The Atacama Cosmology Telescope: A Measurement of the DR6 CMB Lensing Power Spectrum and Its Implications for Structure Growth

The maps were made with the same methodology as in Aiola et al. (2020); they will be described in full detail in S. Naess et al. (2024, in preparation). To summarize briefly, maximum likelihood maps are built at $0\buildrel{\,\prime}\over{.} 5$ resolution using 756 days of data observed in the period 2017 May 10–2021 June 18. Samples contaminated by detector glitches or the presence of the Sun or Moon in the telescope's far sidelobes are cut, but scan-synchronous pickup, like ground pickup, is left in the data since it is easier to characterize in map space.

The maps of each array-frequency band are made separately, for a total of five array-band combinations. For each of these, we split the data into three categories owing to differences in systematics and scanning patterns: night, day-deep, and day-wide. Of these categories, night makes up 2/3 of the statistical power and, as previously stated, is the only data set considered in this analysis.

Each set of nighttime data is split into eight subsets with independent instrument and atmospheric noise noise. These data-split maps are useful for characterizing the noise properties with map differences and for applying the cross-correlation-based estimator described in Section 5.8. In total, for this lensing analysis we use 40 separate night split maps in the f090 and f150 bands.

The data used in this analysis initially cover approximately 19,000 deg² before Galactic cuts are applied and have a total inverse variance of 0.55 nK⁻² for the nighttime data. Figure 3 shows the sky coverage and full survey depth of ACT DR6 nighttime observations.

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In addition to the maps of the CMB sky, ancillary products are produced by the mapmaking algorithm. One such set of products is the "inverse-variance maps" denoted by h , which provide the per-pixel inverse noise variance of the individual array frequencies.

The instrumental beams are determined from dedicated observations of Uranus and Saturn. The beam estimation closely follows the method used for ACT DR4 (Lungu et al. 2022). In short, the main beams are modeled as azimuthally symmetric and estimated for each observing season from Uranus observations. An additional correction that broadens the beam is determined from point-source profiles; this correction is then included in the beam. Polarized sidelobes are estimated from Saturn observations and removed during the mapmaking process. Just as the five observing seasons that make up the DR6 data set are jointly mapped into eight disjoint splits of the data, the per-season beams are also combined into eight per-split beams using a weighted average that reflects the statistical contribution of each season to the final maps (determined within the footprint of the nominal mask used for this lensing analysis). One notable improvement over the DR4 beam pipeline is the way the frequency dependence of the beam is handled. We now compute, using a self-consistent and Bayesian approach, the scale-dependent color corrections that convert the beams from describing the response to the approximate Rayleigh–Jeans spectrum of Uranus to describing the response to the CMB blackbody spectrum. The formalism will be described in a forthcoming paper (M. Hasselfield et al. 2024, in preparation). The CMB color correction is below 1% for the relatively low angular multipole limit ${{\ell }}_{\max }$ used in this paper.

The planet observations are also used to quantify the temperature-to-polarization leakage of the instrument. The procedure again follows the description in Lungu et al. (2022). To summarize, Stokes Q and U maps of Uranus are constructed for each detector array and interpreted as an estimate of the instantaneous temperature-to-polarization leakage. After rotating the Q and U maps to the north pole of the standard spherical coordinate system, an azimuthally symmetric model is fitted to the maps. The resulting model is then converted to a one-dimensional leakage beam in harmonic space: ${B}_{{\ell }}^{T\to E}$ and ${B}_{{\ell }}^{T\to B}$, which relates the Stokes I sky signal to leakage in the E- or B-mode linear polarization field.

Our filter-free, maximum likelihood mapmaking should ideally be unbiased, but that requires having the correct model for the data. In practice, subtle model errors bias the result. The following two main sources of bias have been identified (Naess & Louis 2023).

• 1.  
  Subpixel error: the real CMB sky has infinite resolution, while our nominal maps are made at 05 resolution. While we could have expected this only to affect the smallest angular scales, the coupling of this model error with downweighting of the data to mitigate effects of atmospheric noise leads to a deficit of power on the largest scales of the maps.
• 2.  
  Detector gain calibration: inconsistent detector gains can also cause a lack of power in our maps at f090 and f150 on large angular scales. This inconsistency arises owing to errors in gain calibration at the time-ordered data (TOD) processing stage. The current DR6 maps ⁷⁰ use a preliminary calibration procedure; alternative calibration procedures are currently being investigated to mitigate this effect.

To assess the impact of the loss of power at large angular scales on the lensing power spectrum, a multipole-dependent transfer function ${t}_{{\ell }}^{T}$ is calculated at each frequency by taking the ratio of the corresponding ACT CMB temperature bandpowers ${C}_{{\ell }}^{\mathrm{ACT}\times \mathrm{ACT}}$ and the ACT–Planck (NPIPE) temperature cross-correlation bandpowers ${C}_{{\ell }}^{\mathrm{ACT}\times {\rm{P}}}$:

$$\begin{eqnarray}&&{t}_{{\ell }}^{T}=\displaystyle \frac{{C}_{{\ell }}^{\mathrm{ACT}\times \mathrm{ACT}}}{{C}_{{\ell }}^{\mathrm{ACT}\times {\rm{P}}}}.\end{eqnarray} \tag{ 1 }$$

Here ${C}_{{\ell }}^{\mathrm{ACT}\times \mathrm{ACT}}$ is a noise-free cross-spectrum between data splits, and ${C}_{{\ell }}^{\mathrm{ACT}\times {\rm{P}}}$ is computed by cross-correlating with the Planck map that is nearest in frequency.

A logistic function with three free parameters is fit to the above ${t}_{{\ell }}^{T}$. We then divide the temperature maps in harmonic space by the resulting curve, ${T}_{{\ell }m}\to {T}_{{\ell }m}/{t}_{{\ell }}^{T}$, in order to deconvolve the transfer function. Due to the modest sensitivity of our lensing estimator to low CMB multipoles, deconvolving this transfer function results in only a negligible change in the lensing power spectrum amplitude, ΔA_lens = 0.004 (corresponding to less than 0.2σ). Therefore, we have negligible sensitivity to the details of the transfer function.

We determine calibration factors ${c}_{{A}_{f}}$ at each array-frequency combination A_f of ACT relative to Planck by minimizing differences between the ACT temperature power spectra, ${C}_{\ell }^{\mathrm{ACT}\times \mathrm{ACT},{A}_{f}}$, and the cross-spectrum with Planck, ${C}_{\ell }^{\mathrm{ACT}\times {\rm{P}},{A}_{f}}$, at intermediate multipoles. In these DR6 maps, the transfer functions approach unity as ℓ increases, and they eventually plateau at this value for ℓ > 800 and ℓ > 1250 at f090 and f150, respectively; we therefore use the multipoles 800–1200 at f090 and 1250–1800 at f150 to determine the calibration factors ${c}_{{A}_{f}}$ by minimizing the following χ²:

$$\begin{eqnarray}&&{\chi }^{2}({c}_{{A}_{f}})=\displaystyle \sum _{{\ell }_{b}={\ell }_{b}^{\min }}^{{\ell }_{b}^{\max }}\displaystyle \sum _{{\ell }_{b}^{^{\prime} }={\ell }^{\min }}^{{\ell }_{b}^{\max }}{{\rm{\Delta }}}_{{\ell }_{b}}({c}_{{A}_{f}}){\left[{{\rm{\Sigma }}}^{{A}_{{\rm{f}}}}\right]}_{{\ell }_{b},{\ell }_{b}^{\prime} }^{-1}{{\rm{\Delta }}}_{{\ell }_{b}^{\prime} }({c}_{{A}_{f}}),\end{eqnarray} \tag{ 2 }$$

where the sum is over bandpowers. Here the difference bandpowers are given by

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{{\ell }_{b}}={c}_{{A}_{f}}{C}_{{\ell }_{b}}^{\mathrm{ACT}\times \mathrm{ACT},{A}_{f}}-{C}_{{\ell }_{b}}^{\mathrm{ACT}\times {\rm{P}},{A}_{f}},\end{eqnarray} \tag{ 3 }$$

and ${{\rm{\Sigma }}}_{{{\ell }}_{b},{{\ell }}_{b}^{\prime} }^{{A}_{{\rm{f}}}}$ is their covariance matrix computed analytically, using noise power spectra measured from data, at ${c}_{{A}_{f}}=1$.

The errors we achieve on the calibration factors are small enough that they can be neglected in our lensing analysis; see Appendix C.1 for details.

Polarization efficiencies scale the true polarization signal on the sky to the signal component in the observed polarization maps. Assuming incorrect polarization efficiencies in the sky maps leads to biases in the lensing reconstruction amplitude because our quadratic lensing estimator uses up to two powers of the misnormalized polarization maps; for example, polarization-only quadratic lensing estimators will be biased by the square of the efficiency error ${({p}_{\mathrm{eff}}^{{A}_{f}})}^{2}$.

However, the normalization of the estimator involves dividing the unnormalized estimator, which is quadratic in CMB maps, by fiducial ΛCDM C_ℓ values. If these fiducial ΛCDM spectra are rescaled by the same two powers of the efficiency error, then the estimator will again become unbiased. In other words, as long as we ensure that the amplitude of the spectra used in the normalization is scaled to be consistent with the amplitude of spectra of the data, our estimator will reconstruct lensing without any bias. The physical explanation of this observation is that lensing does not affect the amplitude of the CMB correlations, only their shapes.

To ensure an unbiased polarization lensing estimator, even though the ACT blinding policy in Section 6.3 does not yet allow either a direct comparison of polarization power spectra of ACT and Planck or a detailed comparison of the ACT power spectra with respect to ΛCDM, we employed a simple efficiency self-calibration procedure, which aims to ensure amplitude consistency between fiducial spectra and map spectra. The procedure is explained in detail in Appendix A. In short, we fit for a single amplitude scaling ${p}_{\mathrm{eff}}^{{A}_{f}}$ between our data polarization power spectra and the fiducial model power spectra assumed for the normalization of the estimator. We then simply correct the polarization data maps by this amplitude scaling parameter to ensure an unbiased lensing measurement. We verify in Appendix C.4 that the uncertainties in this correction for the polarization efficiencies are negligible for our analysis.

Point-source-subtracted maps are made using a two-step process. First, we run a matched filter on a version of the DR5 ACT+Planck maps (Naess et al. 2020) updated to use the new data in DR6, and we register objects detected at greater than 4σ in a catalog for each frequency band. The object fluxes are then fit individually in each split map using forced photometry at the catalog positions and subtracted from the map. This is done to take into account the strong variability of the quasars that make up the majority of our point-source sample. Due to our variable map depth, this procedure results in a subtraction threshold that varies from 4 to 7 mJy in the f090 band and from 5 to 10 mJy in the f150 band. An extra map processing step to reduce the effect of point-source residuals not accounted for in the mapmaking step is described in Section 5.2.

Our baseline analysis mitigates biases related to the thermal Sunyaev–Zeldovich (tSZ) effect by subtracting models for the tSZ contribution due to galaxy clusters. We use the Nemo ⁷¹ software, which performs a matched-filter search for clusters via their tSZ signal (see Hilton et al. 2021 for details). We model the cluster signal using the universal pressure profile (UPP) described by Arnaud et al. (2010) and construct a set of 15 filters with different angular sizes by varying the mass and redshift of the cluster model. We construct cluster tSZ model maps for both ACT frequencies by placing beam-convolved UPP model clusters with an angular size corresponding to that of the maximal S/N detection across all 15 filter scales as reported by Nemo, for all clusters detected with S/N greater than 5 on the ACT footprint. This model image is then subtracted from the single-frequency ACT data before coadding. Further details about point-source and cluster template subtraction can be found in MacCrann et al. (2024).

The sky maps are produced at a resolution of $0\buildrel{\,\prime}\over{.} 5$, but because our lensing reconstruction uses a maximum CMB multipole of ${{\ell }}_{\max }=3000$, a downgraded pixel resolution of $1^{\prime}$ is sufficient for the unbiased recovery of the lensing power spectrum and reduces computation time. Therefore, we downgrade the CMB data maps by block-averaging neighboring CMB pixels. Similarly, the inverse-variance maps are downgraded by summing the contiguous full-resolution inverse-variance values.

The sky maps are further processed to reduce the effect of point sources not accounted for in the mapmaking step. As described in Section 3.5, we work with maps in which point sources above a threshold of roughly 4–10 mJy (corresponding to an S/N threshold of 4σ) have been fit and subtracted at the map level. However, very bright and/or extended sources may still have residuals in these maps. To address this, we prepare a catalog of 1779 objects for masking with holes of radius $6^{\prime}$: these include especially bright sources that require a specialized point-source treatment in the mapmaker (see Aiola et al. 2020; Naess et al. 2020), extended sources with S/N > 10 identified through cross-matching with external catalogs, all point sources with S/N > 70 at f150, and an additional list of locations with residuals from point-source subtraction that were found by visual inspection. We include an additional set of 14 objects for masking with holes of radius $10^{\prime}$: these are regions of diffuse or extended positive emission identified by eye in matched-filtered coadds of ACT maps. They include nebulae, Galactic dust knots, radio lobes, and large nearby galaxies. We subsequently inpaint these holes using a constrained Gaussian realization with a Gaussian field consistent with the CMB signal and noise of the CMB fields and matching the boundary conditions at the hole's edges (Bucher & Louis 2012; Madhavacheril & Hill 2020a). This step is required to prevent sharp discontinuities in the sky map that can introduce spurious features in the lensing reconstruction. The total compact-source area inpainted corresponds to a sky fraction of 0.147%. Further, more detailed discussion of compact-object treatment can be found in MacCrann et al. (2024).

To exclude regions of bright Galactic emission and regions of the ACT survey with very high noise, we prepare edge-apodized binary masks over the observation footprint as follows. We start with Galactic emission masks based on 353 GHz emission from Planck PR2, ⁷⁴ rotating and reprojecting these to our Plate-Carrée cylindrical (CAR) pixelization in equatorial coordinates. We use a Galactic mask that leaves (in this initial step) 60% of the full sky as our baseline; we use a more conservative mask retaining initially 40% of the full sky for a consistency test described in Section 6.5.8. From here on we denote the masks constructed using these Galactic masks as 60% and 40% Galactic masks. We additionally apply a mask that removes any regions with rms map noise larger than 70 μK-arcmin in any of our input f090 and f150 maps; this removes very noisy regions at the edges of our observed sky area. Regions with clearly visible Galactic dust clouds and knots are additionally masked, by hand, with appropriately sized circular holes. ⁷⁵ After identifying these spurious features in either match-filtered maps or lensing reconstructions themselves, masking them removes a further sky fraction of f_sky = 0.00138. The resulting final mask is then adjusted to round sharp corners. We finally apodize the mask with a cosine-squared edge roll-off of total width of 3 deg. The total usable area after masking is 9400 deg², which corresponds to a sky fraction of f_sky = 0.23.

