathrm{m_{g}}}$ and $M_{500}-T_{\mathrm{m_{g}}}$
scaling relations

Similarly, the $M_{200}-T_{\mathrm{m_{g}}}$ and $M_{500}-T_{\mathrm{m_{g}}}$
scaling relations are predicted as:

$\displaystyle M_{200}$ $\displaystyle=\left(5.80\pm
1.49\right)\cdot\left[\frac{T_{\mathrm{m_{g}}}}{10^{7}\,\mathrm{K}}\right]^{3/2}\cdot
10^{13}\,\mathrm{M}_{\odot},$ (64)

and, as:

$\displaystyle M_{500}$ $\displaystyle=\left(4.05\pm
0.28\right)\cdot\left[\frac{T_{\mathrm{m_{g}}}}{10^{7}\,\mathrm{K}}\right]^{3/2}\cdot
10^{13}\,\mathrm{M}_{\odot},$ (65)

with each attaining an uncertainty of $25.7\%$ and $7.0\%$, respectively.

####  4.1.3 Comparison of our predictions to observed X-ray relations

Overdensity parameter |  Detection radius |  Excised core |  Concentration |  Dilution | Bounds in $\tau_{1}\equiv T_{\mathrm{X}}[r_{\mathrm{ex}}-r_{\mathrm{det}}]/T_{\mathrm{vir}}$
---|---|---|---|---|---
$\Delta=200$ |  $r_{\mathrm{det}}=0.5r_{200}$ |  $r_{\mathrm{ex}}=0.1r_{200}$ |  $c=5$ |  $d=2$ |  $\tau_{\mathrm{1,min}}=1.146$ |  $\tau_{\mathrm{1,max}}=1.766$
$\Delta=500$ |  $r_{\mathrm{det}}=r_{500}$ |  $r_{\mathrm{ex}}=0.15r_{500}$ |  $c=2.5$ |  $d=1$ |  $\tau_{\mathrm{1,min}}=1.181$ |  $\tau_{\mathrm{1,max}}=1.487$
Table 4: The constraints placed on $\tau_{1}\equiv
T_{\mathrm{X}}[r_{\mathrm{ex}}-r_{\mathrm{det}}]/T_{\mathrm{vir}}$ over the
parameter space of the ideal baryonic cluster halos, each evaluated at two
conventions in the overdensity, $\Delta=200$ and $\Delta=500$, when excising a
central core of halocentric radius $r_{\mathrm{ex}}$. Figure 10: The core-
excised emission-weighted temperature profiles, in scale-free form
$\tau_{1}\equiv
T_{\mathrm{X}}[r_{\mathrm{ex}}-r_{\mathrm{det}}]/T_{\mathrm{vir}}$, for the
ideal baryonic cluster halos, evaluated at fixed detection radii, and traced
over halo inner slopes, $\alpha$. Each panel fixes the core-excised region, in
scale-free form $r_{\mathrm{ex}}/r_{\mathrm{vir}}$, the detection radius, in
scale-free form $r_{\mathrm{det}}/r_{\mathrm{vir}}$, the halo concentration,
$c$, and the dilution, $d$, with these choices corresponding to a particular
choice in the overdensity: the left panel for $\Delta=500$, and the right
panel for $\Delta=200$. Within each box, each colour varies the gas inner
slope, $\varepsilon$, with the solid coloured lines tracing a fraction of
cosmological baryon content of $\eta=0.8$, with the variation between
$\eta=0.6$ and $\eta=1$ so weak that no shaded colour region around these
solid lines is visible.

Where possible, in Figure 9, our predicted scaling relations are compared to
relations devised in the literature of X-ray cluster observations: from
Arnaud2005 (5), Vikhlinin2006 (69) and Vikhlinin2009 (68). Of note, some of
these studies use the cluster’s spectroscopic temperature, $T_{\mathrm{sp}}$,
to fit the scaling relation. This temperature measure is usually assumed to be
comparable to the emission-weighted temperature, $T_{\mathrm{X}}$, hence the
comparison in the left panels of Figure 9; however, $T_{\mathrm{sp}}$ is known
to underestimate $T_{\mathrm{X}}$ (Rasia2005, 61, see, e.g.), and so this
comparison is not one-to-one.

