2.1 Global ICM model

The change in intensity of the cosmic microwave background (CMB) photons as they are scattered by the ICM electrons is given by

(1) \begin{equation}\Delta I_{\nu} \approx i_{0}y\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}\left(x \frac{e^{x}+1}{e^{x}-1}-4\right) \equiv i_{0}yg(x),\end{equation}

where $x = \frac{h \nu}{kT_\mathrm{CMB}}$ , g(x) is the frequency dependence, y is the Compton-y parameter and $i_{0} = \frac{2(k T_\mathrm{CMB})^{3}}{(hc)^{2}}$ is the CMB intensity. We assume the standard non-relativistic approximation to the thermal SZ effect signal for g(x), given by

(2) \begin{equation}g(x) = \frac{x^{4}e^{x}}{(e^{x}-1)^{2}}\left(x \frac{e^{x}+1}{e^{x}-1}-4\right).\end{equation}

The Compton-y parameter is given by

(3) \begin{equation}y(\theta_\mathrm{proj}) = \frac{\sigma_{T}}{m_{e}c^{2}} \int^{\infty}_{-\infty} P_{e}\left(\sqrt{\theta_\mathrm{proj}^{2} + l^{2}}\right) \, \mathrm{d}l,\end{equation}

where $\theta_\mathrm{proj} = \sqrt{{x}^{2} + {y}^{2}}$ is a projected radius with x and y in arcmin representing coordinates on a 2D pixel grid of the sky projection. The parameter $\sigma_{T}$ is the Thomson cross section, $m_{e}$ is the electron mass, and c is the speed of light. The projected radius $\theta_\mathrm{proj}$ describes the radius of each pixel in the grid from the centre of the cluster (defined at (x, y) $= (0,0)$ ). We assume that $P_{e}$ , the thermal pressure in the cluster, is given by the commonly-used Generalised Navarro-Frenk-White (GNFW) pressure profile (Nagai, Kravtsov, & Vikhlinin Reference Nagai, Kravtsov and Vikhlinin2007)

(4) \begin{equation}P_{e}(r) = \frac{P_\mathrm{ei}}{(r/r_{s})^{\gamma}[1+(r/r_{s})^{\alpha}]^{(\beta-\gamma)/ \alpha}},\end{equation}

where $P_\mathrm{ei}$ is the normalisation coefficient, $r_{s}$ is a characteristic radius, and $\alpha$ , $\beta$ , $\gamma$ are shape parameters that describe the slopes of the pressure profiles. We assume the universal shape parameter values given in Arnaud et al. (Reference Arnaud, Pratt, Piffaretti, Böhringer, Croston and Pointecouteau2010). We obtain $P_\mathrm{ei}$ by considering the total integrated Comptonisation parameter, $Y_\mathrm{tot}$ given by integrating

(5) \begin{equation}Y(r) = \frac{\sigma_{T}}{D_{A}^{2}m_{e}c^{2}} \int^{r}_\mathrm{0} P_{e}(r) 4 \pi r^{2} \, \mathrm{d}r,\end{equation}

out to $r=\infty$ . In the GNFW model, $Y_\mathrm{tot}$ has an analytical solution

(6) \begin{equation}Y_\mathrm{tot} = \frac{4 \pi \sigma_{T}}{m_{e}c^{2}} P_\mathrm{ei} D_{A} \theta_{s}^{3} \frac{\Gamma(3-\gamma) \Gamma(\frac{\beta -3}{\alpha})}{\alpha \Gamma(\frac{\beta - \gamma}{\alpha})},\end{equation}

which can be rearranged for $P_\mathrm{ei}$ , given $Y_\mathrm{tot}$ . In this equation, $D_{A}$ is the angular diameter distance to the cluster and $\theta_{s}$ = $r_{s}/D_{A}$ is the angular equivalent of a physical characteristic radius $r_{s} = r_\mathrm{500}/c_\mathrm{500}$ . The radius $r_\mathrm{500}$ surrounds a volume of the cluster which has a mean density 500 times the critical density of the universe at that redshift and is the physical radius of interest. The gas concentration parameter, $c_\mathrm{500}$ , links to a physical description of a cluster with mass (Arnaud et al. Reference Arnaud, Pratt, Piffaretti, Böhringer, Croston and Pointecouteau2010), and $\Gamma$ is the gamma function. Given $Y_\mathrm{500}$ , it is possible to calculate $Y_\mathrm{tot}$ by taking equation (5), integrated out to $\infty$ , and dividing by the same integral, integrated out to $r_{500}$ . This is then multiplied by $Y_\mathrm{500}$ .

A scaling relation is calibrated between $Y_\mathrm{500}$ and $M_\mathrm{500}$ in Planck Collaboration (2014),

(7) \begin{equation}E^{-2/3}(z) \left[\frac{D_{A}^{2}Y_\mathrm{500}}{10^{-4}Mpc^{2}}\right] = 10^{-0.175 \pm 0.011}\left[\frac{M_\mathrm{500}}{6 \times 10^{14}M_\odot}\right]^{1.77 \pm 0.06},\end{equation}

where $M_\mathrm{500}$ is the mass within $r_\mathrm{500}$ and E(z) is the cosmological function used to calculate the Hubble parameter H(z). This relation is used to find $Y_\mathrm{500}$ , given $M_\mathrm{500}$ and z.

We assume the longer baselines of the SKA, insensitive to the large-scale SZ emission, will be used to accurately subtract compact sources so that cluster and cavity detections are not impacted.

2.2 Cavity model

The next step in the model construction is to embed the cavities into the global ICM. In all of the simulated observations in this paper, we consider two cavities. Since we are considering a binary description of cavity gas in terms of (relativistic) thermal or non-thermal distributions, we want to depict how the imprints on the scattered photon spectrum will differ, so that their SZ signals can be distinguished from the global ICM. A general formulation of the SZ effect is described in detail in Enßlin & Kaiser (Reference Enßlin and Kaiser2000) and Colafrancesco, Marchegiani, & Palladino (Reference Colafrancesco, Marchegiani and Palladino2003) which can be adapted to suit the particular scattering electrons’ distribution. A brief description is provided here.

The change in flux density due to the scattering of CMB photons by electrons within the cavities in the optically thin limit is given by,

(8) \begin{equation}\delta i(x) = (j(x)-i(x))\tau_\mathrm{cav},\end{equation}

where i(x) is the shape of the undistorted CMB specific intensity, $i(x)\tau_\mathrm{cav}$ is the flux scattered to other frequencies from x, and $j(x)\tau_\mathrm{cav}$ is the flux scattered from other frequencies to x. Note that this is for single scattering only. The optical depth, $\tau_\mathrm{cav}$ , is given by,

(9) \begin{equation}\tau_\mathrm{cav} = \sigma_{T} \int_\mathrm{cav} n_{e}\, \mathrm{d}l,\end{equation}

where the integration limits cav (notation from Abdulla et al. Reference Abdulla2019) represents the physical boundary that separates the cavity from the global ICM. Equation (8) can also be written as,

(10) \begin{equation}\delta i(x) = \tilde{g}(x)y_\mathrm{cav},\end{equation}

where,

(11) \begin{equation}y_\mathrm{cav} = \frac{\sigma_{T}}{m_{e}c^{2}}\int_\mathrm{cav} P_{e} \, \mathrm{d}l,\end{equation}

(12) \begin{equation}\tilde{g}(x) = (j(x) -i(x)) \frac{m_{e}c^{2}}{\langle k \tilde{T}_{e}\rangle},\end{equation}

