A hybrid approach to cosmic microwave background lensing reconstruction from allsky intensity maps
^{1}
Laboratoire de l’Accélérateur Linéaire (LAL), Univ ParisSud,
CNRS/IN2P3, 91406
Orsay,
France
email: plaszczy@lal.in2p3.fr; lavabre@lal.in2p3.fr
^{2}
Laboratoire de Physique Corpusculaire et de Cosmologie (LPSC),
Université Joseph Fourier Grenoble 1, CNRS/IN2P3, Institut Polytechnique de Grenoble,
53 rue des Martyrs,
38026
Grenoble Cedex,
France
email: perotto@lpsc.in2p3.fr
^{3}
Laboratoire AIM (UMR 7158), CEA/DSMCNRSUniversité Paris Diderot, IRFU, SEDISAP, Service
d’Astrophysique, Centre de Saclay, 91191
GifSurYvette Cedex,
France
email: jstarck@cea.fr
Received:
27
January
2012
Accepted:
18
May
2012
On the basis of realistic simulations, we propose a hybrid method to reconstruct the lensing potential power spectrum, directly on Plancklike cosmic microwave background frequency maps. This involves the use of a large Galactic mask and the treatment of strong inhomogeneous noise. For ℓ ≲ 100, we show that a fullsky inpainting method, which was previously described, still allows a minimal variance reconstruction, with a bias that must be accounted for by a Monte Carlo method but that does not couple to the deflection field. For ℓ ≳ 100, we develop a method based on tiling the cutsky with local 10° × 10° overlapping tangent planes (referred to in the following as patches). We tackle various issues related to their size/position, nonperiodic boundaries, and irregularly sampled data of the planes after the spheretoplane projection. We show that the predominant noise term of the quadratic lensing estimator determined from an apodized patch can still be recovered directly from the data. To prevent any loss of spatial accuracy, we developed a tool that allows the efficient determination of the complex Fourier series coefficients from a bidimensional irregularly sampled dataset, without performing any interpolation. We show that our multipatch approach enables the lensing power spectrum to be reconstructed with a very small bias, thanks to the omission of a Galactic mask and smaller noise inhomogeneities, as well as an almost minimal variance. At each stage, the data quality can be assessed and simple bidimensional spectra compiled, which allows the control of local systematic errors.
Key words: cosmic background radiation / methods: data analysis / largescale structure of Universe
© ESO, 2012
Introduction
Experiments have now reached the sensitivity, in terms of both resolution and noise, to detect the tiny deflection of the cosmic microwave background (CMB) photons (σ_{d} ≃ 2.7′) by the irregular distribution of matter, in their journey from the last scattering surface to Earth. First results on the power spectrum of this deflection field have been reported by the ACT (Das et al. 2011) and SPT (van Engelen et al. 2012) collaborations. The Planck spatial mission should soon provide firm measurements. This information provides access to a new cosmological observable that is sensitive to an epoch (1 ≲ z ≲ 3) much more recent than the CMB decoupling one (z ≃ 1100), giving us a leverarm to lift the socalled geometrical degeneracy (Stompor & Efstathiou 1999), but using one single consistent dataset. In particular, it probes the matter density fluctuations, on scales where the freestreaming of massive neutrinos significantly erases the power spectrum of these fluctuations (Lesgourgues & Pastor 2006), and is expected to help us to determine their total mass by means of global cosmological fits.
On statistical grounds, the properties of the (nearly) optimal quadratic estimator for lensing power reconstruction are now wellunderstood, both in the (infinite) flat sky limit and across the complete sphere (Hu & Okamoto 2002; Okamoto & Hu 2003).
However, real data are affected by contaminants, mostly Galactic dust and point sources in the case of CMB frequency maps, requiring a revised means of lensing reconstruction on a cutsky, which is a nontrivial task. In addition, the scanning strategy of the specific instrument, particularly in the case of Planck, induces some strong spatialnoise inhomogeneities that are not taken into account in the classical estimator of lensing, and must be corrected for by Monte Carlo simulations. In the general case, both effects cannot be distinguished during the reconstruction process.
In a previous study (Perotto et al. 2010), we optimized a sparse inpainting procedure to restore the missing data inside the mask, without significantly biasing the lensing results. We however neglected the noise inhomogeneity. Furthermore, the work was oriented toward componentseparated maps, so that the mask to be filled was rather small (about 10% of the sky).
However, before having to adopt a component separation method that mixes different maps, we wish to investigate in this paper whether the lensing potential can be reconstructed more directly in individual intensity CMB maps, which is indeed a necessary step in assessing the possible systematic errors. For Planck, the channels under consideration correspond to 100, 143, and 217 GHz (Planck Collaboration 2005). This requires the treatment of much larger masks. We will also consider the strong spatialnoise inhomogeneities induced by the scanning strategy.
We revisit the sparse inpainting method in this new configuration (a 30% mask + inhomogeneous noise) and show that i) the estimator under these conditions is strongly biased and ii) a Monte Carlo approach can be used to correct for this bias. We also propose an alternative method (multipatch) that allows us to completely avoid the Galactic region, during the development of which we solved a number of issues related to the pixelizedsphere to plane projection.
