<figure><img src="image_17.png" /><figcaption>**Figure B1.** One- and two-dimensional posterior probability distributions for our model parameters and one derived parameter (the optical depth $\tau$) when fitting a detection of the pKSZ power spectrum measurement at $\ell = 3000$ with a 10 per cent uncertainty and of the 21 cm power spectrum at redshifts of $z = 6.5, 6.5, 7.8$ at $k = 0.1, 0.50, 0.5 \text{ Mpc}^{-1}$ for 1000 h of integration with SKA-Low. For each parameter, the thin black vertical line is the true value used to generate the mock data and the dashed blue vertical line is the median of the distribution.</figcaption></figure>

corner plot in Fig. B1 clearly illustrates this, showcasing the relation between the global history and morphology parameters as well as the derived Thomson optical depth $\tau$. The parameter space is well sampled, with clear correlations seen only between either $z_{re}-z_{end}$ and $\log_{10}(\alpha_0)-\kappa$. A correlation is noticeable between the Thomson optical depth and each of the global history parameters. This is more than expected, as $\tau$ is an integral of the ionization history, defined as a function of $z_{re}$ and $z_{end}$ in equation (14). We discuss the constraints on $\tau$ for our different test cases in Appendix D in more detail.

## APPENDIX C: TESTING THE FORECAST WITH RSAGE

We have shown in Section 4.2 that with as few as three data points (one kSZ, two 21 cm), one can recover the global history of reionization and place some lower limits on the reionization morphology parameters. However, in these cases, both the mock data points and the sampled models were generated using our simplified derivation of the 21 cm power spectrum, ignoring third-order terms and cross-correlations, given in equation (7). In reality, of course, the data will include these extra terms. In this section, we will explore how this discrepancy can impact our forecast results.

We generate two new 21 cm data points corresponding to the 1k2z case of Section 4.2 for 100 h of observations with SKA-Low. This time, the points are generated using the full expression of $P_{21}(k, z)$, including the cross- and higher order terms, given in equation (6). As discussed in Section 4.1, these points have an amplitude between 10

and 25 per cent smaller than what was obtained with the simplified expression.

We fit our four reionization parameters to these new points (note that the kSZ data points remain unchanged) through MCMC sampling and show the resulting two-dimensional posterior distributions in Fig. C1. As expected from our discussions in Section 4.1, the general shape of the joint distributions is unchanged, only the results are now biased. As seen in Fig. 3, the simplified model tends to overestimate the 21 cm power for a given redshift, having an amplitude equivalent to a lower ionization level. For this reason, fitting it to unbiased data leads to an underestimate of the endpoint of reionization by $\Delta z_{end} = 0.12$. The mid-point is even more impacted as it is now biased by $\Delta z_{re} = 0.36$, nine times more than with the simplified mock data. Note that these modified reionization histories result in a large optical depth: $\tau = 0.070 \pm 0.002$. Hence, a way to partially mitigate these biases could be to impose a Gaussian prior on $\tau$. Similarly, the posterior distributions of the morphology parameters are shifted compared to the results with the simplified mock data points.

Although these results exhibit a strong bias in all parameters, this bias is nevertheless well understood and easy to model. A potential way of improvement would be to add a nuisance parameter to the fit, as a pre-factor to the model 21 cm power spectrum. This extra parameter would be marginalized over and would account for the extra power produced by the approximations done in equation (7). Another option would be to precisely quantify this bias and systematically subtract it from the models, allowing one to reproduce
<figure><img src="image_16.png" /><figcaption>**Figure D1.** Posterior distributions of the Thomson optical depth for the range of models discussed within this work (see Tables 3 and 5). The dashed vertical line represents the 'true' value $\tau = 0.065$.