(13) \begin{equation}k \tilde{T}_{e} \equiv \frac{P_{e}}{n_{e}},\end{equation}

(14) \begin{equation}\langle k \tilde{T}_{e}\rangle = \frac{\int n_{e} k \tilde{T}_{e}\, \mathrm{d}l}{\int n_{e} \, \mathrm{d}l}.\end{equation}

Equation (10) is analogous to the non-relativistic thermal case of the global ICM in equation (1). Equation (11) defines the Compton-y parameter, where the segment of the LOS that is being integrated over must penetrate a bubble. The distorted SZ spectrum, $\tilde{g}(x)$ , describes the signal received from the bubbles for different frequencies. It is dependent on the type of gas which is doing the scattering. Some examples of $\tilde{g}(x)$ are shown in Fig. 1. The thermodynamic temperature for the case of a thermal electron population, $k \tilde{T}_{e}$ , is given by equation 13). The concept of temperature is more ambiguous in a non-thermal gas, therefore $k \tilde{T}_{e}$ is the pseudo-temperature of the particles. For a population of non-thermal cosmic ray electrons described by a power-law distribution, $P_{e}$ in equation (13) is specific to the cosmic ray electron pressure and is given by

(15) \begin{equation}P_\mathrm{CR_{e}} = \frac{n_\mathrm{CR_{e}} m_{e} c^{2} \left(\alpha -1\right)}{6\left[p^{1-\alpha}\right]_{p_\mathrm{2}}^{p_{1}}} \left[B_\mathrm{\frac{1}{1+p^{2}}} \left(\frac{\alpha -2}{2}, \frac{3- \alpha}{2}\right)\right]^{p_\mathrm{1}}_{p_\mathrm{2}},\end{equation}

(Pfrommer, Enßlin, and Sarazin Reference Pfrommer, Enßlin and Sarazin2005). This is used to estimate the constant value of $k \tilde{T}_{e}$ for use in equation (12). In equation (15), $\alpha$ is the power-law index of the momentum distribution, B is the incomplete Beta function, $n_{CR_{e}}$ is the cosmic ray electron number density, and $p=\beta_{e}\gamma_{e}$ is the normalised electron momentum (Enßlin & Kaiser Reference Enßlin and Kaiser2000). The quantity $\beta_{e}$ is electron velocity relative to the speed of light and $\gamma_{e}$ is the Lorentz factor, therefore p is dimensionless. The parameters $p_\mathrm{1}$ and $p_\mathrm{2}$ are minimum and maximum momenta of the electrons in the distribution. The notation

(16) \begin{equation}[f(p)]_\mathrm{p_\mathrm{2}}^{p_\mathrm{1}} = f(p_\mathrm{1}) - f(p_\mathrm{2}),\end{equation}

is used in equation (15).

In the literature, it is commonly assumed that the bubbles and surrounding ICM are in pressure equilibrium, as simulations have found that after their inflation, the bubbles will expand until they reach pressure equilibrium with their surroundings (e.g. Weinberger et al. Reference Weinberger, Ehlert, Pfrommer, Pakmor and Springel2017). Therefore, we assume that the shape of the pressure profile within the cavities follows the general shape of the global ICM. This means that the Compton-y parameter within the bubbles, described by equation (11), will be an integral over the GNFW pressure (equation 4).

Figure 1. Dashed lines depict the distorted spectrum of SZ signal from a non-thermal distribution of cosmic ray electrons with minimum momenta $p_\mathrm{1} = 1$ , 10, or 100, maximum momentum $p_\mathrm{2}=10^{5}$ , and $\alpha=6$ . The solid lines depict distortions from thermal distributions of ICM entrained free electrons with temperature $k T_{e} = 10$ , 100 or $1\,000$ keV. The solid black curve is g(x) of the global ICM thermal SZ signal. The different distortions depicted in the curves with different values of $kT_{e}$ and $p_\mathrm{1}$ mean it is possible to distinguish between a thermal and non-thermal signal at different frequencies.

The scattered spectrum j(x) in equations (8) and (12) is given by,

(17) \begin{equation}j(x) = \int_\mathrm{0}^{\infty} K(t)i(x/t) \, \mathrm{d}t,\end{equation}

where K(t) is the photon redistribution function, giving the probability that a photon is scattered to a frequency t times its original frequency. The photon redistribution function is dependent on the type of gas that the photons are scattered off and is given by,

(18) \begin{equation}K(t) = \int^{\infty}_\mathrm{0} f_{e}(p)\mathcal{P}(t;\,p) \, \mathrm{d}p,\end{equation}

where $f_{e}(p)$ is the electron momentum distribution. This is where the differentiation between a thermal and non-thermal distribution comes into the formulation. The photon redistribution function for a mono-energetic electron distribution, $\mathcal{P}(t;\,p)$ , has a compact analytical solution derived in Enßlin & Kaiser (Reference Enßlin and Kaiser2000).

If the characterisation of gas within the bubbles is thermal via ICM entrainment, then $f_{e}(p)$ is given by a relativistic form of the Maxwell-Boltzmann distribution

(19) \begin{equation}f_{e, \mathrm{th}}(p) = \frac{\beta_\mathrm{th}}{K_\mathrm{2}(\beta_\mathrm{th})}p^{2}\exp\!(-\beta_\mathrm{th}\sqrt{1+p^{2}}),\end{equation}

where $K_\mathrm{2}$ is the modified Bessel function and $\beta_\mathrm{th} = m_{e}c^{2}/kT_{e}$ . Note that while massive clusters have global ICM temperatures of $\sim$ 10 keV, and therefore, our global ICM model should technically also follow this form with a mass-dependent temperature, the spectral changes at the frequencies we are considering for the SKA (14.11–37.5 GHz) are very small for these lower temperatures compared to the more significant spectral changes for the $\sim$ 100 keV temperatures that are investigated in the later sections of this paper. This can be seen in Fig. 1, where the blue curve for $\tilde{g}(x)$ at $kT_{e}$ = 10 keV follows the black curve for g(x) very closely at frequencies below $\sim$ 100 GHz, compared to the orange curve for $kT_{e}$ which is more distinct.

A non-thermal population of cosmic ray electrons inhabiting the bubbles will be described by a single power-law electron momentum distribution, given by,

(20) \begin{equation}f_{e, \mathrm{non-th}}(p;\, \alpha, p_\mathrm{1}, p_\mathrm{2}) = \frac{(\alpha -1)p^{- \alpha}}{p_\mathrm{1}^{1-\alpha}-p_\mathrm{2}^{1-\alpha}};\, p_\mathrm{1}\unicode{x003C} p\unicode{x003C} p_\mathrm{2},\end{equation}

where $\alpha$ is the power-law index. The parameter $p_\mathrm{1}$ is considered a free variable, as the amount of scattering will be primarily determined by its value (Colafrancesco et al. Reference Colafrancesco, Marchegiani and Palladino2003). The effect of changing $p_\mathrm{2}$ is small, so its value is kept fixed at $p_\mathrm{2}=10^{5}$ (Colafrancesco et al. Reference Colafrancesco, Marchegiani and Palladino2003).