After rapidly reviewing the various noise contributions to the quadratic estimator (QE) in Sect. 1, and the common simulations used in Sect. 2, we update our fullsky inpainting analysis (hereafter denoted FSinpainting) in Sect. 3. Most of the paper in Sect. 4, then deals with resolving issues related to the projection of a nonperiodic signal from a pixelized sky onto a local patch. In particular, we present a new algorithm (detailed in the Appendix) that allows a fast reconstruction of bandlimited Fourier series coefficients from irregularly sampled data, without performing any interpolation. We then compare both methods, optimize the results in Sect. 5, and argue that a hybrid reconstruction is the most appropriate. In this hybrid approach, the full sky lensing reconstruction presented in Perotto et al. (2010) is used at low ℓ, with an additive Monte Carlo bias correction, while at high ℓ, the new multipatch method is advocated.
This method allows the direct reconstruction of lensing signal from Plancklike CMB frequency maps (namely those at the 100, 143, and 217 GHz). While it would be premature to decide today whether performing a multimap component separation provides a more accurate recovery of the lensing signal, we give some elements of the discussion in the conclusion.
1. A brief review of the quadratic estimator
The gravitational lensing potential φ is a scalar isotropic field (for a review, see e.g. Lewis & Challinor 2006) that spatially remaps the CMB photons according to (1)where d = ∇φ is the deflection field, which has a power spectrum on the sky , or in the flat sky limit. This process slightly breaks the Gaussianity of the CMB field, and estimators have been searched for in order to extract the lensing information using its very local properties
The quadratic estimator was proposed by Hu & Okamoto (2002). For CMB temperature anisotropies, it consists in taking the (weighted) convolution of the observed Fourier modes according to (2)where the normalization A_{K} and filter F are determined by requiring the estimator to be unbiased and have a minimum variance at the leading noise order (socalled N^{(0)}). For an idealized experiment, one gets (3)The filter F involves on the numerator the CMB “true” unlensed powerspectrum (), and on the denominator the “observed” one , which is assumed to be a pure beamdeconvolved Gaussian signal with uncoupled homogeneous noise.
Since this estimator involves only simple operations, it is computable in a few minutes on any standard computer. Its generalization to spherical harmonics across the fullsky was performed in Okamoto & Hu (2003).
The full likelihood estimator was developed by Hirata & Seljak (2003), who demonstrated that it gives results very close to the quadratic one, given the current noise level, but involves much heavier computations.
The covariance of the estimator is related to the true lensing potential spectrum through (4)and remarkably, the noise term is directly related to the estimator normalization Eq. (3) (5)This corresponds to the Gaussian term, in the sense that it comes from the standard disconnected part of the fourpoint correlator that appears when computing the noise and is therefore decoupled from the φ field. Equivalently, it represents the power of the QE when running it on unlensed maps.
A firstorder powerspectrum correction term was soon afterwards discovered by Kesden et al. (2003). This comes from the connected part of the correlator, and hence depends on the φ field itself N^{(1)}(φ).
When actually coding the estimator for the Planck experiment, we still noted a poorly understood bias at low ℓ′s which was finally identified by Hanson et al. (2011) as a nonnegligible secondorder term that can be estimated analytically and was called N^{(2)}(φ). Another way of taking this noise into account is to use a simple “trick” proposed by Bielewicz (Hanson et al. 2011), which consists in inserting the lensed spectrum into the numerator of Eq. (3). This approach was shown to capture even more precisely the secondorder contribution than the N^{(2)} but slightly increases the error in the reconstructed signal.
This is not however the end of the story. Our simulations did not initially incorporate the spatial inhomogeneity of the noise, owing to the Planck scanning strategy. This strategy induces correlations between the different Fourier modes leading to spurious signal reconstruction in the QE. It was shown in Hanson et al. (2009) that the noise inhomogeneity also affects the QE expectation value resulting in a lowℓ bias in the power spectrum that can be analytically estimated under the white noise hypothesis. However, this mean field approach still misses another like term, which is noncomputable analytically but affects the whole lensing spectrum.
Finally, owing to the foreground signal, one can never experimentally make use of the signal across the full sphere. In this case, the spherical harmonics no longer form a “natural” basis and the issue of building a good estimator for lensing is nontrivial. Even the inverse variance weighting of the map (e.g. Smith et al. 2007), which is a computationally very challenging task and sometimes referred to as being “optimal”, does not provide an unbiased estimate of the lensing spectrum, because of the large mode coupling introduced by the mask and the inhomogeneous noise.
To take into account these last two effects, namely the treatment of the masked region and inhomogeneous noise, we add to the estimator covariance a term that can in general depend on the lensing field.
In summary, the deflection estimate variance from applying the QE to data using a given method, includes the following terms: (6)where

