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Separating Angular and Radial Modes with Spherical-Fourier Bessel Power Spectrum on All Scales and Implications for Systematics Mitigation

T0 review · 3 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that radially smooth systematics sit almost entirely in the $n=0$ spherical-Fourier-Bessel modes, so one can cut only those modes and keep large-scale radial information that a $k_{\mathrm{min}}$ cut would discard.

desk verdict A careful methods paper with a real plane-parallel bridge for SFB, but the systematics-retention payoff rests on an unproven assumption that real survey systematics are radially broad and smooth. read the letter →

arxiv 2506.06902 v2 pith:HBWSTTZK submitted 2025-06-07 astro-ph.CO

Robin Y. Wen , Henry S. Grasshorn Gebhardt , Chen Heinrich , Olivier Doré This is my paper
classification astro-ph.CO
keywords sphericalFourier-Besselpowerspectrumangular-radialmodeseparationsystematicsmitigationclusteringwedgeprimordialnon-Gaussianitylarge-scalestructureradialmodes
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Large-scale-structure surveys need to remove observational systematics without throwing away the large-scale modes that carry the rarest cosmological signals, such as local primordial non-Gaussianity. This paper argues that the spherical Fourier-Bessel (SFB) basis, which splits fluctuations into angular multipoles $\ell$ and radial index $n$, makes the separation almost perfect: systematics with broad, smooth radial profiles concentrate in the $n=0$ modes, with diagonal contamination dropping as $k^{-8}$ across radial modes, while the cosmological signals of interest drop only as $k^{-2}$. As a result, one can excise or down-weight just the contaminated modes rather than applying a global $k_{\mathrm{min}}$ cut, as standard power-spectrum-multipole analyses must. The paper also shows that the SFB power spectrum reduces to the clustering wedge $P(k,\mu)$ in the plane-parallel limit, so established wedge-based systematics treatments transfer to the full sky. If the claim holds, future surveys can retain large-scale radial modes that would otherwise be lost, directly improving constraining power for primordial non-Gaussianity and relativistic effects.

What carries the argument

The central object is the discrete SFB basis $g_{n\ell}(x)Y_{\ell m}(\hat{\mathbf n})$, built from spherical Bessel functions satisfying orthonormality over the survey shell $x_{\min}\le x\le x_{\max}$ with the velocity (Neumann) boundary condition, where the derivative of each radial function vanishes at both boundaries. The identity that carries the argument is Eq. (20), mapping each SFB diagonal mode to the clustering wedge $P(k,\mu)$ at the mode's effective distance, plus the asymptotic $d_{n\ell}\sim k^{-4}$ behavior of the radial overlap integrals of the unit function. Together they translate the familiar Cartesian-wedge picture of systematics onto the curved sky and predict the steep $k^{-8}$ leakage suppression that justifies cutting only the $n=0$ modes.

What would settle it

Take the measured three-dimensional systematic template of a wide-field survey (for example, the stellar density map multiplied by the survey redshift distribution), decompose it into discrete SFB coefficients, and compute the diagonal power ratios $C^S_{\ell nn}/C^S_{\ell 00}$. If the fall-off is shallower than approximately $k^{-4}$ or if the $n=1,2$ entries exceed the local-PNG signal at $k<0.01\,h/\mathrm{Mpc}$, the recommendation to cut only $n=0$ modes fails for that survey.

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Extended reading notes

Core claim

The central discovery is that the two indices of the discrete SFB power spectrum carry separate physical information: $\ell$ labels angular oscillations on the sphere, while $n$ counts radial oscillations of the basis function, and $k_{n\ell}$ is the total wavenumber of the mode rather than a radial wavenumber. Once this is established through the transition distance $x^t_{n\ell}$ and the line-of-sight wavenumber $k_{\parallel,n\ell}=n\pi/(x_{\mathrm{max}}-x^t_{n\ell})$, the paper derives a plane-parallel identification: $C^{\mathrm{approx}}_{\ell nn}=P(k=k_{n\ell},\mu=k_{\parallel,n\ell}/k_{n\ell},x_c=x_{\mathrm{eff},n\ell})$, matching the SFB spectrum to the clustering wedge at percent-level accuracy for small angular scales. The systematics result follows from the radial coefficients $d_{n\ell}=\int dx\,x^2g_{n\ell}(x)$, which decay as $k^{-4}$ under the velocity (Neumann) boundary condition, making the diagonal power of an angular systematic decay as $k^{-8}$; only the $n=0$ and $n=1$ modes stay above a percent of the systematic's own largest mode. Stellar contamination, modeled with a realistic stellar template and a uniform radial distribution, is shown to beat a local-PNG signal only at low $\ell$ and $n=0$, whereas in the monopole power spectrum it forces a cut at roughly $k<0.01\,h/\mathrm{Mpc}$. The paper concludes that SFB mode-selective cleaning is the full-sky generalization of clustering-wedge mitigation and should replace blanket $k_{\mathrm{min}}$ cuts in wide-field 3D clustering analyses.