The block-averaging operation used to downgrade the sky maps from $0\buildrel{\,\prime}\over{.} 5$ to $1^{\prime}$ convolves the downgraded map with a top-hat function that needs to be deconvolved. ⁷⁶ We do this by transforming the temperature and polarization maps X to Fourier space, ⁷⁷ giving FFT(X), and dividing by the sinc(f_x ) and sinc(f_y ) functions, where f_x and f_y are the dimensionless wavenumbers ⁷⁸

$$\begin{eqnarray}&&{X}^{\mathrm{pixel}-\mathrm{deconvolved}}=\mathrm{IFFT}\left[\displaystyle \frac{\mathrm{FFT}(X)}{\mathrm{sinc}({f}_{x})\mathrm{sinc}({f}_{y})}\right],\end{eqnarray} \tag{ 4 }$$

where IFFT denotes the inverse (discrete) fast Fourier transform. For simplicity, X without a superscript used in the subsequent sections will refer to the pixel-window-deconvolved maps unless otherwise stated.

Contamination by ground, magnetic, and other types of pickup in the data due to the scanning of the ACT telescope manifests as excess power at constant decl. stripes in the sky maps and thus can be localized in Fourier space. We mask Fourier modes with ∣ℓ_x ∣ < 90 and ∣ℓ_y ∣ < 50 to remove this contamination as in Louis et al. (2017) and Choi et al. (2020). This masking is carried out both in the data and in our realistic CMB simulations. We demonstrate in Appendix D that this Fourier-mode masking reduces the recovered lensing signal by around 10%; we account for this well-understood effect with a multiplicative bias correction obtained from simulations.

In the following section, we describe the method we use to combine the individual array frequency to form the final sky maps used for the lensing measurement.

We first define for each array frequency's data the map-based coadd map c , an unbiased estimate of the sky signal, by taking the inverse-variance-weighted average of the eight split maps m _i :

$$\begin{eqnarray}&&{\boldsymbol{c}}=\displaystyle \frac{{\sum }_{i=0}^{i=7}{{\boldsymbol{h}}}_{i}^{* }{{\boldsymbol{m}}}_{i}}{{\sum }_{i=0}^{i=7}{{\boldsymbol{h}}}_{i}}.\end{eqnarray} \tag{ 5 }$$

Note that in the above equation the multiplication (*) and division denote element-wise operations. These coadd maps provide our best estimate of the sky signal for each array and are used for noise estimation as explained below.

As we will describe in Section 5.8, the cross-correlation-based estimator we use requires the construction of four sky maps d with independent noise. We construct these maps d in the same manner as Equation (5), coadding together split j and j + 4 with j ∈ {0, 1, 2, 3}.

5.6.1. Inverse-variance Coaddition of the Array Frequencies

We combine the different coadded data maps ${{\boldsymbol{d}}}_{{A}_{f}}$ with array frequencies A_f ∈ {PA4 f150, PA5 f090, PA5 f150, PA6 f090, PA6 f150} into single CMB fields ${M}_{{\ell }m}^{X}$, with X ∈ (T, E, B), on which lensing reconstruction is performed. The coadding of the maps is done in spherical harmonic space, ⁷⁹

$$\begin{eqnarray}&&{M}_{\ell m}=\displaystyle \sum _{{A}_{f}}{w}_{\ell }^{{A}_{f}}{d}_{\ell m}^{{A}_{f}}{\left({B}_{\ell }^{{A}_{f}}\right)}^{-1},\end{eqnarray} \tag{ 6 }$$

where

$$\begin{eqnarray}&&{w}_{\ell }^{{A}_{f}}=\frac{{\left({N}_{\ell }^{{A}_{f}}\right)}^{-1}{\left({B}_{\ell }^{{A}_{f}}\right)}^{2}}{\displaystyle \sum _{({A}_{f})}{\left({N}_{\ell }^{{A}_{{\rm{f}}}}\right)}^{-1}{\left({B}_{\ell }^{{A}_{f}}\right)}^{2}}\end{eqnarray} \tag{ 7 }$$

are the normalized inverse-variance weights in harmonic space. These weights, giving the relative contributions of each array frequency, are shown in Figure 4 and are constructed to sum to unity at each multipole ℓ. Note that a deconvolution of the harmonic beam transfer functions ${B}_{\ell }^{{A}_{f}}$ is performed for each array frequency. ⁸⁰ The noise power spectra ${N}_{\ell }^{{A}_{f}}$ are obtained from the beam-deconvolved noise maps of the individual sky maps with the following prescription.

We construct a noise-only map n _i by subtracting the pixel-wise coadd c of each map ⁸¹ from the individual data splits m _i ; this noise-only map is given by

$$\begin{eqnarray}&&{{\boldsymbol{n}}}_{i}={{\boldsymbol{m}}}_{i}-{\boldsymbol{c}}.\end{eqnarray} \tag{ 8 }$$

We then transform the real-space noise-only maps n _i into spherical harmonic space ${n}_{{\ell }m}^{(i)}$ and use these to compute the noise power spectra used for the weights in Equation (7). Since we have k = 8 splits, we can reduce statistical variance by finding the average of these noise spectra: ⁸²

$$\begin{eqnarray}&&{N}_{{\ell }}=\displaystyle \frac{1}{{w}_{2}}\displaystyle \frac{1}{k(k-1)}\displaystyle \frac{1}{2{\ell }+1}\displaystyle \sum _{i}^{k}\displaystyle \sum _{m=-{\ell }}^{{\ell }}{n}_{{\ell }m}^{(i)}{n}_{{\ell }m}^{(i)* },\end{eqnarray} \tag{ 9 }$$

where ${w}_{2}=\int {d}^{2}\hat{{\boldsymbol{n}}}\,{M}^{2}(\hat{{\boldsymbol{n}}})/(4\pi )$ is the average value of the second power of the mask $M(\hat{{\boldsymbol{n}}})$, which corrects for the missing sky fraction due to the application of the analysis mask, as described in Section 5.3. The resulting noise power is further smoothed over by applying a linear binning ⁸³ of Δℓ = 14.

The same coadding operation is performed on simulations containing lensed sky maps and noise maps. The resulting suite of coadded CMB simulations is used throughout our baseline analysis.

5.6.2. Internal Linear Combination Coaddition

Optimal quadratic lensing reconstruction requires as inputs Wiener-filtered X = T, E, and B CMB multipoles and inverse-variance-filtered maps (the latter can be obtained from the former by dividing the Wiener-filtered multipoles by the fiducial lensed power spectra C^fid before projecting back to maps). The filtering step is important because an optimal analysis of the observed CMB sky requires both the downweighting of noise and the removal of masked areas (Hanson et al. 2011).

We write the temperature T and polarization _±2 P ≡ Q ± iU (beam- and pixel-deconvolved) data maps as

$$\begin{eqnarray}&&\left(\begin{array}{l}T\\ {}_{2}P\\ {}_{-2}P\end{array}\right)={ \mathcal Y }\left(\begin{array}{l}{T}_{{\ell }m}\\ {E}_{{\ell }m}\\ {B}_{{\ell }m}\end{array}\right)+\mathrm{noise},\end{eqnarray} \tag{ 10 }$$

where the matrix ${ \mathcal Y }$ contains the spin-weighted spherical harmonic functions to convert the spherical harmonics T_{ℓ m} , E_{ℓ m} , and B_{ℓ m} to real-space maps over the unmasked region. The real-space covariance matrix of the data maps is

$$\begin{eqnarray}&&C={ \mathcal Y }{{\mathbb{C}}}^{\mathrm{fid}}{{ \mathcal Y }}^{\dagger }+{C}_{\mathrm{noise}},\end{eqnarray} \tag{ 11 }$$

where ${{\mathbb{C}}}^{\mathrm{fid}}$ is the matrix of our fiducial lensed CMB spectra with elements

$$\begin{eqnarray}{[{{\mathbb{C}}}^{\mathrm{fid}}]}_{{\ell }m,{{\ell }}^{{\prime} }{m}^{{\prime} }}={\delta }_{{\ell }{{\ell }}^{{\prime} }}{\delta }_{{{mm}}^{{\prime} }}\left(\begin{array}{ccc}{C}_{{\ell }}^{{TT}} & {C}_{{\ell }}^{{TE}} & 0\\ {C}_{{\ell }}^{{TE}} & {C}_{{\ell }}^{{EE}} & 0\\ 0 & 0 & {C}_{{\ell }}^{{BB}}\end{array}\right),\end{eqnarray} \tag{ 12 }$$

and C_noise is the real-space noise covariance matrix. The Wiener-filtered multipoles are then obtained as

$$\begin{eqnarray}&&{X}_{{\ell }m}^{\mathrm{WF}}={{\mathbb{C}}}^{\mathrm{fid}}{{ \mathcal Y }}^{\dagger }{C}^{-1}\left(\begin{array}{c}T\\ {}_{2}P\\ {}_{-2}P\end{array}\right).\end{eqnarray} \tag{ 13 }$$

For our main analysis, we employ an approximate form of the Wiener filter that follows from the rearrangement

$$\begin{eqnarray}\begin{array}{rcl}{{\mathbb{C}}}^{\mathrm{fid}}{{ \mathcal Y }}^{\dagger }{C}^{-1} & = & {\left({\left({{\mathbb{C}}}^{\mathrm{fid}}\right)}^{-1}+{{\mathbb{N}}}^{-1}\right)}^{-1}{{ \mathcal Y }}^{\dagger }{C}^{-1}\\ & = & {\left({\left({{\mathbb{C}}}^{\mathrm{fid}}\right)}^{-1}+{{\mathbb{N}}}^{-1}\right)}^{-1}{{\mathbb{N}}}^{-1}{{ \mathcal Y }}^{\dagger }{\left({ \mathcal Y }{{ \mathcal Y }}^{\dagger }\right)}^{-1}\\ & = & {{\mathbb{C}}}^{\mathrm{fid}}{\left({{\mathbb{C}}}^{\mathrm{fid}}+{\mathbb{N}}\right)}^{-1}{{ \mathcal Y }}^{\dagger }{\left({ \mathcal Y }{{ \mathcal Y }}^{\dagger }\right)}^{-1},\end{array}\end{eqnarray} \tag{ 14 }$$

where ${{\mathbb{N}}}^{-1}\equiv {{ \mathcal Y }}^{\dagger }{C}_{\mathrm{noise}}^{-1}{ \mathcal Y }$. The operation ${{ \mathcal Y }}^{\dagger }{\left({ \mathcal Y }{{ \mathcal Y }}^{\dagger }\right)}^{-1}$ takes the (pseudo)spherical transform of the masked maps, with ${ \mathcal Y }{{ \mathcal Y }}^{\dagger }\,=\mathrm{diag}(1,2,2){\delta }^{(2)}(\hat{{\boldsymbol{n}}}-\hat{{\boldsymbol{n}}}^{\prime} )$. Our approximate form of the Wiener filter takes the form

$$\begin{eqnarray}&&{X}_{{\ell }m}^{\mathrm{WF}}\approx {{\mathbb{C}}}^{\mathrm{fid}}{ \mathcal F }{{ \mathcal Y }}^{\dagger }{\left({ \mathcal Y }{{ \mathcal Y }}^{\dagger }\right)}^{-1}\left(\begin{array}{c}T\\ {}_{2}P\\ {}_{-2}P\end{array}\right),\end{eqnarray} \tag{ 15 }$$

where ${ \mathcal F }$ is the filtering operation applied to the temperature and polarization spherical harmonics. The filters used are diagonal in harmonic space such that each component of X ∈ T, E, B is filtered separately by ${F}_{{\ell }}^{X}=1/({C}_{{\ell }}^{{XX}}+{N}_{{\ell }}^{{XX}})$.

The above diagonal filtering neglects small amounts of mode mixing due to masking, does not account for noise inhomogeneities over the map, and also ignores cross-correlation in ${C}_{{\ell }}^{{TE}}$. However, it has the advantage of allowing the temperature and polarization map to be filtered independently and is a good approximation on scales for which the CMB fields are signal dominated and in situations when the noise level is close to homogeneous, as is the case for ACT DR6. ⁸⁴ This method is also significantly faster than using the more optimal filter in Equation (13), which requires evaluation of the inverse of the covariance matrix C with, for example, conjugate-gradient methods. Therefore, for the main analysis, we employ this diagonal filter.

The inverse-variance filtered maps

$$\begin{eqnarray}&&\bar{{\boldsymbol{X}}}(\hat{{\boldsymbol{n}}})={C}^{-1}\left(\begin{array}{c}T\\ {}_{2}P\\ {}_{-2}P\end{array}\right)\end{eqnarray} \tag{ 16 }$$

are related to the Wiener-filtered multipoles ${X}_{{\ell }m}^{\mathrm{WF}}$ in Equation (15) via

$$\begin{eqnarray}\begin{array}{rcl}\bar{{\boldsymbol{X}}}(\hat{{\boldsymbol{n}}}) & = & {\left({ \mathcal Y }{{ \mathcal Y }}^{\dagger }\right)}^{-1}{ \mathcal Y }{\left({{\mathbb{C}}}^{\mathrm{fid}}\right)}^{-1}{X}_{{\ell }m}^{\mathrm{WF}}\\ & = & \mathrm{diag}\left(1,\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}\right){ \mathcal Y }{\left({{\mathbb{C}}}^{\mathrm{fid}}\right)}^{-1}{X}_{{\ell }m}^{\mathrm{WF}}\\ & = & \mathrm{diag}\left(1,\displaystyle \frac{1}{2},\displaystyle \frac{1}{2}\right){ \mathcal Y }{\bar{X}}_{{\ell }m},\end{array}\end{eqnarray} \tag{ 17 }$$

where, in the last line, ${\bar{X}}_{{\ell }m}={\left({{\mathbb{C}}}^{\mathrm{fid}}\right)}^{-1}{X}_{{\ell }m}^{\mathrm{WF}}$.

In this section, we describe the methodology used to estimate CMB lensing using the quadratic estimator (QE). Our baseline methodology closely follows the pipeline used in Planck Collaboration et al. (2020b), albeit with key improvements in areas such as foreground mitigation (using a profile-hardened estimator that is more robust to extragalactic foregrounds; see Section 5.8.2) and immunity to noise modeling (using the more robust cross-correlation-based estimator described in Section 5.8.3).