is the sought deflection spectrum;

is determined on the data given the knowledge of the underlying true power spectrum;

, and can be computed analytically. Since they depend on the searched field, one may need to setup an iterative determination. In our simulation, we simply use the true deflection spectrum from our test cosmology (Sect. 2) to compute them;

is the bias of the power spectrum estimator, which depends on the inhomogeneous properties of the noise and the way in which we deal with the Galactic contamination (and the coupling of both).
The desired properties of are to have small value, while still keeping the optimal variance for the estimator, and decoupled from the lensing field. In the following, we study this term in two methods using a set of simulations with inhomogeneous noise and a large mask.
We somewhat loosely switch to the multipole notation (ℓ) in the fullsky case, the formal connection being performed in e.g. Hu (2000). We recall that on a square patch of size L × L, the discrete Fourier modes are located on a grid (7)with 35 for L = 10°), and (i,j) being integers. In these units, the power spectrum C_{k} is equivalent to C_{ℓ} in the flat sky limit (White et al. 1999).
2. Simulations
To evaluate the performance of our algorithms, we produced a set of realistic Plancklike frequency maps, i.e. a combination of all channels of a given frequency. The experimental characteristics are the ones published in Planck HFI Core Team (2011) corresponding to almost ten months of datataking.
The most interesting channels for CMB analyses using Planck data, are the 100, 143 and 217 GHz ones, where the Galactic dust contamination increases with frequency but is still subdominant and other Galactic foreground types of emission (such as synchrotron or freefree) which decrease with the frequency band, are weaker than the CMB (Planck Collaboration 2005). The resolution of the instrument, which is crucial to the lensing reconstruction, goes however in the other direction with an average full width at half maximum of the scanning beams of about 9.5′,7.1′, and 4.7′, respectively (Planck HFI Core Team 2011). We chose to focus on the 217 GHz channel, because it is the most challenging for lensing, requiring the largest Galactic mask.
We thus developed the following pipeline for our simulations.
We start with a ΛCDM cosmology {H_{0} = 72, Ω_{b}h^{2} = 0.023, Ω_{CDM}h^{2} = 0.11, Y_{He} = 0.24, N_{eff} = 3.04, τ = 0.09, n_{s} = 0.96, A_{s} = 2.4 × 10^{9}}, which is consistent with the WMAP sevenyear bestfit model (Larson et al. 2011), and run the Boltzmann code CAMB^{1} to produce the corresponding spectra of CMB intensity/polarization anisotropies and lensing potential, using Halofit (Smith et al. 2003) for nonlinear corrections. Both lensed/unlensed spectra are computed with the code. In the following, we focus on temperature maps since this is the bestsuited observable for reconstructing lensing in a Plancklike case. In the following we denote as “fiducial” this true deflection spectrum.
These spectra then feed the LensPix^{2} code, which provides a fullsky high resolution map in the HealPix^{3} scheme (n_{side} = 2048). We use a cutoff ℓ_{max} = 3000. We verified that the resultant lensed spectrum is in excellent agreement with the theoretical ones, up to ℓ ≲ 2750, which is largely sufficient for our analysis. One hundred realizations of these maps were produced. We refer to these maps, which are assumed to represent the data, as the H_{1} set (i.e. lensed).
Starting from the CAMB lensed powerspectra, we also produced one hundred Gaussian realizations using the standard HealPix tools (namely syn_alm_cxx/alm2map_cxx) which help us in debiasing the lensing power spectrum estimate. In the following we will refer to these maps as the H_{0} set (i.e. unlensed).
Maps were then all smoothed by a 4.7′ Gaussian beam using HealPix standard tools.
The Gaussian correlated noise in the maps is generated according to its spectrum measured in Planck HFI Core Team (2011). More precisely, the real and imaginary parts of the spherical harmonics coefficients a_{l,m} are randomly drawn from an independent Gaussian distribution with zero mean and a variance given by the measured spectrum N_{l} using the standard HealPix syn_alm_cxx procedure. We then transform the coefficients into real space, using the alm2map_cxx procedure. Each pixel value in the map is then weighted according to the squareroot of the number of hits in that pixel, preserving the total variance. One hundred of these maps, each with a different seed for the phases, were produced and added to the signal maps.
We then apply a 30% mask obtained by smoothing a higher resolution (857 GHz) map to avoid the leading dust contamination in the Galactic plane.
Since it is not the scope of this study to investigate the systematics errors caused by pointsource residuals, we assume that a pointsource mask with perfect completeness is available. We produce this by combining the Planck Early Release Compact Source Catalog of point sources detected in the 143, 217, and 353 GHz channels and including SunyaevZel’dovich clusters (ESZ) and dust cold cores (CC) (Ade et al. 2011b,a,c), the WMAP sevenyear catalog of point sources with a positive flux in the W band (Gold et al. 2011), and a catalog of IRAS/2MASS IR sources whose flux at 100 μm is above 2 Jy (Beichman et al. 1988; Jarrett et al. 2001). Each catalog entry is masked by a 5σ ≃ 10′ radius disk. This mask covers ≃1.7% of the sky out of the Galactic plane. When combining it with the Galactic one, we are left with a fraction f_{sky} = 0.69 of the sky.
Figure 1 shows one of these simulated maps.
Fig. 1 Example of one of our simulated lensed temperature map, using the procedure described in the text. Units are mK_{CMB}. The gray region corresponds to the Galactic mask we propose to use. A pointsource mask is also included, but barely visible, being more clearly seen in Fig. 5. 

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3. Update on the FSinpainting method
In Perotto et al. (2010), we studied the impact of an inpainting method to fill in, with an appropriate statistical mixture, a rather small mask (cutting a ≃10% fraction of the sky). It was oriented toward componentseparated maps, where such a level of final masking is to be expected. Here, we push the algorithm further to its limits by studying the filling of the large mask defined in Sect. 2 (≃30% of the sky). Furthermore, we add spatially inhomogeneous noise, which most certainly affects the results of the algorithm.
Among the inpainting algorithm implemented within the multiresolution on the sphere (MRS) package^{4}, we found that the most robust results are obtained with the spherical harmonics L_{1} norm minimization using wavelet packet variance regularization (Abrial et al. 2007, 2008).
Each map from the H_{0} and H_{1} sets were inpainted. An example of a restored map is shown in Fig. 2.
Fig. 2 Inpainted map corresponding to filling the Galactic+pointsource mask of Fig. 1. 

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We then apply the Hu & Okamoto quadratic estimator on the fullsky, using the fast spherical harmonic computations provided by the HealPix package, with a multipole cut ℓ_{max} = 2000, since there is no statistical gain in going to higher values given the noise level. The “observed” temperature spectrum that enters the QE filter is estimated for each map, which is a way of “absorbing” the residual spectrum deformation after the mask restoration. We did not adopt the Bielewicz’s trick and therefore insert the theoretical unlensed spectrum into the numerator of the QE filter. Given the resolution and noise, we also analytically computed the N^{(1)} and N^{(2)} terms using the fiducial lensing power spectrum.
The bias size can then be estimated in either the H_{0} or H_{1} simulations. In the former case, one directly measures a bias, after N^{(0)} subtraction, that by construction, does not depend on the potential field. In the latter case, one can estimate the bias with respect to the fiducial model, after the N^{(0)},N^{(1)}, and N^{(2)} corrections have been applied, that can grab some extra contributions. We wish to check the robustness of our correction of the lensing field, by estimating N^{MC} i the H_{0} set. The inpainting process is expected to induce a nonzero lensing coupled bias, since it cannot accurately restore the lensed signal statistic up to the fourpoint correlation function into the masked region. However, this effect can be effectively accounted for by introducing an f_{sky} factor to correct for the lack of power caused by the unrestored lensing modes within the mask, so that the QE variance is given by (8)and the various contributions are shown in Fig. 3.
Fig. 3 Mean deflection spectra reconstructed by applying the FSinpainting method to the lensed maps. “Raw lensing” denotes the spectrum reconstructed directly from the maps. In blue, we show the effect of correcting for the (known) analytical terms N^{(1)} and N^{(2)}. In red, one subtracts the Monte Carlo correction obtained from the set of unlensed maps. The dashed line is the true input spectrum. All points are assigned an error bar corresponding to the sample variance of each map within our Monte Carlo set. 

