Polarization-Only CMB Lensing Reconstruction

• Polarization-only CMB lensing reconstruction infers the deflection field exclusively from polarization (Q/U or E/B) data, converting primordial E modes into lensing-induced B modes.
• Quadratic estimators based on EB mode coupling serve as the standard baseline, while iterative, Bayesian, and machine-learning methods exploit non-Gaussian signals for enhanced precision.
• Empirical demonstrations indicate that these techniques achieve approximately 10% precision in binned multipoles and effectively mitigate foreground and systematic biases in deep surveys.

Polarization-only CMB lensing reconstruction is the inference of the CMB lensing potential $\phi$ or convergence $\kappa$ from polarization data alone, typically the observed Stokes $Q/U$ maps or their $E/B$ decomposition, without using temperature. Its central observable is the lensing-induced mode coupling of primordial $E$ modes into $B$ modes, which makes the $EB$ covariance the dominant internal lensing channel in sufficiently deep polarization surveys. In the regime reached by modern data, and especially once polarization noise approaches $\sim 5$--$6\,\mu$K-arcmin, this problem is no longer well described as exclusively a quadratic-estimator problem: quadratic methods remain the standard baseline, but iterative, Bayesian, joint-MAP, and learned reconstructions become relevant because they can exploit the full non-Gaussian structure of lensed polarization maps (Millea et al., 2020).

1. Physical basis of the polarization signal

Weak gravitational lensing remaps the primordial CMB by the deflection field $d=\nabla\phi$. For a sky direction $\kappa$0 and any field $\kappa$1 or, equivalently, $\kappa$2,

$\kappa$3

The lensing convergence is

$\kappa$4

with harmonic-space relation

$\kappa$5

In the flat sky, the shear components satisfy

$\kappa$6

so polarization-only reconstruction can be formulated in terms of $\kappa$7, $\kappa$8, or shear (Millea et al., 2020).

The distinctive feature of polarization is that, to first order in lensing, the large unlensed $\kappa$9 field is convolved with $Q/U$0 to produce lensing $Q/U$1 modes. A common flat-sky first-order expression is

$Q/U$2

with

$Q/U$3

This induced $Q/U$4 covariance is the most informative polarization-only lensing observable in the Born approximation (Wu et al., 2019).

The practical consequence is that lensing generates nearly all of the small-scale $Q/U$5-mode power in the absence of sizeable primordial $Q/U$6, making polarization-only reconstruction increasingly powerful as experiments resolve lensing $Q/U$7 modes. Under PRISM-like specifications, simulated $Q/U$8 reconstructions were already shown to trace the underlying $Q/U$9-body lensing field down to the smallest angular scales allowed by the setup, with about $E/B$0 precision in binned multipoles over $E/B$1 (Antolini et al., 2013).

2. Quadratic-estimator formalism

The standard polarization-only reconstruction framework uses quadratic estimators built from filtered $E/B$2 and $E/B$3 maps. In flat sky, the dominant estimator is

$E/B$4

with a weight derived from the lensing response and the total filtered power. A common form is

$E/B$5

and the normalization $E/B$6 is chosen so that the estimator is unbiased (Millea et al., 2020).

In survey pipelines this structure is augmented by response calibration, mean-field subtraction, and bias corrections. The SPTpol $E/B$7 analysis formed individual $E/B$8 and $E/B$9 estimators, subtracted a simulation-derived mean field, normalized with analytic and Monte Carlo responses, and estimated the disconnected $E$0 with a realization-dependent procedure while taking $E$1 from simulations (Wu et al., 2019). In that setting, the polarization-only map is an inverse-noise-weighted combination of $E$2 and $E$3, but the $E$4 estimator dominates the sensitivity because unlensed $E$5 power is very small, which keeps the disconnected noise low (Wu et al., 2019).

A related low-$E$6 formulation exists directly in real space. There, the polarization-only reconstruction can be written as local convolutions with compact kernels: an $E$7 convergence estimator and $E$8 or $E$9 shear estimators. The $B$0 shear estimator is particularly important because, in the squeezed limit, $B$1 carries only shear at leading order. The real-space kernels are compact, with most support within $B$2 degrees, which makes them useful on masked and nonuniform maps (Prince et al., 2017).

A recurrent misunderstanding is that the quadratic estimator remains essentially optimal once polarization dominates. The low-noise literature instead shows the opposite: at polarization noise $B$3--$B$4K-arcmin, the $B$5 quadratic estimator is no longer minimum-variance because it does not self-consistently account for lensing-induced non-Gaussianity and the information in higher moments (Millea et al., 2020).

3. Beyond quadratic reconstruction

The most explicit beyond-QE realization on data is the Bayesian map-level approach applied to deep SPTpol polarization maps. Its forward model acts directly on masked $B$6 data: $B$7 with unlensed polarization fields $B$8, beam modes $B$9, polarization angle $EB$0, transfer function $EB$1, leakage templates $EB$2, and noise covariance $EB$3. The joint posterior samples $EB$4, cosmological amplitudes, and nuisance parameters directly; the implementation uses reparameterization, Gibbs steps for scalar parameters, exact conditional Gaussian solves for $EB$5, and Hamiltonian Monte Carlo for $EB$6 (Millea et al., 2020).

