characterizing the foreground amplitude. The spectrum $C_{\ell_b}^{\text{ACT}\times\text{ACT,EE,A}_f}$ is the measured ACT EE spectrum for array frequency $A_f$.

We use the multipole range $\ell_b^{\text{min}} = 1000$ and $\ell_b^{\text{max}} = 1500$ to minimize the impact of temperature-to-polarization leakage, foreground contamination, and effects of the transfer function. A prior on the foreground amplitude $\alpha_{A_f}$, which is dominated by the polarized dust component, is estimated using the measurement of the 353 GHz NPIPE EE power spectrum from Planck. Further details about this prior will be discussed in the upcoming ACT power spectrum analysis paper.

We determine the efficiency scaling $P_{\text{eff}}^{\Lambda_f}$ by finding the maximum posterior value from a simple Gaussian likelihood derived from this $\chi^2$, marginalized over $\alpha_{A_f}$. We also derive an error on the efficiency scaling parameter from the posterior.

## Appendix B

### Further Discussion of Normalization Correction

In this appendix, we provide further details of the normalization correction discussed in Section 5.10. The reconstructed lensing field, $\hat{\phi}_{LM}$, is dependent on the assumed fiducial CMB power spectra through the normalization term, $\mathcal{R}_L^{-1}$. Setting aside the small MC correction for the moment, this dependence can be expressed as

$$ \hat{\phi}_{LM} |_{C_{\ell}^{\text{CMB,fid}}} \approx \frac{\mathcal{R}_L^{-1} |_{C_{\ell}^{\text{CMB,fid}}} \hat{\phi}_{LM} |_{C_{\ell}^{\text{CMB}}}}{\mathcal{R}_L^{-1} |_{C_{\ell}^{\text{CMB}}} \hat{\phi}_{LM} |_{C_{\ell}^{\text{CMB}}}}, \quad (\text{B1}) $$

where $C_{\ell}^{\text{CMB,fid}}$ is the fiducial CMB power spectra used in the lensing reconstruction and $C_{\ell}^{\text{CMB}}$ is the true spectrum that describes the data. The reconstruction $\hat{\phi}_{LM}|_{C_{\ell}^{\text{CMB,fid}}}$ is normalized with the fiducial power spectra and will differ from the unbiased reconstruction $\hat{\phi}_{LM}|_{C_{\ell}^{\text{CMB}}}$ (i.e., normalized with the correct spectrum) if the fiducial spectrum does not match the truth. At the time this article was written, the final ACT DR6 CMB power spectra and their covariance matrices were not yet available. As a result, the baseline normalization was calculated using the full-sky fiducial spectra, $C_{\ell}^{\text{CMB,fid}}$, based on the Planck 2015 TTTEE E cosmology with an updated $\tau$ prior as shown in Calabrese et al. (2017).

While we expect this analytic normalization to be quite accurate, it is a potential concern that the lack of a final measured power spectrum might introduce a small bias: in particular, that our assumption that the CMB power spectrum is consistent with Planck might not be correct at high precision. Fortunately, the preprocessing steps described in Section 5 help to minimize any potential bias that could arise from assuming incorrect fiducial CMB power spectra. Recall that the error in the normalization is small as long as the data power spectra are similar to the fiducial power spectra used in the analysis. By applying the calibration and polarization efficiency determination described in Section 3.3, which are defined relative to Planck, we can ensure that this condition is met because the amplitude of the data spectra is scaled to be in good agreement with the fiducial spectrum used in the normalization.

To confirm quantitatively the effectiveness of our methods for reducing bias, we have tested a scenario in which the ACT DR6 data CMB power spectra match the ACT DR4+WMAP

<figure><img src="image_40.png" /><figcaption>Figure 35. Differences in CMB power spectra used in the lensing normalization. Using the toy model described in Appendix B, we illustrate the difference between the fiducial ACT DR4+WMAP power spectra (as presented in Aiola et al. 2020; Choi et al. 2020) and $C_{\ell}^{\text{CMB,fid}}$ before and after applying calibration and polarization efficiency factors. As shown by the solid curves, the overall amplitudes of the fiducial ACT DR4+WMAP spectra differ from the fiducial spectra by a few percent before correction. The application of calibration and polarization efficiency factors serves to rescale these spectra and bring them more in line with the fiducial spectra used for normalization (dashed lines).</figcaption></figure>

spectra (as presented in Aiola et al. 2020; Choi et al. 2020), although we still assume $C_{\ell}^{\text{CMB,fid}}$ in our normalization. Figure 35 compares the fiducial ACT DR4+WMAP spectra with $C_{\ell}^{\text{CMB,fid}}$ used in our work. In the analysis range, the ACT DR4+WMAP spectra differ by a few percent from $C_{\ell}^{\text{CMB,fid}}$. We follow the procedure described in Sections 3.3 and 3.4 to compute corrections (calibration and polarization efficiency) and apply them to the ACT DR4+WMAP spectra. The dashed lines in the figure show that the differences between the ACT +WMAP and the fiducial spectra are significantly reduced after these corrections are applied.

We calculated the normalization $\mathcal{R}_L^{-1}$ for each of these spectra and compared it to our baseline $\mathcal{R}_L^{-1}$. The results of this comparison are displayed in the left panel of Figure 36. Prior to the correction, the ACT DR4+WMAP $\mathcal{R}_L^{-1}$ was approximately 1.3% lower than the baseline, corresponding to a roughly 1.14σ bias in the lensing power spectrum amplitude $A_{\text{lens}}$. After applying the correction, the difference in $\mathcal{R}_L^{-1}$ was reduced by a factor of around 5, reducing the bias in $A_{\text{lens}}$ to 0.23σ. We anticipate that the ACT DR6 spectra will not be clearly inconsistent with the Planck or ACT DR4+WMAP fiducial spectra, and we thus expect that they will fall somewhere in between these two spectra. Therefore, the actual bias in the data is likely to be less than 0.23σ.

For the lensing-only likelihood analysis considered in this paper, we account for this uncertainty in the true CMB spectra as follows. Since $C_{\ell}^{\text{CMB}}$ is well constrained, the difference between the true CMB power spectra that describe the data, $C_{\ell}^{\text{CMB}}$, and the fiducial $C_{\ell}^{\text{CMB,fid}}$ is expected to be small. We can
<figure><img src="image_41.png" /><figcaption>**Figure 36.** Left: same as Figure 35, but for the normalization factor $R_L^{-1}$. Without the application of calibration and polarization efficiency factors (red solid curve), the mismatch between the power spectrum describing the data and the fiducial spectra leads to the lensing reconstruction being biased high by around 1.3%, corresponding to a bias of around 1.14σ in the inferred lensing amplitude. After applying amplitude corrections to the CMB maps (red dashed curve), this bias is reduced by a factor of 5, resulting in only 0.23σ-level bias in the lensing amplitude. (If we assume that the ACT DR6 spectra fall somewhere in between Planck and ACT DR4 spectra, any biases will be significantly less than 0.23σ.) Right: inflation of lensing bandpower error bars due to uncertainty in the true CMB power spectra, which are needed to normalize the lensing measurement correctly. As described in the text, sampling 1000 ΛCDM spectra from ACT DR4 + Planck chains results in a less than 2.5% increase in error bars and introduces small mode coupling between bandpowers. This change only has a negligible impact on cosmological parameter constraints.</figcaption></figure>

