tSZ clusters in the input map to those in the output needlet ILC Compton-$y$
map, with the results shown in Fig. 17. We find very good agreement between
the two sets of stacked profiles and thereby validate this aspect of our
pipeline. A benefit of having simulations is that we can analyze each
component of the map separately, exploiting the linearity of the ILC method.
Thus, we can explore how just the CIB component contributes to the final tSZ
map. These maps are useful for understanding residual signals in the maps and
for assessing potential biases.

These simulations have a few key limitations: firstly, we only have four sky
realizations, given by the two rotations of the two simulations. Secondly, the
sky power of these simulations does not match the data – whilst the Websky and
Sehgal et al simulations are sophisticated models of the sky, they do not
perfectly capture all the components, leading to differences in the size of
each sky component. For example, the small scale power arising from the
discrete nature of the CIB and radio galaxies is not accurate. Additionally,
there are minor limitations (e.g., delta function passbands and simple beam
treatments) that prevent tests of certain systematic effects. Despite these
limitations, they are invaluable for analyses of non-Gaussian aspects of the
sky.

Figure 17: A validation of the Compton-$y$ map with the Websky simulations. We
compare the stacked NILC Compton-$y$ map with the true input Compton-$y$ map.
The stack is performed on all halos with $M>5\times 10^{14}$ M⊙/$h$ and these
maps are high-pass filtered in the same manner as the data in Fig. 14.

###  A.2 Gaussian Simulations

To complement the non-Gaussian simulations we provide noise models as
described in Ref. (Atkins et al., 2023). These allow the fast generation of
many realizations of the noise in the needlet ILC maps and are ideal for
investigating ensemble properties of the ILC map’s noise.

We generate these noise models using a two stage process: first, we obtain a
model of the sky components present in the input maps. For this purpose we use
the Ref. (Dunkley et al., 2013) model, extended to include power laws for the
Galactic dust, synchrotron, Anomalous Microwave Emission, and free-free, and a
white noise component of arbitrary amplitude and correlation across the maps
to account for the Poisson contribution of point sources. The latter component
is necessary as we remove all detected sources in each map, rather than those
down to a specific flux cut, and this results in each map having a slightly
different level of Poisson power. The strength of this correlation across the
input maps varies due to the different ratio of radio to dusty sources in each
map. Further, we modulate the Galactic dust components by a smoothed version
of the Planck dust intensity map. Without this modulation the large scale
power in the component separated maps is significantly below that of the data,
highlighting how the spatial variations of the Galaxy complicate component
separation.

Second, we use this sky model to simulate observations of the sky. We use the
same methods as in Appendix A.1 to simulate the instrumental noise and beams.
We then process these simulations with the needlet ILC pipeline. From the
resulting needlet ILC maps, we subtract the input realization of the signal of
interest, e.g., for the CMB ILC maps we subtract the CMB power; this gives a
map of the NILC ‘noise’. Using the tools developed in Ref. (Atkins et al.,
2023) we then construct tiled and wavelet noise models. These noise models
allow arbitary numbers of realizations of the NILC ‘noise’ to be generated.
The noise realizations can then be combined with realizations of the signal to
provide a simulated map.

The end-to-end Gaussian simulations produced in generating the noise model
allow further tests of potential biases in our pipeline. As discussed in
Section III.5 and seen in Fig. 3, these simulations can be used to assess the
level of bias in our ILC maps. We find no significant biases across the full
range of scales. In Fig. 18 we compare the power spectra of simulated Planck-
only to Planck & ACT ILC E-mode maps. This demonstrates the value of combining
the ACT data with Planck.

Figure 18: The E-mode auto power spectrum of a simulated Planck-only ILC map
and a simulated Planck & ACT ILC map. It can clearly be seen that the ACT data
allow the E-modes to be probed to significantly smaller scales.

##  Appendix B Harmonic ILC

Instead of working in the needlet domain, the ILC method can be applied in the
harmonic domain – hereafter the harmonic ILC. In this work we use the harmonic
ILC as a baseline to compare against the NILC results, and to build intuition
for the bias reduction methods, as the harmonic frame is conceptually simpler.
In this Appendix we outline the harmonic ILC method and describe the harmonic-
domain implementation of our ILC bias mitigation method.