Given the assumption of consistent pressure profiles inside and outside cavities, the relative change in flux density from the whole cluster can be redefined as the standard thermal SZ contribution from the cluster with a spherical cavity removed, plus any contribution from the general particle population inside the cavity. This is given by,

(21) \begin{equation}\delta i(x) = [y_\mathrm{cl} - y_\mathrm{cav}]g(x) + y_\mathrm{cav}\tilde{g}(x),\end{equation}

where the Comptonisation from the GNFW model with LOS through the entire cluster is $y_\mathrm{cl}$ . The integration limits of $y_\mathrm{cav}$ (equation 11) are defined by the intercepts between the LOS and the cavity boundaries, given by

(22) \begin{equation}z = \pm\sqrt{r_{b}^{2} - ({x}-{x}_{b})^{2} - ({y}-{y}_{b})^{2}} + z_{b},\end{equation}

where $r_{b}$ is the radius of the bubble, x and y are the coordinates on the pixel grid, ${x}_{b}$ and ${y}_{b}$ are the coordinates of the bubble centre on the pixel grid, and $z_{b}$ is the amount that the bubble is shifted in or out of the plane, along the LOS.

Equation (21) can be simplified to:

(23) \begin{equation}\delta i(x) = (y_\mathrm{cl} - fy_\mathrm{cav})g(x),\end{equation}

where

(24) \begin{equation}f \equiv 1 - \frac{\tilde{g}(x)}{g(x)},\end{equation}

is defined by Abdulla et al. (Reference Abdulla2019) as the cavity suppression factor. The function f describes a scaled version of the thermal non-relativistic SZ effect in cavity regions of the ICM, due to the different scattering caused by either relativistic thermal or non-thermal electrons within them. Since f depends on $\tilde{g}(x)$ (equation 12), then it will differ for a thermal or non-thermal cavity gas. The top and bottom images in Fig. 2 depict the suppression factor of a non-thermal and thermal gas, respectively, at different frequencies.

Figure 2. Top figure: The suppression factor (equation 24) of a non-thermal distribution of electrons in the cluster cavities of MS0735 ( $\alpha = 6$ and $p_\mathrm{2} = 10^{5}$ ). Bottom figure: The suppression factor of a thermal distribution of electrons with temperature $kT_{e}$ in keV. The frequencies are those of some SZ instruments considered in this paper: SKA at $\sim$ 15 GHz, CARMA at 30 GHZ, and GBT MUSTANG-2 at 90 GHz. The suppression factor of a non-thermal gas can be negative, meaning an increased SZ signal, compared to a thermal gas which must be $\geq$ 0. At the frequencies shown here, $f\unicode{x003E}1$ is not possible.

The coordinates z in equation (22) are used as the bounds of the integral (3) when $(({x}-{x}_{b})^{2} + ({y}-{y}_{b})^{2}) \unicode{x003C} r_{b}^{2}$ , and the integral is multiplied by f (equation 24) to obtain the effective y-map of the cluster and cavities,

(25) \begin{equation}\begin{split}y = \frac{\sigma_{T}}{m_{e}c^{2}} \int^{\infty}_{-\infty} P_{e}(\sqrt{\theta_\mathrm{proj}^{2} + l^{2}}) \, \mathrm{d}l - \\ f\left(\frac{\sigma_{T}}{m_{e}c^{2}} \int^{+z}_{-z} P_{e}(\sqrt{\theta_\mathrm{proj}^{2} + l^{2}})\, \mathrm{d}l\right).\end{split}\end{equation}

Depending on whether a cluster is being modelled with thermal or non-thermal gas within the bubbles, equations (19) or (20) are used respectively, to calculate f. The y-map is then converted to a signal map in MJy/sr by multiplying equation (25) by the non-relativistic thermal spectral distortion g(x), and $i_\mathrm{0} = \frac{2(k T_\mathrm{CMB})^{3}}{(hc)^{2}}$ . This can subsequently be converted to a map in Jy/pix for use in interferometric simulations.

2.3 Interferometric simulations

To generate simulations of SKA observations, Profile Software (Grainge et al. Reference Grainge, Jones, Pooley, Saunders, Edge, Grainger and Kneissl2002) is used. This software directly mimics the observation process of a real interferometer. The model image of the sky in Jy/pix that is simulated via the process in Sections 2.1 and 2.2 is read from a FITS (Flexible Image Transport System) file. Information including the pixel size in degrees, the cluster location in right ascension (RA) and declination, and the pixels corresponding to that location, is stored in the FITS headers before the file is used as input in Profile. This image represents the sky brightness distribution received by the interferometer. The sky brightness distribution is then modified by the primary beam. The primary beam applied to the simulations was calculated based on an edge-tapered Gaussian aperture illumination function (Grainge, priv. comm.) which resulted in a primary beam with a Gaussian central lobe with full-width at half-maximum (FWHM) given in Table 2. When analyzing the simulations, for simplicity a Gaussian model with the given FWHM was assumed. No calibration errors are assumed in the simulations in this work. The Fourier transform of this map is taken, then sampled at the points in the uv-plane corresponding to the expected layout of the SKA antennas to simulate the visibilities. Since the SKA will consist of two arrays (SKA-Mid and SKA-Low), we specify that the simulations presented in this paper will assume the SKA-Mid (referred to as SKA for the remainder of this paper) layout. We will also involve the MeerKAT antennas in Section 5. See Table 1 for the technical details of these instruments and Table 2 for the proposed frequency bands that are used in this work along side the FWHM of the assumed primary beam. While the SKA bands are not ideal for SZ observations, the SZ effect of galaxy clusters does become visible in the higher end of band 5 which is the frequency range that the simulations in this paper will focus on.

Table 1. Technical information of the SKA-Mid and MeerKAT arrays from Braun et al. (Reference Braun, Bonaldi, Bourke, Keane and Wagg2019) and SKAO (2022).

Table 2. Proposed frequency bands of the SKA alongside the primary beam FWHM assumed in the simulated observations in this work. Band 5+ and 6 would come with a possible expansion of SKA1 (the first phase of the SKA project) (Conway et al. Reference Conway, Beswick, Bourke, Coriat, Ferrari, Jimenez-Serra, Muller and Sargent2020).

Gaussian noise is added to the visibilities at a level determined using the SKAO (SKA Observatory) sensitivity calculator (https://www.skao.int/en/ska-sensitivity-calculators). The noise level used for 14.11 GHz is $0.0061$ Jy (for a single baseline). After the simulation has run and noise has been added, the output is a FITS file of sampled visibilities. A clean algorithm is run to deconvolve the estimate of the observed sky brightness distribution from the dirty image using CASA (Common Astronomy Software Applications). Throughout this work, natural weighting is used to image the interferometric data, giving constant weights to all visibilities since the noise is added at a consistent level to all visibilities.

2.4 Statistical analysis

To determine how well the suppression factor is detected in the clusters involved in this investigation, McAdam software (Feroz et al. Reference Feroz, Hobson and Bridges2009) is used to perform Bayesian inference for parameter estimation and model comparison using nested sampling via MultiNest (Feroz, Hobson, and Bridges Reference Feroz, Hobson and Bridges2009; Feroz & Hobson Reference Feroz and Hobson2008; Feroz et al. Reference Feroz, Hobson, Cameron and Pettitt2019). A single suppression factor across both cavities is assumed. Because our aim is to investigate the SKA’s ability to observe cluster cavities, setting prior distributions that are not informative is important to determine how well the true parameter values can be recovered. This is particularly important to determine if the SKA will be able to measure distinguishable f values for thermal and non-thermal gas. Uniform priors are mostly used, with a wide range from the minimum and maximum values allowed by the models (see Fig. 2). Table 4 lists the priors used.

Table 3. The scale used to quantify a good detection via the Bayesian evidence Z (from Kass & Raftery Reference Kass and Raftery1995), when performing model comparison.