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The bias of the estimator is quite important but corrected for by using the unlensed simulations. This means that the correction does not couple significantly to the lensing spectrum. It however still introduces systematic uncertainties related to our limited knowledge of the instrument. This motivates the development of an alternative method, which completely avoids the masking issue, to the detriment of introducing some new technicalities.
4. The local approach: multipatch
We start with a simple question. How do you derive the power spectrum from a vector of sampled data? There are two approaches:

1.
one solution is to perform a one step fast Fourier transform(FFT), which produces many modes that can then be averaged(binned) later on;

2.
if the low frequency power spectrum is not required, a solution is to slice your sample into bunches, apodize each one, perform the individual FFTs, and take their mean.
Which approach is better? It was found that using the second one, with overlapping segments of data (by ≃50%) provides the nicest (binned) estimates (Oppenheim & Schafer 1975). This is known as the Welsh periodogram. That then happens when some portion of the data is missing? In the first case, one tries to correct for the gaps, possibly by restoring a mixture that has the correct statistical properties given some prior of the signal, i.e. by performing an inpainting. It is much simpler in the multibunch case, where one rejects chunks that overlaps with the gaps, an approach that is efficient as long as there are few of them and they are largely contiguous.
In the following, we apply these ideas to the case of data located on a cutsphere. We extend beyond the power spectrum estimation (which was largely studied in Das et al. 2009) and investigate whether this simple idea can be applied to CMB lensing reconstruction, where the main “gap” is the Galactic plane and the “bunches” are some tangent square planes.
We thus developed a pipeline that allows for a local reconstruction of the CMB lensing in patches. This has the obvious advantage of avoiding the masked regions and should therefore not introduce the large bias due to the mask correlations that appears in a fullsky analysis. Furthermore, the noise inhomogeneity, which adds a sizable contribution to the lensing deflection, is also reduced by working locally.
Working spatially also allows us to easily inspect the quality of the data in different regions of the sky and therefore constrain the experimental systematics. The natural flatsky formalism that is applied can be easily interpreted, and indicators, such as one for lensing isotropy, can be developed.
Statistically, after determining the Fourier complex coefficients for each patch, we use the HuOkamoto quadratic estimator (QE) described in Sect. 1. This has, by construction, a minimum variance so there is no statistical loss in using this approach. However, obviously, no scales below the patch Fourier size can be reconstructed, hence we miss the low multipoles.
4.1. Tiling the cutsphere with patches
The first unknown is the typical size (L × L) of the patches that one must use for lensing. It turns out to be a compromise between several contradictory considerations:

1.
lensing correlates modes over a few degree scale;

2.
the Fourier modes that are to be reconstructed are located at harmonics of in each (k_{x},k_{y}) Fourier direction. For L = 5°,10° and 15°, respectively, this corresponds to k_{0} = 72,36,24, which sets the grid spacing of the measured modes. To derive our final result in reasonably small multipole bins, we therefore chose to adopt a large L value;

3.
when projecting the data from the sphere onto the local tangent plane (using a gnomonic projection), we wish to avoid too much distortion, which implies that we should not use too large L values. The classical L ≲ 20° flatsky upper limit to the flat sky approximation (White et al. 1999) was derived from power spectra considerations and is not necessarily valid for the four point statistics we consider in lensing;

4.
a last consideration is the efficiency of tiling a given cutsky surface with square patches, which causes them to be small. In addition, inspired by the Welsh periodogram, we seek a configuration where the patches overlap by about 50%, so that there is a clear interplay between the patch central positions and their sizes.
These considerations suggests that patches of angular size ≃10° with ≃50% overlap are appropriate. Although it is a manyparameter system, we found that a simple solution is obtained with patches of angular size L = 10° located at the centers of a HealPix n_{side} = 8 map pixels. In this case, each pixel in the sphere falls on average into ≃1.8 patches. We note that the tiling details do not impact the final result, since we performed the same analysis on L = 12° patches (which leads to a pixel on average falling into 2.6 patches) and obtained very similar results.
We then start with 12n_{side}^{2} = 768 patches. Only patches that do not intersect the Galactic mask at all (the reason being explained in the prewhitening section) are then kept, which leaves 395 of them, covering a fraction f_{sky} = 55% of the sky (as represented in Fig. 4). In this configuration, the overlap (the mean number of patches a point of the sphere belongs to) is ≃1.7.
Fig. 4 Example of the tiling of the map shown in Fig. 1 with overlapping 10° × 10° patches, that do not intersect the Galactic masked region. 

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4.2. Preparing the patches
Local pointsource inpainting.
Before extracting the Fourier coefficients, we first need to remove the bright sources from the patches, which are a strong lensing contaminant. This is performed by using the pointsource mask and filling the masked values by an inpainting algorithm. We note that we use an (image) inpainting algorithm that differs from the one described in Sect. 3, because we wish each patch to be treated independently of the others, which is not the case for FSinpainting. We chose a method that has been designed and tuned for weak lensing surveys FastLens^{5}, which consists in minimizing the sparsity of DCT (discrete cosine transform) coefficients for 256 × 256 data blocks.
More precisely, we construct high resolution regular images from the patches using bilinear interpolation. The FastLens code is then run to fill in the pointsource masked areas. The inpainted values are thenbackprojected onto the sphere to obtain again fullsky maps in which each sources belonging to a patch have been filled.
Since the patches overlap, some filled sources sometimes belong to several of them: we then use the inpainted values from the patch whose center is the closest to the source, in order to avoid border effects.
This procedure is applied to the full set of H_{0} and H_{1} simulations. An example is shown in Fig. 5.
Fig. 5 a) Example of a 10° × 10° patch with masked sources in black. b) Inpainting of the sources using the FastLens algorithm. 

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Prewhitening and apodization.
At this stage, we could again project the pixels onto patches and reconstruct the Fourier coefficients (details in Sect. 4.3). However, we note that the bidimensional temperature powerspectra obtained for these patches exhibit a strong leakage along the null Fourier axis (Fig. 6a).
Fig. 6 a) Bidimensional Fourier spectra of one of our simulation at different scales. Upperleft: mean of the squaredamplitude of the Fourier coefficients for all patches i.e. . An isotropic undecimated wavelet transform (“a trous”, see e.g. Starck & Murtagh 2010) is applied to the image and the results for scale one and two are shown in the bottom plots. The upperright one corresponds to the smooth component. One notices a clear leakage along the null axes. b) Same spectra but working on the prewhitened map and applying a KaiserBessel K_{0.5} window function. The leakage along the null axis has clearly disappeared. 