For the harmonic ILC we have

$\displaystyle\hat{s}_{\ell m}=\sum\limits_{i}w_{i}a^{i}_{\ell m}$ (25)

where the weights are defined in Eq. 7. The empirical covariance matrix,
$\hat{\mathcal{C}}$, required to compute the weights is simply the estimated
power spectra

$\displaystyle\hat{\mathcal{C}}_{ij}=\frac{1}{N_{\mathrm{modes}}}\sum\limits_{L-\delta\ell<\ell\leq
L+\delta\ell}\sum\limits_{m}a^{i}_{\ell m}{a^{j}}^{*}_{\ell m},$ (26)

where we compute the power spectrum over a range of $\ell$, quantified by
2$\delta\ell$ and $N_{\mathrm{modes}}$ is the number of modes in each bin. We
average over a small range of $\ell$ to improve the estimate of the covariance
matrix and minimize the ILC bias. To further minimize ILC bias we introduce an
ILC bias reduction method, in a similar manner to those discussed in Section
III.3.2. When computing the ILC weights for mode $m$ we exclude that $m$ mode
from the covariance matrix estimation, Eq. 26. To avoid the computational cost
of estimating the covariance matrix $\ell$ times per scale, we instead split
the modes into $N=10$ subsets (eg., the first $\ell/N$ $m$ modes are assigned
to subset 1 etc) and, for each subset, estimate the covariance matrix with all
modes not in that subset. This allows the ILC bias to be mitigated with
reduced computational overhead, at the cost of a slight increase in the
resulting ILC map’s noise. In fact, whilst the standard method of estimating
the covariance matrix has lower noise, it is biased low. When we correct for
this bias, by dividing out a transfer function, we tend to find almost
identical noise levels to those of our method.

When computing harmonic ILC maps we use the same steps as the needlet ILC
pipeline, described in Section III, including the frequency dependent
responses.

Figure 19: The passbands for the six different ACT DR6 data sets. These are
used in Eq. 15 to compute the responses to each sky signal. The shaded region
denotes the $1\,\sigma$ uncertainty on these measurements.

##  Appendix C Mitigating the large scale ILC bias

As described in Section III.3.2, we mitigate the large scale ILC bias by
isolating a subset of the modes from the needlet band and using the remainder
to compute the ILC weights. In detail this isolation can be performed as
follows: starting from the map of anisotropies at a given needlet scale, we
perform a SHT. We apply a filter that selects a subset of $a_{\ell m}$ modes
and then we perform a second SHT to transform these modes back to the needlet
frame; we call this the ‘data map’. This ‘data map’ is then subtracted from
the original needlet map.131313For computational efficiency we combined these
two steps with the initial transformation into the needlet frame. This saves
one SHT transform.  This new map contains all the modes except the held out
modes and we use this map to compute the ‘covariance maps’ and ILC weights.
The ILC weights are then applied to the ‘data maps’. This procedure is
repeated with the next subset of $a_{\ell m}$ modes etc. The resulting ILC
maps from the subsets are then added back together to obtain the ILC solution
for that needlet scale.

Whilst the approach of removing a subset of $a_{\ell m}$ modes is very similar
to the harmonic ILC, described in Appendix B, there is a slight additional
subtlety in the needlet frame. When we remove $a_{\ell m}$ modes from the
needlet band, the localization of the needlets is altered. If we were to
systematically remove each single $a_{\ell m}$ mode the change in the
localization of the covariance matrix would be negligible. In practice this
would not be effective as effects such as masking and non-Gaussianites mean
that each $a_{\ell m}$ mode is not independent. Thus this would only reduce,
but not eliminate, the ILC bias. Further, removing modes from the needlet
basis requires spherical harmonic transforms, and so this operation comes at a
significantly increased computational cost. Hence, this motivates our choice
to remove small subsets of $a_{\ell m}$ modes across the band.