5.8.1. Standard Quadratic Estimator

A fixed realization of gravitational lenses imprints preferred directions into the CMB, thereby breaking the statistical isotropy of the unlensed CMB. Mathematically, the breaking of statistical isotropy corresponds to the introduction of new correlations between different, formerly independent modes of the CMB sky, with the correlations proportional to the lensing potential ϕ_LM . Adopting the usual convention of using L and M to refer to lensing multipoles and ℓ and m to refer to CMB multipoles, we may write the new, lensing-induced correlation between two different CMB modes ${X}_{{{\ell }}_{1}{m}_{1}}$ and ${Y}_{{{\ell }}_{2}{m}_{2}}$ as follows :

$$\begin{eqnarray}\begin{array}{rcl}\langle {X}_{{{\ell }}_{1}{m}_{1}}{Y}_{{{\ell }}_{2}{m}_{2}}{\rangle }_{\mathrm{CMB}} & = & \displaystyle \sum _{{LM}}{\left(-1\right)}^{M}\left(\begin{array}{ccc}{{\ell }}_{1} & {{\ell }}_{2} & L\\ {m}_{1} & {m}_{2} & -M\end{array}\right)\\ & & \times \,{f}_{{{\ell }}_{1}{{\ell }}_{2}L}^{{XY}}{\phi }_{{LM}}.\end{array}\end{eqnarray} \tag{ 18 }$$

The average 〈 〉_CMB is taken over CMB realizations with a fixed lensing potential ϕ. Here the fields X_{ℓ m} , Y_{ℓ m} ∈ {T_{ℓ m} , E_{ℓ m} , B_{ℓ m} } and the bracketed term is a Wigner 3j symbol. The response functions ${f}_{{\ell }{{\ell }}^{{\prime} }L}^{{XY}}$ for the different quadratic pairs XY can be found in Okamoto & Hu (2003) and are linear functions of the CMB power spectra (the lensed spectra are used to cancel a higher-order correction; Lewis et al. 2011).

The correlation between different modes induced by lensing motivates the use of quadratic combinations of the lensed temperature and polarization maps to reconstruct the lensing field. Pairs of Wiener-filtered maps, X^WF, and inverse-variance-filtered maps, $\bar{X}$, are provided as inputs to a QE that reconstructs an unnormalized, minimum-variance (MV) estimate of the spin-1 component of the real-space lensing displacement field:

$$\begin{eqnarray}&&{}_{1}\hat{d}(\hat{{\boldsymbol{n}}})=-{\displaystyle \sum _{s=0,\pm 2}}_{-s}\bar{X}(\hat{{\boldsymbol{n}}})[{\unicode{x000F0}}_{s}{X}^{\mathrm{WF}}](\hat{{\boldsymbol{n}}}).\end{eqnarray} \tag{ 19 }$$

Here ð is the spin-raising operator acting on spin spherical harmonics and the pre-subscript s denotes the spin of the field. The gradients of the Wiener-filtered maps are given explicitly by

$$\begin{eqnarray}&&\begin{array}{l}\left[{\unicode{x000F0}}_{0}{X}^{\mathrm{WF}}\right](\hat{{\boldsymbol{n}}})\equiv \displaystyle \sum _{{\ell }m}\sqrt{{\ell }({\ell }+1)}{{T}_{{\ell }m}^{\mathrm{WF}}}_{1}{Y}_{{\ell }m}(\hat{{\boldsymbol{n}}}),\\ \left[{\unicode{x000F0}}_{-2}{X}^{\mathrm{WF}}\right](\hat{{\boldsymbol{n}}})\equiv -\displaystyle \sum _{{\ell }m}\sqrt{({\ell }+2)({\ell }-1)}{\left[{E}_{{\ell }m}^{\mathrm{WF}}-{{iB}}_{{\ell }m}^{\mathrm{WF}}\right]}_{-1}{Y}_{{\ell }m}(\hat{{\boldsymbol{n}}}),\\ \left[{\unicode{x000F0}}_{2}{X}^{\mathrm{WF}}\right](\hat{{\boldsymbol{n}}})\equiv -\displaystyle \sum _{{\ell }m}\sqrt{({\ell }-2)({\ell }+3)}{\left[{E}_{{\ell }m}^{\mathrm{WF}}+{{iB}}_{{\ell }m}^{\mathrm{WF}}\right]}_{3}{Y}_{{\ell }m}(\hat{{\boldsymbol{n}}}).\end{array}\end{eqnarray} \tag{ 20 }$$

The displacement field can be decomposed into the gradient ϕ and curl Ω components by expanding in spin-weighted spherical harmonics:

$$\begin{eqnarray}&&{}_{\pm 1}\hat{d}(\hat{{\boldsymbol{n}}})=\mp \displaystyle \sum _{{LM}}{\left(\displaystyle \frac{{\bar{\phi }}_{{LM}}\pm i{\bar{{\rm{\Omega }}}}_{{LM}}}{\sqrt{L(L+1)}}\right)}_{\pm 1}{Y}_{{LM}}(\hat{{\boldsymbol{n}}}).\end{eqnarray} \tag{ 21 }$$

Hence, by taking spin-±1 spherical harmonic transforms of ${}_{\pm 1}\hat{d}(\hat{{\boldsymbol{n}}})$, where ${}_{-1}\hat{d}={}_{1}{\hat{d}}^{* }$, and taking linear combinations of the resulting coefficients, we can isolate the gradient and curl components. The gradient component ϕ_LM contains the information about lensing that is the focus of our analysis. ⁸⁵ The curl Ω_LM is expected to be zero (up to small post-Born corrections; e.g., Pratten & Lewis (2016) and references therein) and can therefore serve as a useful null test, as discussed in Section 6.4.1.

Even in the absence of lensing, other sources of statistical anisotropy in the sky maps, such as masking or noise inhomogeneities, can affect the naive lensing estimator. One can correct such effects by subtracting the lensing estimator's response to such nonlensing statistical anisotropies, which is commonly referred to as the mean field $\langle {\bar{\phi }}_{{LM}}\rangle$. We estimate this mean-field signal by averaging the reconstructions produced by the naive lensing estimator from 180 noiseless ⁸⁶ simulations, each with independent CMB and lensing potential realizations. This averaging ensures that only the response to spurious, nonlensing statistical anisotropy remains (as the masking is the same in all simulations, whereas CMB and lensing fluctuations average to zero). Subtracting this mean field leads us to the following lensing estimator:

$$\begin{eqnarray}&&{\bar{\phi }}_{{LM}}\to {\bar{\phi }}_{{LM}}-\langle {\bar{\phi }}_{{LM}}\rangle .\end{eqnarray} \tag{ 22 }$$

The temperature-only (s = 0) and polarization-only (s = ± 2) estimators ⁸⁷ in Equation (19) are combined at the field level ⁸⁸ to produce the full unnormalized MV estimator.

Expanding the Wiener-filtered fields in terms of the inverse-variance-filtered multipoles ${\bar{X}}_{{\ell }m}$ and extracting the gradient part approximately recovers the usual estimators ${\bar{\phi }}_{{LM}}^{{XY}}$ of Okamoto & Hu (2003), where XY ∈ {TT, TE, ET, EE, EB, BE}. More specifically, the MV estimator presented here is approximately equivalent ⁸⁹ to combining the individual estimators ${\bar{\phi }}_{{LM}}^{{XY}}$ with a weighting given by the inverse of their respective normalization ${({{ \mathcal R }}_{L}^{{XY}})}^{-1}$:

$$\begin{eqnarray}&&{\hat{\phi }}_{{LM}}^{\mathrm{MV}}={\left({{ \mathcal R }}_{L}^{\mathrm{MV}}\right)}^{-1}\displaystyle \sum _{{XY}}{\bar{\phi }}_{{LM}}^{{XY}}.\end{eqnarray} \tag{ 23 }$$

Here ${({{ \mathcal R }}_{L}^{\mathrm{MV}})}^{-1}$ is the MV estimator normalization that ensures that our reconstructed lensing field is unbiased; by construction, it is defined via ${\phi }_{{LM}}={({{ \mathcal R }}_{L}^{{XY}})}^{-1}\langle {\bar{\phi }}_{{LM}}^{{XY}}{\rangle }_{\mathrm{CMB}}$. The normalization is given explicitly by

$$\begin{eqnarray}&&{\left({{ \mathcal R }}_{L}^{\mathrm{MV}}\right)}^{-1}=\displaystyle \frac{1}{{\sum }_{{XY}}({{ \mathcal R }}_{L}^{\mathrm{XY}})}.\end{eqnarray} \tag{ 24 }$$

In the notation adopted here, the unnormalized estimator ${\bar{\phi }}_{{LM}}$ is related to the normalized estimator ${\hat{\phi }}_{{LM}}$ via the normalization ${{ \mathcal R }}_{L}^{-1}$ as ${\hat{\phi }}_{{LM}}={{ \mathcal R }}_{L}^{-1}{\bar{\phi }}_{{LM}}$.

To first approximation, this normalization is calculated analytically with curved-sky expressions from Okamoto & Hu (2003). We generally use fiducial lensed spectra in this calculation (as well as the filtering of Equation (12)), which reduces the higher-order N⁽²⁾ bias to subpercent levels; however, for the TT estimator, we use the lensed temperature-gradient power spectrum ${C}_{{\ell }}^{T{\rm{\nabla }}T}$ to further improve the fidelity of the reconstruction (Lewis et al. 2011). This analytic, isotropic normalization is fairly accurate, but it does not account for effects induced by Fourier-space filtering and sky masking. Therefore, we additionally apply a multiplicative MC correction ${\rm{\Delta }}{A}_{L}^{\mathrm{MC},\mathrm{mul}}$ to all lensing estimators, so that ${\hat{\phi }}_{{LM}}\to {\rm{\Delta }}{A}_{L}^{\mathrm{MC},\mathrm{mul}}{\hat{\phi }}_{{LM}}$. This correction is obtained by first cross-correlating reconstructions from simulations with the true lensing map; we then divide the average of the input simulation power spectrum by the result, i.e.,

$$\begin{eqnarray}&&{\rm{\Delta }}{A}_{L}^{\mathrm{MC},\mathrm{mul}}=\displaystyle \frac{\langle {C}_{L}^{\phi \phi }\rangle }{\langle {C}_{L}^{\hat{\phi }\phi }\rangle }.\end{eqnarray} \tag{ 25 }$$

In practice, this multiplicative MC correction is computed after binning both spectra into bandpowers.

An explanation of the origin of the multiplicative MC correction is provided in Appendix D: it is found to be primarily a consequence of the Fourier-space filtering.

Having obtained our estimate of the lensing map in harmonic space, ${\hat{\phi }}_{{LM}}$, we can compute a naive, biased estimate of the lensing power spectrum. Using two instances of the lensing map estimates ${\hat{\phi }}_{{LM}}^{{AB}}$ and ${\hat{\phi }}_{{LM}}^{{CD}}$, this power spectrum is given by

$$\begin{eqnarray}&&{\hat{C}}_{L}^{\hat{\phi }\hat{\phi }}({\hat{\phi }}_{{LM}}^{{AB}},{\hat{\phi }}_{{LM}}^{{CD}})\equiv \displaystyle \frac{1}{{w}_{4}(2L+1)}\displaystyle \sum _{M=-L}^{L}{\hat{\phi }}_{{LM}}^{{AB}}{\left({\hat{\phi }}_{{LM}}^{{CD}}\right)}^{* },\end{eqnarray} \tag{ 26 }$$

where ${w}_{4}=\int {d}^{2}\hat{{\boldsymbol{n}}}{M}^{4}(\hat{{\boldsymbol{n}}})/(4\pi )$, the average value of the fourth power of the mask $M(\hat{{\boldsymbol{n}}})$, corrects for the missing sky fraction due to the application of the analysis mask. In Equation (37), below, we will introduce a new version of these spectra that ensures that only different splits of the data are used in order to avoid any noise contribution. This will allow us to obtain an estimate of the lensing power spectrum that is not biased by any mischaracterization of the noise in our CMB observations.

Nevertheless, biases arising from CMB and lensing signals still need to be removed from the naive lensing power spectrum estimator. We discuss the subtraction of these biases in Section 5.9.

5.8.2. Profile Hardening for Foreground Mitigation

Extragalactic foreground contamination from Sunyaev–Zel'dovich clusters, the CIB, and radio sources can affect the QE and hence produce large biases in the recovered lensing power spectrum if unaccounted for. For our baseline analysis, we use a geometric approach to mitigating foregrounds and make use of bias-hardened estimators (Namikawa et al. 2013; Osborne et al. 2014; Sailer et al. 2020). As with lensing, other sources of statistical anisotropy in the map such as point sources and tSZ clusters can be related to a response function ${f}_{{{\ell }}_{1}{{\ell }}_{2}L}^{s}$ and a field s_LM describing the anisotropic spatial dependence. Bias-hardened estimators work by reconstructing simultaneously both lensing and nonlensing statistical anisotropies and subtracting the latter, with a scaling to ensure that the resulting estimator has no remaining response to nonlensing anisotropies. Explicitly, the bias-hardened TT part of the lensing estimator is given by

$$\begin{eqnarray}&&{\hat{\phi }}_{{LM}}^{{TT},\mathrm{BH}}=\displaystyle \frac{{\hat{\phi }}_{{LM}}^{{TT}}-{\left({{ \mathcal R }}_{L}^{{TT}}\right)}^{-1}{{ \mathcal R }}_{L}^{\phi ,s}{\hat{s}}_{{LM}}}{1-{\left({{ \mathcal R }}_{L}^{\phi ,s}\right)}^{2}{\left({{ \mathcal R }}_{L}^{{TT}}\right)}^{-1}{\left({{ \mathcal R }}_{L}^{s}\right)}^{-1}},\end{eqnarray} \tag{ 27 }$$

where ${{ \mathcal R }}_{L}^{\phi ,s}$ is the cross-response between the lensing field ${\hat{\phi }}_{{LM}}^{{TT}}$ and the source field ${\hat{s}}_{{LM}}$, and ${({{ \mathcal R }}_{L}^{s})}^{-1}$ is the normalization for the source estimator.

In our case, we optimize the response to the presence of tSZ cluster "sources," and as shown in Sailer et al. (2023), this estimator is also effective in reducing the effect of point sources by a factor of around five. The cross-response function of this tSZ-profile-hardened estimator is given by

$$\begin{eqnarray}&&{{ \mathcal R }}_{L}^{\phi ,\mathrm{tSZ}}=\displaystyle \frac{1}{2L+1}\displaystyle \sum _{{\ell }{{\ell }}^{{\prime} }}\displaystyle \frac{{f}_{{\ell }L{{\ell }}^{{\prime} }}^{\phi }{f}_{{\ell }L{{\ell }}^{{\prime} }}^{\mathrm{tSZ}}}{2{C}_{{\ell }}^{\mathrm{total}}{C}_{{{\ell }}^{{\prime} }}^{\mathrm{total}}},\end{eqnarray} \tag{ 28 }$$

where ${C}_{{\ell }}^{\mathrm{total}}={C}_{{\ell }}^{{TT}}+{N}_{{\ell }}^{{TT}}$ is the total temperature power spectrum including instrumental noise and ${f}_{{\ell }L{{\ell }}^{{\prime} }}^{\mathrm{tSZ}}$ is the response function to tSZ sources. This response function requires a model for cluster profiles; we estimate an effective profile from the square root of the smoothed tSZ angular power spectrum (which is dominated by the one-halo term) obtained from a websky simulation (Sailer et al. 2020).