A closely related generalization is the joint maximum-a-posteriori framework for multiple line-of-sight distortions. In that formulation, polarization-only data are modeled with a forward operator that includes lensing, cosmic birefringence, and patchy screening simultaneously. The gradients are evaluated on delensed, derotated, and descreened maps, and the fields are updated by quasi-Newton optimization. In simulations this improves the cross-correlation coefficient of the reconstructed $EB$7 with the input $EB$8 relative to the quadratic estimator while explicitly reducing mutual contamination between $EB$9, $\sim 5$0, $\sim 5$1, and curl-like modes (Darwish, 5 Mar 2025).

Iterative and delensing-based maximum-likelihood ideas also appear in more specialized settings. For stacked cluster reconstruction, improved quadratic estimators $\sim 5$2 and $\sim 5$3 are applied to delensed maps and iterated toward the maximum-likelihood solution. In those simulations, the polarization-based $\sim 5$4 estimator became competitive with temperature-based reconstruction only if the detector noise for measuring polarization anisotropies is controlled under $\sim 5$5 microK (Yoo et al., 2010).

Machine-learning approaches form a separate beyond-QE branch. "Reconstructing Cosmic Polarization Rotation with ResUNet-CMB" reconstructs $\sim 5$6, $\sim 5$7, $\sim 5$8, and primordial $\sim 5$9 simultaneously from polarization-only $6\,\mu$0 inputs and yields reconstruction variance lower than the standard quadratic estimator for anisotropic rotation while preserving robust $6\,\mu$1 performance (Guzman et al., 2021). "Lensing reconstruction from the cosmic microwave background polarization with machine learning" trains RDLFUnet on lensed $6\,\mu$2 patches and reports lower reconstruction noise than the polarization-only minimum-variance QE at noise below about $6\,\mu$3K-arcmin; at $6\,\mu$4K-arcmin the cumulative SNR is $6\,\mu$5 for RDLFUnet, compared with $6\,\mu$6 for QE (Yan et al., 2023).

4. Empirical demonstrations

Simulation studies established the feasibility of polarization-only reconstruction before deep observational demonstrations. Using Born-approximated ray tracing through Millennium Simulation structures, reconstructed $6\,\mu$7 shear and convergence maps were shown to agree with $6\,\mu$8CDM expectations across the accessible multipole range, with about $6\,\mu$9 precision in $d=\nabla\phi$0 bins for $d=\nabla\phi$1 under PRISM-like instrumental specifications (Antolini et al., 2013).

An early observational milestone was the POLARBEAR cross-correlation with the Herschel CIB. Using polarization-only quadratic reconstruction from two $d=\nabla\phi$2 fields, the analysis obtained $d=\nabla\phi$3 evidence for gravitational lensing of CMB polarization from the $d=\nabla\phi$4 cross-spectrum and $d=\nabla\phi$5 evidence for a lensing $d=\nabla\phi$6-mode signal from the $d=\nabla\phi$7 channel alone. The cross-correlation setting was important because it strongly suppresses additive reconstruction biases uncorrelated with the external tracer (Collaboration et al., 2013).

The SPTpol $d=\nabla\phi$8 lensing measurement established polarization-only reconstruction as a precision internal observable. Restricting to polarization data, it reported

$d=\nabla\phi$9

which the paper described as the most precise polarization-only lensing amplitude constraint to date, with $\kappa$00 statistical significance. In that analysis the $\kappa$01 estimator dominates the polarization sensitivity on large scales, and the polarization-only map has lower reconstruction noise than temperature for $\kappa$02 (Wu et al., 2019).

The deepest demonstration of beyond-QE performance on real data is the Bayesian SPTpol analysis of a $\kappa$03 patch with polarization noise as low as $\kappa$04K-arcmin in the deepest regions. It reported

$\kappa$05

for $\kappa$06, improving the error bar by $\kappa$07 relative to the QE on the same maps, and by $\kappa$08 after marginalizing over the power-spectrum-only contribution through the auxiliary parameter $\kappa$09. In the joint two-parameter fit it found

$\kappa$10

showing that most of the $\kappa$11 information comes from non-Gaussian lensing rather than from the lensed polarization power spectra alone (Millea et al., 2020).

5. Foregrounds, systematics, and bias control

A common misconception is that polarization-only reconstruction is foreground-free. The published analyses support a narrower statement: polarization greatly reduces the dominant temperature foreground biases, but it does not eliminate polarized contaminants. Diffuse Galactic dust is the clearest counterexample. In a three-channel balloon-borne case study centered at $\kappa$12, polarized dust with polarization fractions of a few percent already produced a significant bias in the $\kappa$13-reconstructed convergence spectrum at $\kappa$14. In that setup, template cleaning recovered an unbiased convergence spectrum in all tested cases, whereas parametric component separation required sufficiently accurate knowledge of the dust spectral index to avoid breakdown in low-contrast regimes (Fantaye et al., 2012).