Despite the differences between our model’s predictions and the observational
comparisons in Figure 9, it remains clear that our predictions generally
contain the expected trends, whilst systematically overestimating the
cluster’s halo mass. Only our $M_{500}-T_{\mathrm{m_{g}}}$ prediction, in the
lower right panel, excludes the observed fit from Vikhlinin2006 (69), which
lies $\sim 6\%$ below our lower bound. This small deviation is reasonable, and
likely attributed to systematics involved when comparing the normalisation of
our model and the observational measurement of $r_{500}$ for calculating
$T_{\mathrm{m_{g}}}$.

Moreover, whilst the observed scaling relations are all expected to
underestimate the halo mass, due to the hydrostatic bias, our modelling is
also expected underestimate the halo mass, by the same bias, as in our model
we do not consider non-thermal pressure, which will decrease the temperature
required to attain a given halo mass. As such, the hydrostatic bias itself
cannot explain our systematic overestimates in the scaling relations; instead,
this is attributed to the chosen parameter space and the generous range chosen
in halo structures, driving a larger scatter, which is mitigated in the real
universe as real clusters do not not uniformly sample this parameter space.

####  4.1.4 Excising the central core in the $M_{200}-T_{\mathrm{X}}$ and
$M_{500}-T_{\mathrm{X}}$
scaling relations

In the observational comparisons shown in Figure 9, the cluster region in
which the weighted temperature is measured differs for each study, all
excising some central region, e.g. $0.1r_{500}$, $70$ $\mathrm{kpc}$, or some
other excised halocentric radius, $r_{\mathrm{ex}}$. In our model, excising a
central core has a significant impact on the $T_{\mathrm{X}}$ estimate, and
thus the constraints on $\tau_{1}$, due to its dependence on integrating the
temperature $T^{3/2}$, as in Equation (24), which in the central region, for
non-cool core clusters with high central temperatures, will have a non-
negligible contribution to the value of $T_{\mathrm{X}}$ attained within the
detection radius. For this reason, in Figure 9, we show an additional
prediction for both the $M_{200}-T_{\mathrm{X}}$ and $M_{500}-T_{\mathrm{X}}$
scaling relations, shown in pink, that each excise a central region of
$0.1r_{200}$ and $0.15r_{500}$, respectively. These scaling relations are
constrained by new values of $\tau_{\mathrm{1,min}}$ and
$\tau_{\mathrm{1,max}}$, given in Table 4, as derived in analogue to our
previous analysis, and traced by the scale-free profiles $\tau_{1}\equiv
T_{\mathrm{X}}[r_{\mathrm{ex}}-r_{\mathrm{det}}]/T_{\mathrm{vir}}$ that are
shown in Figure 10. These core-excised scaling relations take the form:

$\displaystyle M_{200}$ $\displaystyle=\left(5.67\pm
1.78\right)\cdot\left[\frac{T_{\mathrm{X}}}{10^{7}\,\mathrm{K}}\right]^{3/2}\cdot
10^{13}\,\mathrm{M}_{\odot},$ (66)

and:

$\displaystyle M_{500}$ $\displaystyle=\left(3.84\pm
0.66\right)\cdot\left[\frac{T_{\mathrm{X}}}{10^{7}\,\mathrm{K}}\right]^{3/2}\cdot
10^{13}\,\mathrm{M}_{\odot},$ (67)

each attaining a significantly lower uncertainty than the previous estimates,
by approximately half, at $31.3\%$ and $17.1\%$, respectively. Thus,
expectedly, by excluding the variance of temperature curves in the central
region, these scaling relations provide a stronger halo mass constraint.
Moreover, these core-excised predictions lower the mid-range value of each
scaling relation, and so become closer to the observational fits, as expected
in this more aligned comparison.

###  4.2 4.2. The $M_{\mathrm{vir}}-Y_{\mathrm{sph}}$ and
$M_{\mathrm{vir}}-Y_{\mathrm{cyl}}$ scaling relations

Figure 11: Our predictions for the halo mass - Sunyaev-Zeldovich scaling
relations, in terms the spherically-integrated Compton parameter,
$M_{200}-Y_{\mathrm{sph}}$ and $M_{500}-Y_{\mathrm{sph}}$, in the left panels,
and the cylindrically-integrated Compton parameter, $M_{200}-Y_{\mathrm{cyl}}$
and $M_{500}-Y_{\mathrm{cyl}}$, in the right panels. The light blue intervals
trace the uncertainties in each of these scaling relations, as quantified by
constraints in the dimensionless parameters, $\zeta_{1}$ and $\zeta_{2}$,
given in Table 3, derived over specified halo parameters. The solid dotted
lines enclosing each interval correspond to the minimum and maximum bounds in
the scaling relation, with the solid central line tracing its mid-range value.
These predictions are shown in comparison to the scaling relations predicted
by Arnaud2010 (6) (the blue dotted lines).