**Figure C1.** Posterior distributions on reionization global (upper panel) and morphology (lower panel) parameters when assuming two measurements of the 21 cm power spectrum at $z = 6.5, 7.8$, and $k = 0.5 \text{ Mpc}^{-1}$ for 100 h of observation with SKA-Low. The mock 21 cm data points used for the fit are generated either with the simplified equation (7), in purple, or with the full equation (6), in orange. Vertical and horizontal black lines correspond to the 'true' values of the parameters, used to generate the mock data.</figcaption></figure>

the results of Section 4.2. However, this idea would be less successful when applied to real data, as this bias is model dependent. From one simulation to another, the amplitude and shape of the cross- and higher order terms missing in equation (7) will vary (Lidz et al. 2007; Georgiev et al. 2022).

## APPENDIX D: FORECAST CONSTRAINTS ON THE THOMSON OPTICAL DEPTH

In Section 2.1.3, we derived the reionization history $x_v(z)$ using the global history parameters $z_{\text{end}}$ and $z_{\text{re}}$ (see equation 14) and can, therefore, derive the Thomson optical depth $\tau$. With this in mind, we can investigate the constraining power of the forecast on $\tau$ from the combined 21 cm and pkSZ power-spectra analysis. Fig. D1 presents the constraint on $\tau$ for each of the cases discussed throughout this work. The dashed line represents a Gaussian centred on the true model value $\tau = 0.0649$ (Table 1) and with standard deviation the uncertainty reported in Planck Collaboration I (2020). First, we note

<figure><img src="image_17.png" /><figcaption>**Figure D1.** Posterior distributions of the Thomson optical depth for the range of models discussed within this work (see Tables 3 and 5). The dashed vertical line represents the 'true' value $\tau = 0.065$.</figcaption></figure>

that, for each mock data set, we are able to recover the true value of the optical depth (values are biased by less than 1 per cent) while achieving a tighter constraining power than with current large-scale CMB data. Despite this, small variations in constraining power are visible between cases.

Specifically, for our fiducial 1k2z case (see Section 4.2.1), there is a visible positive bias, deviating from the true value by a fraction of a per cent. The bias arises from the choice of a fixed $k$-scale for the 21 cm mock data points. Because reionization is a time-dependent and inhomogeneous process, the 21 cm spectrum from the EoR is expected to evolve both with redshift and with $k$-scale. There is a characteristic $k$-scale below which the scale dependence of $P_{21}$ is governed by the matter power spectrum. This characteristic scale evolves as reionization progresses (Furlanetto, McQuinn & Hernquist 2006). Hence, $k$-scales lower than this characteristic scale will be less sensitive to reionization, and vice versa (see figs 5 and 6 Georgiev et al. 2022). While mock data in the latter half of reionization ($z = 6.5, 7.8$) exclude larger values of $\tau$, data limited to $k = 0.5 \text{ Mpc}^{-1}$ slightly favour earlier reionization scenarios. The hiz case (in purple, $k = 0.5 \text{ Mpc}^{-1}$ for $z = 7.8, 10.4$) is biased towards lower values of the Thomson optical depth for similar reasons. The forecast is conducted on data at higher redshift, in the very early stages of reionization. The result is then a per cent bias of the derived optical depth and increased uncertainty, preferring a lower value of $\tau$ synonymous with a later reionization. It is only when the case where both 21 cm mock data is at a fixed $z = 6.5$ but at both high and low scales $k = 0.1, 0.5 \text{ Mpc}^{-1}$ (2k1z in yellow) are considered, that we can recover an unbiased estimate. This is because for the 2k1z mock data, the progress of reionization is encoded differently in each of the $k$-scale of the 21 cm power spectrum (see Section 5.1.2), resulting in a tighter constraint on $\tau$. Naturally, the 3k2z case (seen in yellow), which combines the 1k2z and 2k1z mock data, can further constrain $\tau$ without bias and with a lower uncertainty.

This paper has been typeset from a TeX/LA TeX file prepared by the author.
<figure><img src="image_18.png" /><figcaption>**Figure D1.** Posterior distributions of the Thomson optical depth for the range of models discussed within this work (see Tables 3 and 5). The dashed vertical line represents the 'true' value $\tau = 0.065$.

