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arxiv: 2506.18676 · v1 · submitted 2025-06-23 · 🌌 astro-ph.CO

FuGa3D: Fast full-sky analysis of Galaxy catalogs in 3D

Elina Keih\"anen , Jani Haapala , Valtteri Lindholm , Martin Reinecke , Susan Rissanen , Jussi Valiviita , Akke Viitanen This is my paper

Pith reviewed 2026-05-19 08:17 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords galaxy clusteringcosmic shearcorrelation functionpower spectrumredshift spacefull-sky analysislarge-scale structure
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The pith

FuGa3D defines galaxy positions via two redshifts and angular separation to compute correlations and spectra without assuming a cosmology upfront.

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

The paper presents FuGa3D as a method to compute correlation functions and power spectra for galaxy clustering and cosmic shear from large catalogs. It introduces the redshift-space correlation function in the space of two redshifts and one angular separation, which fully specifies relative galaxy positions under the assumption of no preferred sky direction. This construction allows straightforward derivation of real-space correlation functions, multipoles, and the redshift-space power spectrum as its harmonic counterpart. The approach is demonstrated on mock catalogs and achieves practical run times for simulations containing tens of millions of galaxies.

Core claim

FuGa3D constructs the redshift-space correlation function (RCF) directly in the parameter space of two redshifts and angular separation. Under the assumption of isotropy, these parameters determine relative galaxy positions independently of any cosmological model. From the RCF one can derive the real-space clustering correlation function and its multipoles, while the redshift-space power spectrum follows as the harmonic counterpart and can be evaluated from the discrete galaxy coordinates.

What carries the argument

The redshift-space correlation function (RCF), which stores pair counts in the three-dimensional space of two redshifts and angular separation and serves as the common starting point for all derived correlation and power-spectrum statistics.

If this is right

  • Real-space clustering correlation functions and multipoles follow directly once the RCF is available.
  • The redshift-space power spectrum can be obtained efficiently as the harmonic transform of the RCF using the discrete galaxy positions.
  • The same RCF construction applies to both galaxy clustering and cosmic-shear observables.
  • Processing a 46-million-galaxy catalog requires only tens of node-minutes for clustering or a few node-hours when shear is included.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The cosmology-independent RCF step could simplify consistency checks across different assumed models in the same data set.
  • The method may extend naturally to additional large-scale-structure statistics that can be expressed as functions of redshift pairs and angles.
  • Because the intermediate representation separates geometry from cosmology, it offers a route to test isotropy assumptions directly from the catalog.

Load-bearing premise

There is no preferred direction in the sky, so two redshifts plus one angular separation fully specify the relative position of any galaxy pair independently of the cosmological model.

What would settle it

Run FuGa3D and a standard correlation-code pipeline on the same mock catalog and compare the resulting real-space correlation function and its multipoles at fixed separation scales.

Figures

Figures reproduced from arXiv: 2506.18676 by Akke Viitanen, Elina Keih\"anen, Jani Haapala, Jussi Valiviita, Martin Reinecke, Susan Rissanen, Valtteri Lindholm.

