A Generative Reconstruction of Low-$\ell$ CMB B-Mode Signal using Reverse Diffusion in Deep Learning

TL;DR

This work tackles the challenge of recovering the faint primordial $B$-mode signal at low multipoles in the presence of lensing, foregrounds, and instrument noise. It introduces a score-based diffusion approach using a variance-exploding SDE with a reverse diffusion steered by a neural score model trained on primordial spectra at $r=0.001$, enabling denoising and delensing without explicit foreground templates. The method demonstrates accurate realization-level recovery, unbiased ensemble means, and reconstruction uncertainties that align with cosmic variance, while providing distribution-level fidelity as shown by KL-divergence tests. It also offers a generative tool to simulate new primordial realizations consistent with a given $r$, making it a robust, physics-guided component for upcoming CMB polarization missions like ECHO, CMB-S4, and the Simons Observatory.

Abstract

Detecting primordial B-mode polarization of the Cosmic Microwave Background (CMB) provides a direct probe of inflationary gravitational waves. However, the signal is extremely faint and contaminated by gravitational lensing, instrumental noise, and astrophysical foregrounds. Here we present a score-based diffusion approach, formulated using variance-exploding stochastic differential equations (VE-SDEs), to reconstruct the primordial B-mode angular power spectrum from contaminated observations. The method employs a reverse SDE guided by a score model trained exclusively on random realizations of the primordial low $\ell$ B-mode angular power spectrum corresponding to a fixed tensor-to-scalar ratio $r=0.001$. During inference, the reverse SDE iteratively drives the observed angular power spectrum toward the learned primordial manifold, effectively denoising and delensing the input. The model is tested on simulated observational spectra that include gravitational lensing, complex polarized foreground combinations, and instrumental noise characteristics representative of the proposed ECHO mission. The trained score model learns the underlying statistical distribution of the primordial B-mode field for the given $r$, which acts as a physics-guided prior that can generate new, consistent realizations of the signal. This approach provides a robust framework for primordial signal recovery in future CMB polarization missions.

Figures (11)

• Figure 1: Top panel shows transformation of the data to a noise dominated distribution by slowly adding noise to it using a forward SDE. The bottom panel shows that samples of noise minimized data can be generated by using the reverse SDE method. In both cases the differential equations used are mentioned. The reverse SDE computation requires knowledge of score function.
• Figure 2: Training loss evolution for ScoreNet1D under the weighted denoising score-matching objective (Eq. \ref{['eq:dsm_final']}) over 150 epochs. The loss decreases rapidly during the initial phase and saturates after $\sim$100 epochs, indicating stable convergence of the learned score estimator. Small fluctuations around the plateau arise from stochastic mini-batch sampling of diffusion times and noise realizations.
• Figure 3: Reconstruction of a single low-$\ell$ CMB $B$-mode realization for four distinct foreground configurations (F1–F4; Section \ref{['subsec:testing-data']}). In each panel, the contaminated observed spectrum (green) is dominated by polarized foregrounds and lensing across all multipoles, while the reverse VE--SDE reconstruction (orange) successfully recovers the primordial $B$-mode spectrum (blue) without explicit foreground modeling.
• Figure 4: Direct comparison of the reconstructed (orange) and ground-truth (blue) primordial $B$-mode spectra for the same test realization shown in Fig. \ref{['fig:single_recon']}. The close agreement across all multipoles demonstrates that the reverse VE-SDE reconstruction accurately recovers both the amplitude and shape of the primordial signal.
• Figure 5: Evolution of the reconstructed $D_\ell$ spectrum for a single noisy observation during reverse VE--SDE sampling. Top panel: Initial state at $t=1$, where the spectrum is dominated by noise and carries no primordial structure. Middle panel: Intermediate diffusion stage ($t=0.8$), showing a gradual reduction in amplitude and variance as stochastic noise is suppressed and the spectrum begins to move toward the learned primordial distribution. Bottom panel: Late-stage reconstruction at decreasing diffusion times ($t=0.4,\,0.2,\,0.0$), demonstrating convergence toward the primordial $B$-mode spectrum (black dashed line).