For polarized extragalactic point sources, the mitigation literature is more favorable. In polarization-based quadratic reconstruction for Simons Observatory- and CMB-S4-like experiments, source-hardening and shear-only constructions can suppress point-source-induced biases strongly; for a CMB-S4-like experiment, an optimal linear combination of point-source-hardened estimators reduces the lensing power-spectrum bias by up to two orders of magnitude at a $\kappa$15 noise cost relative to the global minimum-variance estimator (Sailer et al., 2022). In the Bayesian setting, realistic simulations of polarized radio and infrared sources indicate that a CMB-S4-like optimal polarization analysis is insensitive to the expected masked foreground level as long as an accurate foreground power spectrum is included in the covariance. The same study found that keeping the polarized radio foreground power within $\kappa$16 in amplitude yields $\kappa$17 on $\kappa$18 (Qu et al., 2024).

Instrumental systematics also remain relevant. In the SPTpol $\kappa$19 polarization-only analysis, the dominant systematic was polarization calibration uncertainty, $\kappa$20, which contributed $\kappa$21; beam uncertainty contributed $\kappa$22, while $\kappa$23 leakage and global angle rotation were negligible (Wu et al., 2019). The Bayesian SPTpol analysis absorbed polarization calibration, global angle, $\kappa$24 leakage, and beam modes directly into the joint posterior and reduced the systematic uncertainty on $\kappa$25 from polarization calibration from nearly half the statistical error to effectively zero (Millea et al., 2020).

Patchwork reconstructions introduce a distinct class of systematics because long-wavelength modes are degraded independently in each tile. In the patchwork framework, the $\kappa$26-mode power spectrum is biased by baseline uncertainty and $\kappa$27 noise, but the lensing-potential reconstruction remains unbiased if the large-scale $\kappa$28 modes below the blowup scale are removed before applying the estimator (Namikawa et al., 2014). A later systematic study of full-sky patchworks built from $\kappa$29 subpatches found that, at systematic error levels expected in the near future, the lensing potential can still be reconstructed accurately on scales larger than the subpatch size and the subsequent delensing efficiency is not severely degraded (Nagata et al., 2024).

An additional polarization-specific issue is anisotropic cosmic birefringence. Although the usual $\kappa$30 lensing estimator is orthogonal to birefringence at linear order, anisotropic rotation produces an $\kappa$31-like bias in the reconstructed lensing power spectrum. For a CMB-S4-like experiment, a scale-invariant rotation field with standard deviation $\kappa$32 degrees was shown to suppress the small-scale reconstructed lensing power at a level comparable to the effect of $\kappa$33 at $\kappa$34, making rotation an explicit degeneracy source for future polarization-dominated lensing analyses (Cai et al., 2024).

6. Scientific uses and future directions

Polarization-only reconstruction is already linked to two major downstream applications: cross-correlation cosmology and delensing. In the patchwork setting, the reconstructed lensing potential was forecast to enable a $\kappa$35--$\kappa$36 cross-correlation with satellite temperature maps at approximately $\kappa$37 and an $\kappa$38--$\kappa$39 cross-correlation with LiteBIRD-like polarization maps at approximately $\kappa$40 over $\kappa$41. The same reconstructed $\kappa$42 map reduces LiteBIRD lensing $\kappa$43 power by about $\kappa$44 at $\kappa$45 for $\kappa$46K-arcmin and $\kappa$47 (Namikawa et al., 2014).

The parameter-inference role of polarization-only reconstruction is also expanding. The Bayesian SPTpol framework jointly constrained $\kappa$48 and $\kappa$49 while exactly accounting for the correlation between the reconstructed lensing field and the lensed or delensed CMB spectra, and it was presented as immediately extensible to cosmological parameters such as neutrino mass, $\kappa$50, and $\kappa$51 (Millea et al., 2020). In more geometric survey settings, real-space $\kappa$52 shear reconstruction remains attractive because it is local and naturally handles galactic cuts, source holes, and depth variations; in the CMB-S4 regime it is expected to surpass temperature-based reconstruction, with $\kappa$53 shear becoming the leading estimator (Prince et al., 2017).

Specialized applications continue to depend on the same noise threshold. For stacked cluster-mass cross-correlations, polarization-only $\kappa$54 reconstruction becomes competitive with iterative temperature reconstruction only when the polarization detector noise is controlled under $\kappa$55 microK, but it remains attractive because it is robust against the cluster-associated temperature secondaries that complicate $\kappa$56-based methods (Yoo et al., 2010).

The projected gains for future deep surveys are substantial. For SPT-3G, Simons Observatory, and CMB-S4-like polarization maps, the SPTpol Bayesian study forecast improvements reaching approximately $\kappa$57 tighter constraints on $\kappa$58 and up to approximately $\kappa$59 lower effective lensing reconstruction noise than QE (Millea et al., 2020). Joint reconstruction frameworks that solve simultaneously for lensing, birefringence, and patchy screening extend the same logic to multi-distortion analyses and are aimed at robust secondary-anisotropy science, cross-correlations with large-scale structure, and more sensitive searches for primordial $\kappa$60 modes (Darwish, 5 Mar 2025).