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 (e.g. Ogiya & Hahn, Reference Ogiya and Hahn2018). However, this is in tension with observational evidence of dark matter dominated galaxies, such as dwarfs and low-surface brightness galaxies (e.g. Moore, Reference Moore1994; de Blok & McGaugh, Reference de Blok and McGaugh1997; de Blok et al., Reference de Blok, McGaugh, Bosma and Rubin2001; Kuzio de Naray et al., Reference Kuzio de Naray, McGaugh and de Blok2008), 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 (Oh et al., Reference Oh, Brook and Governato2011; Pontzen & Governato, Reference Pontzen and Governato2012; Di Cintio et al., Reference Di Cintio, Brook and Macciò2014b,Reference Di Cintio, Brook and Duttona) 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.

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. Cole & Lacey (Reference Cole and Lacey1996)), 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:

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

such that the mean enclosed mass density is given by the overdensity parameter, $\Delta$ , times the present critical density of the universe, $\rho_\mathrm{crit,0}$ (e.g. White, Reference White2001).

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:

(2) \begin{equation} c \equiv \frac{r_\mathrm{vir}}{r_\mathrm{s}}.\end{equation}

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 (e.g. Bullock et al., Reference Bullock, Kolatt and Sigad2001; Duffy et al., Reference Duffy, Schaye, Kay and Dalla Vecchia2008; Ludlow et al., Reference Ludlow, Navarro and Li2012, Reference Ludlow, Navarro and Angulo2014).

The definition of the concentration parameter, c, depends on the choice in overdensity, $\Delta$ . As $r_{500} \lt r_{200}$ , by definition in Equation (2), this implies an overdensity-dependent concentration, whereby $c_{500} \lt c_{200}$ . 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 $r_{500}/r_{200} \simeq 0.5$ Footnote a, so that $c=2.5$ appropriately describes cluster-scale halos when $\Delta=500$ .

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:

(3) \begin{equation} s \equiv \frac{r}{r_\mathrm{vir}},\end{equation}

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:

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

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:

(5) \begin{equation} u(c, \alpha) \equiv \left[\int _0 ^1 \frac{s^{2-\alpha} \mathrm{d}s}{(1 + cs)^{3-\alpha}}\right]^{-1}.\end{equation}

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 (see, e.g. Frenk et al., Reference Frenk, White and Bode1999).

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:

(7) \begin{equation} d \equiv \frac{r_\mathrm{vir}}{r_\mathrm{dil}}.\end{equation}

From hydrodynamic simulations of $\Lambda \mathrm{CDM}$ cluster halos, this dilution of the gas typically occurs at or near $r_\mathrm{dil} \simeq 0.5 r_\mathrm{200}$ (e.g. Yoshikawa et al., Reference Yoshikawa, Jing and Suto2000). 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:

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

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.

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:

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

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 CMB. We take the value $f_\mathrm{b, cos} = 0.158$ (Planck Collaboration et al., Reference Collaboration, Ade and Aghanim2016a) 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 (see, e.g. Akino et al., Reference Akino, Eckert and Okabe2022). 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:

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

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 are routinely devised in X-ray observations, from fits to the gas density profile, where this fraction is consistently found to be an increasing function of cluster halo mass (e.g. Arnaud et al., Reference Arnaud, Pointecouteau and Pratt2005; Vikhlinin et al., Reference Vikhlinin, Burenin and Ebeling2009). 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 $\eta \simeq 0.67-0.70$ (e.g. Vikhlinin et al., Reference Vikhlinin, Kravtsov and Forman2006; Ettori et al., Reference Ettori, Morandi and Tozzi2009), 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 $\eta \simeq 0.82-0.84$ (e.g. Eckert et al., Reference Eckert, Ettori, Molendi, Vazza and Paltani2013; Morandi et al., Reference Morandi, Sun, Forman and Jones2015; Pratt et al., Reference Pratt, Arnaud, Maughan and Melin2023). 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 $\eta \simeq 1$ (e.g. Morandi et al., Reference Morandi, Sun, Forman and Jones2015; Lyskova et al., Reference Lyskova, Churazov and Khabibullin2023).