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We concluded that this leakage is due to that of the low (k_{x},k_{y}) modes which, for the CMB signal, have the stronger amplitudes, and originates from the sidelobes of the implicit 10° × 10° tophat window used. Instead of using some anisotropic filtering (the lensing itself being a source of anisotropy), we can correct this by prewhitening the map and applying an explicit window.
Prewhitening is a standard means of achieving comparably sized Fourier coefficients (e.g. Das et al. 2009). Since we are interested in a range up to ℓ ≃ 2000 (i.e. which is not too far into the CMB damping tail), we need to approximately scale the spectrum by ℓ^{2}. We therefore simply multiply the spherical harmonic coefficients of the map by ℓ, and return to direct space. In this process, the Galactic values are replaced by zero’s, which results in some ringing around the edges of the mask, which is fortunately damped by the instrument main lobe. This is why we only used patches that do not intersect the mask edges at all, since in practice, they are placed far enough away from the mask frontier. Two illustrative examples are shown in Fig. 7.
Fig. 7 Examples of the 10° × 10° multipatch tiling (in gray) of a prewhitened map, around the borders of the Galactic mask (in green) in two regions of the sky (Galactic coordinates given in degrees). One can discern some very local ringing around the boundaries of the mask and some pointsources located outside the patches and that had therefore not been previously inpainted. 

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We note that the prewhitening procedure was applied to both the “signal” maps (H_{1} set) and the MCcorrection ones (H_{0}), so that if it still had a sizable impact on the lensing reconstruction it would have biased the final lensing estimate. Anticipating a result that is be presented later (Fig. 11), the H_{0} correction is then found to be small, which validates a posteriori the prewhitening procedure (and actually the entire procedure) is harmless to lensing.
From now on we work with these fullsky prewhitened maps for which the harmonic coefficients are of similar order.
Rather than using for each patch the implicit tophat window (which has large sidelobes), we apply an explicit window in the direct space. We work with the family of KaiserBessel functions (Kaiser 1966), which allow us to vary simply the sidetomain lobe ratio, and that is still close to the optimal solution of energy concentration provided by the discrete prolate spheroidal sequence (Slepian 1978; Das et al. 2009).
Each value in the L × L size patch is therefore multiplied by where I_{0} denotes the zeroth order modified Bessel function of the first kind.
The Fourier transform of these windows is^{6}(11)which exhibits how the windows shrinks with α when comparing it to the tophat window in Fourier space: .
We compute numerically the radial power of these windows as: (12)and show them for α = 0.5,1,2 in Fig. 8 in direct and Fourier space.
Fig. 8 Radial power of the KaiserBessel 2D windows with α = 0.5,1,2 for L = 10° in real a) and Fourier b) space. The tophat result is also shown. 

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As is wellknown, diminishing the sidelobes always occurs at the price of increasing the main lobe width (energy conservation). In the following, we describe how we attempted to keep α as small as possible to keep the window strongly peaked since the QE considers the products of modes are convolved by this window in Fourier space (Sect. 4.5).
We checked that after prewhitening and windowing with the Kaiser (α = 0.5) window, the power spectrum leakage disappears as is clear in Fig. 6b. In the following, we therefore use W_{0.5} as an explicit apodization function.
The size of the window in Fourier space, Fig. 8b, fixes the binning. For W_{0.5} on a L = 10° square patch, we use a step of Δk = 40, starting from the first available Fourier mode k_{0} = 35.
4.3. Fourier series estimation of nonequispaced data
We project the prewhitened map onto the local patches, using a gnomonic projection. The HealPix pixel centers then fall onto an irregular bidimensional grid. Can we still perform a spectral analysis?
For a n_{side} = 2048 HealPix map, the mean interpixel separation is about 1.7′, which is not negligible compared to the expected mean deflection of the CMB lensing (≃2.7′), so that an interpolation would induce some large effects. To avoid this interpolation, we therefore developed the socalled “ACT” (Adaptive weight, Conjugate gradient, Toeplitz matrices) algorithm, which allows us to fit the complex Fourier coefficients from a set of irregularly sampled data in a reasonable time (see also Keiner et al. 2009). This method has been proposed for real (1D) data and we generalized it to bidimensional data. We give hereafter the main idea and discuss the technicalities in the Appendix.
We search for the least squares estimates of the a_{k,h} (complex) Fourier coefficients of our bandlimited temperature signal, in the series expansion (13)In the general case, the brute force inversion of the normal equations is prohibitive, but the method takes advantage of the peculiar structure of the Fourier series decomposition to perform operations very efficiently. In our case, we manage to determine the 120 × 120 coefficients of the ≃120 000 data values contained in a 10° × 10° patch of an HealPix n_{side} = 2048 map, in about one minute on a single core computer.
The tool we developed, named FourierToeplitz, has been compared to a standard FFT method, when fitting (actually solving) N × N points with N × N unknowns on a regular grid: our results agree to within machine precision.
This tool opens the road to local analyses of projected spherical maps, which are plagued by interpolation issues. It has been used successfully in computing the spectra and full CMB bispectra in Pires et al. (2012).
4.4. Local power spectra estimates
After running the FourierToeplitz tool, we have an estimate of the complex Fourier coefficients per patch, at wavevector k, located on the regular grid Eq. (7).
By computing the squaredamplitude map, one can study the 2D local power spectrum on the sky, and even though the cosmic variance is large, detect potential experimental problems. By taking the mean of the power spectra for all the patches, one can check for the CMB field isotropy in a simple way.
By plotting the values, with respect to k (i.e. assuming isotropy), one constructs a 1D power spectrum which is equivalent to the famous C_{ℓ} but for the noninteger values k given by Eq. (7). One has a powerful local powerspectrum estimator that solves the issues of masking that is wellsuited to jackknife tests. To get a full determination of the spectrum, one would still have to study the window function, as in Hivon et al. (2002) and Das et al. (2009), but we do not actually need it for the lensing reconstruction since it relies on the observed spectrum. We first need to deconvolve the maps from the main lobe, which is a trivial operation in the Fourier space for a Gaussian shape, and obtain some smooth spectrum. This is obtained by taking the mean power spectrum of the deconvolved Fourier patches, and fitting the coefficients of a generic smooth function to all the data points as explained in Plaszczynski & Couchot (2003). The result is shown for one of our simulation, in Fig. 9.
Fig. 9 Example of a 1D power spectrum used in the lensing estimator. The points are obtained from the bidimensional spectrum (as in Fig. 6b), deconvolved from the beam, and represented with respect to to k. They are fitted to the smooth function in red. Also shown in blue is the fiducial model. 

