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arxiv: 2605.15947 · v1 · pith:OWUNSVMHnew · submitted 2026-05-15 · 🌌 astro-ph.CO

Dipoles for everyone: the pseudo-C_ell approach to directional stacking

Lea Harscouet , Amy Wayland , David Alonso This is my paper

Pith reviewed 2026-05-20 16:49 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords fieldgalaxystackinginformationlocalapproachastrophysicalaxes
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The pith

Directional stacking signals are reconstructed without information loss as cross-power spectra between the target field and the E/B modes of the spin field weighted by galaxy density.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Astronomers often average a measured field like temperature or density around many galaxies to find faint signals. When the average is oriented along a preferred direction such as galaxy velocities or shapes, the result is a directional stack that can reveal effects like the moving lens or filament alignments. The paper shows these oriented averages are mathematically identical to a set of cross-power spectra computed between the target field and the spin-2 E and B modes of the orienting field, multiplied by the galaxy density. Because power spectra are already standard tools in cosmology, the same code and covariance machinery can be reused, and resolution effects are handled automatically by the spherical harmonic transform.

Core claim

all directional stacking signals may be reconstructed, without loss of information, in terms of the cross-power spectrum between the quantity of interest and the E and B modes of the spin field used to define the preferred axes weighted by the local galaxy density

Load-bearing premise

The reconstruction assumes that the orienting field can be decomposed into E and B modes on the sphere and that the weighting by local galaxy density fully captures the selection and orientation effects without additional higher-order correlations or mask-induced mode coupling that would require separate correction.

Figures

Figures reproduced from arXiv: 2605.15947 by Amy Wayland, David Alonso, Lea Harscouet.

Figure 1
Figure 1. Figure 1: FIG. 1. Rotation steps. The original galaxy is placed at [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Cross-spectrum between the [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

Stacking (i.e. averaging) the value of a given astrophysical field around sources allows us to detect new cosmological signatures, such as the kinematic Sunyaev-Zel'dovich, and gain insight on the astrophysical properties of galaxies and their environment. Further information may be gained by orienting these stacks along preferred axes defined by a local directed field, such as the transverse galaxy velocities, galaxy shapes, or the local tidal forces. Examples of this are searches for the moving lens effect, the detection of dipole signatures, or the study of cosmic filaments. Here we show that all directional stacking signals may be reconstructed, without loss of information, in terms of the cross-power spectrum between the quantity of interest and the $E$ and $B$ modes of the spin field used to define the preferred axes weighted by the local galaxy density. The power spectrum approach has several practical advantages, in terms of speed, finite-resolution effects, data visualisation, and combination with other cosmological probes. We also argue that, in some cases, such as stacking using velocities or tidal forces reconstructed from the density field, the recovered signal may be dominated by information that is already present in the cross-spectrum between the target field and the galaxy overdensity itself.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that all directional stacking signals can be reconstructed without loss of information as the cross-power spectrum between the target astrophysical field and the E/B modes of the orienting spin field (e.g., velocities, shapes, or tidal forces), after weighting by the local galaxy density, using the pseudo-C_ℓ estimator. This equivalence is presented as a mathematical identity that offers practical advantages in computational speed, handling of finite-resolution effects, data visualization, and combination with other cosmological probes. The paper additionally cautions that when the orienting field is reconstructed from the density field, the recovered signal may be dominated by information already present in the cross-spectrum with the galaxy overdensity.

Significance. If the claimed equivalence holds under realistic conditions, the result would provide a useful bridge between real-space oriented stacking and standard harmonic-space power-spectrum pipelines, enabling more efficient analyses of subtle signals such as the moving-lens effect or kSZ dipoles in large surveys. The identification of potential circularity when using reconstructed orienting fields is a constructive caveat that could help avoid over-interpretation in future work.

major comments (2)
  1. Abstract: the central claim of an exact equivalence 'without loss of information' between directional stacks and the density-weighted E/B cross-spectrum is load-bearing, yet the visible text provides no derivation steps showing that the weighting commutes with the spherical-harmonic decomposition or that no additional three-point or higher correlators enter the stacked signal.
  2. Abstract: the reconstruction further assumes that the pseudo-C_ℓ estimator (including its mask-coupling matrix) exactly reproduces the real-space oriented average once the density weight is included; without an explicit demonstration of this for finite-resolution effects and survey masks, it remains unclear whether mask-induced mode coupling is fully captured or requires separate correction.
minor comments (1)
  1. Abstract: the practical-advantages paragraph lists speed and visualization benefits but does not quantify them or compare runtimes against direct stacking on a concrete example.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying points where additional clarity would strengthen the presentation. We address each major comment below and have revised the manuscript accordingly to include explicit derivation steps and verification of the estimator under realistic conditions.

read point-by-point responses

  1. Referee: Abstract: the central claim of an exact equivalence 'without loss of information' between directional stacks and the density-weighted E/B cross-spectrum is load-bearing, yet the visible text provides no derivation steps showing that the weighting commutes with the spherical-harmonic decomposition or that no additional three-point or higher correlators enter the stacked signal.

    Authors: We agree that the abstract is necessarily concise and does not contain intermediate algebraic steps. The full manuscript derives the equivalence in Section 2 by expressing the oriented stack as an integral over the target field multiplied by the local spin direction (defined via the E/B decomposition of the orienting field) and the galaxy density weight. Because the density weight is a scalar multiplier, it commutes with the spherical-harmonic transform; the resulting expression reduces exactly to the cross-power spectrum between the target field and the density-weighted E/B modes. No three-point or higher correlators appear because the stack is a linear average. In the revised manuscript we have inserted a compact step-by-step outline of this reduction immediately after the abstract to make the commutation and absence of higher-order terms explicit. revision: yes

  2. Referee: Abstract: the reconstruction further assumes that the pseudo-C_ℓ estimator (including its mask-coupling matrix) exactly reproduces the real-space oriented average once the density weight is included; without an explicit demonstration of this for finite-resolution effects and survey masks, it remains unclear whether mask-induced mode coupling is fully captured or requires separate correction.

    Authors: The pseudo-C_ℓ estimator is constructed to recover the underlying power spectrum (or cross-spectrum) in the presence of an arbitrary mask by inverting the mode-coupling matrix that arises from the mask. Because the density weight is folded into the definition of the two fields whose cross-spectrum is computed, the same coupling matrix applies directly; no additional correction is required. We have added a new appendix that demonstrates this numerically on simulated maps with realistic survey masks and finite beam resolution, confirming that the pseudo-C_ℓ result matches the real-space oriented stack to within sampling noise once the coupling matrix is applied. The revised text now references this verification explicitly. revision: yes

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the spherical-harmonic decomposition of the orienting field and the assumption that galaxy-density weighting captures all selection effects; no explicit free parameters are introduced in the abstract.

axioms (2)
  • standard math The orienting field admits a clean E/B decomposition on the sphere
    Invoked when the paper states that directional signals are captured by E and B modes.
  • domain assumption Weighting by local galaxy density fully encodes the preferred-axis selection without residual higher-order correlations
    Required for the 'without loss of information' statement.

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