Furthermore, we now present the predicted scaling relations of the halo mass
with the integrated SZ observables of galaxy clusters. In analogue, from the
minimum and maximum values of $\zeta_{1}$ and $\zeta_{2}$ detailed in Table 3,
the scaling relations of the halo mass with the integrated Compton parameters,
$Y_{\mathrm{sph}}$ and $Y_{\mathrm{cyl}}$, as devised in Section 2.4 —
Equations (52) and (53) — will be constrained. In Figure 11, these predictions
are shown, with the scaling relations $M_{200}-Y_{\mathrm{sph}}$ and
$M_{500}-Y_{\mathrm{sph}}$ shown in the left panels, and the scaling relations
$M_{200}-Y_{\mathrm{cyl}}$ and $M_{500}-Y_{\mathrm{cyl}}$ shown in the right
panels. Again, these scaling relations and their predicted uncertainties are
given analytically, as detailed below.

####  4.2.1 The $M_{200}-Y_{\mathrm{sph}}$ and $M_{500}-Y_{\mathrm{sph}}$
scaling relations

As before, taking the mid-range value and the associated uncertainty, the
$M_{200}-Y_{\mathrm{sph}}$ scaling relation is predicted as:

$\displaystyle M_{200}$ $\displaystyle=\left(9.54\pm
2.97\right)\cdot\left[\frac{Y_{\mathrm{sph}}}{10^{-6}\,\mathrm{Mpc^{2}}}\right]^{3/5}\cdot
10^{13}\,\mathrm{M}_{\odot},$ (68)

and similarly, the $M_{500}-Y_{\mathrm{sph}}$, as:

$\displaystyle M_{500}$ $\displaystyle=\left(5.58\pm
0.99\right)\cdot\left[\frac{Y_{\mathrm{sph}}}{10^{-6}\,\mathrm{Mpc^{2}}}\right]^{3/5}\cdot
10^{13}\,\mathrm{M}_{\odot},$ (69)

with each attaining an uncertainty of $31.1\%$ and $17.7\%$, respectively.

####  4.2.2 The $M_{200}-Y_{\mathrm{cyl}}$ and $M_{500}-Y_{\mathrm{cyl}}$
scaling relations

Finally, in the same way, the predicted scaling relations
$M_{200}-Y_{\mathrm{cyl}}$ and $M_{500}-Y_{\mathrm{cyl}}$ are given by:

$\displaystyle M_{200}$ $\displaystyle=\left(7.96\pm
2.01\right)\cdot\left[\frac{Y_{\mathrm{cyl}}}{10^{-6}\,\mathrm{Mpc^{2}}}\right]^{3/5}\cdot
10^{13}\,\mathrm{M}_{\odot},$ (70)

and:

$\displaystyle M_{500}$ $\displaystyle=\left(4.68\pm
1.03\right)\cdot\left[\frac{Y_{\mathrm{cyl}}}{10^{-6}\,\mathrm{Mpc^{2}}}\right]^{3/5}\cdot
10^{13}\,\mathrm{M}_{\odot},$ (71)

with each of these scaling relations attaining an uncertainty of $25.2\%$ and
$22.0\%$, respectively.

####  4.2.3 Comparison of our predictions to observed SZ relations

As shown in Figure 11, our predicted scaling relations are compared to the SZ
scaling relations devised in Arnaud2010 (6), for the scaling relation with the
halo mass $M_{500}$. Unlike in the previous comparison, in this comparison the
observed fits can be compared like-for-like with our predictions, as the
integrated Compton parameters fit to each scaling are identical in definition
to the observables chosen in our model. Here, our predicted intervals
perfectly contain the observed fits, illustrating the strong predictive power
of our simple model, and the self-similar expectation for this scaling
relation.

