Matrix Inversion Delensing in CMB

• Matrix inversion delensing is a technique that recovers intrinsic CMB spectra by inverting the mode coupling induced by gravitational lensing.
• It leverages analytic inversion of coupling matrices via a truncated Neumann series for accurate, computationally efficient corrections in the weak lensing regime.
• The method effectively removes lensing B-mode contamination, improving constraints on primordial gravitational waves and tensor-to-scalar ratio estimates.

Matrix inversion delensing refers to a family of techniques that aim to reconstruct the intrinsic, unlensed Cosmic Microwave Background (CMB) temperature and polarization power spectra from their lensed counterparts by explicitly inverting, or perturbatively correcting, the lensing-induced mode coupling. These approaches treat the lensing operation as a linear or near-linear transformation in multipole space, allowing the construction and inversion of coupling matrices to recover the original primordial spectra. Matrix inversion delensing is analytically transparent, computationally tractable in the weak lensing regime, and enables the direct subtraction of the lensing B-mode contamination from observed data, facilitating improved constraints on primordial gravitational wave signals.

1. Theoretical Context and Lensing-Induced Mode Coupling

Gravitational lensing by large-scale structure remaps the unlensed CMB temperature and polarization fields by deflections along the sky, $$T(\hat n) \to \tilde T(\hat n) = T(\hat n+\bm\alpha(\hat n))$$, with $\bm\alpha(\hat n) = \nabla\psi(\hat n)$ for lensing potential $\psi$ (Pal et al., 2013). This remapping induces nontrivial mode coupling in harmonic (multipole) space: lensed multipoles $\tilde a_{\ell m}$ receive contributions from a broad range of unlensed $a_{\ell' m'}$, leading to off-diagonal covariance in the observed spectra. The effect is pronounced for polarization, where E-mode lensing creates B-mode “leakage.”

The lensed angular power spectra $\tilde C_\ell$ are linked to the unlensed $C_\ell$ via convolution with lensing kernels, formulated either by direct analysis of lensed correlation functions and their Hankel transforms (flat sky), or by rotation-matrix formalism (full sky).

2. The Matrix Inversion Delensing Formalism

Pal, Padmanabhan & Pal (2013) introduced a matrix equation relating the lensed and unlensed spectra for each observable $X\in\{TT, EE, BB, TE\}$: $$\tilde{C}_\ell^X = \sum_{\ell'} M^X_{\ell\ell'} C_{\ell'}^X$$ where $$M^X_{\ell\ell'} = \delta_{\ell\ell'} + \delta k^X_{\ell\ell'}$$ encodes the lensing-induced mode coupling (Pal et al., 2013). The core strategy is to invert this system: $$C_\ell^X = (M^X)^{-1}_{\ell\ell'} \tilde{C}_{\ell'}^X$$ In practice, since $\delta k^X$ is small and sharply peaked around the diagonal (i.e., lensing is weak), $M^X$ is well-conditioned and the inverse can be realized efficiently via the Neumann series: $$(M^X)^{-1} = I - \delta k^X + (\delta k^X)^2 - \cdots$$ Second-order truncation is typically sufficient for percent-level accuracy at $\ell\lesssim 1200$.

The kernels $\delta k^X_{\ell\ell'}$ are computed from integrals over Bessel functions (flat sky) or rotation matrices (full sky), weighted by powers of the lensing potential power spectrum $C_L^{\psi\psi}$, with full formulae for temperature, $E$/$B$ combinations, and $TE$ correlation in both coordinate systems (Pal et al., 2013).

3. Numerical Construction, Validation, and Performance

Implementation consists of discretizing multipole space (e.g., $\ell=2$ to $2000$), evaluating the kernel matrices $\delta k^X$ by quadrature, and applying either direct inversion or truncated Neumann expansion to obtain $(M^X)^{-1}$. The delensed spectrum is then computed as $(M^X)^{-1} \tilde{C}^X$.

