Phys. Rev. D 104, 123514 (2021) - Optimal multifrequency weighting for CMB lensing

Optimal multifrequency weighting for CMB lensing

Noah Sailer, Emmanuel Schaan, Simone Ferraro, Omar Darwish, and Blake Sherwin

Phys. Rev. D 104, 123514 – Published 6 December 2021

Abstract

Extragalactic foregrounds in cosmic microwave background (CMB) temperature maps lead to significant biases in CMB lensing reconstruction if not properly accounted for. Combinations of multifrequency data have been used to minimize the overall map variance (internal linear combination, or ILC), or specifically null a given foreground, but these are not tailored to CMB lensing. In this paper, we derive an optimal multifrequency combination to jointly minimize CMB lensing noise and bias. We focus on the standard lensing quadratic estimator, as well as the “shear-only” and source-hardened estimators, whose responses to foregrounds differ. We show that an optimal multifrequency combination is a compromise between the ILC and joint deprojection, which nulls the thermal Sunyaev-Zel’dovich (tSZ) and cosmic infrared background (CIB) contributions. In particular, for a Simons Observatory-like experiment with $ℓ_{\max, T} = 3000$ , we find that profile hardening alone (with the standard ILC) reduces the bias to the lensing power amplitude by 40%, at a 20% cost in noise, while the bias to the cross-correlation with a LSST-like sample is reduced by nearly an order of magnitude at a 10% noise cost, relative to the standard quadratic estimator. With a small amount of joint deprojection the bias to the profile hardened estimator can be further reduced to less than half the statistical uncertainty on the respective amplitudes, at a 20% and 5% noise cost for the auto- and cross-correlation respectively, relative to the profile hardened estimator with the standard ILC weights. Finally, we explore possible improvements with more aggressive masking and varying $ℓ_{\max, T}$ .

Received 4 August 2021
Accepted 23 November 2021

DOI:https://doi.org/10.1103/PhysRevD.104.123514

Physics Subject Headings (PhySH)

Cosmic microwave background

Astrophysical & cosmological simulations

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Noah Sailer ^1,2,*, Emmanuel Schaan ^2,1, Simone Ferraro^2,1, Omar Darwish ³, and Blake Sherwin^3,4

¹Berkeley Center for Cosmological Physics, Department of Physics, University of California, Berkeley, California 94720, USA
²Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley, California 94720, USA
³Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, United Kingdom
⁴Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom

^*nsailer@berkeley.edu

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Vol. 104, Iss. 12 — 15 December 2021

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Figure 1
The top panel shows the temperature power spectrum for different multifrequency combinations, compared to the noiseless lensed CMB (black curve). Compared to the minimum-variance combination (ILC, red), the deprojection of tSZ (orange) comes at a larger cost in noise than that of CIB (green). Jointly deprojecting tSZ and CIB (blue) causes a more than ten-fold noise increase at $ℓ \sim 3000$ , where most of the lensing information resides. We note that the significant increase in power at large scales from joint or tSZ deprojection is primarily due to a large increase in atmospheric noise. The middle and bottom panels show the tSZ and CIB power spectra, respectively, compared to the single-frequency case (150 GHz). For tSZ (middle), the standard ILC and the CIB deprojection cause a significant increase. For CIB (bottom), the ILC very effectively reduces the power; however, the tSZ-deprojection enhances it by more than an order of magnitude.
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Figure 2
As illustrated in the top panel, the standard ILC is the linear combination of multifrequency maps which minimizes the lensing noise, for the standard QE and all the other estimators considered in this paper. As expected from Fig. 1, deprojecting CIB (green) comes at almost no cost in lensing noise, whereas deprojecting tSZ (yellow) multiplies the noise by $\sim 2$ . Despite causing a more than ten-fold increase in temperature noise, joint deprojection only causes a four-fold increase in lensing noise. This can be understood by counting the number of signal-dominated temperature modes. The bottom panel shows the fractional bias to the CMB lensing autospectrum for the various frequency combinations. The gray boxes denote the bandpower errors for the standard ILC. For the ILC, the bias is largely dominated by the enhanced tSZ signal. While the joint deprojection successfully nulls all biases, simply deprojecting tSZ or CIB does not.
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Figure 3
Left: the total $| bias |$ and noise on the lensing amplitude $A_{lens}$ for the various ILC combinations. The biases for the standard QE are shown in blue, shear in red, the point source hardened estimator in green, and the profile hardened estimator in purple. Right: the same as the left panel, but for the amplitude $A_{cross}$ of the cross-correlation with the mock LSST sample. Circles denote the minimum variance (MV) ILC, squares (triangles) denote CIB (tSZ) deprojection, while diamonds denote the simultaneous deprojection of both the CIB and tSZ. The lines connecting MV ILC to either CIB, tSZ or joint deprojection follow Eq. (19), with step sizes $Δ t = 0.2$ . Throughout we sum over $20 < L < 1000$ when calculating the bias and noise for either $A_{lens}$ or $A_{cross}$ , and take $f_{sky} = 0.4$ .
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Figure 4
The same as Fig. 3, but with $| bias (C_{L}^{κ}) |$ replacing $bias (C_{L}^{κ})$ in Eq. (17), and similarly for the cross-correlation with the mock LSST sample.
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Figure 5
The same as Fig. 3, using the more aggressive masking technique described in Appendix pp4. When calculating $σ$ , we neglect any differences in sky coverage due to more of the sky being masked (8% of the sky as opposed to 3% with the fiducial mask used in the main text), as this is a negligible effect.
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Figure 6
Noise vs bias for several $ℓ_{\max, T}$ ’s. For $ℓ_{\max, T} = 3000$ , 3500, and 4000 we plot the line going from the standard ILC to joint deprojection, with the line style denoting the $ℓ_{\max, T}$ . For $ℓ_{\max, T} = 1500$ , 2000, and 2500 we plot the noise and bias at the standard ILC for each estimator (denoted by the circles). Note that for $ℓ_{\max, T} = 1500$ , the Shear estimator’s noise is larger than 10% (3%) for the auto- (cross-) correlation, extending beyond the domain of the plot.
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Physical Review D

covering particles, fields, gravitation, and cosmology

Optimal multifrequency weighting for CMB lensing

Noah Sailer, Emmanuel Schaan, Simone Ferraro, Omar Darwish, and Blake Sherwin

Phys. Rev. D 104, 123514 – Published 6 December 2021

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Physical Review D

covering particles, fields, gravitation, and cosmology

Optimal multifrequency weighting for CMB lensing

Noah Sailer, Emmanuel Schaan, Simone Ferraro, Omar Darwish, and Blake Sherwin

Phys. Rev. D 104, 123514 – Published 6 December 2021

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