###  4.3 4.3. Parameter dependence of the scaling relations

For each of these scaling relations, we now comment on their dependence and
sensitivity to the parameters assumed in our model. In particular, we discuss
the sensitivity of these results to the range in values chosen for each of the
continuous parameters detailed in Table 1: the dark matter halo inner slope,
$\alpha$, the fraction of cosmological baryon content, $\eta$, and the
intracluster gas inner slope, $\varepsilon$, as well as the dependence on the
measurement parameters detailed in our analysis: the detection radius,
$r_{\mathrm{det}}$, and the aperture radius, $R_{\mathrm{ap}}$, each as was
fixed for observational comparison. These parameter dependencies are discussed
in each of the headings below, with reference to each of the four predicted
scaling relations.

In terms of the fixed parameters in Table 1: the concentration, $c$, and the
dilution, $d$, we offer a brief discussion here as to their impact on our
results. The dependence of our predictions on these parameters is best
discussed with reference to Figure 2, tracing the scale-free temperature
profiles of our model, with each panel corresponding to the two overdensity
conventions: $\Delta=500$ in the top panels, with $c=2.5$ and $d=1$, and
$\Delta=200$ in the bottom panels, with $c=5$ and $d=2$. In this figure, the
change in profiles between the top panel and the bottom panel is clearly a
translation shift of each temperature profile toward smaller halocentric
radii, and an amplitude increase for all profiles, except those that exhibit a
peak in temperature beyond the meeting point of the profiles, in which case
the amplitude is decreased. This quantifiable change in the temperature
profiles is mathematically driven by the concentration, which is thus
important for the normalisation of each scaling relation with its measurement
parameter, as this choice in $c$ sets the radial translation, and will thus
impact the calibration of $r_{\mathrm{det}}$ and $R_{\mathrm{ap}}$ to the
virial radii $r_{200}$ and $r_{500}$ in the emission observables. This general
effect can be followed through by comparing the top and bottom panels of
Figures 4, 5, 6 and 7, tracing the profiles of each of these observables.

To mitigate this potential for introducing a quantitative measurement offset,
more accurate modelling could calibrate to specific choices in $c$ on a
cluster-by-cluster basis, as informed by the mass-concentration relation
(bullock2001, 10, 22, 39, 38, 14, e.g.); however, in these relations $c$ is
not a strong function of halo mass in the cluster mass regime, and so any
variation in its values is not expected to propagate drastically from our
predictions. This avenue was not pursued in our study, as fixing these
parameters $c$ and $d$ allowed us to explore the predictions of our model when
instead varying the composition and configuration of clusters, whilst
maintaining a self-similar approach.

####  4.3.1 Dependence of the $M_{\mathrm{vir}}-T_{\mathrm{X}}$ scaling
relation

We first comment on the dependence of the $M_{\mathrm{vir}}-T_{\mathrm{X}}$
scaling relation before considering the excision of a central core, and so we
refer to the top row of Figure 8, which illustrates the dependence of the
parameter $\tau_{1}\equiv T_{\mathrm{X}}(<r_{\mathrm{det}})/T_{\mathrm{vir}}$
on the continuous structural parameters from Table 1. In these panels, the
rate of change of $\tau_{1}$ with respect to the halo inner slope $\alpha$
(the gradient of the curves in these panels) is greatest at the largest,
cuspiest value of $\alpha=1.5$ in the permitted range: setting the value of
$\tau_{\mathrm{1,max}}$ in both regimes. This imposes a strong dependence of
the $M_{\mathrm{vir}}-T_{\mathrm{X}}$ scaling relation on the maximum value of
$\alpha$ chosen in the parameter space. Furthermore, the value
$\tau_{\mathrm{1,min}}$ is set by the cuspiest gas inner slope of
$\varepsilon=1$ within $\alpha\simeq 0$ halo cores. As $\alpha<0$ halos are
unphysical, this minimum value of $\tau_{1}$ is most sensitive to the maximum
value of $\varepsilon$ chosen in the parameter space, implicating the
dependence of the $M_{\mathrm{vir}}-T_{\mathrm{X}}$ scaling relation on the
cuspiest permitted gas profile. For each of these $\tau_{1}$ curves in Figure
8, varying the gas content between $\eta=0.6$ and $\eta=1$ has a negligible
impact on the value of $\tau_{1}$, with this change only barely noticeable for
the cuspiest, $\alpha\simeq 1.5$, gas cores, $\varepsilon=0$; thus, there is
essentially no dependence of this scaling relation on the gas content of the
cluster.

