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## APPENDIX A: AVERAGE DENSITY RATIO FOR WHIM WITH ISOTHERMAL β-PROFILE

Here we derive the average density ratio formulae of equations (7) and (8), alongside the average density ratio asymptote for fixed parameters.

We consider a filament whose WHIM density profile obeys the isothermal β-model:

$$ \rho(r) := \rho(0) \left(1 + \left(\frac{r}{r_c}\right)^2\right)^{-\frac{3}{2}\beta}, \qquad (A1) $$

with $\rho(0)$, $r_c$, and $\beta$ free parameters. A cylinder of radius $r$ and length $l$ has an enclosed mass $m(r)$ and average density $\langle\rho\rangle(r)$ given by

$$ m(r) = \int_{0}^{r} \rho(r') \cdot l \cdot 2\pi r' \, dr'; \quad \langle\rho\rangle(r) = \frac{m(r)}{\pi r^2 l}, \qquad (A2) $$

so that

$$ \langle\rho\rangle(r) = \frac{2\rho(0)}{r^2} \int_{0}^{r} \left(1 + \left(\frac{r'}{r_c}\right)^2\right)^{-\frac{3}{2}\beta} r' \, dr'. \qquad (A3) $$

The integration result depends on $\beta$. For $\beta \neq \frac{2}{3}$, we find

$$ \langle\rho\rangle(r) = \frac{\rho(0)}{1 - \frac{3}{2}\beta} \frac{r_c^2}{r^2} \left( \left(1 + \left(\frac{r}{r_c}\right)^2\right)^{1-\frac{3}{2}\beta} - 1 \right), \qquad (A4) $$

whilst for $\beta = \frac{2}{3}$, we find

$$ \langle\rho\rangle(r) = \rho(0) \frac{r_c^2}{r^2} \ln \left( 1 + \left( \frac{r}{r_c} \right)^2 \right). \qquad (A5) $$

Upon dividing $\langle\rho\rangle(r)$ by $\langle\rho\rangle(R)$, the shared factors $\rho(0) r_c^2$ and $(1 - \frac{3}{2}\beta)^{-1}$ cancel, and we arrive at equations (7) and (8) for the average density ratio.

From equation (A1), we see that for $r \ll r_c$ the density is approximately constant – at level $\rho(0)$. For this reason, one expects that

$$ \lim_{r_c \to 0} \langle\rho\rangle(r) = \rho(0), \qquad (A6) $$

a fact that can be formally verified from equations (A4) and (A5) using L'Hôpital's rule. Because $\beta > 0$, $\rho(r)$ attains its maximum at $r = 0$. For $\beta \neq \frac{2}{3}$, the average density ratio asymptote therefore is

$$ \lim_{r_c \to 0} \frac{\langle\rho\rangle(r)}{\langle\rho\rangle(R)} = \frac{R^2}{r_c^2} \frac{1 - \frac{3}{2}\beta}{\left(1 + \left(\frac{R}{r_c}\right)^2\right)^{1-\frac{3}{2}\beta}} - 1, \qquad (A7) $$

whilst for $\beta = \frac{2}{3}$, we find

$$ \lim_{r_c \to 0} \frac{\langle\rho\rangle(r)}{\langle\rho\rangle(R)} = \frac{R^2}{r_c^2} \ln^{-1} \left( 1 + \left( \frac{R}{r_c} \right)^2 \right). \qquad (A8) $$

## APPENDIX B: LARGE-TO-SMALL-SCALE DENSITY CONVERSION: THE LOW-DENSITY REGIME

In this section, we provide a visual impression of the practical effect of the 1 Mpc-scale IGM density definition introduced in Section 3.2. This definition is key in the conversion from large-scale density $1 + \delta$ to small-scale density $1 + \delta_{IGM}$.

As an example, we consider an NGC 6185 analogon with a 4.2 Mpc-scale relative total matter density $1 + \delta = 2.3 \pm 0.7$. In Fig. B1, we show three ENZO simulation cutouts consistent with this scenario. As can be seen from the top and, to a lesser degree, the bottom panel of Fig. B1, the thus-defined IGM density surrounding the galaxy may represent the IGM density of a broader part of the filament. By comparing Fig. B1 to Fig. 10, it appears that the 1 Mpc-scale IGM density definition introduced in Section 3.2 not only produces reasonable density estimates in group-like environments, but also in lower density filament environments. This is important, because future work may attempt to constrain IGM thermodynamics outside galaxy groups with the methodology presented in this work.
<figure><img src="image_13.png" /><figcaption>**Figure B1.** As in Fig. 10, but for an NGC 6185 analogon that does not occur in a galaxy group, but in a lower density Cosmic Web filament environment. Such a galaxy might have a 4.2 Mpc-scale relative total matter density $1 + \delta = 2.3 \pm 0.7$. As is clear from visual comparison to Fig. 10, ENZO simulation cutouts consistent with this scenario feature both less dense 1 Mpc-scale environments as well as less massive galaxies. Note the more restricted colour bar scale here.</figcaption></figure>

## APPENDIX C: LOWER WHIM TEMPERATURE CONSTRAINTS FROM CURRENT-DAY RADIO DATA

In this section, we demonstrate that our methodology, combined with current-day radio survey data, such as the publicly available LoTSS DR2, in principle allow for more stringent WHIM temperature constraints than those derived in this work – that is, given the availability of a suitable target.