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:

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

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

(12) \begin{equation} \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},\end{equation}

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

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 (Pratt et al., Reference Pratt, Arnaud, Maughan and Melin2023) and eROSITA (Lyskova et al., Reference Lyskova, Churazov and Khabibullin2023) have observed weak cusps of $\varepsilon \simeq 0.4-0.7$ in the central, $r\lesssim 0.1r_{500}$ regions of clusters, in contrast to the central core modelled in the isothermal- $\beta$ profile and assumed in some analytic models for the gas density (e.g. Komatsu & Seljak, Reference Komatsu and Seljak2001). More comprehensive studies have observed a large scatter in the behaviour of the gas profile in this central region (see, e.g. Croston et al., Reference Croston, Pratt and Böhringer2008), with no clear trend between this inner structure of the gas and the cluster’s macroscopic properties.

Table 1. Summary of the five parameters in the ideal baryonic cluster halo model: their symbol, definition and physical values.

Numerical studies have suggested that the behaviour of gas profiles in the central region of clusters are dominated by non-gravitational feedback processes (e.g. McDonald et al., Reference McDonald, Allen and Bayliss2017). 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 (see, e.g. Patej & Loeb, Reference Patej and Loeb2015); 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.

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:

(13) \begin{equation} \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}},\end{equation}

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:

(14) \begin{equation} \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}},\end{equation}

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 light blue shaded region of Figure 1, with the overdensity 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 Pratt et al. (Reference Pratt, Arnaud, Maughan and Melin2023) 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 Lyskova et al. (Reference Lyskova, Churazov and Khabibullin2023) 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.

Figure 1. The intracluster gas density profiles, in scale-free form $\rho_\mathrm{gas}/500 \rho_\mathrm{crit,0}$ , traced over the cluster’s scaled halocentric radius, $r/r_\mathrm{500}$ , as 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 intracluster gas density fits: Pratt et al. (Reference Pratt, Arnaud, Maughan and Melin2023) (the blue-dotted line) and Lyskova et al. (Reference Lyskova, Churazov and Khabibullin2023) (the orange dash-dotted line), with the hydrogen gas density in the latter converted to a gas density.

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:

(15) \begin{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}, \end{equation}

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

(16) \begin{equation} 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,\end{equation}

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 – that is, 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 the 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 (see, e.g. Komatsu & Seljak, Reference Komatsu and Seljak2001). However, observational fits for the gas and temperature profiles of clusters have shown that this polytropic relation fails at large radii (e.g. De Grandi & Molendi, Reference De Grandi and Molendi2002), whilst these polytropic models fail to predict the rise in gas temperature with halo radii in the central region as consistently found observationally (e.g. Vikhlinin et al., Reference Vikhlinin, Kravtsov and Forman2006; Sun et al., Reference Sun, Voit and Donahue2009; Lyskova et al., Reference Lyskova, Churazov and Khabibullin2023). 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:

(17) \begin{equation} 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}.\end{equation}

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:

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

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:

(19) \begin{equation} \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},\end{equation}

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:

(20) \begin{equation} \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.\end{equation}

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:

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

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

(22) \begin{equation} \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},\end{equation}

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

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

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, $f_\mathrm{b, cos}$ .

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, $T_\mathrm{X}(\! \lt r_\mathrm{det})$ , 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:

(24) \begin{equation} T_\mathrm{X}(\! \lt 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},\end{equation}

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:

(25) \begin{equation} T_\mathrm{X}(\! \lt 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},\end{equation}

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:

(26) \begin{equation} \frac{T_\mathrm{X}(\! \lt 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},\end{equation}

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

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, $T_\mathrm{m_g}(\! \lt r_\mathrm{det})$ , 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:

(27) \begin{equation} T_\mathrm{m_g} (\! \lt 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}.\end{equation}

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:

(28) \begin{equation} \frac{T_\mathrm{m_g} (\! \lt 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},\end{equation}

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

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 (Sunyaev & Zeldovich, Reference Sunyaev and Zeldovich1970, Reference Sunyaev and Zeldovich1972):