In terms of the detection radius, $r_{\mathrm{det}}$, it is clear from the
profiles shown in Figure 4 that there is almost no dependence of the value of
$\tau_{1}$ on this choice for values $r_{\mathrm{det}}\gtrsim
r_{\mathrm{vir}}$, except marginal dependence in the case of $\alpha\simeq 0$
halo cores. This independence of the temperature $T_{\mathrm{X}}$ on the
detection radius at large halocentric radii toward $r_{\mathrm{det}}\simeq
r_{200}$ is mathematically expected for NFW-like profiles (Komatsu&Seljak2001,
34, see, e.g.), which is well-reproduced in the panels of Figure 4.

When considering the $M_{\mathrm{vir}}-T_{\mathrm{X}}$ scaling relation when
$T_{\mathrm{X}}$ is measured with an excised core, these aforementioned
parameter dependencies are much more strongly constrained. As seen in Figure
10, tracing $\tau_{1}\equiv
T_{\mathrm{X}}[r_{\mathrm{ex}}-r_{\mathrm{det}}]/T_{\mathrm{vir}}$ over the
same parameter space, it is clear that $\tau_{1}$ becomes almost insensitive
to the gas inner slope, $\varepsilon$, particularly beyond halo cores
$\alpha\simeq 0$. The dependence on the maximal value of $\alpha$ remains, as
the gradient is still steepest at this boundary; however, this gradient
attains a much lower value than previously, and so this dependence is
significantly reduced. Moreover, the dependence on the gas content, as in the
value of $\eta$, is completely invisible in these curves. This reinforces the
importance of core-excision in calibrating the
$M_{\mathrm{vir}}-T_{\mathrm{X}}$ scaling relation, as sensitivities to both
the dark matter and the gas distribution are very strongly reduced.

####  4.3.2 Dependence of the $M_{\mathrm{vir}}-T_{\mathrm{m_{g}}}$ scaling
relation

In the bottom row of Figure 8, the dependence of the parameter $\tau_{2}\equiv
T_{\mathrm{m_{g}}}(<r_{\mathrm{det}})/T_{\mathrm{vir}}$, and thus the
$M_{\mathrm{vir}}-T_{\mathrm{m_{g}}}$ scaling relation, on the halo’s
structural parameters can be assessed. Unlike for $\tau_{1}$, there is no
consistent trend in which the gradient of these curves is maximised for a
particular halo inner slope $\alpha$, despite its value clearly being
sensitive to this parameter. Moreover, in these panels it is clear that
varying the gas inner slope between $\varepsilon=0$ and $\varepsilon=1$ has
minimal impact on the value of $\tau_{2}$, whilst the effect of varying the
gas content between $\eta=0.6$ and $\eta=1$ is completely invisible. As such,
the $M_{\mathrm{vir}}-T_{\mathrm{m_{g}}}$ scaling relation is almost
completely independent of the gas structure and composition over this
parameter space.

Of most sensitivity to this scaling relation is the choice in measurement
parameter, the detection radius, $r_{\mathrm{det}}$. This is illustrated by
Figure 5, where the $\tau_{2}$ profiles do not flatten off for detection radii
at large halocentric radii, unlike the behaviour of the $\tau_{1}$ profiles of
Figure 4. This imposes a strong sensitivity when calibrating the
$M_{\mathrm{vir}}-T_{\mathrm{m_{g}}}$ scaling relation at a fixed detection
radius (i.e. when $r_{\mathrm{det}}=r_{500}$), as, if observational
measurements are not precisely measuring this detection radius in a given
cluster, the assumed correspondence will be compromised.

####  4.3.3 Dependence of the $M_{\mathrm{vir}}-Y_{\mathrm{sph}}$ scaling
relation

To discuss the $M_{\mathrm{vir}}-Y_{\mathrm{sph}}$ scaling relation, we refer
to Figure 6, tracing the SZ parameter $\zeta_{1}\equiv
Y_{\mathrm{sph}}(<r_{\mathrm{det}})/Y_{\mathrm{vir}}$ over the outlined
parameter space. In this figure, varying the gas inner slope between the
values $\varepsilon=0$, $\varepsilon=0.5$ and $\varepsilon=1$ across each row
entails no clear variation in the profiles traced within each box. In
contrast, varying the gas content between $\eta=0.6$ and $\eta=1$ for each of
the coloured curves, each individually tracing a fixed halo inner slope,
$\alpha$, creates a large interval (shown by the shaded colour regions) that
increases in size toward higher detection radii. These intervals are
significantly overlapping, such that the values predicted by varying each of
$\eta$ and $\alpha$ are strongly degenerate. At detection radii
$r_{\mathrm{det}}\simeq r_{\mathrm{vir}}$, the intervals spanned by varying
$\eta$ dominate the contribution to the bounding region of $\zeta_{1}$. These
trends impose a strong dependence of the $M_{\mathrm{vir}}-Y_{\mathrm{sph}}$
scaling relation on the gas content of clusters, but not on the gas profile
itself. As such, if values of $\eta$ for galaxy clusters could be more
narrowly bounded than $\eta\in[0.6,1]$, this scaling relation would be
significantly more constrained.