therefore Taylor expand $R_L^{-1}|_{C_\ell^{\text{CMB}}}$ around the fiducial spectra as

$$ \begin{aligned} R_L^{-1}|_{C_\ell^{\text{CMB}}} &\approx R_L^{-1}|_{C_\ell^{\text{CMB,fid}}} + \frac{\partial R_L^{-1}}{\partial C_\ell^{\text{CMB}}}(C_\ell^{\text{CMB}} - C_\ell^{\text{CMB,fid}}) \\ &\approx R_L^{-1}|_{C_\ell^{\text{CMB,fid}}} \left[ 1 + M_L^\ell |_{C_\ell^{\text{CMB,fid}}}(C_\ell^{\text{CMB}} - C_\ell^{\text{CMB,fid}}) \right], \end{aligned} \quad (B2) $$

where $M_L^\ell|_{C_\ell^{\text{CMB,fid}}} = \partial \ln R_L^{-1} / \partial C_\ell^{\text{CMB}}|_{C_\ell^{\text{CMB,fid}}}$ is the linearized normalization correction matrix. We compute it using finite differences at each $\ell$. After correction for $N_0$, and neglecting $N_1$, this leads to the expected value of the reconstructed lensing power spectrum being related to the true spectrum by

$$ \begin{aligned} \langle C_L^{\hat{\phi}\hat{\phi}} \rangle &\approx \left( \frac{R_L^{-1}|_{C_\ell^{\text{CMB,fid}}}}{R_L^{-1}|_{C_\ell^{\text{CMB}}}} \right)^2 C_L^{\hat{\phi}\hat{\phi}} \\ &\approx \left[ 1 - 2M_L^\ell|_{C_\ell^{\text{CMB,fid}}}(C_\ell^{\text{CMB}} - C_\ell^{\text{CMB,fid}}) \right] C_L^{\hat{\phi}\hat{\phi}}. \end{aligned} \quad (B3) $$

For a lensing-power-spectrum-only likelihood, we need to marginalize over the uncertainty in $C_\ell^{\text{CMB}}$. In Planck Collaboration et al. (2020b), the marginalization is performed analytically using the Plik lite CMB bandpower covariance. At the time of writing this article, the ACT DR6 CMB covariance is not yet available. Instead, we sample 1000 ΛCDM CMB power spectra from the ACT DR4 + Planck parameter chains used in Aiola et al. (2020) to generate an ensemble of smooth power spectrum curves consistent with the ACT DR4 + Planck power spectrum measurements; we then propagate these power spectrum curves through to the lensing normalization and compute the resulting additional dispersion in the lensing bandpowers. As a result, for the lensing-only analysis, we add

an additional term to the covariance matrix:

$$ \tilde{C}_{bb'} = C_{bb'} + C_{bb'}^{\text{CMB}}, \quad (B4) $$

where now $C_{bb'}^{\text{CMB}}$ is the additional covariance induced by the CMB marginalization. In the right panel of Figure 36, we demonstrate the inflation of lensing bandpower error bars resulting from the marginalization over CMB power spectra. This process leads to an increase of at most 2.5% in error bars and the introduction of small correlations between bands. For comparison, we have conducted the marginalization using the ACT DR4+WMAP parameter chains and found a slightly larger increase of up to 3.5%. In the baseline analysis range, the error bars on the lensing amplitude increase from $\sigma(A_{\text{lens}}) = 0.023$ to $\sigma(A_{\text{lens}}) = 0.0256$ when marginalizing over the ACT DR4+Planck chains. In comparison, when marginalizing over the ACT DR4+WMAP chains, the error bars increase to $\sigma(A_{\text{lens}}) = 0.0264$. Although this seems like a nonnegligible change, the impact on cosmological parameters such as $S_8^{\text{CMBL}}$ appears significantly smaller. In particular, the error on the $S_8^{\text{CMBL}}$ only increases from 0.021 to 0.022 when we include the ACT DR4+Planck-based marginalization. One may question the choice of using ACT DR4+Planck chains rather than, for example, only the ACT DR4 chains, as well as the use of smooth ΛCDM curves instead of allowing the bandpowers to vary freely given the likelihood. However, it is important to note that the marginalization approach remains robust as long as the CMB bandpowers have been well characterized, regardless of the specific CMB measurement used. The larger scales, which play an important role in
determining the overall power spectrum amplitude, are already robustly measured by Planck. Therefore, we believe that it is a valid approach to incorporate this information in the marginalization procedure. With regard to the use of smooth $\Lambda$CDM curves, we explicitly test our model dependence by using chains for an extended model (ACT DR4+Planck $\Lambda$CDM +$A_{\text{lens}}^{2\text{pt}}$) in the marginalization. Note that $A_{\text{lens}}^{2\text{pt}}$ is a parameter that characterizes the impact of gravitational lensing on the CMB power spectra and is distinct from the lensing amplitude parameter $A_{\text{lens}}$. We find that using the extended-model chains results in an increase of only 3% on the error bars of the lensing amplitude, $\sigma(A_{\text{lens}})$, compared to the ACT DR4+Planck $\Lambda$CDM case. This negligible change is an illustration of the fact that this marginalization does not introduce significant model dependence into our lensing measurement. Lastly, it is worth highlighting that the ACT DR6 power spectra will have improved constraints on both small and large scales. As a result, future analyses will have the ability to marginalize directly over the ACT-only power spectra.

## Appendix C
Estimation of Instrumental Systematics

### C.1. Calibration and Beam Systematics

Both beam and calibration errors have the potential to bias lensing measurements. One way in which lensing can be affected by such errors is through their coherent rescaling of the measured power spectra across a range of scales. This rescaling then impacts the overall normalization of the measured lensing power spectrum, as discussed in detail in Appendix B. Hence, it is crucial to account carefully for both beam and calibration errors in any lensing analysis to avoid systematic biases.

We run the full lensing pipeline for multiple different realizations of the calibration factor and beam transfer functions, based on their estimated means and uncertainties as discussed in Sections 3.2 and 3.3. We find that the change in the amplitude of the lensing power spectrum (as opposed to a scale-dependent fractional correction) is the dominant effect. In fact, in our tests, we find that the response is, to a good approximation, given by $\Delta \ln C_{\ell}^{\phi\phi} \approx 2\Delta \ln \bar{C}_{\ell}$, where $\Delta \ln \bar{C}_{\ell}$ and $\Delta \ln C_{\ell}^{\phi\phi}$ are fractional changes in the CMB power spectrum averaged over $\ell \in [600, 3000]$ and fractional changes in the lensing power spectrum, respectively. Since quantifying the effect of beams and calibration errors by running the full lensing power spectrum pipeline is computationally prohibitive, we may use the approximation above to quantify their effect.