The decision of how many subsets to use, $N_{\mathrm{subset}}$, and how to
divide the modes into these subsets was made by ensuring the localization of
the covariance matrix is not strongly altered. This is balanced by
computational time (more subsets is slower) and stability (changing masks and
power spectra in simulations should not dramatically change the ILC maps). We
use $N_{\mathrm{subset}}=5$. For our footprint the strong correlations from
the mask led us to assign $\ell<4$ to subset 1, $4<\ell<8$ to subset 2 etc up
to $\ell=20$. Beyond $\ell=20$, the assignment is based on the $m$-mode at
each $\ell$. Modes with $0\leq m<\ell/N_{\mathrm{subset}}$ are assigned to
subset $\ell/20$ MOD $N_{\mathrm{subset}}$. Modes with
$\ell/N_{\mathrm{subset}}\leq m<2\ell/N_{\mathrm{subset}}$ are assigned to
subset $1+\ell/20$ MOD $N_{\mathrm{subset}}$ etc. The adjacent $m$-modes are
assigned to the same subset as these are most tightly coupled by effects such
as masking. Further, by removing different $m$-modes for each $\ell$ we ensure
that the localization of the new covariance matrix closely approximates the
original covariance matrix. This is additionally aided by our smoothing
operation and the fact that we remove only a small subset of the total modes
in the needlet band. One might worry that the small set of modes that the
covariance matrix is applied to correspond to some highly non-local feature in
the maps. However, this is not an issue due to the linearity of the ILC
method: in the standard ILC identical results are obtained if the ILC weights
are applied mode-by-mode or applied to all modes simultaneously. Instead it is
important that the covariance matrix, and hence the weights, remains highly
localized and we ensure this with the choices described above.

##  Appendix D Instrumental Systematic Effects

Figure 20: The absolute change in the stacked cluster profiles, Fig. 14,
caused by the calibration, beam, and passband uncertainties. These effects,
shown in orange, are significantly smaller than the measured scatter in the
stack, shown as a blue band.

The key instrumental systematic effects that impact our analyses arise from
the miscalibration of the input maps and uncertainties in the passbands and
beams. All of these effects can be expressed as changes to the ILC responses,
${r}_{\nu}$ in Section III.3.1, and are important as they control how maps are
combined to extract the signal of interest. If these are misestimated the
resulting ILC map can be biased. In this section we investigate how
significant these biases can be.

The impact of a miscalibrated map is simply a rescaling of the ILC responses
$\mathbf{r}_{\nu}\rightarrow(1+\delta c)\mathbf{r}_{\nu}$, where $\delta c$ is
the amplitude of the miscalibration. Note that the miscalibration factor is
the same for every signal on the sky. On the other hand, the passband
misestimation affects each sky component differently. The impact on each of
the responses can be assessed by recomputing them with altered passbands,
i.e., changing $\tau(\nu)$ in Eq. 20. Beam uncertainty has two impacts:
firstly, it acts as a scale dependent miscalibration, given by the ratio of
the estimated beam to the true beam. Secondly, it alters the color correction
to the response, given in Eq. 20.

To assess the importance of these terms we sample calibration factors,
passbands, and beams within their uncertainties. For the Planck passbands we
shift the central frequencies based on the uncertainties reported in Ref.
(Planck Collaboration Int. LVII, 2020) and use the calibration errors reported
in Ref. (Planck Collaboration Int. LVII, 2020). The Planck beam errors have
been shown to be negligible (Planck Collaboration V, 2020) and so are
neglected here. For ACT DR4 we use the same calibration and beam uncertainties
as are included in the power spectrum analyses (Louis et al., 2017b; Choi et
al., 2020; Lungu et al., 2022) and use the passband errors from Ref.
(Madhavacheril et al., 2020). For DR6 errors we use similar procedures to
compute the equivalent errors, with the details of this provided in an
upcoming ACT paper. The passbands used in this work, with their errors, are
shown in Fig. 19.

In Fig. 20 we show the impact these effects have on the stacked Compton-$y$
profiles. No systematic shift is observed and the spread caused by these
effects is negligible.

This result does not ensure that instrumental systematic effects are
negligible for other science use cases. In particular the beam uncertainties
are scale dependent (and are generally larger on small scales) and the
passband uncertainties will likely become more important in analyses that
require multiple deprojections. Instead, if instrumental effects are a
concern, this analysis should be repeated. Despite these limitations, these
results can be used to estimate the size of the effect.

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