In terms of the measurement parameter in this scaling relation,
$r_{\mathrm{det}}$, in each panel of Figure 6 the values of $\zeta_{1}$ are a
strongly increasing function of increasing $r_{\mathrm{det}}$. As such, these
predictions are strongly sensitive to the chosen value of $r_{\mathrm{det}}$,
implying these relations will be compromised outside of this chosen
correspondence.

####  4.3.4 Dependence of the $M_{\mathrm{vir}}-Y_{\mathrm{cyl}}$ scaling
relation

Finally, Figure 7 allows us to discuss the $M_{\mathrm{vir}}-Y_{\mathrm{cyl}}$
scaling relation, by tracing the SZ parameter $\zeta_{2}\equiv
Y_{\mathrm{cyl}}(<R_{\mathrm{ap}})/Y_{\mathrm{vir}}$ over the outlined
parameter space. The same general trends observed for the parameter dependence
of $\zeta_{1}$ are replicated in these panels — as expected, as both of these
observables reflect the same physical dependence of the Compton parameter,
only integrated over different volumes. As such, the
$M_{\mathrm{vir}}-Y_{\mathrm{cyl}}$ scaling relation is strongly dependent on
the gas content of clusters, but not the gas profile itself. The values of
$\zeta_{2}$ in Figure 7, traced over a two-dimensional aperture radius,
$R_{\mathrm{ap}}$, are strongly increasing with increasing $R_{\mathrm{ap}}$;
therefore, there is again a strong coupling of this scaling relation to this
measurement parameter, with these predictions compromised outside the chosen
values of $R_{\mathrm{ap}}$.

###  4.4 4.4. Limitations of the model

The results presented in this section, for the scaling relations of galaxy
clusters with X-ray and SZ observables, are all dependent upon the assumption
of self-similarity. This is built into the derivation of these scaling
relations, and incorporated into each of the profiles modelled for the ideal
baryonic cluster halos, which are all composed in scale-free, and hence self-
similar, form. In the real universe, the scaling of galaxy clusters will
deviate from self-similarity, particularly with halo mass, toward the low
cluster mass regime, and with high redshift.

####  4.4.1 Low cluster mass limitations

When the temperature of the hot ionised gas falls below $\sim
10^{7}\mathrm{K}$, Bremsstrahlung will no longer dominate the cooling
mechanism of galaxy clusters (Sarazin1988, 62, see, e.g.). Below this regime,
typically at cluster masses of less than $M_{200}\sim$ a few $\times
10^{14}\mathrm{M}_{\odot}$, the cooling function in our definition of
$T_{\mathrm{X}}$, from Equation (25), will become an increasingly unreliable
model of the gas emission, breaking the assumed self-similarity in this
observable. As such, at these low cluster masses, our predictions for the
$T_{\mathrm{X}}-M_{\mathrm{vir}}$ scaling relation will become unreliable.
Without any explicit dependence on the cooling mechanism of the gas or its
emission processes, our predictions for the
$T_{\mathrm{m_{g}}}-M_{\mathrm{vir}}$ scaling relation, and both of the SZ
scaling relations, are not expected to strongly deviate from self-similarity
in the low cluster mass regime. This self-similarity is corroborated by
observational constraints, over a wide range of cluster masses, whereby slight
deviations from self-similarity at the low cluster mass regime can be
sufficiently explained in terms of a decrease in the gas fraction, increasing
gas clumpiness, or changes in the slope of the gas’ pressure profile
(Ettori2015, 25, 26, e.g.). These differences all reflect changes in the gas’
composition or increasing non-gravitational feedback processes in the low
cluster mass regime, all of which could be accounted for by adjusting the
value of $\eta$, or introducing non-thermal pressure (in modifying the gas’
pressure slope), whilst maintaining our general analytic approach.

