Using inpainting to construct accurate cut-sky CMB estimators

The direct evaluation of manifestly optimal, cut-sky cosmic microwave background (CMB) power spectrum and bispectrum estimators is numerically very costly, due to the presence of inverse-covariance filtering operations. This justifies the investigation of alternative approaches. In this work, we mostly focus on an inpainting algorithm that was introduced in recent CMB analyses to cure cut-sky suboptimalities of bispectrum estimators. First, we show that inpainting can equally be applied to the problem of unbiased estimation of power spectra. We then compare the performance of a novel inpainted CMB temperature power spectrum estimator to the popular apodized pseudo-$C_l$ (PCL) method and demonstrate, both numerically and with analytic arguments, that inpainted power spectrum estimates significantly outperform PCL estimates. Finally, we study the case of cut-sky bispectrum estimators, comparing the performance of three different approaches: inpainting, apodization and a novel low-l leaning scheme. Providing an analytic argument of why the local shape is typically most affected we mainly focus on local-type non-Gaussianity. Our results show that inpainting allows us to achieve optimality also for bispectrum estimation, but interestingly also demonstrate that appropriate apodization, in conjunction with low-l cleaning, can lead to comparable accuracy.

Figure 1

The variances of PCL power spectrum estimates divided by full-sky variance for various degrees of apodization. We plot exact variances as obtained from $10^5$ Monte-Carlo (MC) samples as well as variances predicted by analytic approximations to the PCL covariance matrices. The approximations perform very well with minor deviations only at the lowest $l$ and, in the case of the unapodized mask, highest $l$. Also plotted are Fisher errors obtained under the assumption of a constant power spectrum $C_l=C_0$ (flat magenta line). The inset magnifies the region at low $l$ where PCL variances approach the cosmic variance limit as expected based on the fact that the temperature CMB can be viewed as nearly band limited when studying the largest scales due to the large amount of power in low-$l$ modes.

Figure 2

Real part of the masked spherical harmonic $\mathrm{Re}\{Y_{105}\}$ (left) and the corresponding inpainted spherical harmonic $\mathrm{Re}\{Y_{105}^{\mathrm{I}}\}$. In both cases the boundary of the mask is highlighted.

Figure 3

Density plots of coupling matrices in the case of the unapodized U73 mask (top left) and inpainting (top right). The bottom panel shows a cross section ${\mathrm{\Pi }}_{l_{0}l}$ through coupling matrices for various smoothing scales and inpainting at $l_0=500$. The asymmetry of the inpainting coupling is clearly visible. While low-high coupling is heavily suppressed compared to all PCL cases, high-low coupling is in fact enhanced as explained in the main text.

Figure 4

The variances of inpainted power spectrum estimates divided by full-sky variance. For comparison the PCL variances for the unapodized and S03 mask and the Fisher errors obtained under the assumption of a constant power spectrum are included as well. As in Fig. 1 we plot exact variances as obtained from $10^5$ MC samples as well as variances predicted by analytic approximations to the covariance matrices. Just like the PCL approximations, the analytic approximation to the inpainting variance discussed in Sec. 2g performs very well. The inset magnifies the low-$l$ region. Note that upon close inspection very minor differences at high $l$ between the PCL variances shown here and in Fig. 1 are visible that arise because of the different choices of $N_{\mathrm{side}}$ that cause resolution effects in the harmonic transforms and the apodization scheme.

Figure 5

Plot of the ratio of the variance of PCL estimates in the case of the S03 mask and the inpainted power spectrum estimates.

Figure 6

The relative difference between the covariance approximations and variances obtained from MC sampling the case of PCL estimation using the S03 mask and inpainted power spectrum estimates. The shaded area indicates $2\sigma$ error bars on the variances expected from $10^5$ MC samples. The only notable differences occur at very low $l$.

Figure 7

The ratios ${\overline{C}}_l/C_l$ for PCL estimation using the unapodized and S03 mask along with ${\overline{C}}_l^{\mathrm{I}}/C_l$ for inpainting. The plot also shows ${\overline{C}}_l$ in the case of inpainting to highlight the impact of the ${\mathrm{\Pi }}_{ll}/{\mathrm{\Pi }}_{ll}^{\mathrm{I}}$ scaling entering ${\overline{C}}_l^{\mathrm{I}}$.

Figure 8

Plot of the reduced bispectra of the mask (left) and the local shape (right) based on a modal reconstruction and weighted with the large-angle solution for the constant shape (see main text). Note that the modal reconstruction is only able to capture a small component of the full, highly oscillatory mask bispectrum. The reconstruction is based on relatively slowly varying basis functions that describe physical reduced bispectra and is thus ideally suited for visualizing the components of the mask bispectrum that are relevant for constraining cosmology. Each bispectrum has been normalized relative to its variance and the color scale is linear, with warm tones being positive and cool tones negative.

Figure 9

The efficiency of the local estimator for the different apodization scales in the $l_{\mathrm{cl}}=70$ (orange) and uncleaned (blue) case. Also shown is the prediction based on the assumption of constant spectra (green dotted line) and the optimal limit (dashed black line).

Figure 10

Images and kernels of the projection operators ${\mathit{P}}_C$. The successive oblique projections lead to a large eigenvalue $\lambda$.