To quantify the typical effect of beam and calibration errors on lensing, we proceed as follows. We do not attempt to incorporate beam and calibration errors into our statistical error but instead will treat the typical error as a systematic that we argue is negligible. First, as our calibration factors are computed with respect to the Planck maps, they may not be independent across array frequencies, and this correlation can increase the overall bias. To investigate this issue, we estimate their correlations by jointly sampling calibration factors from a likelihood of the following form (up to an irrelevant constant):

$$-2 \ln \mathcal{L}(\{c_{A_f}\}|\{\Delta_{\ell}^{A_f}\}) = \sum_{\ell=\ell_{\min}}^{\ell_{\max}} \sum_{A_f, A_f'} \Delta_{\ell}^{A_f} [\Sigma_{\ell}^{-1}]^{A_f A_f'} \Delta_{\ell}^{A_f'}, \quad (\text{C1})$$

where $\Delta_{\ell}^{A_f} = c_{A_f}^2 C_{\ell}^{\text{ACT} \times \text{ACT}, A_f} - c_{A_f} C_{\ell}^{\text{ACT} \times P, A_f}$ is the difference between the auto-power spectrum of ACT array frequency $A_f$ and the cross-power between ACT and the Planck map closest in frequency, after scaling the ACT array-frequency map by the calibration factor $c_{A_f}$. The indices $A_f$ and $A_f'$ run through the ACT array frequencies used in our analysis, and $\Sigma_{\ell}$ is an approximate analytic covariance matrix for the $\Delta_{\ell}^{A_f}$ (assumed diagonal in $\ell$) computed using the fiducial CMB theory spectrum and measured noise spectrum. Our analysis shows that the 90 GHz channels are strongly correlated at a level of 51%, while the 150 GHz channels exhibit lower correlations ranging from 10% to 18%. This difference can be attributed to the lower noise in the 90 GHz channels. Furthermore, the correlation between the 90 and 150 GHz channels is negligible because calibration factors are fitted at different multipole scales for the two frequency bands. We draw 100 independent MC samples of the calibration and beam, based on their estimated means and uncertainties (see Sections 3.2 and 3.3 and Lungu et al. 2022 for details), as well as accounting for the estimated correlations among calibration factors. We then process raw data maps using these samples, following the preprocessing procedure described in Section 5, and remeasure the CMB power spectra. Figure 37 (left) shows the fractional change in the coadded temperature power spectrum. We find that the scatter in the power spectrum is less than 0.2% in our baseline analysis range ($\ell \in [600, 3000]$).

We then convert the error in the power spectra to an error in the lensing power spectra using the approximation above. We take the rms ($1\sigma$) shift as an estimate of the typical systematic bias from ignoring beam and calibration uncertainty. We show the derived systematic biases for calibration and beam in the right panel of Figure 37. We conclude that these are subdominant sources of error compared to our 2.3% statistical uncertainty in our lensing power spectrum measurement.

In particular, for calibration, we find $\Delta A_{\text{lens}} = 0.00432 (0.18\sigma)$ and $\Delta A_{\text{lens}} = 0.00419 (0.09\sigma)$ for the MV and MVPol estimators, respectively. Similarly, for beam uncertainty, we find $\Delta A_{\text{lens}} = 0.00113 (0.05\sigma)$ and $\Delta A_{\text{lens}} = 0.00110 (0.02\sigma)$ for MV and MVPol, respectively.

### C.2. Temperature-to-polarization Leakage Verification

As described in Section 3.2, we find evidence for some degree of temperature-to-polarization ($T \to P$) leakage in our measurements. This leakage is estimated to be small but could affect the accuracy of our polarization-based lensing estimators, mainly due to its effect on normalization. To quantify its effect, we add additional leakage terms to the polarization multipoles for each array frequency:

$$\begin{aligned} \tilde{X}_{\ell m} &= X_{\ell m} + X_{\ell m}^{\text{leakage}} \\ &= X_{\ell m} + B_{\ell}^{T \to P} T_{\ell m}, \end{aligned} \quad (\text{C2})$$

where $X_{\ell m}$ are either the baseline E or B multipoles, $B_{\ell}^{T \to P}$ is the estimated leakage beam, and $T_{\ell m}$ are the baseline T multipoles. The left panel of Figure 37 shows an example of the leakage beams in our analysis. Since the baseline data, $X_{\ell m}$, already include the $T \to P$ leakage effect, our modified maps, $\tilde{X}_{\ell m}$, include twice the amount of leakage compared to a scenario with no leakage. We
<figure><img src="image_42.png" /><figcaption>**Figure 37.** Left: mean and $1\sigma$ confidence intervals of the reprocessed coadded temperature power spectrum compared to the baseline. The scatter is due to uncertainties in calibration factors (shown in red) and instrument beams (shown in blue). The confidence intervals are computed using the 100 MC samples of calibration and beams applied to raw maps processed in the same way as the baseline. The grayed-out regions are outside of the baseline analysis range. Right: similar to the left panel, but for the lensing power spectrum. This is computed using the same 100 samples but with the fractional error in the CMB power spectrum propagated to the lensing power spectrum with the approximation $\Delta \ln C_L^{\phi\phi} \approx 2\Delta \ln \bar{C}_l$ discussed in the text. The grayed-out regions are outside of the baseline analysis range.</figcaption></figure>

run our analysis pipeline on these modified maps and find that the shift in the estimated lensing power spectrum is less than 0.1% compared to our MV baseline (see the right panel of Figure 37). In terms of our $\Delta A_{\text{lens}}$ metric (Equation (44)), we find $\Delta A_{\text{lens}} = -0.00021 (-0.01\sigma)$ and $\Delta A_{\text{lens}} = -0.00052 (-0.01\sigma)$ for MV and MVPol, respectively.

Preliminary analysis (at the time of preparing this paper) of the ACT DR6 CMB power spectra has revealed an additional leakage for a specific array frequency. This leakage is consistent with a constant temperature-to-polarization leakage in the form of $\hat{E}_{\ell m} = E_{\ell m} + \alpha T_{\ell m}$, where $\alpha = 0.0035$. While this feature is yet to be confirmed, we have developed a toy model to investigate its potential impact. In this toy model, we assume that temperature maps leak into polarization maps (both *E* and *B*) at all array frequencies with a constant coefficient of $\alpha = 0.0035$. However, it should be noted that this feature has only been observed in a specific array frequency. Thus, it is unlikely that it would affect all channels equally, and this model likely represents a worst-case scenario. We run our analysis pipeline on this toy model and find $\Delta A_{\text{lens}} = -0.00461 (-0.19\sigma)$ and $\Delta A_{\text{lens}} = -0.00887 (-0.18\sigma)$ for MV and MVPol, respectively.

