Dipoles for everyone: the pseudo-$C_\ell$ approach to directional stacking

Technical Summary: Dipoles for everyone: the pseudo-Cℓ approach to directional stacking

Problem Statement
Directional stacking (or oriented stacking) is a technique used in cosmology to detect faint signals by averaging astrophysical fields around sources, aligned along preferred axes defined by local directed fields (e.g., transverse galaxy velocities, galaxy shapes, or tidal forces). While effective for extracting signatures like the kinematic Sunyaev-Zel'dovich (kSZ) effect, moving lens effects, or dipole/quadrupole patterns in the Cosmic Microwave Background (CMB) and large-scale structure (LSS), traditional real-space stacking has limitations. Real-space methods can be computationally intensive, sensitive to finite-resolution effects, and difficult to combine analytically with other cosmological probes. Furthermore, the mathematical connection between real-space directional stacks and the underlying cosmological power spectra has not been fully formalized for general spin-$s$ fields, particularly regarding the separation of parity-even and parity-odd components.

Methodology
The authors develop a formalism that maps directional stacking estimators directly to the cross-power spectrum between the target field and a constructed "spin-s momentum" field.

1. Formalism Derivation:

   • The paper defines a directional stacking estimator where a target field $X(\hat{n})$ (e.g., CMB temperature) is rotated such that a source at $\hat{n}_i$ moves to the North Pole, and a local spin-$s$ field $\tilde{v}_i$ (defining the preferred direction) aligns with the local $x$-axis.
   • The rotated field is weighted by $e^{is\phi'}$ (where $\phi'$ is the azimuthal angle) and binned radially.
   • By expanding the target field in spherical harmonics and utilizing the properties of Wigner D-matrices under rotation, the authors derive that the stacking estimator $\bar{X}(\theta_d)$ is mathematically equivalent to a weighted sum of the cross-power spectrum $C_{\pi X}^\ell$ between the target field and a complex momentum field $s\pi$.
   • The momentum field $s\pi(\hat{n})$ is constructed by weighting each source by the value of the spin-$s$ field $\tilde{v}$ defining the direction.
   • Crucially, the cross-spectrum is decomposed into E-mode and B-mode components ($C_{\pi X}^E$ and $C_{\pi X}^B$). The real part of the stacking estimator (cosine-weighted) corresponds to the E-mode, while the imaginary part (sine-weighted) corresponds to the B-mode.

2. Numerical Validation:

   • The authors use the CoLoRe code to generate mock galaxy catalogs and maps of lensing convergence and the Integrated Sachs-Wolfe (ISW) effect.
   • They reconstruct transverse velocities and tidal fields from the mock data to define the preferred directions.
   • They compare the results of the harmonic-space estimator (using the NaMaster code and pseudo-$C_\ell$ techniques) against traditional real-space directional stacks.

3. Real Data Application:

   • Velocity-Oriented Dipoles ($s=1$): Using BOSS CMASS galaxies and Planck data, the authors measure the dipole signature of galaxy overdensity, lensing convergence, and CMB temperature aligned with reconstructed transverse velocities.
   • Filament-Oriented Quadrupoles ($s=2$): Using the same galaxy sample and the Planck Compton-$y$ map, they measure the quadrupole signature of the thermal Sunyaev-Zel'dovich (tSZ) effect aligned with the reconstructed transverse tidal field.

4. Information Content Analysis:

   • The paper investigates whether directional stacking using fields reconstructed from the galaxy overdensity (e.g., velocities derived from density) contains information beyond the simple cross-correlation between the target field and the galaxy overdensity.
   • They decompose the momentum field into a linear term (proportional to density) and a non-linear term (involving the bispectrum). By evaluating the reconstructed velocities at random positions versus real galaxy positions, they isolate the bispectrum contribution.

Key Results

• Equivalence: The authors demonstrate that directional stacking signals can be reconstructed without loss of information from the cross-power spectrum between the target field and the spin-weighted momentum field. The real-space stack is accurately reproduced from the harmonic-space measurements using the derived weighting function $W_\ell^s(\theta_d)$.
• E/B Mode Separation: The formalism naturally separates the signal into E-modes (parity-even) and B-modes (parity-odd). In the validation tests, E-modes were detected at high significance, while B-modes were consistent with zero, as expected for fields derived from scalar potentials.
• Real Data Measurements:
  • Dipoles: High-significance detections (SNR $\sim$ 35 for galaxy overdensity, SNR $\sim$ 6 for lensing convergence) were found for the dipole signal aligned with transverse velocities. The CMB temperature dipole was not significantly detected (SNR $\sim$ 1), likely due to noise and uncorrelated primary anisotropies, though the signal was qualitatively similar to previous real-space results.
  • Quadrupoles: A quadrupole signal in the tSZ effect aligned with tidal forces was detected with SNR $\sim$ 10, higher than previous reports, attributed to the larger sky area used.
• Information Dominance: In cases where the directing field (velocity or tidal) is reconstructed from the galaxy overdensity itself, the authors find that the signal is dominated by the cross-spectrum between the target field and the galaxy overdensity (the linear term). The bispectrum-like contribution (non-linear term) is subdominant. This implies that for such specific cases, the directional stack does not add significant new information beyond the standard power spectrum, unless the bispectrum term can be isolated (e.g., by subtracting the random-position correlation).

Significance and Claims
The paper claims to provide a general framework for expressing directional stacking as a power spectrum measurement, offering several practical advantages over real-space methods:

• Efficiency: Power spectrum estimators are typically faster than real-space correlations.
• Statistical Properties: Power spectrum uncertainties are largely uncorrelated, facilitating easier goodness-of-fit tests and analytical error estimation.
• Resolution: Catalog-based estimators allow measurements on arbitrarily small scales, avoiding finite-resolution effects inherent in real-space binning.
• Integration: The approach allows directional probes to be seamlessly combined with other cosmological probes in joint analyses using consistent model choices and scale cuts.

The authors emphasize that while the method recovers the same information as real-space stacking, its harmonic-space formulation makes it a more robust tool for future high-precision cosmology, particularly when combining with other LSS probes. They also note that if the directing field is independent of the galaxy overdensity (e.g., intrinsic galaxy shapes or proper motions), the directional stack would contain complementary information not present in the standard galaxy-target cross-spectrum.