Pal et al. calibrated this pipeline against outputs from CAMB with WMAP 7-year and 9-year best-fit parameters for $\Lambda$CDM+TENS, confirming that, for $\ell \lesssim 1000$, the reconstructed unlensed spectra matched input to $10^{-4}$–$10^{-3}$ relative discrepancy (Pal et al., 2013). At higher $\ell$ (e.g., $\ell\sim1500$), errors increase to $\sim 0.3$–$0.9$\%. Application to WMAP 9-year unbinned $TT$ data revealed that the relative lensing correction $(\tilde C^{TT} - C^{TT})/\tilde C^{TT}$ is $2$–$3$\% for $\ell\lesssim800$.

Neglected higher-order corrections, such as omitted $(\delta k)^3$ terms, induce errors $\lesssim 10^{-3}$ at $\ell\lesssim1200$ and up to $10^{-2}$ by $\ell\sim1600$. Nonlinear lensing and noise are not included in the baseline method; the approach assumes perfect knowledge of the lensing potential and no instrumental error.

4. Relation to Inverse-Lensing and LenseFlow Approaches

Matrix inversion delensing is distinct from “inverse-lensing” techniques that seek to invert the lensing remapping operator at the map or pixel level by solving non-linear equations in real space, as developed in more recent work (Chen et al., 26 Feb 2025, Millea et al., 2017). The matrix-inversion method operates directly at the two-point (power spectrum) level, reformulating lensing as a near-linear operation in harmonic space.

Inverse-lensing approaches—such as the Newton–Raphson inversion and ODE-based LenseFlow—execute iterative map-based delensing, reconstructing the unlensed field by numerically solving for the inverse deflection field or by integrating flow equations backward in time (Millea et al., 2017, Chen et al., 26 Feb 2025). These methods are complementary: matrix inversion is analytic and optimal for full-sky, idealized spectra, whereas pixel-based inversion is more flexible in the face of incomplete sky coverage, spatially varying noise, and non-Gaussianities.

5. Application to CMB Delensing and Primordial B-Modes

The primary application of matrix inversion delensing is to remove lensing-induced power from observed $B$-modes, enabling the isolation of any intrinsic, primordial B-mode signal. This facilitates improved estimation of the tensor-to-scalar ratio $r$, a target of current and next-generation CMB experiments.

Pal et al. demonstrated that, assuming zero intrinsic $B$-mode power, their method recovers BB power consistent with that produced by pure $E$-mode lensing, peaking at $\sim 0.1~\mu{\rm K}^2$ at $\ell\sim1000$ (Pal et al., 2013). The delensed $TT$ and $EE$ spectra are corrected at the $2$–$3$\% level for $\ell\lesssim 800$, and the TE correction is a few percent at $\ell\sim1000$.

A plausible implication is that extensions of matrix inversion delensing could be incorporated into joint Bayesian or iterative delensing pipelines, leveraging improved lensing reconstructions from quadratic estimators and external tracers.

6. Limitations and Prospects

Key limitations of the original matrix inversion approach include the neglect of higher-order lensing corrections, the assumption of perfect knowledge of $C_L^{\psi\psi}$, and non-inclusion of noise and beam effects. These constraints restrict the method’s direct applicability to idealized scenarios; for analysis pipelines applied to real data, additional steps to propagate reconstruction uncertainties and observational effects are necessary.

The method’s analytic tractability renders it a valuable tool for validating map-based delensing codes, benchmarking mode-coupling kernels, and understanding lensing corrections in parameter estimation. The outlook, as suggested by (Pal et al., 2013), is that once accurate lensing reconstruction is available—e.g., from quadratic estimators applied to $E$/$B$ maps—the same matrix-based framework can be used to isolate residual primordial $B$-mode power with high precision.

7. Comparative Summary of Delensing Methods

Approach	Key Operation	Data Domain
Matrix Inversion	Invert mode-coupling matrix	Power Spectra
Inverse-Lensing	Map-based positional remap	Real space
LenseFlow	ODE integration (forward/inverse)	Pixelized maps

Matrix inversion delensing provides computationally efficient, analytic correction of lensing distortions at the two-point function level, complementing more general but computationally intensive map-based approaches suited for high-fidelity ground and space-based experiments.