### C.3. Polarization Angle Verification

At the time of writing, a precise characterization of the absolute polarization angle of the DR6 data set is not yet available. However, tests differencing DR4 and DR6 EB spectra have found that the EB power spectrum from the DR6 data set is consistent with that of the DR4 data set. This suggests that the DR6 polarization rotation angle should be consistent with the DR4 estimate of $\Phi_p = -0.07 \pm 0.09$ deg. In order to assess the impact of a nonzero polarization angle on our results, we artificially rotate the *Q*/*U* maps of our data set

by $\Phi_p = -0.16$ deg (the mean minus $1\sigma$) and reanalyze the resulting maps using our analysis pipeline. The resulting lensing auto-spectrum estimates for these rotated maps are shown in the right panel of Figure 38 as the blue dashed curve. As demonstrated in the figure, our bandpowers are not significantly affected by the rotation angle; the rotation results in minimal changes to our $\Delta A_{\text{lens}}$ statistics. We find $\Delta A_{\text{lens}} = -0.00025 (-0.01\sigma)$ and $\Delta A_{\text{lens}} = 0.00070 (0.01\sigma)$ for MV and MVPol, respectively.

### C.4. Polarization Efficiency

As previously discussed in Section 5, we apply a correction to our polarization maps based on the mean of the measured polarization efficiencies. Following this correction, our baseline and polarization-only estimators show good agreement (see Figure 13). To assess the robustness of our correction method, we conduct an approximate test in which the estimated polarization efficiency was artificially lowered by $1\sigma$. It is important to note that the deviations from the mean polarization efficiency are not anticipated to correlate strongly across array frequencies, and as such, it is unlikely that all efficiencies would be biased in the same manner. Therefore, this test represents a very conservative upper limit. The resulting lensing auto-spectrum estimates for these maps are shown in the right panel of Figure 38 as the green dashed curve. We find $\Delta A_{\text{lens}} = -0.00896 (-0.37\sigma)$ and $\Delta A_{\text{lens}} = -0.01884 (-0.38\sigma)$ for MV and MVPol, respectively. Given that these values were obtained from conservative estimates, we anticipate that the actual data bias is much smaller than the values presented. We therefore include them in our summary table as conservative upper limits.
<figure><img src="image_43.png" /><figcaption>**Figure 38.** Left: example of the $T \to P$ leakage beams (discussed in Section 3.2). We calculate the impact of these leakage beams on the lensing bandpowers, following the method outlined in Appendix C.2, and find $\Delta A_{\text{lens}} = -0.00021 (-0.01\sigma)$ and $\Delta A_{\text{lens}} = -0.00052 (-0.01\sigma)$ for MV and MVPol, respectively. Right: effects of polarization angle, $T \to P$ leakage, and polarization efficiency on lensing power spectrum estimates. See Appendices C.2, C.3, and C.4 for further information. The effects from polarization angle and $T \to P$ leakage are negligible. The impact of polarization efficiency is larger (green dashed line), although it is important to note that the result here represents a very conservative upper limit. The bias to the lensing amplitude for each case can be found in Table 3 in the main text.</figcaption></figure>

<figure><img src="image_44.png" /></figure>

## Appendix D
### Pipeline Verification and Origin of Multiplicative Normalization Correction

We demonstrated the performance of our pipeline using simulations in Figure 6. The multiplicative MC correction constitutes a 10%–15% effect; however, it is well understood and arises almost entirely from the fact that the Fourier modes of $|\ell_x| < 90$ and $|\ell_y| < 50$ are filtered away. To demonstrate this, Figure 39 shows the ratio of the cross-spectrum between a lensing reconstruction and the input lensing map with the input

<figure><img src="image_45.png" /><figcaption>**Figure 39.** Cross-spectrum between the reconstructed lensing map from simulated CMB maps and the lensing map of the input simulation, $C_L^{\phi\phi}$, divided by the input lensing auto-spectrum $C_L^{\phi\phi}$. It can be seen in the red curve that the Fourier-space filter, which helps to remove ground pickup, causes a multiplicative suppression of the reconstructed lensing map by 10%–20%. Without this filter (blue curve), the lensing suppression is absent, except at low multipoles where mask effects become important. This well-understood multiplicative suppression due to Fourier-space filtering is simply corrected using a multiplicative factor derived from simulations.</figcaption></figure>

auto-spectrum from simulations. With no Fourier-space mask applied, we find that this ratio is quite close to unity (with the departures at large scales expected from the couplings induced by the mask); in contrast, the same ratio computed on maps where the Fourier-space mask is applied is lower by 10%–20%.

## Appendix E
### Lensing Power Spectrum Biases

This appendix describes in more detail the algorithms used to compute the lensing power spectrum biases; we focus on the computation of $N_0$, $N_1$ and additive MC biases.

#### E.1. Realization-dependent $N_0$

Lensing power spectrum estimation aims to probe the connected four-point function that is induced by lensing. However, the naive lensing power spectrum estimator $\hat{C}_L^{\phi\phi}$ [$\bar{X}, \bar{Y}, \bar{A}, \bar{B}$] $\sim$ XYAB also contains disconnected contributions arising from Gaussian fluctuations (e.g., $\langle XA \rangle \langle YB \rangle$), which are nonzero even in the absence of lensing. These contributions, which are typically referred to as the Gaussian or $N_0$ bias and which can be understood as a bias arising from the “noise” in the lensing reconstructions, must be subtracted to recover an unbiased estimator of lensing.

The Gaussian ($N_0$) bias is estimated using an algorithm involving different pairings of data and two sets of independent simulation maps, denoted with superscripts S and S' (Namikawa et al. 2013; Planck Collaboration et al. 2020b):

$$ \Delta C_L^{\text{Gauss}} = \langle \bar{C}_L^\chi[XY^S, AB^S] + \bar{C}_L^\chi[X^S Y, AB^S] + \bar{C}_L^\chi[X^S Y, A^S B] + \bar{C}_L^\chi[XY^S, A^S B] - \bar{C}_L^\chi[X^S Y^S, A^S B^S] - \bar{C}_L^\chi[X^S Y^S, A^S B^S] \rangle_{S,S'} \quad (E1) $$

This estimator can be obtained from the Edgeworth expansion of the lensing likelihood; it has the useful feature
that it corrects for a mismatch between two-point functions of the data and of simulations. The estimator achieves this by also using the two-point function of the data, rather than simulations alone, when calculating this Gaussian bias; the combination employed above can be shown to be insensitive to errors in the simulation two-point function, to first order in the fractional error. Furthermore, this estimator helps to reduce the correlation between different lensing bandpowers (Hanson et al. 2011), as well as the correlation of the lensing power spectra with the primary CMB spectra (Schmittfull et al. 2013). Finally, the negative of the last two terms of Equation (E1), obtained purely from simulations, constitutes the MC N0, or MCN0. This term corresponds to the Gaussian reconstruction noise bias of the lensing bandpowers averaged over many CMB and lensing reconstructions; we use this term in Appendix E.3 to estimate the size of the additive MC bias.

We use 480 different realizations of noiseless simulation pairs S, S' to calculate this bias for the real measurement. Noiseless simulations can be used here because we are using the cross-correlation-based estimator, and the absence of noise helps to reduce the number of simulations required to achieve convergence. We verify that increasing the number of realizations from 240 to 480 did not substantially affect our results (with $\Delta A_{\text{lens}} = 5 \times 10^{-6}$). We hence conclude that 240 realizations are sufficient for convergence, though we use 480 simulations in our data and null tests to be conservative.