Table 4. True values of cluster MS0735 parameters used in the model of the SZ signal map. The cavity positions ( ${x_{b}}$ and ${y_{b}}$ ) are given as offsets from the cluster center at (0,0). Also listed are the priors used in the Bayesian analysis. Cavity parameters are derived from Chandra X-ray observations (Vantyghem et al. Reference Vantyghem, McNamara, Russell, Main, Nulsen, Wise, Hoekstra and Gitti2014). Subscripts 1 and 2 refer to the northern and southern bubble, respectively and $r_{b}$ is the bubble radius.

To quantify the strength of the detection of the bubbles the Bayesian evidence is used. The Jeffrey’s scale (Jeffreys Reference Jeffreys1961) is often used in Astrophysics to quantify detection via the evidence, and it is also laid out in Kass & Raftery (Reference Kass and Raftery1995). The scale ranks values of the Bayes factor, which is the ratio of the evidences of two models $Z_\mathrm{1}/Z_\mathrm{2}$ , where a higher value represents a better fit of model 1 to the data, over model 2. The McAdam software outputs $\ln\!(Z)$ . Therefore, the scale in Kass & Raftery (Reference Kass and Raftery1995) converted to $\ln\!(Z_\mathrm{1}/Z_\mathrm{2})$ is given in Table 3. In this work, the model associated with $Z_\mathrm{1}$ is a model of a cluster with bubbles, and a cluster with no bubbles is associated with $Z_\mathrm{2}$ .

The final stage of the analysis is to determine how well an observation constrains the suppression factor within the cavities. To determine if the posteriors are a genuine representation of the actual constraints on the parameters that one can infer from the data, given our model, we employ a test taken from Harrison et al. (Reference Harrison, Sutton, Carvalho and Hobson2015). This test involves obtaining the total probability mass contained in the highest probability density (HPD) region of the posteriors. Given a data point x sampled from a distribution f(x), the total probability mass, $\zeta(x)$ , in the HPD is given by,

(26) \begin{equation}\zeta(x) = \int_{f(u)\geq f(x)}f(u) \, \mathrm{d^{n}}u,\end{equation}

where the region of integration is defined by the constraint $f(u)\geq f(x)$ , which is the HPD. The HPD has the property that any data point sampled outside of it will always have a lower probability than the samples within. If a multidimensional variable $\mathbf{x}$ is considered, the mapping to the probability mass $\zeta (\mathbf{x})$ is given by $\mathbb{R}^{n} \rightarrow [0,1]$ , and is obtained by calculating $\zeta$ contained within the HPD and having $\mathbf{x}$ as its boundary.

The null hypothesis is that the data $\mathbf{x}$ are sampled from the distribution $f(\mathbf{x})$ . Therefore, the proposed validation procedure is then to make use of the fact that under the null hypothesis, the probability mass $\zeta$ is uniformly distributed in the range [0,1]. A proof of this can be found in Harrison et al. (Reference Harrison, Sutton, Carvalho and Hobson2015).

This is utilised in this analysis by testing if the true parameters $\theta$ of a simulated cluster can be sampled accurately from the predicted posteriors. The expression to compute the probability mass becomes,

(27) \begin{equation}\zeta (\theta) = \int_{p(u|\hat{\theta}) \geq p(\theta|\hat{\theta})} p(u|\hat{\theta})\, \mathrm{d^{n}}u,\end{equation}

(Legin et al. Reference Legin, Hezaveh, Perreault-Levasseur and Wandelt2023) for sample u, where $\hat{\theta}$ is the observation and $p(\theta|\hat{\theta})$ is the posterior. Then for multiple realisations of a simulated SKA observation that has been analysed via MultiNest, the values of $\zeta(\theta)$ from each realisation should be uniformly distributed if the posteriors are a truthful representation of the actual constraints on the parameters from the data. The empirical cumulative distribution function (ECDF) for $\zeta(\theta)$ is then expected to be a linear plot, which we refer to as the posterior validation curve. The linear posterior validation curve implies that the true parameter value should be within the 99% confidence interval of 99% of the posterior realisations, within the 68% confidence interval of 68% of the posteriors, etc. If the curve falls below this expected line, then the errors of the posteriors of the different realisations have been underestimated, and the posteriors are not a good representation of the measurements of the data. If the errors have been overestimated, the curve will fall above the expected line, indicating that the parameter constraints fall closer to the true value than expected if the posterior was being dominated by the likelihood.

Throughout the analysis in this work, either 20 or 50 realisations of each simulated SKA observation are performed. For the first model we tested (Section 3), we inspected the posterior validation curves and shapes of the individual posteriors for the 50 realisations each of a range of different observing times to determine the required observation length for an accurate and informative constraint on the suppression factor. We found that an average (over the 50 realisations) log-evidence difference $\ln\!(Z_1/Z_2) \gtrsim 10$ and average suppression factor posterior width $\sigma \lesssim 0.3$ for a given observing time satisfied these requirements. These metrics were therefore used to define the required observation lengths for subsequent models, with the exception of Section 3.3 where $\sigma \lesssim 0.2$ was used instead given the lower suppression factor in the model.

3.1 Modelling MS0735 with CARMA constraints

Figure 3. CLEANed images of simulated SKA observations of the model cluster with non-thermal cavities, where $f_{14}=0.989$ and $p_\mathrm{1}=100$ (constrained by CARMA), representing an estimate of the sky brightness distribution seen by the SKA at $14.11$ GHz. Top panel (left to right): HA = 0.5 days=1, HA = 0.5 days = 2, HA = 1 days = 1. Centre panel (left to right): HA = 1 days = 2, HA = 2 days = 1, HA = 1 days = 3. Bottom: HA = 2 days = 2. These images were CLEANed with CASA, using a Gaussian taper of $3\,000\lambda$ to down-weight the longest baselines and $uvrange\unicode{x003C}10\,000 \lambda$ to reduce contributions from very small amplitude signal. A circular mask is also applied around the centre of the image with a radius of 30 pix to isolate the cluster and the bubbles for the clean algorithm. The red contours represent the true bubble positions and radius (as an offset).

Galaxy cluster MS 0735.6+ 7421 (MS0735), located at $z=0.216$ and with mass $M_\mathrm{500}=8 \times 10^{14}\,\mathrm{ M}_{\odot}$ (Gitti et al. Reference Gitti, McNamara, Nulsen and Wise2007), is chosen as the first candidate for simulating X-ray cavity observations with the SKA. This cluster hosts the largest and most energetic X-ray cavities currently known, making it an excellent observation target. The redshift of MS0735 corresponds to a scale of 3.503 kpc/arcsec, and we observe at $14.11$ GHz.

Previous observations of the SZ signal from cluster MS0735 by CARMA and MUSTANG-2 have been used to constrain some of the cavity parameters for the model image of the sky. Their measured suppression factors are used to determine the corresponding temperature $kT_{e}$ or minimum momentum $p_\mathrm{1}$ within the cavities used in our models. We begin with the CARMA constraints. Adopting the notation $f_{\nu}$ where $\nu$ is the observation frequency, CARMA measured $f_{30}=0.98$ (assuming the cavity minor axis in the elliptical model from Abdulla et al. (Reference Abdulla2019) falls along the LOS) which corresponds to $p_\mathrm{1} \approx 100$ in the case of a non-thermal gas, and $kT_{e} \approx 10\,000$ keV for a thermal gas. These produce $f_{14} = 0.977$ and $f_{14} = 0.989$ for thermal and non-thermal gas, respectively. Because these suppression factors are so similar, the analysis going forward is for non-thermal bubbles with suppression factor $f_{14}=0.989$ . Later in this section we discuss the SKA’s potential to distinguish between a thermal and non-thermal bubble scenario. The other parameters that are used in the model image of cluster MS0735 are listed in Table 4. Alongside these are the prior types and values that are used in the Bayesian analysis. Note that non-physical values are included in the prior range for the suppression factor because the true suppression factor is very close to 1 ( $f_{14} = 0.989$ ), therefore it is very likely that the measured value would fall in the non-physical prior range due to noise from the observation.

