CMB delensing with deep learning

TL;DR

The lensing CMB sky map and full-sky angular power spectrum processed by the UNet++ algorithm are very close to those of the CMB without lensing effects, and the error is more than 10 times smaller than that given by the QE algorithm.

Abstract

The cosmic microwave background (CMB) stands as a pivotal source for studying weak gravitational lensing. While the lensed CMB aids in constraining cosmological parameters, it simultaneously smooths the original CMB's features. The angular power spectrum of the unlensed CMB showcases sharper acoustic peaks and more pronounced damping tails, enhancing the precision of inferring cosmological parameters that influence these aspects. Although delensing diminishes the $B$-mode power spectrum, it facilitates the pursuit of primordial gravitational waves and enables a lower variance reconstruction of lensing and additional sources of secondary CMB anisotropies. In this work, we explore the potential of deep learning techniques, specifically the U-Net++ algorithm, to play a pivotal role in CMB delensing. We analyze three fields, namely $T$, $Q$, and $U$ sky maps, present the angular power spectra of the CMB delensed $TT$, $EE$, $BB$, and $TE$, and compare them with the unlensed CMB angular power spectra. Our findings reveal that the angular power spectrum of the lensed CMB, processed by U-Net++, closely aligns with that of the unlensed CMB. Thus, U-Net++ based CMB delensing proves to be effective in mitigating the impacts of weak gravitational lensing, paving the way for enhancing the CMB delensing power spectrum in forthcoming CMB experiments. The code utilized for this analysis is available on GitHub.

Figures (8)

• Figure 1: CMB XX angular power spectrum, $X\in\{{\rm T}, {\rm E}, {\rm B}\}$. Top Panel: The red solid line represents the lensed TT spectrum, while the light red dashed line corresponds to the unlensed TT spectrum. The green solid line denotes the EE spectrum under the influence of lensing, whereas the light green dashed line shows the EE spectrum without lensing effects. The blue solid line indicates the lensing-modified BB spectrum, and the light blue dashed line depicts the BB spectrum in the absence of lensing effects. The parameters for generating the temperature angular power spectrum are $A_s = 2.1 \times 10^{-9}$ and $r = 0.005$. The black solid line displays the noise level of the temperature detectors, and the grey solid line represents that of the polarization detectors. Bottom Panel: It illustrates the relative differences between these highly similar spectra. The red dashed line marks the ratio of the unlensed TT spectrum to the lensed TT spectrum; the green dashed line indicates the ratio of the unlensed EE spectrum to the lensed EE spectrum; and the blue dashed line shows the ratio of the unlensed BB spectrum to the lensed BB spectrum.
• Figure 2: A half-orthogonal view projection of the sky map.Gray sequentially labels pixels with four neighbors, while larger black numbers highlight those with only three.
• Figure 3: The T, E and B sky map patch. Each patch has a size of $214.86~{{\rm deg}}^2$. The center of the sky patches is $(l, b) = (101.25{\rm ^\circ}, 19.471{\rm ^\circ})$, where $l$ and $b$ are the Galactic longitude and Galactic latitude, respectively. From left to right, the sky patches are T, E and B map, respectivly. The top row of patches displays sky map patches affected by lensing effects, including noise and instrumental artifacts, which are used to construct the training dataset. The second row presents the original, unlensed sky map patches, which serve as the basis for generating the label dataset. The unit is ${\rm \mu K}$.
• Figure 4: Unet++ network architecture. Each node in the graph represents a convolution block, downward arrows indicate down-sampling, upward arrows indicate up-sampling, dot arrows indicate skip connections, and the dot box indicates the four outputs. UNet++ combines UNets of different depths into a unified architecture. All substructures share the same encoder, but have their own decoders. Then skip connections are dropped, and every two neighboring nodes are connected with a short skip connection, enabling the deeper decoder to send supervisory signals to the shallower decoder. Finally, by connecting the decoders, a densely connected skip connection is generated so that the dense features propagate along the skip connection, resulting in more flexible feature fusion at the decoder nodes. Thus, each node in the UNet++ decoder combines multiscale features of the same resolution from all its preceding nodes from a horizontal perspective, and integrates multiscale features of different resolutions from its preceding nodes from a vertical perspective.This multiscale feature aggregation in UNet++ gradually synthesizes the segmentation, resulting in improved accuracy and fast convergence.
• Figure 5: Loss function evolution per network over epochs. From top to bottom, represent the results of training for T, E, and B maps, respectively. The dark blue solid line indicates the training set loss function evolution and the light blue solid line indicates the validation set loss function evolution.