Skew-spectra: a generalization to spin-$s$

1. Problem Statement

The analysis of modern cosmological datasets (e.g., from DESI, LSST, Euclid, Roman) requires extracting non-Gaussian information to constrain cosmological parameters beyond what is possible with standard two-point statistics (power spectra). While higher-order statistics like the bispectrum (three-point correlation function) contain this information, they face significant challenges:

• Computational Cost: Estimating the full bispectrum is computationally expensive, scaling poorly with data volume.
• Theoretical Modeling: Modeling the bispectrum in the non-linear regime is difficult, often requiring complex emulators or simulations.
• Weak Lensing Specifics: Many weak lensing higher-order statistics (e.g., peak counts, Minkowski functionals) rely on mass-mapping (inverting shear to convergence). This process is prone to spurious signals due to survey masks (e.g., Kaiser-Squires inversion artifacts) and requires inpainting, introducing systematics and uncertainty.
• Spin-s Complexity: Existing skew-spectrum methods were largely limited to scalar fields. Cosmological observables like CMB polarization and weak lensing shear are spin-s fields (specifically spin-2), requiring a generalized formalism.

2. Methodology

The authors propose a generalization of skew-spectra to arbitrary spin-s fields. The skew-spectrum is defined as the cross-power spectrum between an original field and the square of a field (or a product of two fields).

Key Theoretical Steps:

• Spin-s Formalism: The authors utilize the decomposition of tensor fields on the sphere into irreducible representations of $SO(2)$, specifically using Spin-Weighted Spherical Harmonics (SWSH). Fields are decomposed into E-modes (parity even) and B-modes (parity odd).
• Construction of Skew-Spectra:
  • They consider the product of two spin-s fields, $a \cdot b$, which results in a new field with spin $s_a + s_b$.
  • They derive the harmonic coefficients ($E$ and $B$ modes) of this "spin-squared" map in terms of the original fields' coefficients.
  • They calculate the cross-power spectrum between the original field and the squared field: $\langle F^a_{\ell m} (G^{b \cdot c}_{\ell m})^* \rangle$.
• Relation to Bispectrum: The authors mathematically demonstrate that these skew-spectra are linear combinations of the angular bispectrum ($B_{\ell_1 \ell_2 \ell_3}$). The relationship involves sums over Wigner-3j symbols, which enforce selection rules (triangular conditions and parity conservation).
• Handling Parity: The formalism distinguishes between parity-even and parity-odd configurations. For example, the cross-spectrum of an E-mode with a squared E-mode ($E \times E^2$) receives contributions only from specific parity combinations of the bispectrum.
• Alternative "E/B-Squared" Approach: In Appendix B, they discuss an alternative method where one squares the E and B modes directly (scalar quantities) rather than the spin-s fields. While this isolates single bispectrum types more cleanly, the authors note it suffers from non-locality issues (leakage) when squared, making mask correction difficult.

3. Key Contributions

• Generalization to Spin-s: The primary contribution is the derivation of skew-spectra for arbitrary spin-s fields, enabling the application of this efficient statistic to CMB polarization and weak lensing shear (spin-2).
• Avoidance of Mass-Mapping: By working directly with the shear field ($\gamma$) rather than converting to convergence ($\kappa$), the method bypasses the need for mass-mapping algorithms (like Kaiser-Squires), thereby avoiding associated inversion artifacts and systematics.
• Infrastructure Reuse: The method leverages the highly developed, efficient infrastructure for calculating power spectra (e.g., Pseudo-$C_\ell$ estimators) to estimate bispectrum information. This allows for fast computation and robust handling of complex survey masks.
• Analytical Framework: The paper provides the explicit mathematical relations (Eqs. 25–28) connecting skew-spectra to the underlying angular bispectrum for various combinations of E and B modes.

4. Results

The authors applied their formalism to Weak Lensing within the $\Lambda$CDM framework:

• Statistics Defined: They defined specific skew-spectra involving the shear field ($\gamma$) and galaxy clustering ($\delta_g$), such as $E_\gamma \times E_{\gamma^2}$, $E_\gamma \times E_{\gamma^* \gamma}$, and cross-correlations with galaxy density ($g$).
• Theoretical Predictions: Using the BiHalofit fitting function for the 3D matter bispectrum and the Limber approximation, they generated theoretical predictions for these statistics.
• Configuration Dependence:
  • The statistics show strong dependence on triangle configurations in harmonic space. Squeezed configurations ($\ell_1 \ll \ell_2 \approx \ell$) generally yield the largest amplitudes, followed by folded configurations, with equilateral shapes being the smallest.
  • The Wigner-3j symbols act as "compression factors," suppressing or enhancing power depending on the $\ell$ configuration and the filter limits ($\ell_{min}, \ell_{max}$).
• Cosmological Sensitivity: The authors varied key cosmological parameters ($\sigma_8, \Omega_m, S_8, n_s$).
  • The skew-spectra showed clear sensitivity to these parameters.
  • Notably, the $E_\gamma \times E_{\gamma^* \gamma}$ statistic exhibited larger relative differences compared to the fiducial model than the $E_\gamma \times E_{\gamma^2}$ statistic, particularly at higher multipoles.
  • The dependence on cosmology appears smooth, suggesting these statistics are suitable for interpolation in inference pipelines.

5. Significance

• Efficiency: This approach offers a computationally efficient alternative to full bispectrum estimation, scaling similarly to power spectra ($O(\ell_{max}^3)$ vs $O(\ell_{max}^6)$ for optimal estimators) while capturing non-Gaussian information.
• Robustness: By avoiding mass-mapping, it provides a more robust path for analyzing weak lensing data from future surveys (LSST, Euclid) where systematics and masks are critical limiting factors.
• Future Pipeline Integration: The statistics are designed to be compatible with existing power spectrum analysis tools (like Pseudo-$C_\ell$), facilitating their immediate integration into cosmological analysis pipelines.
• Broader Applicability: While demonstrated on weak lensing, the formalism is general enough to be applied to CMB polarization and other spin-s fields, potentially improving constraints on primordial non-Gaussianity and neutrino masses.

In conclusion, the paper establishes a rigorous, efficient, and robust framework for extracting non-Gaussian information from spin-s cosmological fields, bridging the gap between the theoretical richness of the bispectrum and the practical efficiency of power spectrum analysis.