Fig. 3 shows the CLEANed images of 7 different observation lengths from Profile simulations (one of the 50 simulations that were produced). The observation times tested are 1, 2, 4, 6, and 8 h over either 1, 2 or 3 days, defined by the hour angle (HA) and number of observing days. Note that an hour angle of 1 h, for example, will mean that the telescope will begin observing one hour before the source crosses the meridian of the telescope and will end one hour after it has crossed this point, giving a total of 2 h observing time in one day. To make sure that the bubbles are extracted from the noise, a Gaussian taper weighting (uv taper parameter in CASA) is applied to the uv data with a width of $3\,000\lambda$ . This is used to reduce the weighting of the longest baselines so that the random fluctuations due to noise where the signal amplitude is very small are not as prominent. A circular mask is also applied around the centre of the image with a radius of 30 pix to enclose the cluster and the bubbles. The uvrange is set to $\unicode{x003C}$ 10 000 $\lambda$ because the bubbles of cluster MS0735 are relatively large and the redshift is relatively low, the signal on the longest baseline lengths will be very faint. Fig. 4 shows the uv-coverage of the SKA from these simulations.

Figure 4. The uv coverage in units of $\lambda$ (with a cut-off of 10 000 $\lambda$ ) of a simulated 8 h SKA observation.

The SZ signal from the cluster itself is mostly detected with each observation time, and the SZ contrast from the non-thermal bubbles becomes more clear the longer the SKA observes for. The CLEANed images show a first approximation via the image sky brightness that the SKA will be able to detect the MS0735 bubbles, but statistical analysis on the uv data will indicate how accurate the suppression factor constraint is. Therefore, model comparison and parameter estimation are performed directly in uv space. The data are prepared for Bayesian inference by binning the output visibilities from the simulation (with a uv cut-off of $10\,000 \lambda$ ), with grid cells of size $130 \lambda$ for $14.11$ GHz observations. Bin sizes are determined based on the aperture illumination function (Fourier transform of the primary beam) size, following Marshall (Reference Marshall2003).

The posterior validation curves resulting from the 50 suppression factor posteriors of each observation time are shown in Fig. 5. To have a realistic chance of observing the MS0735 cavities with the SKA in the future, an observation time of no longer than 6 or 8 h should be able to constrain the suppression factor, as the demand for observing time is expected to be high. The posterior validation curve is a good fit to the null hypothesis curve/expected ECDF (black line) for both the 6 and 8 h observations which implies that any deviations from the true value are consistent with the estimated error. Fig. 6 depicts a histogram of the estimated suppression factors for the 50 realisations of the 8 h observation, alongside the true suppression factor value. This indicates that the estimates are scattered fairly randomly around the input. The top left image of Fig. 5 suggests that an observation time of 1 h is overestimating the error, therefore 1 h can be ruled out as a viable observation time. The posterior validation curves of both 2 h observations (top centre and right images in Fig. 5) are a better fit to the null hypothesis than the 1 h observation and therefore deviations from the true value are more consistent with the error, and the fit is even comparable to the 6 and 8 h observations. Therefore, a 2 h observation would likely produce accurate measurements. Note that while the HA = 0.5, days = 2 case seems to be a better fit to the posterior validation curve than some of the longer observation times, we assume that this is likely due to a random fluctuation in the simulated noise. The posterior validation curve for the 4 h observation over 2 days (left and centre images in the centre panel of Fig. 5) is a good fit to the null hypothesis. It is possible that the error is being underestimated with a 4 h observation over 1 day (centre image in centre panel of Fig. 5), as the curve falls below the null hypothesis for low values of $\zeta$ . However, $\sim$ 68% of the posteriors contain the true value of the suppression factor within their 68% confidence interval, indicating that the true value is being detected as expected for an accurate measurement.

Figure 5. Suppression factor posterior validation curves of the non-thermal MS0735 cavities ( $f_{14}=0.989$ ) produced from 50 simulations of each observation time. The error is overestimated for the 1 h observation. The 2 h curves fit well to the null hypothesis, therefore the error of the 50 posteriors is a good representation of the true constraint. Although the curves of the 4 h observations are a worse fit to the null hypothesis, $\sim$ 68% of the 50 posteriors contain the true value of f in their 68% confidence interval, which is expected from a good constraint. The curves for a 6 and 8 h observation fit well, and the suppression factor has been well constrained.

The estimated suppression factors for 50 realisations of simulated 8-h observations of the model MS0735 non-thermal cavities show no bias when compared to the input suppression factor.

As described in Section 2.4 the results from these observations are used to quantify a good constraint on the suppression factor. Fig. 7 depicts the average error $\sigma$ of the 50 suppression factor posteriors for each observation time, alongside the average values of $\ln (Z_{1}/Z_{2})$ . Since one hour is ruled out as a viable observation time, this graph shows that $\ln\!(Z_{1}/Z_{2}) \approx 10$ is needed to represent a good suppression factor detection. The error that corresponds to this value is $ \sigma \approx 0.3$ . The following results in this paper will use these thresholds to describe a good detection and an informative constraint on the suppression factor.

Next, we discuss the possibility of observing the suppression factor if it is thermal, and whether it could be distinguished from non-thermal suppression. Because of the similarity between the non-thermal and thermal suppression factors at $14.11$ GHz ( $f_{14} = 0.989$ and $f_{14} = 0.977$ , respectively), given the temperature and momentum values constrained by CARMA, it would be reasonable to assume that the above results also apply to the thermal scenario. However, Abdulla et al. (Reference Abdulla2019) conclude that their suppression factor could define a thermal plasma of temperature ‘several hundred to thousands of keV’. Therefore, although their measured suppression factor corresponds to $\sim$ 10 000 keV, there could be a lower limit of $\sim$ 1 000 keV. It is possible that the temperature constrained by the suppression factor model is not exact, as there are many different features of the cluster that could shift the observed suppression away from the true suppression (which depends only on the distribution of electrons). For example, the ellipticity of the bubbles, the possibility of a shock affecting the SZ signal, mixtures of thermal and non-thermal electrons, and any differences in the pressure profile inside the bubble could all affect the observed f if they are not accounted for. It is then possible that $kT_{e}\approx1\,000$ keV and $p_{1}=100$ would give the same CARMA observed $f_{30}$ (even if they correspond to different true f in the thermal and non-thermal models), if there are additional influences that are not accounted for.

3.2 Testing whether thermal and non-thermal SZ suppression can be distinguished

Because of the above explanation, we next investigate whether a thermal and non-thermal suppression could be distinguished by the SKA if the CARMA constraint $kT_{e}=1\,000$ keV is true, rather than $10\,000$ keV which gives an f almost identical to the non-thermal case at $14.11$ GHz. If this is true, it is possible the plasma type could be discovered in future observations.