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4.5. Local deflection estimates
After determining the complex Fourier coefficients for each patch, and the observed/true C_{ℓ}’s, we may apply the Hu & Okamoto flatsky estimator to obtain the (noisy) potential maps in the Fourier domain, Eq. (2). But before going on to the lensing spectrum estimate, we need to review the noise of the estimator on an apodized patch since the standard QE has not been derived in this case.
The quadratic term upon which the estimator in Eq. (2) is build, is affected in a nontrivial way by the apodization procedure. Metcalf & White (2007, Appendix B) show that it introduces:

1.
an “aliasing” effect due to the overlap of the windows in Fourierspace that affects only the low ℓ’s modes;

2.
some complicated “smoothing” of the lensing potential.
We do not try to build an optimal estimator from the elaborate expression. We note instead that the apodization process scales the lensing Gaussian noise merely by a constant factor, that can be understood in the following way.
On the basis of Hivon et al. (2002) and Efstathiou (2004), one can show that, assuming that the window is well peaked in Fourier space such that the spectrum does not vary too much over it, the twopoint correlation function of an apodized Gaussian field T^{apo} along with its variance can be approximated by where (16)and W(x,y) is the window function in direct space.
These approximations are accurate for large k values (Efstathiou 2004), which corresponds, given our window size to k ≳ 100.
In the following, we use windows normalized by , so that the reconstructed power spectrum, the only entity that varies with apodization in the filter/normalization of the QE, Eq. (3), is mainly unchanged (Eq. (14)). We checked for instance that applying a W_{0.5} window would change the lensing normalization A_{k} by less than 1%.
The Gaussian noise in the apodized case can be written as the variance in the QE applied to the apodized unlensed sky: (17)Substituting Eq. (15) into this equation and using the normalized window, one obtains (18)What is the accuracy of this approximation? We ran the QE on the unlensed simulations, computed the mean lensing spectrum and compared it to the ideal case given by Eq. (5). The result in Fig. 10 shows that the ratio is indeed reasonably flat for values of ℓ ≳ 100. Fitting the mean value of this ratio in the high ℓ region gives a result very close to the analytical value .
Fig. 10 Color plots showing the ratio of the nonapodized N^{(0)} term Eq. (5) to the reconstructed lensing variance in the patches apodized by W_{0.5} for each of the 100 maps of our unlensed (H_{0}) set. A constant term, whose value is depicted in the upper box, is fitted to the ℓ ≥ 1000 part. 

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We performed the exercise for several windows, including the often used Hanning one (e.g. Oppenheim & Schafer 1975), and give our results in Table 1. They all show excellent agreement with the simple scaling factor.
Comparison of the lensing excess Gaussian noise due to various apodization windows.
This leads us to rewrite the QE total covariance Eq. (6) in the apodized case, as (19)and propose the simple estimator for an apodized patch (20)where the integral is performed in small size rings over the discrete Fourier grid.
This proposed estimator, Eq. (20), is then tested on our H_{1} simulations in order to assess its bias/variance. We note that the value of the scale factor does not need to be known very precisely. What matters is that the same factor is used in the data (here H_{1}) and the Monte Carlo correction (here H_{0}).
We now have all in hand to compute the deflection powerspectra. This is performed for each map of our Monte Carlo set, in the following way:

1.
from the Fourier coefficients obtained on each patch, we form the 2DFourier potential map using Eq. (2). The normalization (and N^{(0)} term) is computed in the standard way, using Eqs. (5) and (3) and the true/observed spectrum as given in Fig. 9;

2.
for each patch, we form the noisecorrected deflection power map of where f = 0.86 in our case;

3.
we accumulate the 395 power maps and evaluate their mean and variance;

4.
we compute the inversevariance weighted average in rings of constant ΔK = 40 width (starting at K_{0} = 35). The binned values are reported at the mean of the different modes K_{i,j} locations within the ring.
We compute these spectra in the H_{1} set, which still have noise contributions from the N^{(1)}, N^{(2)} and N^{MC} terms. The N^{MC} term (the “bias”) is taken from the H_{0} simulations as the mean difference from 0 of the reconstructed spectra following the same procedure.
The mean spectrum is shown in Fig. 11, where one sees the various contributions.
Fig. 11 Mean deflection spectra reconstructed by the multipatch method from the lensed maps. “Raw lensing” denotes the spectrum measured directly on the maps as described in the text without performing any bias correction. In blue, we show the effect of subtracting the (known) analytical terms N^{(1)} and N^{(2)}. In red, one accounts for the Monte Carlo correction obtained for the set of all unlensed maps. The dashed line is the fiducial input spectrum. All points are assigned an error bar corresponding to the variance in the Monte Carlo simulations per sky map. The first bin for the rawlensing estimate is located outside the plot (at a value of 3.5 × 10^{7}). The same range as in Fig. 3 has been used for proper comparison. 

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The initial (raw) lensing spectrum estimate already has a very little bias, thanks to our lack of use of any Galactic data and to a reduced local noise inhomogeneity.
We show that the multipatch method leads, after a small correction, to an unbiased estimate of the deflection over the whole ℓ ∈ [75,2000] range. The very first bin [35,75] is also unbiased but receives a stronger correction from the H_{0} MC, which is due to the breakdown of the flatsky limit (ℓ cannot be identified to k in this case) and to the apodization window having an overlap integral that extends to approximately 100 (see Fig. 8b).
We note that unlensed simulations accurately correcting the lensed ones means that the full reconstruction process does not induce any (significant) couplings to the underlying lensing potential. The variance in the estimator is discussed in the next part.
Finally, we note that working on patches has a number of other benefits:

1.
we derive bidimensional maps of the lensing potential, so thatas in Fig. 6, we can more easily test the deflectionfield isotropy. An example is show in Fig. 12;

2.
for each patch, we can check for unexpected systematic error effects that would provide an excessive lensing signal (as missed sources);

3.
with knowledge of all sources of noise, one can apply a Wiener filter to each patch to reconstruct the lensing potential maps that can then be crosscorrelated to other cosmological probes of the matter, such as Galactic weak lensing (cosmic Shear) or cosmic infrared background, which are only generally measured over a small region of the sky.
Fig. 12 Example of an isotropy check of the lensing potential. For one of our lensed maps, we evaluate the “raw lensing” estimator (i.e. that does not include any MC correction) for each patch, and take their mean. We represent the power map after N^{(0)} subtraction, and smoothed by an “a trous” transform, as in the upper right part of Fig. 6b. 