In Fig. C1, we show that given typical LoTSS DR2 noise levels, analoga of NGC 6185’s GRG but with surface brightnesses that are four times lower, are unsuitable targets. However, in Fig. C2, we show that analoga that are two or three times fainter can be accurately analysed. For each analogon, we repeat the Metropolis-Hastings MCMC procedure described in Section 3.1. A comparison between the middle panels of Figs 6 and C2 reveals that lobe geometry inference remains stable far into the low signal-to-noise regime. We find that the three times fainter analogon yields an IGM temperature estimate of $T_{\text{IGM}} = 4_{-2}^{+3} \times 10^6$ K, or $k_B T_{\text{IGM}} = 0.3_{-0.1}^{+0.3}$ keV.

<figure><img src="image_14.png" /><figcaption>**Figure C1.** A simulated 90 arcsec image at $v_{\text{obs}} = 144$ MHz of a four times fainter analogon of NGC 6185’s GRG. Given the LoTSS DR2 90 arcsec noise level $\sigma \sim 2$ Jy deg$^{-2}$, such a GRG is unlikely to be recognized, correctly deconvolved, and analysed.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Figure B1.** As in Fig. 10, but for an NGC 6185 analogon that does not occur in a galaxy group, but in a lower density Cosmic Web filament environment. Such a galaxy might have a 4.2 Mpc-scale relative total matter density $1 + \delta = 2.3 \pm 0.7$. As is clear from visual comparison to Fig. 10, ENZO simulation cutouts consistent with this scenario feature both less dense 1 Mpc-scale environments as well as less massive galaxies. Note the more restricted colour bar scale here.</figcaption></figure>

## APPENDIX C: LOWER WHIM TEMPERATURE CONSTRAINTS FROM CURRENT-DAY RADIO DATA

In this section, we demonstrate that our methodology, combined with current-day radio survey data, such as the publicly available LoTSS DR2, in principle allow for more stringent WHIM temperature constraints than those derived in this work – that is, given the availability of a suitable target.

In Fig. C1, we show that given typical LoTSS DR2 noise levels, analoga of NGC 6185’s GRG but with surface brightnesses that are four times lower, are unsuitable targets. However, in Fig. C2, we show that analoga that are two or three times fainter can be accurately analysed. For each analogon, we repeat the Metropolis-Hastings MCMC procedure described in Section 3.1. A comparison between the middle panels of Figs 6 and C2 reveals that lobe geometry inference remains stable far into the low signal-to-noise regime. We find that the three times fainter analogon yields an IGM temperature estimate of $T_{\text{IGM}} = 4_{-2}^{+3} \times 10^6$ K, or $k_B T_{\text{IGM}} = 0.3_{-0.1}^{+0.3}$ keV.

<figure><img src="image_14.png" /><figcaption>**Figure C1.** A simulated 90 arcsec image at $v_{\text{obs}} = 144$ MHz of a four times fainter analogon of NGC 6185’s GRG. Given the LoTSS DR2 90 arcsec noise level $\sigma \sim 2$ Jy deg$^{-2}$, such a GRG is unlikely to be recognized, correctly deconvolved, and analysed.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Figure C2.** Bayesian radio galaxy lobe model fits in the sense of Section 3.1 applied to simulated analoga of NGC 6185's GRG, but with lobes that are two (left-hand column) and three (right-hand column) times fainter. This is a visual demonstration of the fact that (minimum energy and equipartition) lobe pressures can be robustly extracted from analoga of NGC 6185's GRG whose surface brightnesses are a factor of order unity lower. The figure is similar to Fig. 6, but has a different colour bar scaling. Alongside 90" images at $v_{\text{obs}} = 144$ MHz of the two and three times fainter analoga of NGC 6185's GRG (top row), we show 90 arcsec MAP estimates from the Bayesian radio galaxy lobe model (middle row) and residual images (bottom row) obtained by subtracting the middle row images from the top row images.</figcaption></figure>

This paper has been typeset from a TeX/LATEX file prepared by the author.

© 2022 The Author(s)

Published by Oxford University Press on behalf of Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
<figure><img src="image_15.png" /><figcaption>**Figure C2.** Bayesian radio galaxy lobe model fits in the sense of Section 3.1 applied to simulated analoga of NGC 6185's GRG, but with lobes that are two (left-hand column) and three (right-hand column) times fainter. This is a visual demonstration of the fact that (minimum energy and equipartition) lobe pressures can be robustly extracted from analoga of NGC 6185's GRG whose surface brightnesses are a factor of order unity lower. The figure is similar to Fig. 6, but has a different colour bar scaling. Alongside 90" images at $v_{\text{obs}} = 144$ MHz of the two and three times fainter analoga of NGC 6185's GRG (top row), we show 90 arcsec MAP estimates from the Bayesian radio galaxy lobe model (middle row) and residual images (bottom row) obtained by subtracting the middle row images from the top row images.</figcaption></figure>

This paper has been typeset from a TeX/LATEX file prepared by the author.

© 2022 The Author(s)

Published by Oxford University Press on behalf of Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.