The suppression factor for $kT_{e}=1\,000$ keV at $14.11$ GHz is $f_{14}=0.796$ . A y-map of cluster MS0735 is modelled using this value. 50 Profile simulations for each of the 7 observation times are made, to mimic the SKA observations of the SZ signal. Fig. 8 shows the average value of $\ln\!(Z_{1}/Z_{2})$ and the average error, $\sigma$ , of the 50 posteriors. The posterior validation curves for each observation time are not shown, as they are visually very similar to the non-thermal case, and the important features are quantified in Fig. 8.

Figure 6. The estimated suppression factors for the 50 realisations of simulated 8 h observations of the model MS0735 non-thermal cavities. This shows that the estimates are scattered randomly around the input value (the true suppression factor, represented by the yellow dashed line).

Figure 7. The average error, $\sigma$ , and $\ln\!(Z_{1}/Z_{2})$ of the 50 simulated SKA observations of non-thermal MS0735 cavities with $f_{14}=0.989$ , for each tested observation time. The average $\ln\!(Z_{1}/Z_{2})$ increases with an increasing observation time, showing that the cavities become better detected. According to the scale in Table 3, there is very strong detection for each observation time, except for HA = 0.5, days = 1, where $\ln\!(Z_{1}/Z_{2})\unicode{x003C}5$ . The error clearly decreases with an increasing observation time, showing the constraint on f becomes much more informative. Note that while $\ln\!(Z_{1}/Z_{2})\unicode{x003E}5$ represents very strong detection of the cavities, it does not directly correspond to an accurate constraint on the suppression factor which instead requires $\ln\!(Z_{1}/Z_{2})\unicode{x003E}10$ .

For both the 6 and 8 h observations, the average $\ln\!(Z_{1}/Z_{2})\unicode{x003E}10$ and $\sigma\unicode{x003C}0.3$ , which is the threshold for a good detection. We conclude from these results that a 6 and 8 h observation time is sufficient to accurately constrain the suppression factor in the bubbles of galaxy cluster MS0735, in the thermal (lower limit of CARMA constrained $kT_{e}$ ) case. Since the 1 h observation time for the non-thermal case is not sufficient, the same is assumed for the thermal bubbles which have a lower suppression factor, and are therefore harder to detect. Note that although a smaller suppression factor means there is more signal coming from the bubbles, the SZ contrast between bubbles and global ICM is smaller, which is what makes the detection of f more difficult. We conclude above that a 2 h time could be viable for the non-thermal case. However, for the 2 h thermal observation, $\ln\!(Z_{1}/Z_{2})\unicode{x003C}10$ , which quantifies a poor constraint on f. An observation of thermal bubbles will need longer than 2 h. For the more promising 4 h observation, the average errors are both less than 0.30, which corresponds to the $\ln\!(Z_{1}/Z_{2})$ threshold. Therefore, a 4 h observation of the thermal suppression factor in cluster MS0735 will give a well constrained value of f for most observations.

To explore the SKA’s ability to distinguish between a thermal and non-thermal scenario, given $kT_{e}=1\,000$ keV or $p_{1}=100$ , the average of mean values of the 50 posteriors for each observation time are compared in Table 5. The 1, 2, 4, and 6 h observation times have thermal and non-thermal average mean values within $1\sigma$ of each other. It is therefore unlikely that these observation times of the MS0735 cavities will result in distinguishable thermal and non-thermal f values. The 8 h observation time is able to produce measurements of the suppression factors that are just outside of $1\sigma$ from one another. While this shows that there is a slightly better chance of distinguishing between thermal and non-thermal suppression, a difference of $1\sigma$ still has a high probability of being due to random chance. It is therefore unlikely that the SKA will be able to distinguish between a thermal and non-thermal suppression factor in cluster MS0735 when observing for 8 h or less.

Table 5. Average of the mean measured value of f and its error $\sigma$ of the 50 posterior constraints for MS0735 bubbles with $p_{1}=100$ ( $f_{14}=0.989$ ) in the non-thermal case, and $kT_{e}=1\,000$ keV ( $f_{14}=0.796$ ) in the thermal case. The 1, 2, 4 and 6 h observations have thermal and non-thermal average mean values within $1\sigma$ of each other and cannot be distinguished. The 8 h observation gives f constraints that are just outside of $1\sigma$ from one another.

Figure 8. The average error, $\sigma$ , and $\ln\!(Z_{1}/Z_{2})$ of the 50 simulated SKA observations of thermal MS0735 cavities with $f_{14}=0.796$ , for each tested observation time. The average $\ln\!(Z_{1}/Z_{2})$ increases with an increasing observation time, showing that the cavities become better detected. According to the scale in Table 3, there is very strong detection for each observation time, except for HA = 0.5, days = 1, where $\ln\!(Z_{1}/Z_{2})\unicode{x003C}5$ . However, as described for the non-thermal case, this may not directly correspond to an accurate constraint on the suppression factor. The error clearly decreases with an increasing observation time, showing the constraint on f becomes much more informative.

To conclude, while the temperature and momentum constrained by CARMA from the suppression factor model ( $p_{1}=100$ and $kT_{e}=10\,000$ keV) do not yield distinguishable thermal and non-thermal $f_{14}$ values, the limited accuracy of the model allows for other possible values ( $p_{1}=100$ and $kT_{e}=1\,000$ keV) that correspond to suppression factors that the SKA has the potential to distinguish (albeit only at $1\sigma$ level) at 14.11 GHz.

3.3 The smallest observable suppression factor

As explained in Section 1 the constraints on the suppression factor measured by the GBT/MUSTANG-2 camera differ to those constrained by CARMA. The next investigation will explore how well the SKA can detect and distinguish thermal and non-thermal suppression factors of MS0735 cavities if they really are the values constrained by MUSTANG-2, and will in turn determine the smallest observable suppression factor in cluster MS0735.

Orlowski-Scherer et al. (Reference Orlowski-Scherer2023) constrain suppression factors ranging from $f_{90}=0.39$ – $0.95$ and we create a simulated image of the cluster for five values of f within this range: $f_{90}=0.39, 0.5, 0.6, 0.7, 0.8$ . These values are used to constrain the possible values of $p_{1}$ given a non-thermal gas in the bubbles, and $kT_{e}$ given a thermal gas. The following analysis is assuming a non-thermal gas. We obtain the corresponding suppression factors at the SKA frequency ( $14.11$ GHz), $f_{14}=0.13, 0.29, 0.42, 0.57, 0.72$ . Note that the range of suppression factors found by MUSTANG-2 correspond to $p_{1} \approx 1-15$ and $kT_{e} \approx 100-3\,000$ keV. These temperature and momentum constraints differ to those found via CARMA observations, where $p_{1} \approx 100$ and $kT_{e} \approx 10\,000$ keV. Orlowski-Scherer et al. (Reference Orlowski-Scherer2023) investigated the impact of changing many assumptions, leading to a broad range of possible suppression factors compared to Abdulla et al. (Reference Abdulla2019). Therefore, the discrepancies are likely due to the simple model that ties the observed suppression factor back to a $p_{1}$ or $kT_{e}$ value, when it might not be the true suppression factor.

Figure 9. Images CLEANed via CASA showing simulated 8 h SKA observations of cluster MS0735, with non-thermal cavities based on MUSTANG-2 constraints ( $f_{90}=0.39, 0.5, 0.6, 0.7, 0.8$ ). These measurements constrain $p_{1}$ values, yielding f values at the SKA frequency of $f_{14}=0.13$ , $0.29$ , $0.42$ , $0.57$ , $0.72$ , depicted in these images. Top (left to right): $f_{14}=0.13$ , $f_{14}=0.29$ , $f_{14}=0.42$ . Bottom (left to right): $f_{14}=0.57$ , $f_{14}=0.72$ . The images are produced using a Gaussian taper of $3000\lambda$ to down-weight the longest baselines, and a uv cut-off of $10\,000\lambda$ to reduce contributes from very small amplitude signal. A circular mask is also applied around the center of the image with a radius of 30 pix to isolate the cluster and the bubbles for the clean algorithm. The red contours represent the true bubble positions and radius (as a offset).

