CMB delensing with deep learning

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.

💡 Research Summary

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This paper investigates the use of deep learning, specifically the U‑Net++ architecture, for delensing the Cosmic Microwave Background (CMB). Weak gravitational lensing by large‑scale structure smooths the acoustic peaks of the CMB temperature and polarization power spectra and converts part of the dominant E‑mode polarization into a B‑mode component, thereby obscuring the faint primordial B‑mode signal from inflationary gravitational waves. Traditional delensing techniques, such as the quadratic estimator (QE), reconstruct the lensing potential from the observed CMB maps and then attempt to undo its effect, but they leave significant residuals, especially at small angular scales where the lensing signal is strongest.

The authors generate a suite of realistic full‑sky simulations using publicly available tools (Lenspyx, CAMB, LensPix, HEALPix). They adopt the Planck 2018 ΛCDM cosmology and vary the scalar amplitude (Aₛ) and tensor‑to‑scalar ratio (r) to produce 30 distinct CMB realizations. Each realization includes Gaussian beam smoothing (FWHM ≈ 8.3 arcmin) and white instrumental noise, mimicking the conditions of upcoming ground‑based and satellite experiments. The maps are rendered at Nside = 2048 (≈ 4 arcmin pixel size), providing high‑resolution data for both temperature (T) and polarization (Q, U).

To apply a convolutional neural network to spherical data, the authors adopt a sky‑map segmentation scheme: the HEALPix sphere is divided into many overlapping planar patches, each treated as a 2‑D image. These patches serve as inputs to a U‑Net++ model, which extends the classic U‑Net by adding dense skip pathways and nested decoder branches. This design enables the network to capture multi‑scale features and to reuse low‑level spatial information throughout the deep layers, which is crucial for learning the subtle, non‑linear transformations induced by lensing.

Training is performed on 80 % of the simulated data, with the remaining 20 % reserved for validation. The loss function is a simple L2 (mean‑squared‑error) between the network output (the predicted delensed map) and the ground‑truth unlensed map. Optimization uses the Adam optimizer with a learning rate of 1 × 10⁻⁴, batch size 32, and 200 epochs. Regularization techniques such as batch normalization and dropout are employed to mitigate over‑fitting.

After training, the model is applied to the held‑out test set. The authors compute the angular power spectra (TT, EE, BB, TE) of the delensed maps and compare them to both the original lensed spectra and the true unlensed spectra. The U‑Net++‑delensed spectra match the unlensed spectra to within a few parts in 10⁴ across a wide multipole range (ℓ ≈ 30–3000). By contrast, the QE‑delensed spectra retain residual lensing power, especially at ℓ > 1500, where the U‑Net++ error is more than ten times smaller. In the BB channel, the reduction of lensing‑induced power is particularly striking, implying that the deep‑learning approach could substantially improve the sensitivity to primordial gravitational waves with tensor‑to‑scalar ratios as low as r ≈ 0.001.

The paper emphasizes several key insights. First, deep learning can learn the complex, non‑Gaussian mapping from lensed to unlensed CMB maps without explicit modeling of the lensing potential, outperforming the analytically derived QE in realistic noise conditions. Second, the U‑Net++ architecture’s dense skip connections are well‑suited for preserving fine‑scale temperature and polarization structures while removing the large‑scale smoothing introduced by lensing. Third, the method is computationally efficient at inference time: once trained, a full‑sky delensing operation can be performed in a matter of minutes on a modern GPU, far faster than iterative QE pipelines.

However, the study also acknowledges limitations. The training data are pure simulations; real observations contain foreground emissions (Galactic dust, synchrotron), beam asymmetries, correlated noise, and systematic effects not represented in the current dataset. Generalizing the model to such complexities will require either augmenting the training set with realistic foreground simulations or adopting transfer‑learning strategies. Moreover, the current implementation processes each planar patch independently, which may introduce edge artifacts; future work could explore spherical convolutional networks that operate directly on the HEALPix sphere.

The authors make their code publicly available on GitHub, facilitating reproducibility and encouraging the community to extend the approach to more realistic scenarios, incorporate multi‑frequency data, and explore model compression for deployment on large‑scale CMB analysis pipelines. They conclude that deep‑learning‑based delensing, exemplified by U‑Net++, offers a promising path toward the high‑precision CMB measurements required by next‑generation experiments such as LiteBIRD, CMB‑S4, and the Simons Observatory.