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

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:

(30) \begin{equation} y \equiv \frac{\sigma_\mathrm{T}}{m_\mathrm{e}c_\gamma^2} \int _{\mathcal{V}} p_\mathrm{e} \mathrm{d}\mathcal{V},\end{equation}

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, $c_\gamma$ Footnote b, 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:

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

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

(32) \begin{equation} 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}.\end{equation}

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, $Y_\mathrm{sph} (\! \lt r_\mathrm{det})$ , Compton parameter, measured within the sphere of halocentric radius of $r_\mathrm{det}$ , the associated SZ signal is given by:

(33) \begin{equation} Y_\mathrm{sph} (\! \lt 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.\end{equation}

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

(34) \begin{equation} Y_\mathrm{cyl}(\! \lt 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];\end{equation}

or, equivalently, by subtracting from the total spherically integrated parameter at the halocentric cluster boundary, $r_\mathrm{b}$ , as (Arnaud et al., Reference Arnaud, Pratt and Piffaretti2010):

(35) \begin{equation}\begin{aligned} Y_\mathrm{cyl}(\! \lt R_\mathrm{ap}, r_\mathrm{b}) &= Y_\mathrm{sph}(\! \lt r_\mathrm{b}) \\[5pt] & - \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.\end{aligned}\end{equation}

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 $r_\mathrm{b} \simeq 5 r_{500}$ , as predicted in hydrodynamic simulations (see, e.g. Arnaud et al., Reference Arnaud, Pratt and Piffaretti2010).

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:

(36) \begin{equation} 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},\end{equation}

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:

(37) \begin{equation} \frac{Y_\mathrm{sph}(\! \lt 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.\end{equation}

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:

(38) \begin{equation} S \equiv \frac{R}{r_\mathrm{vir}}.\end{equation}

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

(39) \begin{equation}\begin{aligned} \frac{Y_\mathrm{cyl}(\! \lt S_\mathrm{ap}, s_\mathrm{b})}{Y_\mathrm{vir}} &= \frac{Y_\mathrm{sph}(\! \lt s_\mathrm{b})}{Y_\mathrm{vir}} \\[5pt] & - 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,\end{aligned}\end{equation}

as subtracted from the total, scale-free spherically integrated signal from Equation (37), and where all 3-dimensional, dimensionless halo radii are denoted by a lower-case s, and all 2-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 $r_\mathrm{b} = 5 r_{500}$ when $\Delta=500$ , we will assume this is equivalently approximated to $r_\mathrm{b} = 2.5 r_{200}$ when $\Delta=200$ , approximately consistent with numerical predictions (e.g. Lau et al., Reference Lau, Nagai, Avestruz, Nelson and Vikhlinin2015).

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 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 $\mathrm{Mpc} \, \mathrm{km}^{2}\,\mathrm{s}^{-2} \, \mathrm{M}_\odot ^{-1}$ ), $\rho_\mathrm{crit,0}$ (measured in $\mathrm{M}_\odot \, \mathrm{Mpc}^{-3}$ ), $k_\mathrm{B}$ (measured in $\mathrm{M}_\odot \, \mathrm{km}^{2} \, \mathrm{s}^{-2} \, \mathrm{K}^{-1}$ ), and $m_\mathrm{p}$ (measured in $\mathrm{M}_\odot$ ), such that, by dimensionality, this relation will take the form:

(40) \begin{equation} 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},\end{equation}

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:

(41) \begin{equation} 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},\end{equation}

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:

(42) \begin{equation} 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},\end{equation}

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:

(43) \begin{equation} 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}.\end{equation}

In these scaling relations above, the dimensionless parameters $\tau_1$ and $\tau_2$ are each defined as:

(44) \begin{equation} \tau_1 \equiv \frac{T_\mathrm{X}(\! \lt r_\mathrm{det})}{T_\mathrm{vir}}, \quad \tau_2 \equiv \frac{T_\mathrm{m_g}(\! \lt r_\mathrm{det})}{T_\mathrm{vir}},\end{equation}

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:

(45) \begin{equation} 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},\end{equation}

and for the cylindrically integrated Compton parameter, the form:

(46) \begin{equation} 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}.\end{equation}

In these relations, the dimensionless parameters $\zeta_1$ and $\zeta_2$ are each defined as:

(47) \begin{equation} \zeta_1 \equiv \frac{Y_\mathrm{sph}(\! \lt r_\mathrm{det})}{Y_\mathrm{vir}}, \quad \zeta_2 \equiv \frac{Y_\mathrm{cyl}(\! \lt R_\mathrm{ap})}{Y_\mathrm{vir}},\end{equation}

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

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:

(48) \begin{equation} \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},\end{equation}

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 $h=0.6751$ (Planck Collaboration et al., Reference Collaboration, Ade and Aghanim2016a), 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:

(49) \begin{equation} \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},\end{equation}

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

(50) \begin{equation} \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},\end{equation}

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 $\tau_1$ .

Table 2. The constraints placed on $\tau_1 \equiv T_\mathrm{X}(\! \lt r_\mathrm{det})/T_\mathrm{vir}$ and $\tau_2 \equiv T_\mathrm{m_g}(\! \lt r_\mathrm{det})/T_\mathrm{vir}$ over the parameter space of the ideal baryonic cluster halos, each evaluated at two conventions in the overdensity, $\Delta=200$ and $\Delta=500$ .

Table 3. The constraints placed on $\zeta_1 \equiv Y_\mathrm{sph}(\! \lt r_\mathrm{det})/Y_\mathrm{vir}$ and $\zeta_2 \equiv Y_\mathrm{cyl}(\! \lt R_\mathrm{ap})/Y_\mathrm{vir}$ over the parameter space of the ideal baryonic cluster halos, each evaluated at two conventions in the overdensity, $\Delta=200$ and $\Delta=500$ .

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:

(51) \begin{equation} \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},\end{equation}

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:

(52) \begin{equation} \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},\end{equation}

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

(53) \begin{equation} \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},\end{equation}

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 $\zeta_2$ .

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: $r_\mathrm{det} = 0.5 r_{200}$ , $c=5$ , $d=2$ .

2. 2. The $M_{200} - Y_\mathrm{cyl}$ scaling relation:

   with parameters: $R_\mathrm{ap} = 0.5 r_{200}$ , $r_\mathrm{b} = 2.5 r_{200}$ , $c=5$ , $d=2$ .

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: $R_\mathrm{ap} = r_{500}$ , $r_\mathrm{b} = 5 r_{500}$ , $c=2.5$ , $d=1$ .

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.

The temperature profile of the ideal baryonic cluster halos

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:

(54) \begin{equation} \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),\end{equation}

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

(55) \begin{equation}\begin{aligned} \mathcal{I}(s, c, \alpha, \eta, d, \varepsilon)&\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}} \\[5pt] & \hspace{-15mm} + \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\}.\end{aligned}\end{equation}

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 (e.g. Vikhlinin et al., Reference Vikhlinin, Kravtsov and Forman2006; Sun et al., Reference Sun, Voit and Donahue2009; Ghirardini et al., Reference Ghirardini, Eckert and Ettori2019b; Lyskova et al., Reference Lyskova, Churazov and Khabibullin2023).

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 cluster’s scaled halocentric radius, $r/r_\mathrm{500}$ , as 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 Ghirardini et al. (Reference Ghirardini, Eckert and Ettori2019b), 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 Arnaud et al. (Reference Arnaud, Pratt and Piffaretti2010) (the purple dotted line, in the bottom panel).

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:

(56) \begin{equation} \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).\end{equation}

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 Ghirardini et al. (Reference Ghirardini, Eckert and Ettori2019b):- 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 Arnaud et al. (Reference Arnaud, Pratt and Piffaretti2010), 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 $r\lesssim 0.5 r_{500}$ . Towards 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 towards large halocentric radii is theoretically expected, due to non-thermal pressure contributors, such as turbulence and shocks, increasing towards the outer radii of real clusters, as studied in non-radiative hydrodynamic simulations (e.g. Nelson et al., Reference Nelson, Lau and Nagai2014; Angelinelli et al., Reference Angelinelli, Vazza and Giocoli2020) and inferred observationally from measuring the hydrostatic bias of cluster masses (e.g. Eckert et al., Reference Eckert, Ghirardini and Ettori2019; Sayers et al., Reference Sayers, Sereno and Ettori2021). 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.