In the formalism presented here, the appropriately normalized MV estimator, with the temperature estimator part "hardened" against tSZ, is obtained by first subtracting the standard temperature lensing estimator from the MV estimator and then adding back the profile-hardened temperature estimator, i.e.,

$$\begin{eqnarray}\begin{array}{rcl}{\hat{\phi }}_{{LM}}^{\mathrm{MV},\mathrm{BH}} & = & {\left({{ \mathcal R }}_{L}^{\mathrm{MV}}\right)}^{-1}\left[\displaystyle \frac{{\hat{\phi }}_{{LM}}^{\mathrm{MV}}}{{\left({{ \mathcal R }}_{L}^{\mathrm{MV}}\right)}^{-1}}-\displaystyle \frac{{\hat{\phi }}_{{LM}}^{{TT}}}{{\left({{ \mathcal R }}_{L}^{{TT}}\right)}^{-1}}\right.\\ & & +\,\left.\displaystyle \frac{{\hat{\phi }}_{{LM}}^{{TT},\mathrm{BH}}}{{\left({{ \mathcal R }}_{L}^{{TT}}\right)}^{-1}}\right].\end{array}\end{eqnarray} \tag{ 29 }$$

Both the investigation of foreground mitigation in MacCrann et al. (2024), summarized in this paper in Section 7.1, and the foreground null tests discussed in Section 6 show that this baseline method can control the foreground biases on the lensing amplitude A_lens to levels below 0.2σ, where σ is the statistical error on this quantity.

5.8.3. Cross-correlation-based Quadratic Estimator

The lensing power spectrum constructed using the standard QE is sensitive to assumptions made in simulating and modeling the instrument noise used for calculating the lensing power spectrum biases. This is despite the use of realization-dependent methods, as described in Appendix E.1 (which discusses power spectrum bias subtraction). Hence, in practice, we construct our lensing power spectrum using lensing maps ${\hat{\phi }}_{{LM}}^{({ij}),{XY}}$ reconstructed from different data splits, indexed by i and j, which have independent noise. Using the shorthand notation of QE(X^A , Y^B ) for the QE (see Equation (19)) operating on two sky maps X^A and Y^B , ${\hat{\phi }}_{{LM}}^{({ij}),{XY}}$ is defined as

$$\begin{eqnarray}&&{\hat{\phi }}_{{LM}}^{({ij}),{XY}}=\displaystyle \frac{1}{2}[\mathrm{QE}({X}^{i},{Y}^{j})+\mathrm{QE}({X}^{j},{Y}^{i})].\end{eqnarray} \tag{ 30 }$$

Note that this is symmetric under interchange of the splits.

We use this cross-correlation-based estimator from Madhavacheril et al. (2020b) with independent data splits to ensure that our analysis is immune to instrumental and atmospheric noise effects in the mean-field and N₀ (Gaussian) biases (introduced below in Section 5.9). This makes our analysis highly robust to potential inaccuracies in simulating the complex atmospheric and instrumental noise in the ACT data.

The coadded, standard lensing estimator, equivalent to Equation (23), which uses all the map-split combinations, is given by

$$\begin{eqnarray}&&{\hat{\phi }}_{{LM}}^{{XY}}=\displaystyle \frac{1}{{4}^{2}}\displaystyle \sum _{{ij}}{\hat{\phi }}_{{LM}}^{({ij}),{XY}}.\end{eqnarray} \tag{ 31 }$$

The corresponding estimate of the power spectrum from XY and UV standard QEs is then

$$\begin{eqnarray}&&{C}_{L}^{\hat{\phi }\hat{\phi }}[{XY},{UV}]=\displaystyle \frac{1}{{4}^{4}}\displaystyle \sum _{{ijkl}}{\hat{C}}_{L}^{\hat{\phi }\hat{\phi }}\left({\hat{\phi }}_{{LM}}^{({ij}),{XY}},{\hat{\phi }}_{{LM}}^{({kl}),{UV}}\right).\end{eqnarray} \tag{ 32 }$$

This is modified by removing any terms where the same split is repeated to give the cross-correlation-based estimator:

$$\begin{eqnarray}&&{C}_{L}^{\hat{\phi }\hat{\phi },\times }[{XY},{UV}]=\displaystyle \frac{1}{4!}\displaystyle \sum _{i\ne j\ne k\ne l}{\hat{C}}_{L}^{\hat{\phi }\hat{\phi }}\left({\hat{\phi }}_{{LM}}^{({ij}),{XY}},{\hat{\phi }}_{{LM}}^{({kl}),{UV}}\right).\end{eqnarray} \tag{ 33 }$$

In this way, only lensing maps constructed from CMB maps with independent noise are included, so noise mismodeling does not affect the mean-field estimation, and any cross-powers between lensing maps that repeat splits (and hence contribute to the Gaussian noise bias) are discarded.

We can accelerate the computation of Equation (33) following Madhavacheril et al. (2020b). We introduce the following auxiliary estimators using different combinations of splits:

$$\begin{eqnarray}&&{\hat{\phi }}_{{LM}}^{\times ,{XY}}={\hat{\phi }}_{{LM}}^{{XY}}-\displaystyle \frac{1}{16}\displaystyle \sum _{i=1}^{4}{\hat{\phi }}_{{LM}}^{({ii}),{XY}},\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&{\hat{\phi }}_{{LM}}^{(i),{XY}}=\displaystyle \frac{1}{4}\displaystyle \sum _{j=1}^{4}{{\hat{\phi }}^{({ij}),{XY}}}_{{LM}},\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&{\hat{\phi }}_{{LM}}^{(i)\times ,{XY}}={\hat{\phi }}_{{LM}}^{(i),{XY}}-\displaystyle \frac{1}{4}{\hat{\phi }}_{{LM}}^{({ii}),{XY}},\end{eqnarray} \tag{ 36 }$$

in terms of which the cross-correlation-based estimator may be written as

$$\begin{eqnarray}&&\begin{array}{l}{C}_{L}^{\hat{\phi }\hat{\phi },\times }[{XY},{UV}]=\displaystyle \frac{1}{4!}\left[\Space{0ex}{3.6ex}{0ex}256{\hat{C}}_{L}^{\hat{\phi }\hat{\phi }}\left({\hat{\phi }}_{{LM}}^{\times ,{XY}},{\hat{\phi }}_{{LM}}^{\times ,{UV}}\right)\right.\\ \quad -\,64\displaystyle \sum _{i=1}^{4}{\hat{C}}_{L}^{\hat{\phi }\hat{\phi }}\left({\hat{\phi }}_{{LM}}^{(i)\times ,{XY}},{\hat{\phi }}_{{LM}}^{(i)\times ,{UV}}\right)\\ \quad +\,\left.4\displaystyle \sum _{i\leqslant j}{\hat{C}}_{L}^{\hat{\phi }\hat{\phi }}\left({\hat{\phi }}_{{LM}}^{({ij}),{XY}},{\hat{\phi }}_{{LM}}^{({ij}),{UV}}\right)\right].\end{array}\end{eqnarray} \tag{ 37 }$$

Finally, the baseline lensing map we produce, which again avoids repeating the same data splits in the estimator, is given by

$$\begin{eqnarray}&&{\hat{\phi }}_{{LM}}^{{XY}}=\displaystyle \frac{1}{6}\displaystyle \sum _{i\lt j}{\hat{\phi }}_{{LM}}^{({ij}),{XY}}.\end{eqnarray} \tag{ 38 }$$

The resulting lensing map is shown in CAR projection in Figure 5, with the map filtered to highlight the signal-dominated scales.

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Naive lensing power spectrum estimators based on the autocorrelation of a reconstructed map are known to be biased owing to both reconstruction noise and higher-order lensing terms. This is also true for the cross-correlation-based lensing power spectrum in Equation (37), despite its insensitivity to noise. To obtain an unbiased lensing power spectrum from the naive lensing power spectrum estimator, we must subtract the well-known lensing power spectrum biases: the N₀ and N₁ biases, as well as a small additive MC bias. The bias-subtracted lensing power spectrum is thus given by

$$\begin{eqnarray}&&{\hat{C}}_{L}^{\phi \phi ,\times }={C}_{L}^{\hat{\phi }\hat{\phi },\times }-{\rm{\Delta }}{C}_{L}^{\mathrm{Gauss}}-{\rm{\Delta }}{C}_{L}^{{N}_{1}}-{\rm{\Delta }}{C}_{L}^{\mathrm{MC}}.\end{eqnarray} \tag{ 39 }$$

These biases can be understood in more detail as follows. The N₀ or Gaussian bias, ${\rm{\Delta }}{C}_{L}^{\mathrm{Gauss}}$, is effectively a lensing reconstruction noise bias. Equivalently, since the lensing power spectrum can be measured by computing the connected part of the four-point correlation function of the CMB, ${\rm{\Delta }}{C}_{L}^{\mathrm{Gauss}}$ can be understood as the disconnected part that must be subtracted off the full four-point function; these disconnected contractions are produced by Gaussian fluctuations present even in the absence of lensing. The N₀ bias is calculated using the now-standard realization-dependent N₀ algorithm introduced in Namikawa et al. (2013) and Planck Collaboration et al. (2014). This algorithm, which combines simulation and data maps in specific combinations to isolate the different contractions of the bias, is described in detail in Appendix E.1. The use of a realization-dependent N₀ bias reduces correlations between different lensing bandpowers and also makes the bias computation insensitive to inaccuracies in the simulations.

The N₁ bias subtracts contributions from "accidental" correlations of lensing modes that are not targeted by the QE (see Kesden et al. 2003 for details; the nomenclature arises because the N₁ bias is first order in ${C}_{L}^{\phi \phi }$, unlike the N₀ bias, which is zeroth order in the lensing spectrum). The N₁ bias is computed using the standard procedure introduced in Story et al. (2015) and described in Appendix E.2.

Finally, we absorb any additional residuals arising from nonidealities, such as the effects of masking, in a small additive MC bias ${\rm{\Delta }}{C}_{L}^{\mathrm{MC}}$ that is calculated with simulations. We describe the computation of this MC bias in detail in Appendix E.3.

The unbiased lensing spectrum, scaled by L²(L + 1)²/4, is binned in bandpowers with uniform weighting in L. Details regarding the bins and ranges adopted in our analysis can be found in Section 6.1.

To illustrate the sizes of the different bias terms subtracted, we plot them all as a function of scale in Figure 6. The fact that the additive MC bias is small is an important test of our pipeline and indicates that it is functioning well. The procedures laid out above constitute our core full-sky lensing pipeline, which enables the unbiased recovery of the lensing power spectrum after debiasing.

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Prior to normalization, the quadratic lensing estimator probes not just the lensing potential ϕ; it is instead sensitive to a combination ${\phi }_{L,M}\times {{ \mathcal R }}_{L}{| }_{{C}_{{\ell }}^{\mathrm{CMB}}}$, where the response ${{ \mathcal R }}_{L}{| }_{{C}_{{\ell }}^{\mathrm{CMB}}}$ is a function of the true CMB two-point power spectra. Applying the normalization factor ${{ \mathcal R }}_{L}^{-1}{| }_{{C}_{{\ell }}^{\mathrm{CMB},\mathrm{fid}}}$, where ${C}_{{\ell }}^{\mathrm{CMB},\mathrm{fid}}$ are the fiducial CMB power spectra assumed in the lensing reconstruction, attempts to divide out this CMB power spectrum dependence and provide an unbiased lensing map. If the power spectra describing the data are equal to the fiducial CMB power spectra (i.e., ${C}_{{\ell }}^{\mathrm{CMB}}={C}_{{\ell }}^{\mathrm{CMB},\mathrm{fid}}$), the estimated lensing map is indeed unbiased. Otherwise, the estimated lensing potential is biased by a factor ${{ \mathcal R }}_{L}^{-1}{| }_{{C}_{{\ell }}^{\mathrm{CMB},\mathrm{fid}}}/{{ \mathcal R }}_{L}^{-1}{| }_{{C}_{{\ell }}^{\mathrm{CMB}}}$.

In early CMB lensing analyses, it was assumed that the CMB power spectra were determined much more precisely than the lensing field, so that any uncertainty in the CMB two-point function and in the normalization could be neglected; however, with current high-precision lensing measurements, the impact of CMB power spectrum uncertainty must be considered. We use as our fiducial CMB power spectra the standard ΛCDM model from Planck 2015 TTTEEE cosmology with an updated τ prior as in Calabrese et al. (2017). In Appendix B, we describe in detail our tests of the sensitivity of our lensing power spectrum measurements to this assumption; we summarize the conclusions below.

We analytically compare the amplitude of the lensing power spectrum A_lens when changing the fiducial CMB power spectra described above to the best-fit model CMB power spectra for an independent data set, namely ACT DR4+WMAP (Aiola et al. 2020); we account for the impact of calibration and polarization efficiency characterization in this comparison. Doing this, we find a change in A_lens of only 0.23σ, comfortably subdominant to our statistical uncertainty. An important reason why this change is so small is that our preprocessing procedures, which involve calibration and polarization efficiency corrections relative to the Planck spectra, drive the amplitudes of the spectra in our data closer to our original fiducial model. This result reassures us that the CMB power spectra are sufficiently well measured, by independent experiments, not to degrade our uncertainties on the lensing power spectrum significantly.

Nevertheless, we additionally account for uncertainty in the CMB power spectra in our cosmological inference from the lensing measurements alone (i.e., when not also including CMB anisotropy measurements) by adding to the covariance matrix a small correction calculated numerically from an ensemble of cosmological models sampled from a joint ACT DR4+Planck chain (see Aiola et al. 2020 for details). This results in a small increase in our errors (by approximately 3% for the lensing spectrum bandpower error bars), although the changes to the cosmological parameter constraints obtained are nearly negligible. ⁹⁰

We obtain the bandpower covariance matrix from N_s = 792 simulations. We do not subtract the computationally expensive realization-dependent RDN0 from all the simulations when evaluating the covariance matrix. Instead, we use an approximate, faster version, referred to as the semianalytic N₀, which we describe briefly below in Section 5.11.1.

To account for the fact that the inverse of the above covariance matrix is not an unbiased estimate of the inverse covariance matrix, we rescale the estimated inverse covariance matrix by the Hartlap factor (Hartlap et al. 2007):

$$\begin{eqnarray}&&{\alpha }_{\mathrm{cov}}=\displaystyle \frac{{N}_{s}-{N}_{\mathrm{bins}}-2}{{N}_{s}-1},\end{eqnarray} \tag{ 40 }$$

where N_bins is the number of bandpowers.

5.11.1. Semianalytic N₀

5.11.2. Covariance Verification

5.11.3. Covariance Matrix Results and Correlation between Bandpowers

The correlation matrix for our lensing power spectrum bandpowers, obtained using a set of 792 simulations, can be seen in Figure 7. We find that correlations between different bandpowers are small, with off-diagonal correlations typically below 10%.