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5. Comparison of the methods
In Fig. 13, we compare the bias, in each ℓ bin, of the FSinpainting and multipatch methods. They were obtained using unlensed simulations, and were indeed shown to correct the deflection power measure in lensed maps. We recall that the binning that we used starts at ℓ = 35 (first multipatch accessible mode for a 10° × 10° patches) and has a width Δℓ = 40. Two extra bins were added to FSinpainting [2,13] and [14,34] .
Fig. 13 a) Bias of the methods computed from our simulations as the mean of the spurious deflection power on the unlensed set. For FSinpainting, it corresponds to in Eq. (8), whereas for multipatch, it is the result of applying the modified QE to H_{0} as described in Sect. 4.5. b) Same plot with a log scale to emphasize the low ℓ’s. No mode below 35 is available to the multipatch method. 

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The standard deviation in each bin from the H_{1} set is shown in Fig. 14. The Fisher error estimate for the QE (21)is also depicted. We recall that the multipatch method has a lower sky coverage (f_{sky} = 0.55) than the inpainted one (f_{sky} = 0.69) owing to the procedure for tiling the cutsky.
Fig. 14 a) Standard deviation among the methods computed from our simulations, as the spread in each bin of the lensing estimators for the 100 H_{1} set. b) Same plot with a log scale to emphasize low ℓ’s. 

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From these plots, it appears that

the multipatch approach clearly shows less bias, except forthe very first bin [35,75] . It cannot reconstruct modes below these values;

for large ℓ ≳ 100, both estimators follow the naive Fisher error estimates, the FSinpainting one having a slightly smaller variance than for multipatch, owing to the larger sky coverage.
We feel that the small relative statistical loss (≃10%) of the multipatch method is largely compensated for by the gain on systematic errors. As explained in Sect. 4, the reconstruction can be controlled and visually inspected at each step, but most importantly the systematic errors due to the bias of a necessarily imperfect simulation are minimized.
We therefore advocate the use of an hybrid method consisting in the multipatch approach for ℓ ≳ 100 and FSinpainting for lower ℓ’s.
6. Conclusion
We have investigated two methods to reconstruct the lensingdeflection power spectrum from Plancklike CMB frequency maps, using a large Galactic cut and including some strong noise inhomogeneity. The first one, FSinpainting, which was previously presented in Perotto et al. (2010), is still found to be efficient in this extreme configuration as long as one corrects for a large yet lensingindependent bias using Monte Carlo simulations. We have developed a wellsuited method to deal with large masks, based on tiling the cutsky with 10° × 10° patches and performing local analyses. This has required us to solve some problems related to nonperiodic boundary conditions and Fourier coefficients determination for irregularly sampled 2D data. For this purpose, we have developed the FourierToeplitz tool, which allows the fast exact fitting of the Fourier series coefficients in irregularly sampled 2D data. This is a valuable tool for other analyses that require a high level of precision at the spatial location.
Both methods have been demonstrated for realistic Plancklike simulations of the 217 GHz CMB channel. It was found that the multipatch approach has a very low bias in the whole 100 ≤ ℓ ≤ 2000 range, thanks to the avoidance of the Galactic plane, and lower local noise inhomogeneity. It allows us at each step to check for experimental systematic errors and perform local images of both the temperature and deflection bidimensional power spectra. Its final variance is only marginally larger than a fullsky method, and could be improved by a smarter strategy for tiling the cutsky sphere. The final result is insensitive to the precise position of the patches and of their overlap. To perform some cosmological fits using the reconstructed spectrum, the interbin correlation would still have to be measured accurately, and included in the likelihood, since we have measured some (15 ± 10)% correlation level, with our Δℓ = 40 binning. This requires a large number of simulations (≃1000).
In the ℓ ≤ 100 range, we advocated using the FSinpainting method, which provides the minimal variance estimate in the cut sky. Since our simulations did not include a Galactic contaminant, this boundary could slightly shift. However, using our 30% Galactic mask, we checked by adding a simulated Galactic component to our maps, that its net effect on the reconstructed deflection power was extremely negligible (on real data, some template would be subtracted).
These results open the road to measuring CMB lensing directly in Plancklike CMB maps, without even performing a component separation of the foregrounds. This is not exactly true for the FSinpainting method, where one must have clean boundaries at the mask frontier. However, this can be obtained by a simple template subtraction measured in the high frequency channels. For the multipatch method, one can still perform the analysis without “undusting” the map, by choosing appropriate CMBdominated patches; we checked in simulations that a subdominant amount of dust contamination does not affect the lensing deflection spectrum.
It is not our goal to come to a decision on whether a component separation method is a more accurate means of lensing reconstruction, since it is an area that remains under active development. We note however that the statistical gain offered by using a larger fraction of the sky can be counterbalanced by a higher N^{(0)} term due to a larger (combined) lobe of the instrument or a higher final noise level (see Eq. (21)). Adding the high/low frequency channels will also “bring back” some additional infrared/radio sources that need to be masked out, lowering the final statistical gain of the combined map.
Working directly on intensity maps allows us to perform various sanity tests by checking the consistency between the reconstructions from different frequency maps, which is a way of assessing the robustness of the estimate against either experimental uncertainties or physical contamination – as from possible SZlensing or unresolved radiosourcelensing correlations. The reconstructions of each frequency maps can also be minimumvariance combined, which offers a robust reconstruction that allows for a high level of systematic control. Finally, this is a wellsuited approach to crosscorrelation studies with other mass tracers, by selecting frequencies that are unaffected by contaminants that may induce extra correlations. For instance, one may wish to use only frequencies over 100 GHz (with negligible unresolved radiosource contamination), while studying correlations with external radio surveys, or below 217 GHz for a CIBlensing correlation estimate.
Appendix: FourierToeplitz tool
In order to not lose accuracy in determining the Fourier coefficients from a sample of irregularly sampled points, we developed a tool for fitting these coefficients in a reasonable time.
We start with the 1D case, where we have implemented the “second generation” algorithm proposed in Feichtinger et al. (1995).
We define f to be a function sampled on any support { t_{i} } . In a given interval (0,T), the function can be expanded into a Fourier series Assuming that it has a bandlimited spectrum, so that we can limit the number of Fourier modes to the problem is to determine the a_{k} coefficients given the sampled values f_{i}.
We consider the reduced variable. If the number of samples u_{i} is N, one writes the N equations This is a linear system of N equations with 2M + 1 unknowns. The wellknown normal equations obtained from least squares minimization are where F is the column vector of sampled values and G is the matrix with elements g_{kl} = e^{2iπkul}. The solution is in general computationally heavy using standard methods.
Here, the interesting point is that the generic term of the system is of the type: which is a Toeplitz matrix. One solves the system using the conjugategradient algorithm (the matrix is Hermitian and positive), which consists in performing successive matrixvector products. One then pads the Toeplitz matrix with zeros to obtain a circulant matrix, since the product of a circulant matrix with a vector can be computed efficiently using an FFT.
We extended this method to the 2D case using the formalism of the Kronecker products of matrices and the properties of the separability of FFTs. This allows for the fast determination of the a_{kh} coefficients in the Fourier expansion In our case, a 10 × 10° patch from an HealPix n_{side} = 2048 map, contains about 120 000 irregularly sampled values.
The modes are harmonics of , so we need to determine 120 × 120 (half is negative) of them to obtain all modes below l_{max} = 2000. This is performed in about one minute on a single core. The conjugate gradient converges in about seven iterations, without using any special preconditioner, so we did not add the adaptive weight scheme.
Acknowledgments
We acknowledge the use of CAMB (Lewis et al. 2000), HealPix (Górski et al. 2005), LensPix (Challinor & Lewis 2005), MRS (Starck et al. 2006), FastLens (Pires et al. 2009), and FuturCMB2 (Perotto et al. 2006) packages. We thank Simon Prunet for the precise derivation of the effective number of degrees of freedom in Hivon et al. (2002) and Martin Reinecke for some dedicated HealPix C++ developments.
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All Tables
Comparison of the lensing excess Gaussian noise due to various apodization windows.
All Figures
Fig. 1 Example of one of our simulated lensed temperature map, using the procedure described in the text. Units are mK_{CMB}. The gray region corresponds to the Galactic mask we propose to use. A pointsource mask is also included, but barely visible, being more clearly seen in Fig. 5. 