Furthermore, the parameter choices within our model were all strongly informed
by observational fits to the intracluster gas density, which represent only
the X-ray emitting ionised gas component of galaxy clusters, at temperatures
above $\sim$ a few $\times 10^{6}\mathrm{K}$. As such, the intracluster gas
below such temperatures is not well constrained observationally, and could
become significant toward low cluster masses. For example, the gas fraction of
galaxy clusters is observed to fall below $\eta=0.6$, our chosen minimum
value, at cluster masses less than $M_{200}\sim$ a few $\times
10^{13}\mathrm{M}_{\odot}$ (Dev et al. 2024, Submitted). Whether lower
temperature ionised gas, below detection capabilities (sometimes known as the
‘warm’ gas component) compensates for these lower gas fractions in low mass
clusters is not confidently known.

####  4.4.2 High redshift limitations

In our analysis, the form of the cluster’s scale-free profiles were all
devised in terms of a redshift $z=0$ normalisation, scaled by the cluster’s
present-day virial parameters, each defined in terms of the present-day
critical density, $\rho_{\mathrm{crit,0}}$. This analysis would need to be
corrected for application to high redshift; in particular, re-scaling each
profile by redshift-dependent virial parameters, in terms of the critical
density $\rho_{\mathrm{crit}}(z)$ at redshift $z$, which will increase toward
earlier times. This will result in a comparatively higher gas density, and a
subsequently higher gas temperature and pressure; however, the analytic form
of the scale-free profiles will remain unadjusted with this re-scaling.
Instead, when maintaining the halo mass conventions $M_{200}$ and $M_{500}$,
each cluster at a given halo mass will be able to maintain denser, and thus
hotter, gas; this will shift the scaling relations to predict a comparatively
smaller halo mass at a given cluster temperature or SZ signal. Beyond this
linear adjustment, the parameters in our model were all informed by low-
redshift observations, and redshift evolution of some of these parameters — in
particular, the halo concentration bullock2001 (10, 22, 39, 38) — is expected.
As such, a wider parameter space would need to be physically motivated to
encompass high redshift clusters.

More limiting for the application of our model, toward higher redshift, is the
more dynamic and chaotic state of galaxy clusters at earlier times. In
particular, our model predicts the scaling of clusters in virial and
hydrostatic equilibrium; at high redshift, both of these equilibria are
expected to be more commonly violated, as attributed to higher merger rates
(Genel2009, 30, see, e.g.), in the case of virial equilibrium, and higher gas
accretion rates (Pizzardo2023, 54, see, e.g.), in the case of hydrostatic
equilibrium. For these such clusters, our model will not reliably predict
their scaling relations, and thus our approach will not be applicable to a
larger population of galaxy clusters toward earlier times.

##  5 5\. Conclusion

This study has explored the relationship between a galaxy cluster’s dark
matter halo mass, the density profile of its dark matter and intracluster gas
components, and its intracluster gas emission properties. In particular, we
have demonstrated that an analysis of the cluster’s emission profiles, in an
idealised framework — referred to as the ideal baryonic cluster halos — can be
used to place constraints on the cluster’s scaling relations with these
observables. These predictions are thus independent of any numerical models,
simulation calibrations or empirical best-fits.

As a fundamental component of this analysis, we have constructed a competitive
tool-kit for modelling the density profile of galaxy clusters. As entirely
determined by a minimal set of physically-grounded parameters — the dark
matter halo inner slope, $\alpha$, the halo concentration parameter, $c$, the
fraction of cosmological baryon content, $\eta$, the dilution parameter, $d$
and the gas inner slope, $\varepsilon$ — this analytic model captures the
expected structural properties of galaxy clusters, whilst modelling its dark
matter and intracluster gas structural profiles in a decoupled and scale-free
formalism. We hope this tool-kit can be applied to upcoming X-ray and SZ
surveys when modelling the radial profiles of clusters, to better inform these
fits in terms of physically-understood parameters.