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For our baseline analysis, we use the lensing multipoles 40 < L < 763 with the following nonoverlapping bin edges for N_bins = 10 bins at [40, 66, 101, 145, 199, 264, 339, 426, 526, 638, 763]. The baseline multipole range 40 < L < 763 was decided prior to unblinding. This range is informed by both the results of the null tests and the simulated foreground estimates. The scales below L = 40 are removed owing to large fluctuations at low L observed in a small number of null tests; these scales are difficult to measure robustly since the simulated mean field becomes significantly larger than the signal, although the cross-based estimator relaxes simulation accuracy requirements on the statistical properties of the noise. The L_max limit is motivated by the results of the foreground tests on simulations performed in MacCrann et al. (2024), where at ${L}_{\max }=763$ the magnitude of fractional biases in the fit of the lensing amplitude is still less than 0.2σ (0.5%), although biases rise when including smaller scales. This upper range is rather conservative, and hence we also provide an analysis with an extended cosmology range up to ${L}_{\max }=1300$, although we note that this extended range was not determined before unblinding and that instrumental systematics have only been rigorously tested for the baseline range. (We also note that the null test PTEs and simulated foreground biases still appear acceptable in the extended range, although, again, we caution that we only carefully examined the extended-range null tests after we had unblinded.)

In any null test, we construct a set of null bandpowers d ^null, which (after appropriate bias subtraction) should be statistically consistent with zero. For map-level null tests, d ^null are the bandpowers obtained by performing lensing power spectrum estimation on CMB maps differenced to null the signal, while for bandpower-level null tests they are given by differences of reconstructed lensing power spectra, ${\rm{\Delta }}{C}_{L}^{\hat{\phi }\hat{\phi }}$. We test consistency of the null bandpowers with zero by calculating the χ² with respect to null:

$$\begin{eqnarray}&&{\chi }^{2}=({{\boldsymbol{d}}}^{\mathrm{null}}){}^{\top }{{\mathbb{C}}}^{-1}\,{{\boldsymbol{d}}}^{\mathrm{null}}.\end{eqnarray} \tag{ 41 }$$

The relevant covariance matrix ${\mathbb{C}}$ for each null test is estimated by performing the exact same analysis on 792 simulations, ensuring that all correlations between the different data sets being nulled are correctly captured. The PTE is calculated from the χ² with 10 degrees of freedom as we have 10 bandpowers in the baseline range. (We also consider and compute PTEs for our extended scale range, which has 13 degrees of freedom.)

We adopt a blinding policy that is intended to be a reasonable compromise between reducing the effect of confirmation bias and improving our ability to discover and diagnose issues with the data and the pipeline efficiently. We define an initial blinded phase after which, when predefined criteria are met, we unblind the data.

In the initial blinded phase, as part of our blinding policy, we agree in advance to abide by the following rules.

• 1.  
  We do not fit any cosmological parameters or lensing amplitudes to the lensing power spectrum bandpowers. While we allow debiased lensing power spectra to be plotted, we do not allow them to be compared or plotted either against any theoretical predictions or against bandpowers from any previous CMB lensing analyses, including the Planck analyses. In this way, we are blind to the amplitude of lensing at the precision needed to inform the S₈ tension and constrain neutrino masses, but we can still rapidly identify any catastrophic problems with our data—although none were found.
• 2.  
  During the blinded phase, we also allow unprocessed lensing power spectra without debiasing to be plotted against theory curves or Planck bandpowers for L > 200. The justification for this is that the unprocessed spectra are dominated on small scales by the Gaussian N₀ bias and hence not informative for cosmology, although they allow for useful checks of bias subtraction and noise levels. When analyzing bandpowers of individual array-frequency reconstructions, we allow unprocessed spectra to be plotted over all multipoles, because individual array-frequency lensing spectra are typically noise bias dominated on all scales. ⁹³

We calculate PTE values of bandpowers in our map-level null tests (see Section 6.4) and for differences of bandpowers in our consistency tests (Section 6.5) during the blinded phase. For the power spectra of the CMB maps themselves (as opposed to those of lensing reconstructions), we follow a blinding policy that will be described in an upcoming ACT DR6 CMB power spectrum paper.

After unblinding, all these restrictions are lifted and we proceed to the derivation of cosmological parameters. We require the following criteria to be satisfied before unblinding.

• 1.  
  All baseline analysis choices made in running our pipeline, such as the range of CMB angular scales used, are frozen.
• 2.  
  No individual null test PTE should lie outside the range 0.001 < PTE < 0.999.
• 3.  
  The distribution of PTEs for different null tests should be consistent with a uniform distribution (verified via a Kolmogorov–Smirnov test, with the caveat that this neglects correlations).
• 4.  
  The number of null and consistency tests that fall outside the range 0.01 < PTE < 0.99 should not be significantly inconsistent with the expectations from random fluctuations.
• 5.  
  The comparison of the sum of χ² for several different types of tests against expectations from simulations should fall within 2σ of the simulation distributions.

The PTE ranges we accept are motivated by the fact that we calculate ${ \mathcal O }(100)$ PTEs but not ${ \mathcal O }(1000)$.

6.3.1. Post-unblinding Change

This subsection describes null tests in which we apply the full lensing power spectrum estimation pipeline to maps that are expected to contain no signal in the absence of systematic effects. In all cases except for the curl reconstruction in Section 6.4.1, this typically involves differencing two variants of the sky maps at the map level (hence nulling the signal) and then proceeding to obtain debiased lensing power spectra from these null maps. To adhere closely to the baseline lensing analysis, we always prepare four signal-differenced maps and make use of the cross-correlation-based estimator.

We describe each of our map-level null tests in more detail in the sections below with a summary of the χ² and associated PTE found in Table 1.

6.4.1. Curl

The lensing deflection field d can be decomposed into gradient and curl parts based on the potentials ϕ and Ω, respectively, i.e., in terms of components d_i = ∇_i ϕ + _i ^j ∇_j Ω, where Ω is the divergence-free or "curl" component Ω of the deflection field and ϕ is again the lensing potential. (Here _ij is the alternating tensor on the unit sphere.) The curl Ω is expected to be zero at leading order and therefore negligible at ACT DR6 reconstruction noise levels (although a small curl component induced by post-Born and higher-order effects may be detectable in future surveys; Pratten & Lewis 2016). However, systematic effects do not necessarily respect a pure gradient-like symmetry and hence could induce a nonzero curl-like signal. An estimate of this curl field can thus provide a convenient diagnostic for systematic errors that can mimic lensing. Furthermore, curl reconstruction also provides an excellent test of our simulations, our pipeline, and our covariance estimation.

We obtain a reconstruction of this curl field in the same manner as described in Section 5.8.1, by taking linear combinations of the spin-1 spherical harmonic transform of the deflection field. The bias estimation steps are then repeated in the same way as for the lensing estimator. The result for this null test is shown in Figure 8 for the MV coadded result, ⁹⁴ which is the curl equivalent of our baseline lensing spectrum. This test has a PTE of 0.37, in good agreement with null. We also show curl null test results for the temperature-only (TT) version of our estimator in Figure 8.

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The consistency of our curl measurement with zero provides further evidence of the robustness of our lensing measurement. Intriguingly, the curl null test was not passed for the TT estimator in Planck, and instead (despite valiant efforts to explain it) a 4.1σ deviation ⁹⁵ from zero has remained, located in the range 264 < L < 901 (Planck Collaboration et al. 2020a); see Figure 8. Our result provides further evidence that this nonzero curl is not physical in origin.

For completeness, we also compute curl tests associated with all other null tests described in the subsequent sections; we summarize the results and figures in Appendix I. These results also show that there is no evidence of curl modes found even in subsets of our data.

6.4.2. Noise-only Null Tests: Individual Array-frequency Split Differences

6.4.3. Map-level Frequency-difference Test

We prepare frequency-differenced null maps by subtracting the beam-deconvolved f150 split maps from the f090 split maps. The resulting difference maps are passed into the lensing reconstruction pipeline with the filters, normalization, and bias-hardening procedure the same as used for the baseline reconstruction, which combines f150 and f090. This filter choice weights different scales in the null maps in the same way as for our baseline lensing measurement, which ensures that null test results can be directly compared with our baseline lensing results. The null lensing power spectrum ${C}_{L}^{\mathrm{null}}$ is given schematically by

$$\begin{eqnarray}\begin{array}{rcl}{C}_{L}^{\mathrm{null}} & = & \langle \mathrm{QE}({T}^{90}-{T}^{150},{T}^{90}-{T}^{150})\\ & & \times \,\mathrm{QE}({T}^{90}-{T}^{150},{T}^{90}-{T}^{150})\rangle .\end{array}\end{eqnarray} \tag{ 42 }$$

This measurement is a rigorous test for our mitigation of foregrounds: the effect of foregrounds such as CIB and tSZ is expected to be quite different in these two frequency channels (with f090 more sensitive to tSZ and less to CIB), so we do not expect full cancellation of foregrounds in the difference maps. In particular, this null test targets the residual foreground-only trispectrum of the lensing maps; we compare our results with the levels expected from simulations in MacCrann et al. (2024). In addition, this map-level null test is also sensitive to beam-related differences between the two frequency channels.

As shown in Figure 10, these null tests are consistent with zero, with PTEs of 0.67 and 0.84 for MV and TT, respectively; no evidence for unmitigated foreground contamination is found.

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6.4.4. Frequency-nulled map$\,\times \,{\hat{\phi }}^{\mathrm{MV}}$

To perform an additional, similarly powerful test of foregrounds, we cross-correlate the null reconstruction from the frequency-difference maps, obtained as in the previous null test, with the baseline reconstruction ${\hat{\phi }}^{\mathrm{MV}};$ i.e., schematically, we compute

$$\begin{eqnarray}&&{C}_{L}^{\mathrm{null}}=\left\langle \mathrm{QE}({T}^{90}-{T}^{150},{T}^{90}-{T}^{150})\times {\hat{\phi }}^{\mathrm{MV}}\right\rangle .\end{eqnarray} \tag{ 43 }$$

This measurement is sensitive mainly to the foreground bispectrum ⁹⁶ involving two powers of foreground residuals and one power of the true convergence field. To a lesser extent, given the small residual foreground biases remaining in ${\hat{\phi }}^{\mathrm{MV}}$, the test is also sensitive to a foreground trispectrum contribution. The null test results in Figure 10 show good consistency with zero, with a passing PTE of 0.61 and 0.93 for MV and TT, respectively.

Since the foreground bias probed by this test is the dominant one on large scales, the consistency of this test with null is a particularly powerful test of foreground mitigation in our analysis.

6.4.5. Array-frequency Differences

This section describes tests that aim to assess whether lensing spectrum bandpowers from variations of our analysis, or subsets of our data, are consistent with each other. For each variation or subset, we subtract the resulting debiased lensing power spectrum from our baseline debiased lensing power spectrum; both spectra are obtained with our standard methodology described in Section 5.8. We obtain a covariance matrix for this difference by repeating this analysis (with semianalytic debiasing described in Section 5.11.1) on simulations. We then use the nulled bandpower vector and its covariance matrix to check for consistency with zero. We summarize the results of these tests in Table 2; in this table, we also utilize the statistic ΔA_lens to quantify the magnitude of any potential bias to the lensing amplitude produced by the departure of the null test bandpowers from zero, i.e.,

$$\begin{eqnarray}&&{\rm{\Delta }}{A}_{\mathrm{lens}}=\displaystyle \frac{{\sum }_{{{bb}}^{{\prime} }}{\hat{C}}_{{L}_{b}}^{\mathrm{null}}\,{{\mathbb{C}}}_{{{bb}}^{{\prime} }}^{-1}\,{\hat{C}}_{{L}_{b}}^{\phi \phi }}{{\sum }_{{{bb}}^{{\prime} }}{\hat{C}}_{{L}_{b}}^{\phi \phi }\,{{\mathbb{C}}}_{{{bb}}^{{\prime} }}^{-1}\,{\hat{C}}_{{L}_{b}}^{\phi \phi }}.\end{eqnarray} \tag{ 44 }$$

Here ${\hat{C}}_{{L}_{b}}^{\phi \phi }$ is the baseline lensing power spectrum and ${{\mathbb{C}}}_{{{bb}}^{{\prime} }}$ is the baseline covariance matrix. The ΔA_lens results are summarized in Figure 12.

For all of the null tests discussed in subsequent sections, we present plots that show the lensing bandpowers in the top panel, with the baseline analysis in red boxes. Additionally, we include a subpanel showing differences of bandpowers divided by the baseline MV errors σ_L .

6.5.1. Temperature−Polarization Consistency

6.5.2. Bandpower-level Frequency-difference Test

We compare the lensing power spectrum derived from f090 and f150 data alone to our baseline analysis in Figure 14.

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The null bandpowers are formed by taking the difference

$$\begin{eqnarray}&&{C}_{L}^{\mathrm{null}}={C}_{L}^{\hat{\phi }\hat{\phi },90\,\mathrm{GHz}}-{C}_{L}^{\hat{\phi }\hat{\phi },150\,\mathrm{GHz}},\end{eqnarray} \tag{ 45 }$$

where ${C}_{L}^{\hat{\phi }\hat{\phi },90\,\mathrm{GHz}}$ is the lensing spectrum reconstructed with the f090 data only, i.e., PA5 f090 and PA6 f090, and ${C}_{L}^{\hat{\phi }\hat{\phi },90\mathrm{GHz}}$ is obtained by reconstructing the data at f150 only (from the PA4 f150, PA5 f150, and PA6 f150 array frequencies). For the coaddition of the data we use the same noise weights (up to normalization) as in the baseline analysis. For the reconstruction we use the same filters used for the baseline analysis. This null test is sensitive to all foreground contributions (including both bispectrum and trispectrum terms). However, compared to the map-level frequency-difference null test above, this measurement has larger errors, since the lensed CMB is not nulled at the map level. Our results in Figure 14 show good agreement of the lensing reconstruction obtained from different frequencies. The curl is also shown in Appendix I.8.

6.5.3. Consistency with CIB Deprojection Analysis

The companion paper MacCrann et al. (2024) finds that a CIB-deprojected version of the analysis shows similar performance to our baseline analysis in mitigating foreground biases to a negligible level without incurring a large S/N penalty. We therefore perform a consistency check between this alternative, multifrequency-based foreground mitigation method and the geometry-based profile hardening method that is our baseline.

MacCrann et al. (2024) describe the production of CIB-deprojected temperature maps by performing a harmonic-space constrained ILC (hILC) of the DR6 coadded temperature map and the high-frequency data from Planck at 353 and 545 GHz. The high-frequency Planck channels are chosen because the CIB is much brighter and the primary CMB information is subdominant at high frequencies; these high-frequency maps are hence valuable foreground monitors that can be used while still keeping our analysis largely independent of CMB measurements from Planck.