The same RDN0 calculation should also be carried out for each simulation used to obtain the covariance matrix, with the “data” corresponding to the relevant simulated lensed CMB realization. However, in practice, it is computationally unfeasible to estimate this RDN0 bias for each of the 792 simulations used for the covariance matrix, due to the very large number of reconstructions required for each RDN0 computation. Therefore, we resort to an approximate version of the realization-dependent $N_0$, referred to as the semianalytic $N_0$; we discussed this in detail in Section 5.11.1.

## E.2. $N_1$ Bias

In addition to the Gaussian bias, another, smaller bias term arises from “accidental” correlations of lensing modes that are not targeted by the QE, as described in detail in Kesden et al. (2003). Since this bias is linear in $C_L^{\phi\phi}$, it is denoted the $N_1$ bias.

We calculate the $N_1$ bias by using 90 pairs of simulations with different CMB realizations but common lensing potential maps denoted ($S_\phi, S'_\phi$) and 90 pairs of simulations with different CMB and lensing potential ($S, S'$) (we have carried out convergence tests as in Appendix E.1 to ensure that 90 simulations are sufficient):

$$ \Delta C_L^{N_1} = \langle \bar{C}_L^\chi [X^{S_\phi} Y^{S'_\phi}, A^{S_\phi} B^{S'_\phi}] \\ + \bar{C}_L^\chi [X^{S_\phi} Y^{S'_\phi}, A^{S'_\phi} B^{S_\phi}] \\ - \bar{C}_L^\chi [X^{S_\phi} Y^{S'_\phi}, A^{S'_\phi} B^{S'_\phi}] \\ - \bar{C}_L^\chi [X^{S_\phi} Y^{S'_\phi}, A^{S'_\phi} B^{S_\phi}] \rangle_{S,S',S_\phi,S'_\phi} \quad (E2) $$

We do not include biases at orders higher than $N_1$, since biases such as $N^{(3/2)}$ due to nonlinear large-scale structure growth and post-Born lensing effects are still negligible at our levels of sensitivity (Beck et al. 2018).

## E.3. Additive MC Correction

Although subtraction of the Gaussian and $N_1$ biases results in a nearly unbiased lensing power spectrum, we calculate an additive MC bias $C_L^{\text{MC}}[XY, AB]$ with simulations to absorb any residual arising from nonideologies that have only been approximately captured, such as the effects of masking. As can be seen in Figure 6, this bias term is very small. This justifies why we may model this bias term as additive rather than multiplicative: any true functional form of the MC correction could be simply Taylor expanded to give an additive term. This additive MC bias is given by taking it to be the difference of the fiducial lensing spectrum $C_L^{\phi\phi}$ and the average of 480 (mean-field subtracted) cross-only lensing power spectra obtained from simulations, i.e.,

$$ \Delta C_L^{\text{MC}} = C_L^{\phi\phi} - (\Delta A_L^{\text{MC,mul}} \langle C_L^\chi \rangle_{\text{sim}} - \text{MCN0} - \Delta C_L^{\text{N1}}). \quad (E3) $$

Here $\Delta A_L^{\text{MC,mul}}$ is the multiplicative bias term defined in Section 5.8.1.

# Appendix F

## Cross-correlation-based Estimator: Motivation from Noise-only Null Tests

A novelty in our lensing measurement presented here compared to past results is the use of the cross-correlation-based lensing estimator, where we estimate the lensing bandpowers from four independent data splits as outlined in Section 5.8.1. At a cost of modestly reducing the S/N by avoiding using the data with nonindependent noise, we are completely immune to the assumptions made in modeling or simulating the instrument noise. Accurate noise modeling is important for the lensing analysis for calculating the Gaussian disconnected bias (RDN0), which receives a contribution from both the CMB signal and foregrounds, as well as the instrument noise. Our original plan was to use the standard QE, as our simulated results agree with the data within 5% at the CMB two-point power spectrum level and the RDN0 algorithm itself is designed to self-correct for small differences between data and simulations. However, using this standard estimator, we were unable to pass the noise-only null test, which directly tests the robustness of our pipeline and the accuracy of our noise simulations. This noise-only null test consists of the following: For each data array, we prepare four splits of data map {$X_0, X_1, X_2, X_3$} from the original eight splits by weighting them with their respective inverse-variance maps. We then form the following signal-nulled data map: $[(X_0 + X_1) - (X_3 + X_4)]/2$. The same processing is done for a set of 300 simulations used to calculate the mean field and the RDN0. We do not need to estimate the $N_1$ here, as this term is zero owing to the absence of the lensing signal.

The left panel of Figure 40 shows the resulting null test for the array PA5 f090 obtained with noise simulation drawn from three different types of noise models: directional wavelet, isotropic wavelet, and tiled while using the same reconstruction filter and normalization. We see that the “null” bandpowers are dependent on the details of the noise modeling, and in this particular case, only the tile model passes the test. This scatter of results suggests that an accurate noise model required for lensing is hard to construct for ground-based CMB surveys covering a significant portion of the sky, with the main challenges coming from the combination of the atmosphere and the ACT scanning strategy.
<figure><img src="image_46.png" /></figure>

<figure><img src="image_47.png" /><figcaption>**Figure 40.** Left: noise-only null test using the standard QE for the three different types of noise simulations (directional wavelet, isotropic wavelet, and tiled) for the array PA5 f090. Right: noise-only null test using the cross-correlation-based estimator for the three different types of noise simulations (directional wavelet, isotropic wavelet, and tiled) for the array PA5 f090. It can be seen that the cross-correlation-based estimator avoids biases arising from noise mismodeling. We also note that, with the cross-correlation-based estimator, the errors and PTEs are similar for all noise simulations, indicating that the covariance matrix is stable for this estimator.</figcaption></figure>

Space-based surveys like Planck avoid these complications from the atmosphere, and many previous ground-based measurements cover a smaller region at lower precision, which allows the standard lensing power spectrum estimator to be used. It is worth noting that we are investigating improvements to the noise modeling on the basis of this null test result.

Switching to the four cross estimator and performing the same null test as in Section 6.4.2 means that this noise-only null does not depend on noise for the mean field and RDN0, and hence the null test is independent of the details of the noise modeling. This can be seen in the null bandpowers in the right panel of Figure 40, obtained from the same noise sims. The null bandpowers are now consistent with zero. We also obtain nearly the same PTE values independent of the noise simulation type used, indicating that our covariance matrix is stable.