As a result of this study, we demonstrated that the halo masses $M_{200}$ and
$M_{500}$ can be predicted with an uncertainty of $57.3\%$ and $41.6\%$,
respectively, from the emission-weighted temperature observable,
$T_{\mathrm{X}}$, and that these uncertainties reduce by a factor of
approximately half, to $31.3\%$ and $17.1\%$, respectively, when a central
core is excised from $T_{\mathrm{X}}$. Similarly, we predicted these halo
masses with a systematically tighter constraint of $25.7\%$ and $7.0\%$,
respectively, when recovered from the mean gas mass-weighted temperature
observable, $T_{\mathrm{m_{g}}}$. Within the parameter space, we demonstrated
that the maximum value permitted for the halo inner slope, taken as
$\alpha=1.5$, as well as the maximum value permitted for the gas inner slope,
taken as $\varepsilon=1$, impose the strongest parameter dependence for the
$M_{200}-T_{\mathrm{X}}$ and $M_{500}-T_{\mathrm{X}}$ scaling relations. These
parameter dependencies are substantially reduced after excising a central core
from the temperature $T_{\mathrm{X}}$, verifying the importance of this
excision in constraining these relations. In contrast, for the
$M_{200}-T_{\mathrm{m_{g}}}$ and $M_{500}-T_{\mathrm{m_{g}}}$ scaling
relations, this sensitivity to the cluster’s underlying structure is minimal,
producing the resulting stronger constraint on the halo mass. However, for
these latter relations, there is a strong dependence on the detection radius,
$r_{\mathrm{det}}$, within which $T_{\mathrm{m_{g}}}$ is measured. As the
observable $T_{\mathrm{X}}$ is essentially independent of $r_{\mathrm{det}}$
when taken at typical values, its scaling relations are correspondingly
independent of this measurement.

Moreover, we predicted the halo masses $M_{200}$ and $M_{500}$ from the
cluster’s SZ observables: recovered with uncertainties of $31.1\%$ and
$17.7\%$, respectively, from the spherically-integrated Compton parameter,
$Y_{\mathrm{sph}}$, and with uncertainties of $25.2\%$ and $22.0\%$,
respectively, from the cylindrically-integrated Compton
parameter,$Y_{\mathrm{cyl}}$, such that both SZ observables produce comparable
constraints on the halo mass. Within the parameter space, we showed that these
scaling relations are insensitive to the form of the gas profile, but instead
are driven by the gas content: captured by the fraction of cosmological baryon
content $\eta$, with these scaling relations strongly dependent on the minimum
$\eta=0.6$ and maximum $\eta=1$ values chosen in our model. Furthermore, we
demonstrated that these scaling relations implicate a strong association with
the measurement parameters, either the spherical detection radius,
$r_{\mathrm{det}}$, or the cylindrical aperture radius, $R_{\mathrm{ap}}$, as
both SZ observables are a strong function of increasing integration volume.

In conclusion, comparing these predictions for the recovery of the halo mass
from a cluster’s emission observables, the mean gas mass-weighted temperature,
$T_{\mathrm{m_{g}}}$, and both integrated Compton parameters,
$Y_{\mathrm{sph}}$ and $Y_{\mathrm{cyl}}$, all provide the strongest
constraints, each recovering the halo mass within $\sim 20-30\%$; comparable
constraints are also attained for the emission-weighted temperature,
$T_{\mathrm{X}}$, only after excision of the central core. Furthermore, if the
detection radius can be accurately measured , $T_{\mathrm{m_{g}}}$ is expected
to permit the strongest constraint on the halo mass, as exemplified by an
uncertainty of merely $7\%$ when this observable is measured within
$r_{\mathrm{det}}=r_{500}$. Whilst entailing a larger uncertainty,
$T_{\mathrm{X}}$ evades a dependence on accurately knowing the detection
radius, which may be preferable observationally when such accuracy is not
available.

Whilst dark matter halos, galaxy clusters and their emission properties remain
the focus of increasingly resolved hydrodynamic simulations, we have shown
that an analytical framework can be utilised to inform and complement
observational constraints on the properties of these complex and dynamic
structures. We hope that our predictions will contribute toward improved halo
mass estimates, exciting further constraints on the Halo Mass Function and in
turn the prospect of untangling the nature of dark matter.

This paper is the second in a series of papers predicting the scaling
relations of dark matter halos. In future, we will endeavour to analytically
investigate the non-thermal pressure composition of galaxy clusters, which was
neglected in this study, to improve this model and contribute toward better
understanding the hydrostatic bias.

AS acknowledges the support of the Australian Government Research Training
Program Fees Offset; the Bruce and Betty Green Postgraduate Research
Scholarship; and The University Club of Western Australia Research Travel
Scholarship. AS and CP acknowledge the support of the ARC Centre of Excellence
for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number
CE170100013.

CB gratefully acknowledges support from the Forrest Research Foundation.

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