Performing the hILC requires the use of the total auto- and cross-spectra for all the input maps; these are measured directly from the data and are smoothed with a Savitzky–Golay filter (Savitzky & Golay 1964; window length 301 and polynomial order 2), to reduce "ILC bias" (see, e.g., Delabrouille et al. 2009) arising from fluctuations in the spectrum measurements. We also generate 600 realizations of these maps, using the Planck NPIPE noise simulations provided by Planck Collaboration et al. (2020c); these are used for the N₀ subtraction, mean-field correction, and covariance matrix estimation. The deprojected temperature maps are then used in our lensing reconstruction and lensing power spectrum estimation (along with the same polarization data as are used in the baseline analysis and the same cross-correlation-based estimator with profile hardening).

As seen in Figure 15, the results show that the bandpowers are consistent with the baseline analysis with a PTE of 0.11. This implies that CIB contamination is not significant in our lensing analysis. Given the similarity of the spectral energy distributions of the CIB and Galactic dust, these results also provide evidence against significant biases from Galactic dust contamination. At the same time, since CIB deprojection increases the amount of tSZ in our maps (Abylkairov et al. 2021; Kusiak et al. 2023), the stability of our results also suggests that tSZ is mitigated well.

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6.5.4. Shear Estimator

6.5.5. Array Differences

In addition to testing the consistency of the different array frequencies at the map level, we further test their consistency by comparing the lensing bandpowers obtained from each array. For the filtering operation, we use a filter with a noise level consistent with the given array instead of the coadded baseline noise level; this choice was made in order to increase the S/N for each single-array spectrum (which is not always high) and increase the sensitivity of the test. This bandpower null test provides a broader assessment of the presence of possible multiplicative biases that could create inconsistencies among the different array frequencies. The results in Table 2 and Figure 16 show that there is no evidence of such effects in our data.

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6.5.6. Cross-linking

A pixel of the maps is said to be well cross-linked when it has been observed by different scans that are approximately orthogonally oriented at the location of the pixel on the sky. The DR6 scan strategy produces adequate cross-linking over the survey area except for a narrow region around decl. = − 35°. The poorly cross-linked region has significantly more correlated noise than the rest of the patch.

We isolate a region with poor cross-linking, with the footprint borders shown in green in Figure 5. We compare the bandpowers obtained from this region (using a filter consistent with the coadd noise of this small noisy patch) with those from our baseline analysis in Figure 17 and find no statistically significant difference in the bandpowers obtained from both regions, with a null PTE = 0.94.

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6.5.7. Multipole-range Variation

We compare our baseline MV reconstruction, which uses the CMB with multipoles in the range 600 < ℓ < 3000, against reconstructed lensing spectra obtained from different CMB scale ranges: 500 < ℓ < 3000, 600 < ℓ < 3000, 800 < ℓ < 3000, 1000 < ℓ < 3000, 1500 < ℓ < 3000, 300 < ℓ < 2500, and 300 < ℓ < 2000. This multipole-range variation tests the following: (i) the consistency of lensing spectra when including a more or less extended range of CMB modes and, more specifically, (ii) the impact of extragalactic foregrounds such as CIB and tSZ, which should increase as we increase the maximum multipole; and (iii) the impact of any ground pickup, transfer function, or Galactic foreground systematics, which should worsen when decreasing the minimum multipole.

The lensing bandpowers and the null bandpowers can be seen in Figure 18. We observe excellent consistency with the baseline bandpowers.

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6.5.8. Dust Mask Variation

6.5.9. North Galactic versus South Galactic

6.5.10. Fourier-space Filter Variations

6.5.11. Temporal Null Tests

In this section, we present an overview of all our null test results. We first summarize the results of the previous sections using the overall distribution of PTEs, shown in Figure 19. Our conclusion is that the distribution of the PTEs of all the null tests is consistent with a uniform distribution for both the baseline and extended range (passing at 10% and 74% with the Kolmogorov–Smirnov test, respectively).

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Beyond performing individual null tests, a powerful check for subdominant systematic errors is to analyze suitable groups of null tests that probe similar effects. To that end, we group all the null tests discussed previously into five categories: tests focusing on noise properties, different angular scales, different frequencies/foregrounds, isotropy, and instrument-related systematics. We then compare (i) the sum of all χ² values within each category and (ii) the worst (i.e., largest) χ² within each category from the data with the distribution of such statistics obtained in simulations, as shown in Figures 9, 11, 20, 21, and 22. For each test, we consider both gradient lensing modes and curl modes. In the figures, the blue lines show the sum or worst χ² statistic for the data and the dotted red lines show the 2σ limits from the ensemble of simulated measurements of the relevant statistic. We do not see strong evidence of a systematic effect from the above analysis. Only the worst-χ² statistics for the noise-only and instrument null test sets slightly exceed the 2σ limits from simulations. These tests correspond to the coadded noise-only test (Section 6.4.2) and the map-level null for the PA4 f150 and PA5 f090 array frequencies (Section 6.4.5), respectively.

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For the following reasons, we do not consider the array-frequency null test failures related to PA4 f150 and PA5 f090 concerning.

• 1.  
  The array-frequency map difference tests are very hard to pass, as the signal-variance contribution to the error bars is absent; this implies, first, that the null test errors can be much smaller than the measurement errors and, second, that the requirements on the fidelity of the noise simulations are much more stringent than needed for the standard lensing spectrum measurement. We also note that, since we do not subtract an N₀ bias from these null tests, a failure could simply indicate that the CMB power spectrum of the two different maps being nulled is inconsistent. This does not necessarily imply an inconsistency in lensing, because in our lensing power spectrum analysis the realization-dependent bias subtraction methodology should absorb small changes in the CMB power spectra.
• 2.  
  Some of these worst-performing map-level null tests involve the array frequency PA4 f150. We further checked for possible inconsistencies of this array with the others by performing bandpower-level null tests in Section 6.5.5; these tests should be more sensitive to multiplicative biases affecting different data sets than the map-level test and, furthermore, include the signal part of the covariance matrix. In these targeted tests, we found no evidence of a systematic difference in PA4 f150 compared to the other array frequencies.
• 3.  
  In addition, the array PA4 f150 has the least weight in our coadd data; as can be seen in Figure 4, it only contributes less than 10% at each CMB multipole to the total coadded sky map used in our analysis.
• 4.  
  Foregrounds are not a likely cause of the PA4 f150 − PA5 f090 null failure since the much more sensitive f090 − f150 coadd map null test passes.

MacCrann et al. (2024) focus exclusively on the systematic impact of extragalactic foregrounds; we briefly summarize that work here. We characterize and validate the mitigation strategies employed in our lensing analysis via a three-pronged approach.

• 1.  
  We refine and test the mitigation strategies on two independent microwave-sky simulations: those from Sehgal et al. (2010), and the websky simulations (Stein et al. 2020). We demonstrate that the baseline approach taken here, i.e., finding/subtracting S/N > 4 point sources, finding and subtracting models for S/N > 5 clusters, and using a profile-hardened QE (see Section 5.8.2 for details), performs better than or equally well to the other tested mitigation strategies, with subpercent fractional biases to ${C}_{L}^{\phi \phi }$ on both simulations within most of the analysis range (see the left panel of Figure 2 in MacCrann et al. 2024, reproduced here as Figure 23). We also find that for both simulations biases in the inferred lensing power spectrum amplitude, A_lens, are below 0.2σ in absolute magnitude and below 0.25σ when extending the analysis range to ${L}_{\max }=1300$, as shown in the right panel of Figure 23. In addition, we demonstrate that a variation of our analysis where we include Planck 353 and 545 GHz channels in a harmonic ILC while deprojecting a CIB-like spectrum also performs well (see the dashed lines in Figure 23); this motivated also running this variation on the real DR6 data to ensure that our measurement is robust to the CIB.
• 2.  
  Since the results described above to some extent rely on the realism of the extragalactic microwave-sky simulations, we also demonstrate our robustness to foregrounds using targeted null tests that leverage the fact that the foreground contamination is frequency dependent, while the CMB lensing signal is not. These null tests are also presented in this paper (Figures 10 and 14).
• 3.  
  Finally, we demonstrate that the DR6 data bandpowers are consistent when we use either CIB-deprojected maps, which should reduce CIB contamination but to some extent increase tSZ contamination (see Section 6.5.3), or the shear estimator (see Section 6.5.4).

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We investigate the effects of various instrumental systematic factors on the lensing auto-spectrum measurement and present the summary here. These include (i) miscalibration, (ii) beam uncertainty, (iii) temperature-to-polarization leakage, (iv) polarization efficiency, and (v) polarization angles. A comprehensive evaluation of these factors and their impact on the measurement is presented in Appendix C. Here we summarize the estimated systematic biases to the lensing power spectrum in terms of ΔA_lens, the typical bias in the lensing amplitude. This is calculated from Equation (44), with ${\hat{C}}_{{L}_{b}}^{\mathrm{null}}$ replaced by ${\rm{\Delta }}{\hat{C}}_{{L}_{b}}^{\phi \phi }$, the shift to the power spectrum due to a systematic effect. Note that the ΔA_lens is computed for the baseline range (L ∈ [40, 763]). Table 3 summarizes the ΔA_lens values of the systematics considered in this study. The estimated levels of instrument systematics would not produce a significant bias to our lensing measurements. We now briefly discuss these systematic effects in turn.

The lensing power spectrum can be impacted by miscalibration of the sky maps and beam uncertainties. To address this, we have estimated error budgets for both factors using both exact and approximate methods, and we found no significant bias (Appendix C.1). Summary statistics for ΔA_lens, given the estimated uncertainties in the calibration and beam transfer functions, are reported in Table 3. Furthermore, the consistency of patch-based isotropy tests in Appendix I.2 shows that there is no evidence of spatially varying beams at levels relevant for lensing.

As discussed in Section 3.2, measurements on Uranus provide evidence of temperature-to-polarization leakage in the ACT data. We estimate the impact of this by adding leakage terms to the polarization maps and analyzing their response; see Appendix C.2 for details. These results indicate only a small effect in our lensing measurement, as summarized in Table 3.

Some evidence has been found for a larger temperature-to-polarization leakage in the maps than specified by the nominal Uranus leakage model, based on power spectrum differences between detector arrays. We analyze a simple model for such leakage in Appendix C.2 and propagate this through our lensing measurement for the extreme case of the same leakage for all array frequencies. The bias on A_lens is again a small fraction of the statistical error (see Table 3).

The absolute polarization angle error in the ACT DR6 data set is found to be consistent with the previous ACT DR4 data set, and rotating the Q/U maps by an amount equal to the estimated DR4 angle plus its uncertainty (a total Φ_P = − 016) has minimal impact on the measurement (Table 3, with further details in Appendix C.3).

The polarization efficiency correction that we apply to the polarization maps was discussed in Section 3.4; see also Appendix A for a full description. It is based on the mean of the measured values for each array, obtained by fitting the ACT DR6 EE spectra to the fiducal model. A test where the estimated polarization efficiency is lowered by 1σ, where σ is the mean of the per-array statistical errors from the fitting, shows no significant impact (Table 3 and Appendix C.4). We note that this is a very conservative test, as it assumes that the errors in the fitted polarization efficiences are fully correlated across array frequencies.

Finally, we note that a broader, simulated investigation of the impact of instrument systematics on CMB lensing was performed in Mirmelstein et al. (2021); although the simulation choices are not an exact match to ACT DR6, it is reassuring that the resulting systematic biases reported there appear negligible at the levels of precision considered in our analysis.

This analysis uses the first science-grade version of the ACT DR6 maps, namely the dr6.01 maps. Since these maps were generated, we have made some refinements to the mapmaking that improve the large-scale transfer function (discussed in Section 3.3) and polarization noise levels, and we include data taken in 2022. We expect to use a second version of the maps for further science analyses and for the DR6 public data release. While we caution that we cannot, with absolute certainty, exclude changes to the lensing power spectrum with future versions of the maps, we are confident that our results are robust and stable for the reasons discussed below.

First, as discussed in Section 3.3, there are nonidealities such as transfer function and leakage effects in the analyzed version of the maps. In principle, one could worry that the source of this transfer function problem also affects lensing measurements. However, these effects appear primarily at low multipoles below ℓ ∼ 1000. Fortunately, the lensing estimator is insensitive to effects at low multipoles, which is the reason for the negligible change found when correcting for the transfer function (as discussed in Section 3.3).

Second, after refinements to the mapmaking were implemented, improving the large-scale transfer functions and polarization noise levels, a small region of sky was mapped with the old and new mapmaking procedures to allow comparison. We measure lensing from the old and new maps and construct lensing power spectrum bandpowers and map-level null tests as described in Section 6. In the comparison of the lensing power spectra, the new maps produce a lower spectrum by 0.3σ (of the measurement error on this region) or 3.5%; while at the time of publication noise simulations from the new mapping procedure were not available to us to assess the significance of the difference, this variation appears consistent with random scatter. How do you make this judgment without noise simulations? By determining the degree of correlation of the two CMB maps, maybe? In addition, residuals in the map-level null test are entirely negligible (below ${C}_{L}^{\phi \phi }/1000$), ruling out significant non-Gaussian systematics that differ between the map versions.

Finally, several of our null tests should be sensitive to the known nonidealities in the maps. For example, transfer function effects appear worse in f150 than in f090; related systematics might therefore affect the f090 − f150 frequency null tests or array-frequency null tests. Similarly, systematics affecting low or high multipoles differently should affect the multipole-range stability tests shown in Figure 18. Despite having run a suite of more than 200 null tests, no significant evidence for systematics was found; this, in particular, gives us confidence in our results.

Our baseline lensing power spectrum measurement is shown in Figure 1 and Table 4. In addition to this combined temperature and polarization (MV) measurement, we also provide temperature-only and polarization-only lensing bandpowers in Figure 24 and their respective A_lens values in Figure 25. A compilation of the most recent lensing spectra made by different experiments is shown in Figure 26. Our measurement reaches the state-of-the-art precision obtained by the latest Planck NPIPE analysis (Carron et al. 2022); compared with other ground-based measurements of the CMB lensing power spectrum, our result currently has the highest precision. Furthermore, our polarization-only lensing amplitude estimate, determined at 19.7σ, is the most precise among measurements of its kind, surpassing the 12.7σ and 10.1σ from NPIPE (Carron et al. 2022) and SPTpol (Wu et al. 2019), respectively.