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In the text 
Fig. 2 Inpainted map corresponding to filling the Galactic+pointsource mask of Fig. 1. 

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In the text 
Fig. 3 Mean deflection spectra reconstructed by applying the FSinpainting method to the lensed maps. “Raw lensing” denotes the spectrum reconstructed directly from the maps. In blue, we show the effect of correcting for the (known) analytical terms N^{(1)} and N^{(2)}. In red, one subtracts the Monte Carlo correction obtained from the set of unlensed maps. The dashed line is the true input spectrum. All points are assigned an error bar corresponding to the sample variance of each map within our Monte Carlo set. 

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In the text 
Fig. 4 Example of the tiling of the map shown in Fig. 1 with overlapping 10° × 10° patches, that do not intersect the Galactic masked region. 

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In the text 
Fig. 5 a) Example of a 10° × 10° patch with masked sources in black. b) Inpainting of the sources using the FastLens algorithm. 

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In the text 
Fig. 6 a) Bidimensional Fourier spectra of one of our simulation at different scales. Upperleft: mean of the squaredamplitude of the Fourier coefficients for all patches i.e. . An isotropic undecimated wavelet transform (“a trous”, see e.g. Starck & Murtagh 2010) is applied to the image and the results for scale one and two are shown in the bottom plots. The upperright one corresponds to the smooth component. One notices a clear leakage along the null axes. b) Same spectra but working on the prewhitened map and applying a KaiserBessel K_{0.5} window function. The leakage along the null axis has clearly disappeared. 

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In the text 
Fig. 7 Examples of the 10° × 10° multipatch tiling (in gray) of a prewhitened map, around the borders of the Galactic mask (in green) in two regions of the sky (Galactic coordinates given in degrees). One can discern some very local ringing around the boundaries of the mask and some pointsources located outside the patches and that had therefore not been previously inpainted. 

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In the text 
Fig. 8 Radial power of the KaiserBessel 2D windows with α = 0.5,1,2 for L = 10° in real a) and Fourier b) space. The tophat result is also shown. 

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In the text 
Fig. 9 Example of a 1D power spectrum used in the lensing estimator. The points are obtained from the bidimensional spectrum (as in Fig. 6b), deconvolved from the beam, and represented with respect to to k. They are fitted to the smooth function in red. Also shown in blue is the fiducial model. 

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In the text 
Fig. 10 Color plots showing the ratio of the nonapodized N^{(0)} term Eq. (5) to the reconstructed lensing variance in the patches apodized by W_{0.5} for each of the 100 maps of our unlensed (H_{0}) set. A constant term, whose value is depicted in the upper box, is fitted to the ℓ ≥ 1000 part. 

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In the text 
Fig. 11 Mean deflection spectra reconstructed by the multipatch method from the lensed maps. “Raw lensing” denotes the spectrum measured directly on the maps as described in the text without performing any bias correction. In blue, we show the effect of subtracting the (known) analytical terms N^{(1)} and N^{(2)}. In red, one accounts for the Monte Carlo correction obtained for the set of all unlensed maps. The dashed line is the fiducial input spectrum. All points are assigned an error bar corresponding to the variance in the Monte Carlo simulations per sky map. The first bin for the rawlensing estimate is located outside the plot (at a value of 3.5 × 10^{7}). The same range as in Fig. 3 has been used for proper comparison. 

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In the text 
Fig. 12 Example of an isotropy check of the lensing potential. For one of our lensed maps, we evaluate the “raw lensing” estimator (i.e. that does not include any MC correction) for each patch, and take their mean. We represent the power map after N^{(0)} subtraction, and smoothed by an “a trous” transform, as in the upper right part of Fig. 6b. 

Open with DEXTER  
In the text 
Fig. 13 a) Bias of the methods computed from our simulations as the mean of the spurious deflection power on the unlensed set. For FSinpainting, it corresponds to in Eq. (8), whereas for multipatch, it is the result of applying the modified QE to H_{0} as described in Sect. 4.5. b) Same plot with a log scale to emphasize the low ℓ’s. No mode below 35 is available to the multipatch method. 

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In the text 
Fig. 14 a) Standard deviation among the methods computed from our simulations, as the spread in each bin of the lensing estimators for the 100 H_{1} set. b) Same plot with a log scale to emphasize low ℓ’s. 

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In the text 