Profile simulations (CLEANed via CASA) in Fig. 9 show the SKA observed MS0735 non-thermal bubbles that are based on the minimum momentum constrained by the five MUSTANG-2 constrained suppression factors. This gives an initial indication that the bubbles are not observed for the lowest suppression factors. Each simulation is for an 8 h observation only, so that the lowest possible detectable suppression factor can be investigated at the maximum viable observation time. 50 realisations of each observation are simulated, and Bayesian analysis is performed.

The average value of $\ln\!(Z_{1}/Z_{2})$ and error, $\sigma$ , on the f constraint for the 50 realisations are given in Fig. 10. Again, using previously mentioned thresholds, bubbles with $f_{14}=0.13$ , $f_{14}=0.29$ , and $f_{14}=0.42$ are not detected with an 8 h observation time as the average $\ln\!(Z_{1}/Z_{2})\unicode{x003C}10$ . Note that in this case, a value of $\ln\!(Z_{1}/Z_{2})\unicode{x003E}10$ corresponds to an error $\unicode{x003C}0.2$ rather than $\unicode{x003C}0.3$ . This indicates that the error will need to be lower to represent a good constraint of a small suppression factor. This can also be understood by recognising that if the error is $0.3$ and the true value is $0.42$ , then $f \approx 1.4 \sigma$ which is almost only one error bar away from $f=0$ (representing no bubble detection). So a smaller error is needed to pinpoint the true suppression. Observations of $f_{14}=0.57$ and $f_{14}=0.72$ both result in $\ln\!(Z_{1}/Z_{2})\unicode{x003E}10$ and $\sigma\unicode{x003C}$ 0.2. Given these results, we conclude that the SKA can accurately measure the suppression factor of the MS0735 cavities if they are inhabited by non-thermal electrons with momentum that is constrained by MUSTANG-2, and that corresponds to suppression factors of $f_{14}=0.57$ and $f_{14}=0.72$ .

The five thermal suppression factor values at $14.11$ GHz that correspond to the range of cavity electron temperatures constrained by MUSTANG-2 are: $f_{14} = 0.27, 0.38, 0.49, 0.60, 0.73$ . Using the results from the non-thermal case, the SKA would be able to detect $f_{14}=0.60, f_{14}=0.73$ and likely $f_{14}=0.49$ , as these are larger than the corresponding non-thermal suppression factors that are well detected. Note that this conclusion is based on the results from the non-thermal simulations, and simulations with these exact suppression factor values were not performed. The thermal and non-thermal values at the SKA frequency are very similar, therefore we would not expect a differentiation between the two cases. This conclusion is supported by the following statistic: for the five non-thermal suppression factors observed by the SKA, the corresponding thermal suppression factor values fall within the 68% confidence interval of at least 68% of the 50 posteriors. Because of this, if the temperature and momentum within the cavities are indeed what was constrained from the MUSTANG-2 observations, the SKA would likely observe a value of f that could represent both a thermal and a non-thermal gas. However, as described earlier, the suppression factor model ties the true suppression factor to a $kT_{e}$ and $p_{1}$ , not the observed. Therefore, different temperature and momentum values in the MS0735 bubbles are possible, and distinguishing between the thermal and non-thermal scenarios remains a possibility.

3.4 Can the SKA measure the suppression factor of the MS0735 cavities?

In summary, the investigation of thermal and non-thermal suppression factors in the MS0735 cavities, considering existing constraints on electron temperature and momentum, has revealed that they will be observable by the SKA during an 8-h observation, provided $f_{14} \unicode{x003E} 0.42$ . If the cavities are described by the temperature $kT_{e} \approx 10\,000$ keV, and momentum $p_{1} \approx 100$ constrained by CARMA, then an observation time as low as 4 h will measure the suppression factor. It is also possible that a thermal and non-thermal suppression factor could be distinguished with an observation time greater than 8 h, if instead $kT_{e} \approx 1\,000$ keV which is the lower end of the CARMA constrained temperature values. The range of $kT_{e}$ and $p_{1}$ constrained by MUSTANG-2, however, imply that the suppression factors of the two gas types are too close to be distinguishable, if the (simple) suppression factor model is accurate.

Figure 10. The average error, $\sigma$ , and $\ln\!(Z_{1}/Z_{2})$ of the 50 realisations of 8 h observations of each possible value of non-thermal suppression factor ( $f_{14}=0.13$ , $0.29$ , $0.42$ , $0.57$ , $0.72$ ) determined from MUSTANG-2 constrained $p_{1}$ values. The average $\ln\!(Z_{1}/Z_{2})$ increases with an increasing value of f, showing that the cavities become better detected with more suppression. According to the scale in Table 3, there is poor cavity detection when $f_{14}=0.13$ and $f_{14}=0.29$ , as $\ln\!(Z_{1}/Z_{2}) \unicode{x003C} 5$ , but cavities with the remaining $f_{14}$ values are strongly detected (note that a strong detection of the cavities does not directly imply an accurate measurement of the suppression factor; a value of $\ln\!(Z_{1}/Z_{2})\unicode{x003E}10$ needed). The error clearly decreases with an increasing value of $f_{14}$ , showing the constraint on f becomes much more informative if the SZ signal in the cavities is more suppressed.

3.5 Cavity line of sight position effects

An interesting effect on the suppression factor that was investigated in Orlowski-Scherer et al. (Reference Orlowski-Scherer2023) is the position of the bubbles along the LOS. It is likely that, in reality, the axis of the two AGN jets is not parallel to the plane of the sky, but is rotated and is inflating bubbles that are shifted along the LOS. The LOS geometry of the bubbles in cluster MS0735 is not known, therefore Orlowski-Scherer et al. (Reference Orlowski-Scherer2023) investigate the influence on the suppression factor constraint, when increasing the angle $\theta$ from the plane of the sky. They find that the constrained value will increase with $\theta$ . This is because the bubbles will move into more tenuous regions of the global ICM (given the radial dependence of pressure), therefore $y_\mathrm{cav}$ in equation (23) will decrease. The suppression factor f must then increase to counteract the decrease in $y_\mathrm{cav}$ , and to construct the signal received from MUSTANG-2.

This effect is also observed for constraints on the suppression factor by the SKA. We investigate this by exploring the effect of different LOS position priors on the MS0735 suppression factor. An 8 h SKA observation is simulated to detect the signal of MS0735 constructed using $p_{1}=100$ in the non-thermal case. The bubbles are placed in the plane of the sky in the model image ( $\theta=0$ ), and the true suppression factor is $f_{14}=0.989$ . Bayesian inference is run on the data observed by the SKA for five different cases. First, an informative prior that the bubbles are in the plane of the sky is given. Then for each subsequent run, delta priors on $z_{b}$ (in equation 22) are given: $z_{b} = 0.04, 0.08, 0.14, 0.24, 0.53$ Mpc for the northern bubble and $z_{b} = -0.05, -0.10, -0.17, -0.30,$ $-0.64$  Mpc for the southern bubble. These correspond to LOS angles of $\theta = 15, 30, 45, 60, 75$ . Note that asymmetry is due to the asymmetric ${x}_{b}$ and ${y}_{b}$ positions of the bubbles in the cluster. This analysis shows what would happen to the suppression factor constraint if we had an observed signal but did not know where the bubbles were along the LOS, so specified them to be at different (potentially incorrect) positions.