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Table 4. Lensing Bandpowers in the Baseline Range, for Our MV Estimator

$[{L}_{\min }\quad {L}_{\max }]$	Lb	${10}^{7}{[L(L+1)]}^{2}{C}_{L}^{\hat{\phi }\hat{\phi }}/4$
[40 66]	53.0	2.354 ± 0.157
[67 101]	83.5	1.822 ± 0.096
[102 145]	123	1.368 ± 0.068
[146 199]	172	1.000 ± 0.054
[200 264]	231.5	0.758 ± 0.047
[265 339]	301.5	0.617 ± 0.048
[340 426]	382.5	0.439 ± 0.039
[427 526]	476	0.409 ± 0.032
[527 638]	582	0.249 ± 0.027
[639 763]	700.5	0.156 ± 0.026

Note. Values are given for ${10}^{7}{[L(L+1)]}^{2}{C}_{L}^{\hat{\phi }\hat{\phi }}/4\,=\,{10}^{7}{C}_{L}^{\hat{\kappa }\hat{\kappa }}$ averaged over noninterlapping bins, with bin edges given in the first column and band centers L_b in the second.

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We estimate the lensing amplitude parameter A_lens by fitting our baseline bandpower measurements to a theory lensing power spectrum predicted from the best-fit ΛCDM model from the Planck 2018 baseline likelihood plikHM TTTEEE lowl lowE, allowing the amplitude of this lensing power spectrum to be a free parameter in our fit. ⁹⁷ We find

$$\begin{eqnarray}&&{A}_{\mathrm{lens}}=1.013\pm 0.023\quad (68 \% \,\mathrm{limit})\end{eqnarray} \tag{ 46 }$$

from the baseline multipole range L = 40–763, in good agreement with the lensing spectrum predicted by the Planck 2018 ΛCDM model (A_lens = 1). The scaled model is a good fit to the lensing bandpowers, with a PTE for χ² of 13%. (The equivalent PTE for our bandpowers without rescaling the Planck model by A_lens is 17%. Note that the PTE is higher here because we have one more degree of freedom.) We also find

$$\begin{eqnarray}&&{A}_{\mathrm{lens}}=1.016\pm 0.023\end{eqnarray} \tag{ 47 }$$

using an extended multipole range of 40 < L < 1300. The fit remains good over this extended range, with a PTE of 12%. ⁹⁸ For comparison, the latest CMB lensing analysis from Planck NPIPE obtained A_lens = 1.004 ± 0.024 (Carron et al. 2022). ⁹⁹ For measurements performed purely with temperature and polarization we obtain the lensing amplitudes of A_lens =1.009 ± 0.037 and A_lens = 1.034 ± 0.049, respectively. The consistency of the A_lens amplitude obtained from MV, TT, and MVPOL is illustrated in Figure 25. Our measured lensing bandpowers are therefore fully consistent with a ΛCDM cosmology fit to the Planck 2018 CMB power spectra, with no evidence for a lower amplitude of structure.

It is also important to emphasize that the agreement of our late-time measurements with the structure growth predicted by the CMB power spectra also holds for CMB power spectra measured by other experiments (not just Planck). Our lensing measurements are also consistent with a ΛCDM model fit to independent CMB power spectra measurements from ACT DR4 + WMAP (Aiola et al. 2020): fitting to a rescaling of the best-fit ACT DR4 + WMAP ΛCDM model prediction yields an amplitude of lensing of A_lens = 1.005 ± 0.023. We shall explore the consequences of our measurements for structure growth further in the next section.

We obtain cosmological constraints by constructing a lensing likelihood function ${ \mathcal L }$ assuming Gaussian errors on ${\hat{C}}_{L}^{\phi \phi }$ obtained from the MV estimator:

$$\begin{eqnarray}&&-2\mathrm{ln}{ \mathcal L }\propto \displaystyle \sum _{{{bb}}^{{\prime} }}\left[{\hat{C}}_{{L}_{b}}^{\phi \phi }-{C}_{{L}_{b}}^{\phi \phi }({\boldsymbol{\theta }})\right]{{\mathbb{C}}}_{{{bb}}^{{\prime} }}^{-1}\left[{\hat{C}}_{{L}_{{b}^{{\prime} }}}^{\phi \phi }-{C}_{{L}_{{b}^{{\prime} }}}^{\phi \phi }({\boldsymbol{\theta }})\right],\end{eqnarray} \tag{ 48 }$$

where ${\hat{C}}_{{L}_{b}}^{\phi \phi }$ is the measured baseline lensing power spectrum, ${C}_{{L}_{b}}^{\phi \phi }({\boldsymbol{\theta }})$ is the theory lensing spectrum evaluated for cosmological parameters θ , and ${{\mathbb{C}}}_{{{bb}}^{{\prime} }}$ is the baseline covariance matrix. We discussed the construction of the covariance matrix in Section 5.11, while verification of the Gaussianity of the lensing bandpowers can be found in Appendix H. Further corrections to this likelihood are applied when considering joint constraints with CMB power spectra, as described in our companion paper, Madhavacheril et al. (2024). These account for the dependence of the normalization of the lensing bandpowers on the true CMB power spectra and of the N₁ correction on both the true CMB and lensing power spectra. For the lensing-only constraints presented in this paper, we account for uncertainty in the CMB power spectra by sampling 1000 flat-ΛCDM CMB power spectra from ACT DR4 + Planck and propagating these through the lensing normalization; the scatter in the normalization leads to an additional broadening of the bandpower covariance matrix. For further details, see our earlier discussion in Section 5.10 and also Appendix B.

We now consider constraints on the basic ΛCDM parameters—cold dark matter and baryon densities Ω_c h² and Ω_b h², the Hubble constant H₀, the optical depth to reionization τ, and the amplitude and scalar spectral index of primordial fluctuations A_s and n_s —from our lensing measurements alone. These parameters are varied with priors as summarized in Table 5; these are the same priors assumed in the most recent Planck lensing analyses (Planck Collaboration et al. 2020b; Carron et al. 2022). Since lensing is not sensitive to the CMB optical depth, we fix this at τ = 0.055 (Planck Collaboration et al. 2016). We fix the total neutrino mass to be consistent with the normal hierarchy, assuming one massive eigenstate with a mass of 60 meV.

Table 5. Priors Used in the Lensing-only Cosmological Analysis of This Work

Parameter	Prior
$\mathrm{ln}({10}^{10}{A}_{s})$	[2, 4]
H0	[40, 100]
ns	${ \mathcal N }(0.96,0.02)$
Ωb h2	${ \mathcal N }(0.0223,0.0005)$
Ωc h2	[0.005, 0.99]
τ	0.055

Note. Uniform priors are shown in square brackets, and Gaussian priors with mean μ and standard deviation σ are denoted ${ \mathcal N }(\mu ,\sigma )$. The priors adopted here are identical to those used in the lensing power spectrum analysis performed by the Planck team (Planck Collaboration et al. 2020b).

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Weak-lensing observables in cosmology depend on both the late-time amplitude of density fluctuations in terms of σ₈ and the matter density Ω_m ; there is an additional dependence on the Hubble parameter H₀. In Figure 27 we show the CMB-lensing-only constraints derived from our spectrum measurement; these follow, as in previous lensing analyses, a narrow line in the space spanned by σ₈–H₀–Ω_m . In our companion paper, Madhavacheril et al. (2024), we argue that this line-shaped posterior arises because CMB lensing on large and small scales constrains two different combinations of σ₈–H₀–Ω_m . These two constraint planes intersect in a constraint "line," which explains the form of the posterior seen in the figure. The DR6 lensing data provide the following constraint on this three-dimensional σ₈–H₀–Ω_m parameter space:

$$\begin{eqnarray}&&\displaystyle \frac{{\sigma }_{8}}{0.8}{\left(\displaystyle \frac{{{\rm{\Omega }}}_{m}}{0.3}\right)}^{0.23}{\left(\displaystyle \frac{{{\rm{\Omega }}}_{m}{h}^{2}}{0.13}\right)}^{-0.32}=0.9938\pm 0.0197.\end{eqnarray} \tag{ 49 }$$

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This line-like degeneracy projects into constraints within a narrow region in the σ₈–Ω_m plane, as shown in Figure 2. The best-constrained direction corresponds approximately to a determination of ${\sigma }_{8}{{\rm{\Omega }}}_{m}^{0.25}$. Constraining this parameter combination with our data, we obtain

$$\begin{eqnarray}&&{\sigma }_{8}{{\rm{\Omega }}}_{m}^{0.25}=0.606\pm 0.016.\end{eqnarray} \tag{ 50 }$$

This translates to a constraint on the CMB lensing equivalent of the usual S₈ parameter, which we define as

$$\begin{eqnarray}&&{S}_{8}^{\mathrm{CMBL}}\equiv {\sigma }_{8}{\left(\displaystyle \frac{{{\rm{\Omega }}}_{m}}{0.3}\right)}^{0.25},\end{eqnarray} \tag{ 51 }$$

of

$$\begin{eqnarray}&&{S}_{8}^{\mathrm{CMBL}}=0.818\pm 0.022\,\,(0.830\pm 0.020).\end{eqnarray} \tag{ 52 }$$

In the constraint shown above, the result for the baseline analysis is shown first, followed by the constraint from the extended range of scales in parentheses. These can be compared with the value expected from Planck CMB power spectrum measurements assuming a ΛCDM cosmology. Extrapolating the Planck CMB anisotropy measurements to low redshifts yields a value of ${S}_{8}^{\mathrm{CMBL}}=0.823\pm 0.011$. This is entirely consistent with our direct ACT DR6 lensing measurement of this parameter. As in the case of A_lens, this agreement is not limited to comparisons with Planck; similar levels of agreement are achieved with ACT DR4 + WMAP CMB power spectrum measurements, which give a constraint of ${S}_{8}^{\mathrm{CMBL}}=0.828\pm 0.020$. Our measurement is also consistent with the direct ${S}_{8}^{\mathrm{CMBL}}$ result from NPIPE lensing, ${S}_{8}^{\mathrm{CMBL}}=0.809\pm 0.022$. This consistency of ${S}_{8}^{\mathrm{CMBL}}$ between extrapolations from CMB power spectra, which probe primarily z ∼ 1100, and direct measurements with CMB lensing at lower redshifts z ∼ 0.5–5 can be seen in Figure 28.

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Our constraint on ${S}_{8}^{\mathrm{CMBL}}$ is robust to the details of our analysis choices and data sets. In Figure 29, we present the marginalized posteriors of ${S}_{8}^{\mathrm{CMBL}}$ obtained using different variations of our analysis. Since levels of extragalactic foregrounds are significantly lower in polarization than in temperature, the consistency between our baseline analysis and the analyses using temperature data and polarization data alone suggests that foreground contamination is under control. While our baseline analysis incorporates the modeling of nonlinear scales using the nonlinear matter power spectrum prescription of Mead et al. (2016), we also present constraints that use linear theory only. The consistency of this result with our baseline shows that we are mainly sensitive to linear scales. Finally, we present constraints using our pre-unblinding method of inpainting clusters instead of masking, as discussed in Section 6.3.1. This method only results in a shift of 0.15σ in ${S}_{8}^{\mathrm{CMBL}}$.

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In Figure 30 we also show constraints on ${S}_{8}^{\mathrm{CMBL}}$ (and Ω_m ) with the sum of neutrino masses freed and marginalized over, instead of being fixed at the minimum allowed mass set by the normal hierarchy, 60 meV. ¹⁰⁰ In this case, our constraint of ${S}_{8}^{\mathrm{CMBL}}=0.797\pm 0.024$ becomes tighter than constraints from Planck 2018 CMB power spectra, ${S}_{8}^{\mathrm{CMBL}}=0.811\pm 0.027$. The fact that our constraints are only slightly weakened when marginalizing over a free neutrino mass, whereas Planck constraints on ${S}_{8}^{\mathrm{CMBL}}$ are significantly degraded, is expected. Since our lensing results originate from low redshifts, minimal extrapolation to z = 0 (where ${S}_{8}^{\mathrm{CMBL}}$ is evaluated) is required, which makes our constraints comparatively insensitive to neutrino mass. In constrast, for CMB power spectrum constraints, extrapolation over a wide redshift range from z ∼ 1100 to z = 0 is required, which implies that the constraints have significant sensitivity to neutrino mass.

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The DR6 CMB lensing measurement from the ground contains information that is, to some extent, independent of the space-based measurement obtained with the Planck satellite. The two measurements have different noise and different instrument-related systematics; their sky coverage and their angular scales also only have partial overlap ¹⁰¹ over a sky fraction of 67% used in the Planck analysis.

This partial independence motivates not just comparing the two measurements but, given that we find good consistency between the two, combining them to obtain tighter constraints. The excellent agreement between both measurements can be seen in Figure 31, where for illustrative purposes we show also in blue the combined (with an inverse-variance weight) lensing bandpowers from ACT DR6 and Planck. ¹⁰²

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In this paper, we combine the two lensing spectra at the likelihood level, taking into account the small correlation between the two data sets in order to obtain further improved constraints on S₈. For the NPIPE lensing measurements, we use the published NPIPE lensing bandpowers. ¹⁰³ The NPIPE part of the covariance matrix is obtained using a set of 480 NPIPE lensing reconstructions. ¹⁰⁴

9.3.1. Covariance Matrix between ACT and Planck Lensing Spectra

Although we expect the ACT and Planck data sets to have substantially independent information, the sky overlap and scale overlap are large enough that we cannot, without further investigation, neglect the correlation between these two data sets. We therefore compute the joint covariance, proceeding as follows. We start with the same set of 480 full-sky FFP10 CMB simulations used by NPIPE to obtain the Planck part of the covariance matrix. We apply the appropriate ACT DR6 mask and obtain lensing reconstructions using the standard (not cross-correlation-based ¹⁰⁵ ) estimator with a filter that has the same noise level as the DR6 analysis. We do not add noise to the ACT CMB maps, as we expect the instrument noise of Planck and ACT to be entirely independent; the ACT noise should hence not enter the ACT–Planck covariance (when using the cross-correlation-based estimator for ACT). We use these 480 reconstructed ACT bandpowers along with the corresponding NPIPE lensing bandpowers to obtain the off-diagonal ACT–Planck elements of the joint bandpower covariance matrix for the two data sets.

The correlation between the two data sets is small, as we will show, but the computation of this larger covariance matrix causes challenges in ensuring convergence with a modest number of simulations. To reduce the impact of fluctuations and ensure convergence of the resulting covariance matrix, we use a criterion that allows us to simulate only the most correlated bandpowers. This is done by only simulating the terms in the covariance matrix that we expect to be significantly different from zero and nulling all other elements. To determine which elements can be safely nulled, we compute the covariance matrix between ACT and Planck in an unrealistic scenario of maximum overlap in area and multipole by analyzing a set of 396 ACT CMB simulations with the same ACT analysis mask and multipole range 600 < ℓ < 3000 (instead of the NPIPE analysis mask covering 67% of the sky and multipole range 100 < ℓ < 2048) but with a filter, used for the inverse-variance and Wiener filtering of the CMB maps, with noise levels appropriate for NPIPE. The resulting covariance matrix only shows a significant correlation among the bins where the ACT and NPIPE bandpowers overlap in multipole L. (This is unsurprising because, even within ACT, different-L bandpowers are only minimally correlated, and we expect even smaller correlations between Planck and ACT bandpowers at a different L.) From this maximally overlapping covariance matrix we choose the elements where the absolute value of the correlation between the bandpowers is less than 0.15 and zero these elements in the final ACT × Planck covariance matrix, as these are likely due to noise fluctuations rather than physical correlation between the bandpowers. ¹⁰⁶ The resulting covariance matrix is shown in Figure 32.