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:

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

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.

Figure 4. The emission-weighted temperature profiles for the ideal baryonic cluster halos, in scale-free form $\tau_1 \equiv T_\mathrm{X}(\! \lt r_\mathrm{det})/T_\mathrm{vir}$ , 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 $\eta = 1$ .

When calculating the index $\Gamma$ over the parameter space of ideal baryonic cluster halos, we find that $\Gamma \gt 5/3$ 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 $\Gamma \gt 5/3$ 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, that is, $\varepsilon = 0^+$ , 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 $0.01 r_\mathrm{vir} \leq r \leq r_\mathrm{vir}$ , with all halos converging towards 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 (see, e.g. Ghirardini et al., Reference Ghirardini, Ettori, Eckert and Molendi2019a).

The emission-weighted temperature of the ideal baryonic cluster halos

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:

(58) \begin{equation} \frac{T_\mathrm{X}(\! \lt 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} }}.\end{equation}

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

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

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

(59) \begin{equation} \frac{T_\mathrm{m_g}(\! \lt 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} } }.\end{equation}

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

Figure 5. The mean gas mass-weighted temperature profiles for the ideal baryonic cluster halos, in scale-free form $\tau_2 \equiv T_\mathrm{m_g}(\! \lt r_\mathrm{det})/T_\mathrm{vir}$ , 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 $\eta = 1$ .

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:

(60) \begin{equation}\begin{aligned} \frac{Y_\mathrm{sph}(\! \lt s_\mathrm{det}, c, \alpha, \eta, d, \varepsilon)}{Y_\mathrm{vir}} &= 3 \eta \, \mathcal{U}(c, \alpha, d, \varepsilon) \\[5pt] & \hspace{-10mm} \times \int _0 ^{s_\mathrm{det}} \mathcal{I}(s, c, \alpha, \eta, d, \varepsilon) s^2 \mathrm{d} s.\end{aligned}\end{equation}

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

Figure 6. The spherically integrated Compton parameter for the ideal baryonic cluster halos, in scale-free form $\zeta_1 \equiv Y_\mathrm{sph}(\! \lt r_\mathrm{det})/Y_\mathrm{vir}$ , 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 $\eta = 1$ .

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 by:

(61) \begin{equation}\begin{aligned} \frac{Y_\mathrm{cyl}(\! \lt S_\mathrm{ap}, s_\mathrm{b}, c, \alpha, \eta, d, \varepsilon)}{Y_\mathrm{vir}} &= 3 \eta \, \mathcal{U}(c, \alpha, d, \varepsilon) \,\,\, \times \\[5pt] & \hspace{-37mm} \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].\end{aligned}\end{equation}

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

Figure 7. The cylindrically integrated Compton parameter for the ideal baryonic cluster halos, in scale-free form $\zeta_2 \equiv Y_\mathrm{cyl}(\! \lt R_\mathrm{ap})/Y_\mathrm{vir}$ , 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 $\eta = 1$ .

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.

Figure 8. The emission-weighted and mean gas mass-weighted temperature profiles, in scale-free form $\tau_1 \equiv T_\mathrm{X}(\! \lt r_\mathrm{det})/T_\mathrm{vir}$ and $\tau_2 \equiv T_\mathrm{m_g}(\! \lt r_\mathrm{det})/T_\mathrm{vir}$ , 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 $\eta = 1$ .

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 $\zeta_1 \equiv Y_\mathrm{sph}(\! \lt r_\mathrm{det})/Y_\mathrm{vir}$ and $Y_\mathrm{cyl}(\! \lt R_\mathrm{ap})/Y_\mathrm{vir}$ at each desired detection and aperture radius, respectively, the bounds summarised in Table 3 are calculated.