Figure 1
Figure 1. Figure 1: Partially filled base pixels are discarded from analysis, to en￾sure uniform coverage over the full survey area. Empty base pixels are marked by red crosses. Pixels with an empty pixel as neighbor, either directly (blue) or diagonally (purple), are interpreted as being on the sur￾vey boundary. Both the empty pixels and their neighbors are discarded from further analysis. represents an uncorrelated distribu… view at source ↗
Figure 2
Figure 2. Figure 2: Real-space 2PCF ξ(r) from MICE simulation, for different as￾sumed cosmologies and redshift ranges. The correlation functions have been constructed from the same input RCF. elements of the input RCF into ξ(r, µ) of ξ(r) with rr weight￾ing, and, when requested, project onto Legendre polynomials to extract multipoles. Because of the discretization, where galaxies are assigned to the centers of their respectiv… view at source ↗
Figure 3
Figure 3. Figure 3: 2PCF monopole from FuGa3D, and from exact pair counting (Corrfunc). The input catalog is the full MICE simulation in z = 1.0 − 1.2. FuGa3D results are shown for three resolution settings (see main text for details). The lower panel shows the difference with respect to the Corrfunc result. 20 0 20 2 r 2 Quadrupole 2 and 2 Fuga Nside=512 Fuga Nside=1024 Fuga Nside=2048 Corrfunc 0 50 100 150 200 r [Mpc] 0.05 … view at source ↗
Figure 4
Figure 4. Figure 4: Same as [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows a similar comparison for the EE shear spec￾trum. Here, the uncorrected pseudo-spectra cannot be compared directly, since there is no common normalization. Increasing the resolution further beyond the point where there is only one galaxy per pixel, scales the spectrum down as inversely propor￾tional to the pixel size squared, ∝ 1/N 4 side. To bring the spectra to a common normalization, we scale each … view at source ↗
Figure 7
Figure 7. Figure 7: Angular clustering correlation forz ∈ [1.0, 1.1], by direct compu￾tation (Nside=2048), or from power spectrum (ℓmax=8000). The bottom panel shows the difference. these two different methods agrees very well, further validating our code. 5.5. Considerations of speed [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: CPU cost of RCF computation, as a function of pixel resolution Nside, and different redshift resolutions z∆, for clustering (top), or for clustering and shear (bottom). The redshift resolution is, from top down, z∆=0.0005 (blue), z∆=0.001 (yellow), z∆=0.0025 (green), zdelta=0.005 (red). The dashed lines connect points where Nside and z∆ correspond to the same spatial resolution (from left to right, 6 Mpc, … view at source ↗
Figure 10
Figure 10. Figure 10: CPU cost of harmonic-space analysis of FuGa3D, for 630 redshift-shell pairs. Shown is the cost of computing the redshift-space pseudo-spectrum for ℓmax=2000–8000 for full (circles) and small (tri￾angles) MICE in z = 1.0 − 1.2. The redshift resolution was fixed at z∆=0.005. The squares indicate the cost of constructing the mode-mode coupling kernel to correct for the survey footprint. shear analysis includ… view at source ↗
read the original abstract

We present FuGa3D, a code for fast computation of correlation functions and power spectra for galaxy survey observables, including galaxy clustering and cosmic shear. We define the redshift-space correlation function (RCF) as the correlation function defined in the parameter space of two redshifts and an angular separation angle. Assuming that there is no preferred direction in the sky, these parameters fully define the relative position of two galaxies, independently of the assumed cosmological model. Once the RCF is constructed, it is easy to compute derived correlation metrics, such as the real-space clustering correlation function and its multipoles. We further define the redshift-space power spectrum as the harmonic counterpart of the RCF, and show that it can be computed efficiently using the discrete galaxy coordinates. We validate the code with simulated mock catalogs. Computing the RCF and the two-point correlation function at 1.5 Mpc (3 Mpc) resolution for a MICE simulation with 46 million galaxies, took 47 node-min for clustering only, and 7.3 node-hours with shear analysis included.

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 / 2 minor

Summary. The manuscript introduces FuGa3D, a computational code for fast full-sky analysis of galaxy catalogs. It defines a redshift-space correlation function (RCF) directly in the observable parameter space of two redshifts (z1, z2) and angular separation θ, which under the assumption of isotropy is claimed to fully specify relative galaxy positions independently of any cosmological model. From the RCF the authors derive the real-space two-point correlation function and its multipoles, as well as a redshift-space power spectrum via harmonic transform of the discrete galaxy coordinates. The method is validated on MICE mock catalogs containing 46 million galaxies, with reported wall-clock times of 47 node-minutes for clustering-only RCF and two-point function computation at 1.5–3 Mpc resolution and 7.3 node-hours when cosmic shear is included.

Significance. If the central construction truly permits extraction of standard real-space and multipole statistics without introducing hidden dependence on a fiducial cosmology, the approach could offer a practical advantage for model-independent analyses of upcoming wide-field surveys. The concrete performance numbers on a 46-million-galaxy mock constitute a verifiable strength, and the joint treatment of clustering plus shear broadens applicability. However, the overall significance remains conditional on resolving the cosmology-mapping step in the derivation of real-space quantities.