**Figure C1.** Posterior distributions on reionization global (upper panel) and morphology (lower panel) parameters when assuming two measurements of the 21 cm power spectrum at $z = 6.5, 7.8$, and $k = 0.5 \text{ Mpc}^{-1}$ for 100 h of observation with SKA-Low. The mock 21 cm data points used for the fit are generated either with the simplified equation (7), in purple, or with the full equation (6), in orange. Vertical and horizontal black lines correspond to the 'true' values of the parameters, used to generate the mock data.</figcaption></figure>

the results of Section 4.2. However, this idea would be less successful when applied to real data, as this bias is model dependent. From one simulation to another, the amplitude and shape of the cross- and higher order terms missing in equation (7) will vary (Lidz et al. 2007; Georgiev et al. 2022).

## APPENDIX D: FORECAST CONSTRAINTS ON THE THOMSON OPTICAL DEPTH

In Section 2.1.3, we derived the reionization history $x_v(z)$ using the global history parameters $z_{end}$ and $z_{re}$ (see equation 14) and can, therefore, derive the Thomson optical depth $\tau$. With this in mind, we can investigate the constraining power of the forecast on $\tau$ from the combined 21 cm and pkSZ power-spectra analysis. Fig. D1 presents the constraint on $\tau$ for each of the cases discussed throughout this work. The dashed line represents a Gaussian centred on the true model value $\tau = 0.0649$ (Table 1) and with standard deviation the uncertainty reported in Planck Collaboration I (2020). First, we note

<figure><img src="image_19.png" /><figcaption>**Figure D1.** Posterior distributions of the Thomson optical depth for the range of models discussed within this work (see Tables 3 and 5). The dashed vertical line represents the 'true' value $\tau = 0.065$.</figcaption></figure>

that, for each mock data set, we are able to recover the true value of the optical depth (values are biased by less than 1 per cent) while achieving a tighter constraining power than with current large-scale CMB data. Despite this, small variations in constraining power are visible between cases.

Specifically, for our fiducial 1k2z case (see Section 4.2.1), there is a visible positive bias, deviating from the true value by a fraction of a per cent. The bias arises from the choice of a fixed $k$-scale for the 21 cm mock data points. Because reionization is a time-dependent and inhomogeneous process, the 21 cm spectrum from the EoR is expected to evolve both with redshift and with $k$-scale. There is a characteristic $k$-scale below which the scale dependence of $P_{21}$ is governed by the matter power spectrum. This characteristic scale evolves as reionization progresses (Furlanetto, McQuinn & Hernquist 2006). Hence, $k$-scales lower than this characteristic scale will be less sensitive to reionization, and vice versa (see figs 5 and 6 Georgiev et al. 2022). While mock data in the latter half of reionization ($z = 6.5, 7.8$) exclude larger values of $\tau$, data limited to $k = 0.5 \text{ Mpc}^{-1}$ slightly favour earlier reionization scenarios. The hiz case (in purple, $k = 0.5 \text{ Mpc}^{-1}$ for $z = 7.8, 10.4$) is biased towards lower values of the Thomson optical depth for similar reasons. The forecast is conducted on data at higher redshift, in the very early stages of reionization. The result is then a per cent bias of the derived optical depth and increased uncertainty, preferring a lower value of $\tau$ synonymous with a later reionization. It is only when the case where both 21 cm mock data is at a fixed $z = 6.5$ but at both high and low scales $k = 0.1, 0.5 \text{ Mpc}^{-1}$ (2k1z in yellow) are considered, that we can recover an unbiased estimate. This is because for the 2k1z mock data, the progress of reionization is encoded differently in each of the $k$-scale of the 21 cm power spectrum (see Section 5.1.2), resulting in a tighter constraint on $\tau$. Naturally, the 3k2z case (seen in yellow), which combines the 1k2z and 2k1z mock data, can further constrain $\tau$ without bias and with a lower uncertainty.

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