The posterior distributions of the suppression factor for each of these LOS positions are shown in Fig. 11, where the true suppression factor value is represented by the vertical dashed lines. There is a cut off of the prior range at 2, which is causing the constraints of the largest line-of-sight angles to be restricted, with a less accurate shape. Similarly to the conclusions made regarding the LOS position for MUSTANG-2 observations in Orlowski-Scherer et al. (Reference Orlowski-Scherer2023) (see Figure 2 in their study), the SKA will measure larger suppression factors for a prior fixed at a larger value of LOS angle. This emphasises the importance of knowing where the bubbles are along the LOS when analysing the observed signal. Incorrect assumptions of their position will lead to incorrect suppression factor measurements. We discuss this further in Section 6 and suggest future investigations to help disentangle the degeneracy between suppression factor and LOS position.

Figure 11. Posteriors of the constraints on non-thermal f. The simulated data were generated with the bubbles at $z_{b}=0$ (in the plane of the sky), but analysed by assuming $z_{b} = 0.04, 0.08, 0.14, 0.24, 0.53$ Mpc (northern bubble) and $z_{b} = -0.05, -0.10, -0.17, -0.30,$ $-0.64$ Mpc (southern bubble) as delta priors. These shifts are related to angles $\theta = 15, 30, 45, 60, 75$ . The dashed line represents the true value of $f_{14}=0.989$ . This shows that if no prior information on the LOS position of the observed bubbles is known, it is possible they are in the plane of the sky and the measured f will be smaller, or that they are along the LOS and the measured f will be larger. Clearly, it is important to have prior information of the LOS position, so that the true f can be discovered.

An important aspect of this analysis is that, for the SKA frequency ( $14.11$ GHz), a value of $f_{14}\unicode{x003E}1$ in the cavities of MS0735 is not physical (e.g. the blue curves in Fig. 2). The original simulated y-map had a suppression factor of $f_{14}=0.989$ and thus the LOS angle shifts the measured suppression factor above 1. This indicates that if the signal obtained from the MS0735 cavities via the SKA implies a suppression factor of $f_{14}=0.989$ , under the assumption that the bubbles are located in the plane of the sky, then it would be unlikely that the bubbles are actually shifted some angle along the LOS. In the analysis of real cluster signals, priors that are based on the suppression factor values obtained from plots of f against $kT_{e}$ and $p_{1}$ , should be used. This will put limits on the possible LOS positions of the bubbles for that given signal.

Since the line-of-sight geometry is an added dimension to the model of the suppression factor within cluster cavities, it is important to consider the implications for being able to distinguish between the thermal and non-thermal scenarios. Fig. 12 depicts the mean value and error of the suppression factor for the different LOS angles, measured with the Bayesian approach in the thermal and non-thermal case for MS0735 ( $kT_{e}=1\,000$ keV and $p_{1}=100$ , respectively). While for each angle, there is still some difference between the thermal and non-thermal suppression factors (although with most of the errors overlapping), and the thermal values stay below the non-thermal ones, there will be degeneracy between the two scenarios for the smaller angles. For example, if the SKA were to measure a suppression factor of $\sim$ 0.9, it is unclear whether this suppression factor represents thermal gas inhabiting bubbles which are at an angle of 45 degrees along the line-of-sight, or a non-thermal gas at an angle of 30 degrees. It is important for future observations to recognise this as an added challenge in determining the gas type. Orlowski-Scherer et al. (Reference Orlowski-Scherer2023) also note that this investigation reinforces the need for multi-wavelength SZ observations to disentangle the effects of different pressure support scenarios from the effect of LOS geometries.

Figure 12. Suppression factor, f, as a function of the line-of-sight angle $\theta$ , with $\theta=0$ being in the plane of the sky and $\theta=90$ lying along the z-axis. The yellow data points represent measurement of the suppression factor given a non-thermal cavity gas with $p_{1}=100$ (as constrained for MS0735 by CARMA in Abdulla et al. Reference Abdulla2019). The blue data points represent cavities with a thermal gas with $kT_{e}=1\,000$ keV (the lower end of CARMA constraints). This shows that there is some degeneracy between thermal/non-thermal gas and LOS position, enforcing the importance of having prior LOS information so that the cavity gas type can be more accurately constrained.

Although the measured value of the suppression factor increases if the delta prior values on the LOS angle are higher (for the same observed signal), we discovered that detection of cavities and suppression factor constraints becomes much more difficult, the further along the LOS the cavities are positioned. Fig. 13 depicts simulated SZ signal of cluster MS0735 based on non-thermal CARMA constraints in the cavities ( $p_{1}=100$ ) for LOS angles 0, 45, 60, 70, 75. It is clear the SZ contrast between cavity and ICM seems to decrease, despite the suppression factor staying the same in each image. This could be misinterpreted as the suppression factor decreasing, when in reality, it is because the LOS integration will cover more of the cluster core with strong SZ signal, so the signal received is enhanced. The bubbles will instead be suppressing a weaker ICM region and will be less visible. This will affect the observation time required by the SKA to constrain accurately the true suppression factor. We performed an 8 h SKA simulated observation of the non-thermal MS0735 cavities in the plane of the sky and at a LOS angle of 70 degrees, as well as a 40 h (HA = 4, days = 5) observation of the bubbles at 70 degrees to test this theory. The average $\ln\!(Z_{1}/Z_{2})$ for the observations are: $\ln\!(Z_{1}/Z_{2})=$ 56.75 for an 8 h observation of bubbles in the plane of the sky, $\ln\!(Z_{1}/Z_{2})=$ 1.55 for an 8 h observation of bubbles at $\theta=70$ , $\ln\!(Z_{1}/Z_{2})=$ 17.01 for a 40 h observation of bubbles at $\theta=70$ . It is clear that detection deteriorates if the bubbles are moved along the LOS, but improves with a longer observation time. A poor constraint via the SKA in the future could cause confusion regarding whether the suppression factor is too small to be detected, or whether it is large but hard to detect because the bubbles are out of the plane of the sky.

Table 6. The positional parameters of the cavities used in this investigation. The coordinates of their centres on the model map of the sky, ${x_{b}}$ and ${y_{b}}$ (offset from the cluster center at (0,0)), are based on the MS0735 bubbles, which have a radius $\sim 100$ kpc. The positions of bubbles with radii 40, 60, 80 kpc are adjusted proportionally based on the percentage difference between their radius and the radius of the MS0735 bubbles. Subscripts 1 and 2 refer to the northern and southern cavity respectively.

Figure 13. Model signal maps depicting non-thermal MS0735 cavities, at different angles along the LOS (the CARMA constraint of $p_{1}=100$ is used in each image and therefore $f_{14}=0.989$ ). Top panel (left to right): $\theta = 0$ , $\theta = 45$ , $\theta = 60$ . Bottom panel (left to right): $\theta = 70$ , $\theta = 75$ . This could be misinterpreted as the suppression factor decreasing. In reality, the decrease in SZ contrast is because the LOS integration will cover more of the strong SZ signal at the cluster core, with the bubbles suppressing a weaker ICM region when shifted along the LOS. Therefore, longer observation times will be required to obtain an accurate measurement of f, if the bubbles are not in the plane of the sky.