## Appendix G
### Semianalytic $N_0$

The semianalytic $N_0$ bias can be computed as follows:

$$ \Delta C_L^{\text{Gauss,diag}}[\bar{X}, \bar{Y}, \bar{W}, \bar{Z}] = \frac{\mathcal{R}_L^{XY-1}\mathcal{R}_L^{WZ-1}}{\Delta^{XY}\Delta^{ZW}} \left[ -\zeta_L^{XY,WZ}[C_\ell - \hat{C}_\ell] + \zeta_L^{XY,ZW}[\hat{C}_\ell] \right], \quad (G1) $$

where $\Delta^{XX} = 2$, $\Delta^{EB} = \Delta^{TB} = 1$. Here $\zeta$ is equal to the integral of the lensing normalization if $X=Y=W=Z$, and it depends on $C_\ell^{\text{fid}}$, the theoretical power spectra, and $\hat{C}_\ell$, the realization-dependent CMB spectrum. Writing this out explicitly, it has the form

$$ \zeta_1^{XY,ZW}[C] = \int \frac{d^2\mathbf{L}}{(2\pi)^2} f_{\ell,L}^{XY} f_{\ell,L}^{ZW} F_L^X F_L^Z F_{L'}^Y F_{L'}^W (C_L^{XZ} C_{L'}^{YW} + C_L^{XW} C_{L'}^{YZ}), \quad (G2) $$

where $F_\ell^X = 1/(C_\ell^X + N_\ell^X)$ is the total diagonal power spectrum used for the filter.<fnref n="109" />

## Appendix H
### Gaussianity of the Bandpowers

The likelihood we use in the cosmological analysis is built with the assumption that our bandpowers are Gaussian distributed. A priori, this assumption should not hold at arbitrary precision given the fact that even our lensing reconstruction map arises from a quadratic function of the (nearly Gaussian) observed CMB fields. However, the large number of effectively independent modes in our bandpowers suggests that the central limit theorem will drive the distribution of our bandpowers toward a Gaussian. We test this assumption using a set of 400 simulated lensing reconstructions by investigating the distribution of the resulting bandpowers. We choose the lowest and highest bins as representative examples; the distributions of these bandpowers

<figure><img src="image_48.png" /><figcaption>**Figure 41.** To test that a Gaussian likelihood for the lensing power spectrum bandpowers is sufficient, we plot histograms of the distribution of $C_L^{\hat{\phi}\hat{\phi}}$ bandpowers obtained from simulations for a bin with $L_B = 53$ (top panel) and $L_B = 700.5$ (bottom panel). In both cases, the distribution of the bandpowers closely follows a Gaussian distribution with the expected properties (i.e., a mean and width given by the bandpower mean and standard deviation). No evidence for departures from Gaussianity is found.</figcaption></figure>

<footnote n="109">¹⁰⁹ For simplicity we showed the flat-sky expression here, but in the pipeline this is done using the curved-sky formalism.</footnote>
are shown in Figure 41. The Kolmogorov–Smirnov statistic (Massey 1951) shows that both distributions are well described by Gaussians, with PTEs of 0.6 and 0.9, respectively.

# Appendix I
## Additional Null Tests

In the main text, for brevity we focused on summary statistics characterizing our ensemble of null tests and on some of the most crucial individual tests. In this appendix, we explain and show results from the additional null tests we have performed with our data.

### 1.1. Coadded Split Differences

A stringent noise-only null test, in the sense that it has small errors, is obtained by coadding in harmonic space the eight splits of each array with the same weights used to combine the data and forming four null maps by taking $X^{i null} = \text{split}_i - \text{split}_{i+4}$. The filter and normalization used for this test are the same as those used for the baseline lensing analysis.

The gradient null is shown in the left panel of Figure 42, and the curl is shown in the fifth panel of Figure 47. While the curl passes with a PTE of 0.85, the PTE for the gradient test is low, 0.02; however, given the large number of tests we have run, the fact that the failure does not exhibit any obvious systematic trend, and the fact that the magnitude of the residuals is negligibly small compared to the signal, we do not consider this concerning.

### 1.2. Isotropy

We test for consistency of our lensing measurements across different regions of the sky by dividing our sky into 50 patches $10^\circ \times 25^\circ$ nonoverlapping patches.<fnref n="110" /> We compare the lensing power spectrum obtained in each patch, debiased and with errors estimated from simulations using our baseline procedure described in Section 5.8. The tests show that our bandpowers are consistent with the assumption of statistical isotropy in the lensing signal and lensing power spectrum. This test is particularly targeted toward identifying spurious residual point sources in the map; higher spatial resolution and smaller patches are helpful for this. It complements well other tests of isotropy with larger regions: for example, the comparison of north versus south patches where we similarly did not find any evidence for anisotropy. Furthermore, effects such as beam asymmetry and small time variations can all lead to spatially varying beams, which could induce mode coupling similar to lensing.

The bandpower consistency across the different regions also provides evidence that there are no problematic beam variations across the analysis footprint.

The ensemble of 50 isotropy lensing is shown in Figure 43. These have PTE values consistent with a uniform distribution; the distribution of the combined 100 lensing and curl null

bandpowers passes the K-S statistic with a PTE of 0.66 as shown in Figure 44.

### 1.3. Time-split Difference Map Level

We difference maps made from observations during the period 2017–2018 from those constructed from the period 2018–2021. This test targets potential instrument systematics that may be different between these two phases and also makes sure that the beam, calibrations, and transfer functions used are consistent across different observation periods.

The gradient null bandpowers in the right panel of Figure 42 are marginally consistent with zero with a PTE of 0.05; the curl in Figure 47 is also consistent with zero with a PTE of 0.48. To obtain accurate PTEs and debiasing in the bandpower test described in Appendix 1.4 below, we produce special noise simulations capturing the characteristics of the two time splits using *mnms* (Atkins et al. 2023).

### 1.4. Time-split Bandpower-level Test

We difference lensing spectra made from 2017–2018 observations from the spectra obtained from data taken during the period 2018–2021. This test targets systematic variations of beam, calibration, and transfer functions with observing time. We apply the same filter and normalization used for our baseline lensing analysis and find no evidence of systematic differences between the two time splits. The bandpower difference is shown in the left panel of Figure 45; it has a passing PTE of 0.33.

### 1.5. PWV Split Differences

We obtain lensing spectra from two sets of sky maps that are made to contain observations at high and low PWV, respectively; a null difference of these two spectra can test for instrument systematics, e.g., systematics that have any dependence on the detector optical loading and to the level of atmospheric fluctuations. The noise levels of the high-PWV and low-PWV maps are noticeably different; hence, for the Wiener and inverse-variance filtering of the maps, the filter is built using a power spectrum whose noise is given by the average noise power of both maps. The null test results are shown in the right panel of Figure 45 with a PTE of 0.89. To obtain accurate PTEs and debiasing in this null test, we draw special noise simulations capturing the characteristics of the two PWV splits using *mnms*.

### 1.6. Marginalized Posteriors for Cosmology Runs

Here we provide a detailed presentation of the complete posterior contours for the cosmology runs that are presented in this paper. The constraints obtained from our DR6 lensing baseline and extended range are depicted in the left panel of Figure 46 in blue and red, respectively. Likewise, the posterior plots for the baseline and extended range when incorporating Planck NPIPE lensing bandpowers are shown in the right panel of Figure 46.