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Finally, we broaden the covariance matrix to account for marginalization over the uncertainties in the CMB lensing power spectra, as described earlier for the ACT part in Section 9.1. This step is applied consistently to both the ACT and the NPIPE parts of the covariance matrix and is described in detail in Appendix B.

The parameter constraints on σ₈ and Ω_m derived from the combination of ACT and Planck lensing are shown in Figure 33. As expected, the joint constraint in red is in good agreement with the ACT DR6-only constraint and with the Planck CMB power spectrum prediction in black.

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The joint ACT DR6 + Planck lensing constraints in the ${S}_{8}^{\mathrm{CMBL}}$–Ω_m plane are shown in Figure 34. We obtain the following constraint on ${S}_{8}^{\mathrm{CMBL}}$ from the ACT–Planck combined data set:

$$\begin{eqnarray}&&{S}_{8}^{\mathrm{CMBL}}=0.813\pm 0.018\,\,(0.822\pm 0.017),\end{eqnarray} \tag{ 53 }$$

where the constraint in parentheses is from the extended lensing multipole range. ¹⁰⁷

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Our lensing power spectrum bandpowers are consistent with the ΛCDM prediction over a range of scales. Good consistency with ΛCDM is found even in the extended range of scales where foregrounds and nonlinear structure growth could be more relevant.

As previously discussed, our results are highly relevant for the S₈ tension. A powerful test of structure formation is to extrapolate, assuming ΛCDM structure growth but no other free parameters, a model fit to the Planck CMB power spectrum at (mostly) early times down to low redshifts and then compare this extrapolation with direct measurements of ${S}_{8}^{\mathrm{CMBL}}\,\equiv {\sigma }_{8}{({{\rm{\Omega }}}_{m}/0.3)}^{0.25}$ or σ₈. Intriguingly, some lensing and galaxy clustering measurements seem to give lower values of S₈ or σ₈ than predicted by Planck at the (2–3)σ level, including KiDS, DES, and HSC (Heymans et al. 2013, 2021; Asgari et al. 2021; Krolewski et al. 2021; Abbott et al. 2022; Loureiro et al. 2022; Philcox & Ivanov 2022; Dalal et al. 2023; Li et al. 2023), although this conclusion is not universal (eBOSS Collaboration 2021). Our measurement of lensing and structure growth is independent of Planck and also does not rely at all on galaxy survey data, with the associated challenges in modeling and systematics mitigation.

As described in the previous section, we find ${S}_{8}^{\mathrm{CMBL}}\,=0.818\pm 0.022$ as our baseline result from ACT. This is in full agreement with the expectation based on the Planck CMB anisotropy power spectra and ΛCDM structure growth. We emphasize that the agreement between precise lensing measurements at z ∼ 0.5 − 5 and predictions from CMB anisotropies probing primarily z ∼ 1100 is a remarkable success for the ΛCDM model. From a fit of ΛCDM at the CMB last scattering surface, our standard cosmology predicts, with no additional free parameters, cosmic structure formation over billions of years, the lensing effect this produces, and the non-Gaussian imprint of lensing in the observed CMB fluctuations; our measurements match these predictions at the 2% level.

Unlike several galaxy lensing measurements and other large-scale structure probes, we find no evidence of any suppression of the amplitude of structure.

Our lensing power spectrum and the resulting parameters also agree with the results of the Planck lensing power spectrum measurement. Since both ACT lensing power spectra and CMB power spectra are in good agreement with Planck measurements, this disfavors systematics in the Planck measurement as an explanation of the S₈ tension. This also means that we may combine with the Planck lensing measurement to obtain an even more constraining measurement ${S}_{8}^{\mathrm{CMBL}}=0.813\pm 0.018$, again in agreement with the Planck or ACT DR4 + WMAP CMB power spectra.

We note that while we appear to find a higher value of ${S}_{8}^{\mathrm{CMBL}}$ than the S₈ measured by several weak-lensing surveys, the two quantities differ in the power of Ω_m and can hence, in principle, be brought into better agreement by a lower value of Ω_m < 0.3 (although the matter density is very well constrained by other probes); in any case, the discrepancy does not reach high statistical significance. Our companion paper (Madhavacheril et al. 2024) discusses this comparison in more detail by combining these data sets with baryon acoustic oscillation (BAO) data to constrain Ω_m to ≈0.3.

We expect the agreement of our lensing power spectrum with the Planck CMB power spectrum extrapolation to disfavor resolutions to the S₈ tension that involve new physics having a substantial effect at high redshifts (z > 1) and on linear scales, where our lensing measurement has highest sensitivity.

However, this need not imply that statistical fluctuations or systematics in other lensing and LSS measurements are the only explanation for the reported tensions. The possibility remains that new physics suppresses structure only at very low redshifts, z < 1, or on nonlinear scales, k > 0.2h Mpc⁻¹, to which our CMB lensing measurements are much less sensitive than current cosmic shear, galaxy–galaxy lensing, or galaxy clustering constraints. An example of such physics could be the small-scale matter power spectrum suppression proposed in Amon & Efstathiou (2022) or He et al. (2023) (along with systematics in CMB lensing cross-correlation analyses); modified gravity effects that become important at only very low redshifts are another example, although these need to be consistent with expansion and redshift-space distortion measurements.

More information on possible structure growth tensions can be obtained from ACT lensing measurements not just through combination with external data, as in our companion paper, but also by cross-correlations with low-redshift tracers. Such cross-correlations will allow structure growth to be tracked tomographically as a function of redshift down to z < 1, providing for powerful tests of new physics. Several tomographic cross-correlation analyses with the lensing data presented here are currently in progress.

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 ${\rm{\Delta }}\mathrm{ln}{C}_{L}^{\phi \phi }\approx 2{\rm{\Delta }}\mathrm{ln}{\bar{C}}_{{\ell }}$, where ${\rm{\Delta }}\mathrm{ln}{\bar{C}}_{{\ell }}$ and ${\rm{\Delta }}\mathrm{ln}{C}_{L}^{\phi \phi }$ are fractional changes in the CMB power spectrum averaged over ℓ ∈ [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):

$$\begin{eqnarray}&&-2\mathrm{ln}{ \mathcal L }(\{{c}_{{A}_{f}}\}| \{{{\rm{\Delta }}}_{\ell }^{{A}_{f}}\})=\displaystyle \sum _{\ell ={\ell }_{\min }}^{{\ell }_{\max }}\displaystyle \sum _{{A}_{f},{{\rm{A}}}_{f}^{^{\prime} }}{{\rm{\Delta }}}_{\ell }^{{A}_{f}}{\left[{{\rm{\Sigma }}}_{\ell }^{-1}\right]}^{{A}_{f}{{\rm{A}}}_{f}^{^{\prime} }}{{\rm{\Delta }}}_{\ell }^{{{\rm{A}}}_{f}^{{\prime} }},\end{eqnarray} \tag{ C1 }$$

where ${{\rm{\Delta }}}_{\ell }^{{A}_{f}}={c}_{{A}_{f}}^{2}{C}_{\ell }^{\mathrm{ACT}\times \mathrm{ACT},{A}_{f}}-{c}_{{A}_{f}}{C}_{\ell }^{\mathrm{ACT}\times {\rm{P}},{A}_{f}}$ 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 ${A}_{f}^{{\prime} }$ run through the ACT array frequencies used in our analysis, and Σ_ℓ is an approximate analytic covariance matrix for the ${{\rm{\Delta }}}_{\ell }^{{A}_{f}}$ (assumed diagonal in ℓ) 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 (ℓ ∈ [600, 3000]).

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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 (1σ) 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 ΔA_lens = 0.00432 (0.18σ) and ΔA_lens = 0.00419 (0.09σ) for the MV and MVPol estimators, respectively. Similarly, for beam uncertainty, we find ΔA_lens = 0.00113 (0.05σ) and ΔA_lens =0.00110 (0.02σ) for MV and MVPol, respectively.

As described in Section 3.2, we find evidence for some degree of temperature-to-polarization (T → 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:

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{X}}_{{\ell }m} & = & {X}_{{\ell }m}+{X}_{{\ell }m}^{\mathrm{leakage}}\\ & = & {X}_{{\ell }m}+{B}_{{\ell }}^{T\to P}{T}_{{\ell }m},\end{array}\end{eqnarray} \tag{ C2 }$$

where X_{ℓ m} are either the baseline E or B multipoles, ${B}_{{\ell }}^{T\to P}$ is the estimated leakage beam, and T_{ℓ 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_{ℓ m} , already include the T → P leakage effect, our modified maps, ${\tilde{X}}_{{\ell }m}$, include twice the amount of leakage compared to a scenario with no leakage. We 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 ΔA_lens metric (Equation (44)), we find ΔA_lens = − 0.00021 (−0.01σ) and ΔA_lens = − 0.00052 (−0.01σ) 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 α = 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 α = 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 ΔA_lens =− 0.00461 (−0.19σ) and ΔA_lens = − 0.00887 (−0.18σ) for MV and MVPol, respectively.

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 Φ_p = − 0.07 ± 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 Φ_p = − 0.16 deg (the mean minus 1σ) 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 ΔA_lens statistics. We find ΔA_lens = − 0.00025 (−0.01σ) and ΔA_lens = 0.00070 (0.01σ) for MV and MVPol, respectively.

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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 1σ. 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 greeen dashed curve. We find ΔA_lens = − 0.00896 (−0.37σ) and ΔA_lens = − 0.01884(−0.38σ) 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.

Lensing power spectrum estimation aims to probe the connected four-point function that is induced by lensing. However, the naive lensing power spectrum estimator ${\hat{C}}_{L}^{\hat{\phi }\hat{\phi }}[\bar{X},\bar{Y},\bar{A},\bar{B}]\sim {XYAB}$ also contains disconnected contributions arising from Gaussian fluctuations (e.g., 〈XA〉〈YB〉), which are nonzero even in the absence of lensing. These contributions, which are typically referred to as the Gaussian or N₀ 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 (N₀) bias is estimated using an algorithm involving different pairings of data and two sets of independent simulation maps, denoted with superscripts S and ${S}^{{\prime} }$ (Namikawa et al. 2013; Planck Collaboration et al. 2020b):

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{C}_{L}^{\mathrm{Gauss}} & = & \langle {\bar{C}}_{L}^{\times }[{{XY}}^{S},{{AB}}^{S}]+{\bar{C}}_{L}^{\times }[{X}^{S}Y,{{AB}}^{S}]\\ & & +\,{\bar{C}}_{L}^{\times }[{X}^{S}Y,{A}^{S}B]+{\bar{C}}^{\times }[{{XY}}^{S},{A}^{S}B]\\ & & -\,{\bar{C}}_{L}^{\times }[{X}^{S}{Y}^{{S}^{{\prime} }},{A}^{S}{B}^{{S}^{{\prime} }}]-{\bar{C}}_{L}^{\times }[{X}^{S}{Y}^{{S}^{{\prime} }},{A}^{{S}^{{\prime} }}{B}^{S}]{\rangle }_{S,{S}^{{\prime} }}.\end{array}\end{eqnarray} \tag{ E1 }$$

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 $S,{S}^{{\prime} }$ 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 ΔA_lens = 5 × 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 N₀, referred to as the semianalytic N₀; we discussed this in detail in Section 5.11.1.

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 N₁ bias.

We calculate the N₁ bias by using 90 pairs of simulations with different CMB realizations but common lensing potential maps denoted $({S}_{\phi },{S}_{\phi }^{{\prime} })$ and 90 pairs of simulations with different CMB and lensing potential $(S,{S}^{{\prime} })$ (we have carried out convergence tests as in Appendix E.1 to ensure that 90 simulations are sufficient):

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{C}_{L}^{{N}_{1}} & = & \langle {\bar{C}}_{L}^{\times }[{X}^{{S}_{\phi }}{Y}^{{S}_{\phi }^{{\prime} }},{A}^{{S}_{\phi }}{B}^{{S}_{\phi }^{{\prime} }}]\\ & & +\,{\bar{C}}_{L}^{\times }[{X}^{{S}_{\phi }}{Y}^{{S}_{\phi }^{{\prime} }},{A}^{{S}_{\phi }^{{\prime} }}{B}^{{S}_{\phi }}]\\ & & -\,{\bar{C}}_{L}^{\times }[{X}^{S}{Y}^{{S}^{{\prime} }},{A}^{S}{B}^{{S}^{{\prime} }}]\\ & & -\,{\bar{C}}_{L}^{\times }[{X}^{S}{Y}^{{S}^{{\prime} }},{A}^{{S}^{{\prime} }}{B}^{S}]{\rangle }_{S,{S}^{{\prime} },{S}_{\phi },{S}_{\phi }^{{\prime} }}.\end{array}\end{eqnarray} \tag{ E2 }$$

We do not include biases at orders higher than N₁, 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).

Although subtraction of the Gaussian and N₁ biases results in a nearly unbiased lensing power spectrum, we calculate an additive MC bias ${C}_{L}^{\mathrm{MC}}[{XY},{AB}]$ with simulations to absorb any residual arising from nonidealities 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.,

$$\begin{eqnarray}&&{\rm{\Delta }}{C}_{L}^{\mathrm{MC}}={C}_{L}^{\phi \phi }-({\rm{\Delta }}{A}_{L}^{\mathrm{MC},\mathrm{mul}}\langle {C}_{L}^{\times }{\rangle }_{\mathrm{sim}}-\mathrm{MCN}0-{\rm{\Delta }}{C}_{L}^{{{\rm{N}}}_{1}}).\end{eqnarray} \tag{ E3 }$$

Here ${\rm{\Delta }}{A}_{L}^{\mathrm{MC},\mathrm{mul}}$ is the multiplicative bias term defined in Section 5.8.1.

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 = split_i − 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.

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We test for consistency of our lensing measurements across different regions of the sky by dividing our sky into 50 patches 10° × 25° nonoverlapping patches. 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.

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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 I.4 below, we produce special noise simulations capturing the characteristics of the two time splits using mnms (Atkins et al. 2023).

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.

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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.

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.

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Extensions to the flat-ΛCDM cosmology with parameters such as the spatial curvature Ω_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 ${S}_{8}^{\mathrm{CMBL}}=0.814\pm 0.022$, consistent with the baseline constraints ${S}_{8}^{\mathrm{CMBL}}=0.818\pm 0.022$.

We present a more comprehensive summary of the curl results for the different null test, including a compilation of the PTEs and χ² values in Table 6. Our analysis reveals no significant evidence of systematic biases in our measurements.

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