Accurate and efficient likelihood modeling for large-scale CMB data

TL;DR

This work addresses the challenge of accurate likelihood modeling for large-scale CMB data, where $C_\ell$ estimators exhibit non-Gaussian statistics under partial sky coverage. It introduces and benchmarks three HL-based approaches—HL, cross-spectra HL (cHL), and the novel marginalized HL (mHL)—using simulations of three data splits across different noise regimes and fiducial assumptions. The results show that while HL is exact in the full-sky with correct noise, it suffers from bias when noise bias is misestimated; mHL consistently offers the best agreement with pixel-based likelihood, particularly under cut-sky and noise-mismatch conditions, with cHL showing biases that grow with the number of cross-spectra. The findings advocate using a multi-field mHL framework for robust, accurate parameter inference of $ au$ and $r$ in large-scale CMB polarization analyses, with practical implications for upcoming missions and future sky-cover constraints.

Abstract

Accurate parameter estimation from cosmic microwave background data requires reliable likelihood modeling, particularly at large angular scales where angular power spectrum estimators exhibit non-Gaussian statistics. We present a novel approach, based on the Hamimeche-Lewis formalism, that marginalizes over auto-spectra, thus reducing residual biases from noise misestimation and partial sky coverage. We validate our approach by simulating three independent CMB channels, or data splits, in a multi-field setting, comparing to the pixel-based likelihood ground truth estimates for the optical depth $τ$ and the tensor-to-scalar ratio $r$. We benchmark our method against the main power spectrum based alternatives available in the literature, showing that it outperforms all of them in terms of accuracy, while remaining fast and computationally efficient.

Figures (10)

• Figure 1: Auto- (red) and cross-spectra (blue) for the signal-dominated, tensor-detection, and noise-dominated cases, respectively. Solid lines indicate the mean over the $N_{\rm data}$ simulations, while shaded regions refer to the $1\sigma$ scatter around the mean and dotted ones to the corresponding offset $O_\ell$. As a reference, we also show the fiducial power spectrum with and without noise bias with black solid and dashed lines.
• Figure 2: Posterior distributions from pixel-based (gold), HL (red), mHL (blue), and cHL (green) likelihoods under full-sky assumptions. Dashed (dotted) curves indicate under- (over-) estimation of the noise bias. Vertical black lines denote the true parameter values.
• Figure 3: Dependence of cHL on the number of cross-spectra used. Results are shown for one, two, and three spectra for the tensor-detection case in full-sky.
• Figure 4: Posterior distributions from pixel-based (gold), HL (red), mHL (blue), and cHL (green) likelihoods for $f_{\rm sky} = 40\%$. The columns correspond to signal-dominated, tensor-detection, and noise-dominated cases. In the first three rows, dashed (dotted) curves correspond to under- (over-) estimated noise bias. Vertical black lines mark the true parameter values ($\tau = 0.06$, $r=0.01$ and $r=0$). The bottom row provides a quantitative comparison of the solid-line posteriors above, together with the single-field HL (maroon). Crosses show MAP estimates, while shaded bars indicate 68% and 95% credible intervals.
• Figure 5: Comparison between pixel-based (gold), HL (red), mHL (blue), and cHL (green) likelihoods with $f_{\rm sky} = 40\%$ using only channels A and B. In this case, cHL and oHL are identical. The columns correspond to signal-dominated, tensor-detection, and noise-dominated cases. The black vertical lines represent the true values of the parameters ($\tau = 0.06$, $r=0.01$ and $r=0$). The bottom row provides a quantitative comparison of the posteriors above, together with the single-field HL (maroon). Crosses show MAP estimates, while shaded bars indicate 68% and 95% credible intervals.