major comments (2)
  1. [Section describing derivation of real-space CF and multipoles from the RCF] The claim that the RCF parameters (z1, z2, θ) fully define relative positions independently of cosmology is load-bearing for the paper’s headline advantage. Deriving the real-space correlation function nevertheless requires mapping observed redshifts to comoving separations via the integral χ(z), which depends on Ωm, H0 and other parameters. The manuscript does not state whether a fiducial cosmology is adopted for this conversion or whether the final estimator is constructed to remain invariant under changes to that fiducial model. This step must be made explicit (ideally with an equation or pseudocode) before the independence assertion can be accepted.
  2. [Validation section (MICE mock results)] Validation is reported only via run times on the MICE mock; no quantitative error analysis, covariance estimation, or tests under realistic survey masks and selection functions are described. Because the central performance and utility claims rest on these numerical results, the absence of such diagnostics weakens the ability to judge robustness for actual survey data.
minor comments (2)
  1. [Abstract] The abstract states that the redshift-space power spectrum is the 'harmonic counterpart' of the RCF and is computed from discrete coordinates; a short sentence or reference to the underlying spherical-harmonic transform algorithm would improve immediate clarity.
  2. [Methods] Notation for the RCF and its derived quantities should be introduced with a compact table or equation block early in the methods to avoid later ambiguity when multipoles and power spectra are discussed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment below and indicate the changes we will make to the manuscript.

read point-by-point responses

  1. Referee: [Section describing derivation of real-space CF and multipoles from the RCF] The claim that the RCF parameters (z1, z2, θ) fully define relative positions independently of cosmology is load-bearing for the paper’s headline advantage. Deriving the real-space correlation function nevertheless requires mapping observed redshifts to comoving separations via the integral χ(z), which depends on Ωm, H0 and other parameters. The manuscript does not state whether a fiducial cosmology is adopted for this conversion or whether the final estimator is constructed to remain invariant under changes to that fiducial model. This step must be made explicit (ideally with an equation or pseudocode) before the independence assertion can be accepted.

    Authors: We agree that this step requires explicit clarification. The RCF is defined directly in the observable space of (z1, z2, θ) and is therefore independent of cosmology by construction. Derivation of the real-space correlation function and multipoles does require mapping redshifts to comoving distances via χ(z) evaluated in a fiducial cosmology; the resulting quantities therefore depend on that choice, as is conventional. We will revise the manuscript to state this explicitly, adding the relevant equation for the χ(z) conversion together with pseudocode for the derivation from the RCF. This will distinguish the model-independent RCF from the fiducial-dependent derived statistics. revision: yes

  2. Referee: [Validation section (MICE mock results)] Validation is reported only via run times on the MICE mock; no quantitative error analysis, covariance estimation, or tests under realistic survey masks and selection functions are described. Because the central performance and utility claims rest on these numerical results, the absence of such diagnostics weakens the ability to judge robustness for actual survey data.

    Authors: We acknowledge that the validation would be strengthened by additional quantitative diagnostics. The current results focus on computational performance for a 46-million-galaxy catalog, but we agree that error analysis, covariance estimation, and tests with masks and selection functions are important for assessing robustness. We will expand the validation section in the revised manuscript to include these elements. revision: yes

Circularity Check

0 steps flagged

No circularity: direct computational procedure from observables

full rationale

The paper defines the redshift-space correlation function (RCF) directly from discrete galaxy coordinates in observable parameters (redshifts z1, z2 and angular separation θ). Derived quantities such as the real-space correlation function and multipoles are obtained from this RCF via standard transformations that do not rely on fitted parameters or self-referential assumptions within the paper. No equations or steps reduce the output to the input by construction, and validation uses external mock catalogs. The central claim of cosmology independence for the RCF rests on the isotropy assumption applied to observables, which is stated explicitly without circular reduction. This is a self-contained computational method with no load-bearing self-citations or ansatzes that presuppose the result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the isotropy assumption stated in the abstract and on the numerical correctness of the harmonic transform applied to discrete points; no free parameters or new physical entities are introduced.

axioms (1)
  • domain assumption There is no preferred direction in the sky, so two redshifts and one angle fully specify relative galaxy positions independently of cosmology.
    Direct quote from abstract; this premise allows the RCF to be constructed without a fiducial cosmology.

pith-pipeline@v0.9.0 · 5747 in / 1319 out tokens · 40388 ms · 2026-05-19T08:17:45.636411+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    We define the redshift-space correlation function (RCF) as the correlation function defined in the parameter space of two redshifts and an angular separation angle. Assuming that there is no preferred direction in the sky, these parameters fully define the relative position of two galaxies, independently of the assumed cosmological model.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Once the RCF is constructed, it is easy to compute derived correlation metrics, such as the real-space clustering correlation function and its multipoles... adopt a cosmological model to convert redshifts to physical distances

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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