<footnote n="110">¹¹⁰ The whole footprint is divided into $10^\circ \times 25^\circ$ rectangular patches, but some patches at the edge of the map include masked, zeroed regions outside our analysis footprint. This is why $50 \times 250 = 12,500 \text{ deg}^2$ does not correspond to the exact area of 9400 deg$^2$; the fact that some regions (e.g., patch 33) only include a small unmasked area also explains why such regions' spectra have larger errors.</footnote>
<figure><img src="image_49.png" /><figcaption>**Figure 42.** Left: all array frequencies coadd difference map noise-only bandpowers. Noise-only test obtained by coadding first the individual array frequencies and taking the difference of the coadded splits. The low PTE of 0.02 is not concerning, as the null bandpowers do not exhibit a systematic trend and the size of the residual is small compared to the lensing signal. Right: null bandpowers obtained from differencing the time-split 1 and time-split 2 maps.</figcaption></figure>

## 1.7. Sensitivity of Lensing-only Constraints to $\Omega_k$

Extensions to the flat-$\Lambda$CDM cosmology with parameters such as the spatial curvature $\Omega_k$ are considered in our companion paper (Madhavacheril et al. 2024). In principle, the lensing-only constraints could be significantly affected by fully freeing curvature since the CMB lensing power spectrum amplitude can be affected by curvature; however, typical curvature values preferred by current constraints, such as those in our companion paper, are sufficiently close to flatness that the lensing-only constraints are minimally affected by assuming such curvature values. For example, assuming the

maximum posterior value for the curvature from Madhavacheril et al. (2024) only minimally shifts our lensing-only constraints to $S_8^{\text{CMBL}} = 0.814 \pm 0.022$, consistent with the baseline constraints $S_8^{\text{CMBL}} = 0.818 \pm 0.022$.

## 1.8. Curl Results Summary

We present a more comprehensive summary of the curl results for the different null test, including a compilation of the PTEs and $\chi^2$ values in Table 6. Our analysis reveals no significant evidence of systematic biases in our measurements.
<figure><img src="image_50.png" /><figcaption>**Figure 43.** Null tests differencing the lensing power spectrum measured in 50 different patches (regions) of the sky from the baseline lensing power spectrum measured on 9400 deg². No significant evidence is found for deviations from statistical isotropy of the lensing signal. This test also shows that there is no evidence for spatially varying beams at levels significant for our lensing measurement. Each patch spans 10° × 25°, with patch 0 starting at the lower right corner and going from right to left in steps of 25°.</figcaption></figure>
<figure><img src="image_51.png" /><figcaption>**Figure 44.** Distribution of the PTEs for the 100 isotropy bandpower difference and curl tests. The distribution is consistent with a uniform distribution, passing the K-S statistic with a PTE of 0.66.</figcaption></figure>

<figure><img src="image_52.png" /><figcaption>**Figure 45.** Left: comparison of lensing spectra from time-split 1 and time-split 2 maps. Right: the lensing power spectrum measured with PWV high and PWV low maps. No significant difference between the two measurements is found.</figcaption></figure>
<figure><img src="image_53.png" /><figcaption>**Figure 46.** Left: marginalized posteriors for the combined ACT DR6 lensing constraints in the baseline range (red) and the constraints obtained using the extended range of scales (blue). Right: marginalized posteriors for the combined ACT DR6 and Planck NPIPE Lensing constraints.</figcaption></figure>

<table><caption>**Table 6**
Summary of Results for the Curl Versions of Our Null Tests</caption><thead><tr><th>Curl Null Test</th><th>χ<sup>2</sup></th><th>(PTE)</th></tr></thead><tbody><tr><td>PA4 f150 noise-only</td><td>5.2</td><td>(0.88)</td></tr><tr><td>PA5 f090 noise-only</td><td>13.9</td><td>(0.18)</td></tr><tr><td>PA5 f150 noise-only</td><td>8.1</td><td>(0.62)</td></tr><tr><td>PA6 f090 noise-only</td><td>11.6</td><td>(0.31)</td></tr><tr><td>PA6 f150 noise-only</td><td>19.1</td><td>(0.04)</td></tr><tr><td>Coadded noise</td><td>5.4</td><td>(0.85)</td></tr><tr><td>600 &lt; l<sub>CMB</sub> &lt; 2000</td><td>14.2</td><td>(0.16)</td></tr><tr><td>600 &lt; l<sub>CMB</sub> &lt; 2500</td><td>15.1</td><td>(0.13)</td></tr><tr><td>800 &lt; l<sub>CMB</sub> &lt; 3000</td><td>13.1</td><td>(0.22)</td></tr><tr><td>1500 &lt; l<sub>CMB</sub> &lt; 3000</td><td>13.8</td><td>(0.18)</td></tr><tr><td>CIB deprojection</td><td>8.6</td><td>(0.57)</td></tr><tr><td>PA4 f150 – PA5 f090</td><td>14</td><td>(0.16)</td></tr><tr><td>PA4 f150 – PA5 f150</td><td>9.5</td><td>(0.48)</td></tr><tr><td>PA4 f150 – PA6 f090</td><td>8.0</td><td>(0.63)</td></tr><tr><td>PA4 f150 – PA6 f150</td><td>9.6</td><td>(0.48)</td></tr><tr><td>PA5 f090 – PA5 f150</td><td>15.3</td><td>(0.12)</td></tr><tr><td>PA5 f090 – PA6 f090</td><td>5.2</td><td>(0.88)</td></tr><tr><td>PA5 f090 – PA6 f150</td><td>6.7</td><td>(0.75)</td></tr><tr><td>PA5 f150 – PA6 f090</td><td>14.7</td><td>(0.14)</td></tr><tr><td>PA5 f150 – PA6 f150</td><td>12.0</td><td>(0.29)</td></tr><tr><td>PA6 f090 – PA6 f150</td><td>11.7</td><td>(0.30)</td></tr><tr><td>f090 – f150 map MV</td><td>8.0</td><td>(0.63)</td></tr><tr><td>f090 – f150 map TT</td><td>4.4</td><td>(0.92)</td></tr><tr><td>TT f150</td><td>13.5</td><td>(0.19)</td></tr><tr><td>TT f090</td><td>6.0</td><td>(0.82)</td></tr><tr><td>MV f150</td><td>18.0</td><td>(0.05)</td></tr><tr><td>MV f090</td><td>4.3</td><td>(0.93)</td></tr><tr><td>MV coadd</td><td>11.7</td><td>(0.37)</td></tr><tr><td>TT coadd</td><td>6.8</td><td>(0.75)</td></tr><tr><td>MVPOL coadd</td><td>13.2</td><td>(0.21)</td></tr><tr><td>North region</td><td>9.6</td><td>(0.47)</td></tr><tr><td>South region</td><td>14.7</td><td>(0.14)</td></tr><tr><td>40% mask</td><td>8.3</td><td>(0.41)</td></tr><tr><td>Aggressive ground pickup</td><td>10.8</td><td>(0.37)</td></tr><tr><td>Poor cross-linking region</td><td>9.0</td><td>(0.54)</td></tr></tbody></table>
<figure><img src="image_54.png" /><figcaption>**Figure 47.** Compilation of curl null test bandpowers. No significant evidence for any curl signal is found. These results not only test for systematic contamination but also provide additional validation of our covariance matrix estimates.</figcaption></figure>

# Appendix J

## Post-unblinding Change: From Inpainting to Subtracting Clusters

As described in Section 6.3.1, although our initial procedure for removing clusters involved inpainting a region around the cluster, we decided, based on simulation results, that

subtracting a model template was more stable; we therefore switched to this procedure after unblinding. The two versions of the lensing spectra, which use inpainting and model subtraction, respectively, are seen in Figure 48. As discussed in Section 6.3.1, the resulting change to $S_8^{\text{CMBL}}$ is found to be negligible.
<figure><img src="image_55.png" /><figcaption>**Figure 48.** DR6 lensing bandpowers compared: bandpowers from cluster mitigation with inpainting (green) can be compared with bandpowers from model subtraction (red). The difference only leads to negligible shifts in cosmological parameters.</figcaption></figure>

## Appendix K
Lensing Power Spectrum Redshift Kernel

The lensing kernel that is usually presented represents how sensitive we are to structure at a given redshift. It is given by the formula

$$W^{\kappa}(z) = \frac{3}{2} \Omega_{m} H_{0}^{2} \frac{(1+z)}{H(z)} \frac{\chi(z)}{c} \left[ \frac{\chi(z_{\star}) - \chi(z)}{\chi(z_{\star})} \right]. \quad (K1)$$

It enters the lensing convergence equation

$$\kappa(\hat{n}) = \int_{0}^{\infty} dz W^{\kappa}(z) \delta_{m}(\chi(z) \hat{n}, z), \quad (K2)$$

and it is shown as the dashed blue curve in Figure 49.

However, this kernel does not tell us where lensing we probe in the measured power spectrum comes from. To answer this question, we first recall that the lensing power spectrum can be approximated by the redshift integral

$$C_{\ell}^{\phi\phi} = \int dz W(z) P_{\text{NL}}(\ell/\chi(z)), \quad (K3)$$

where $W(z) = [H/c\chi^2]W^\kappa(z)^2$ and $P_{\text{NL}}$ is the nonlinear matter power spectrum, typically evaluated with HALOFIT (Takahashi et al. 2012). At each multipole, the relative contribution of a given redshift to the power spectrum amplitude is the logarithmic derivative, namely,

$$\frac{d \ln C_{\ell}^{\phi\phi}}{dz} = W(z) P_{\text{NL}}(\ell/\chi(z)). \quad (K4)$$

Note that having the log in the sum is correct because we estimate $A_{\text{lens}}$ for each bandpower, which is $C_{\ell}/C_{\ell}^{\text{fiducial}}$, and

<figure><img src="image_56.png" /><figcaption>**Figure 49.** Comparison of the lensing kernel and weighted $C_{\ell}^{\phi\phi}$ integrand.</figcaption></figure>

then average that $A_{\text{lens}}$ estimate over bandpowers, so it makes sense when thinking about contributions to $C_{\ell}$ to divide by $C_{\ell}^{\text{fiducial}}$, i.e., effectively average $\frac{d \ln C_{\ell}^{\phi\phi}}{dz}$ over multipoles. This quantity depends on both redshift and multipole and is shown on a 2D grid in the left panels of Figure 50. We used CLASS (Blas et al. 2011; Lesgourgues 2011a, 2011b) to compute this quantity efficiently. Our lensing power spectrum measurements are such that we are more or less sensitive to some $\ell$-range. So to give an accurate picture of where the signal measured by the lensing power spectrum comes from, we want to weight (K4) in an $\ell$-dependent way. We chose to weight it by $[C_{\ell}^{\phi\phi}/\sigma_{\ell}]^2$, so the smaller the errors are, the bigger the weight is. Finally, we obtain the error-weighted lensing kernel by taking the average over all $\ell$ values at each $z$, computing $\Sigma_{\ell}[C_{\ell}^{\phi\phi}/\sigma_{\ell}]^2 d \ln C_{\ell}^{\phi\phi}/dz$. This quantity is shown as the red line in Figure 49. And the cumulative is the orange line.

From the cumulative we can read off several interesting redshifts: 50% of the signal is from $1.16 < z < 4.09$, 67% of the signal is from $0.88 < z < 5.44$, and 95% of the signal comes from $z < 9.6$. While the mean redshift from which our lensing comes is $z = 2.16$, the peak of the red curve—i.e., the redshift that contributes most strongly to our $C_{\ell}^{\phi\phi}$—is slightly below 1, at $z = 0.96$.

Other noticeable features are the visibility of the BAO in the integrand, which is washed out in the Limber integral, and the importance of the nonlinear evolution, which gives more weight to low redshift at high multipole. This is shown in Figure 50.
<figure><img src="image_57.png" /><figcaption>**Figure 50.** Lensing power spectrum integrand, impact of nonlinear evolution (left) vs. linear (right).</figcaption></figure>

## ORCID iDs

Mathew S. Madhavacheril https://orcid.org/0000-0001-6740-5350

Irene Abril-Cabezas https://orcid.org/0000-0003-3230-4589

Simone Aiola https://orcid.org/0000-0002-1035-1854

Jason E. Austermann https://orcid.org/0000-0002-6338-0069

Nicholas Battaglia https://orcid.org/0000-0001-5846-0411

Elia Stefano Battistelli https://orcid.org/0000-0001-5210-7625

Tanay Bhandarkar https://orcid.org/0000-0002-2971-1776

Steve K. Choi https://orcid.org/0000-0002-9113-7058

Susan E. Clark https://orcid.org/0000-0002-7633-3376

William Coulton https://orcid.org/0000-0002-1297-3673

Simon Dicker https://orcid.org/0000-0002-1940-4289

Peter Doze https://orcid.org/0000-0002-2408-0813

Adriaan J. Duivenvoorden https://orcid.org/0000-0003-2856-2382

Jo Dunkley https://orcid.org/0000-0002-7450-2586

Rolando Dünner https://orcid.org/0000-0003-3892-1860

Gerrit Farren https://orcid.org/0000-0001-5704-1127

Yilun Guan https://orcid.org/0000-0002-1697-3080

Mark Halpern https://orcid.org/0000-0002-1760-0868

Brandon Hensley https://orcid.org/0000-0001-7449-4638

Matt Hilton https://orcid.org/0000-0002-8490-8117

Adam D. Hincks https://orcid.org/0000-0003-1690-6678

Kevin M. Huffenberger https://orcid.org/0000-0001-7109-0099

John P. Hughes https://orcid.org/0000-0002-8816-6800

Giovanni Isopi https://orcid.org/0000-0002-8458-0588

Kenda Knowles https://orcid.org/0000-0002-8452-0825

Arthur Kosowsky https://orcid.org/0000-0002-3734-331X

Felipe Menanteau https://orcid.org/0000-0002-1372-2534

Kavilan Moodley https://orcid.org/0000-0001-6606-7142

Tony Mroczkowski https://orcid.org/0000-0003-3816-5372

Sigurd Naess https://orcid.org/0000-0002-4478-7111

John Orlowski-Scherer https://orcid.org/0000-0003-1842-8104

Lyman A. Page https://orcid.org/0000-0002-9828-3525

Emmanuel Schaan https://orcid.org/0000-0002-4619-8927

Neelima Sehgal https://orcid.org/0000-0002-9674-4527

David N. Spergel https://orcid.org/0000-0002-5151-0006

Lukas Wenzl https://orcid.org/0000-0001-5245-2058

Edward J. Wollack https://orcid.org/0000-0002-7567-4451

Zhilei Xu https://orcid.org/0000-0001-5112-2567

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