Load-bearing premise

The localization argument hinges on the velocity boundary condition for the radial basis and on the assumption that real survey systematics are radially broad and smooth; under the alternative potential boundary condition the drop is only $k^{-4}$, and a radially narrow systematic contaminates many more radial modes.

Editorial extensions

If this is right

  • A wide-field survey that adopts the SFB power spectrum can remove just the low-$\ell$, $n=0$ modes contaminated by stellar foregrounds and keep the $n=1$ and $n=2$ radial modes at $k<0.01\,h/\mathrm{Mpc}$ that a monopole analysis would discard.
  • Because the SFB spectrum is the full-sky generalization of the clustering wedge, existing wedge-based treatments for fiber collisions, astrophysical foregrounds, and interlopers carry over to surveys without a global line of sight.
  • Local primordial non-Gaussianity and general-relativistic signals, which scale as $k^{-2}$ in SFB space, remain measurable in higher radial modes even when a radially smooth systematic dominates the lowest modes.
  • The plane-parallel mapping permits one-loop effective-field-theory calculations of the Cartesian $P(k,\mu)$ to be converted into SFB power spectra, extending SFB analyses to quasi-linear scales.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural follow-up the paper leaves implicit is a Fisher forecast quantifying how much an $n=0$-only cut improves $f_{\rm NL}$ constraints relative to a $k_{\rm min}$ cut; the stellar-contamination example suggests the gain is large.
  • The boundary-condition dependence of the leakage suggests one could choose or design radial basis functions to maximize systematics separation rather than treating the Neumann condition as fixed.
  • For systematics that are narrow in radius and therefore spread across $n$, the paper's own reasoning points to a hybrid strategy: SFB mode cuts for broad systematics plus real-space masking or deprojection of the narrow radial features.
  • The steep $k^{-8}$ versus $k^{-2}$ contrast also implies an empirical pattern test: at fixed $\ell$, measuring the fall-off of the SFB power spectrum across $n$ in data would flag systematics before templates are needed.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. The paper studies the discrete spherical Fourier-Bessel (SFB) power spectrum as a tool for separating angular and radial modes. It clarifies that the radial index n labels radial oscillations, defines the LOS Fourier mode k_||,nℓ via the transition distance, and proposes Eq. (20), which approximates the diagonal SFB spectrum by the clustering wedge P(k, μ) at an effective distance. It then analyzes additive angular systematics with separable radial profiles R(x), showing that leakage into higher n modes drops as k^-8 for uniform profiles under the velocity boundary condition and more slowly for other profiles. Using a Gaia-based stellar contamination template, it argues that only n=0 and low-ℓ modes need to be removed in an SFB analysis, whereas a power-spectrum-multipole analysis requires a k_min cut. The paper concludes that SFB permits surgical mode cuts and is a promising framework for systematics mitigation in upcoming surveys.

Significance. If the localization claim holds, this is a substantial methodological contribution: it connects the SFB power spectrum to the familiar clustering wedge formalism, provides an analytic approximation that is validated at percent level at high ℓ, and offers a concrete route to retaining large-scale radial modes for primordial non-Gaussianity and relativistic-effect measurements. The k^-8 scaling is derived analytically and checked against exact SFB computations, and the paper is honest about wide-angle limitations and boundary-condition dependence. The main gap is the empirical support for the radial-profile assumption underlying the practical mode-retention claim; the mathematical framework itself appears sound and non-circular.

major comments (3)
  1. [Sec. V B, Eq. (27), Fig. 7] The mode-retention recipe "cut n=0 only" is not established as a general consequence of broad radial profiles. For the exponential and broad-Gaussian profiles in Fig. 7 the diagonal systematic drops only as k^-4, not k^-8, and the figure does not give the amplitudes or widths of these profiles, so one cannot determine how many n modes remain above the PNG signal. The paper needs a quantitative selection criterion (for example, the largest n for which C_sys/C_PNG exceeds a given threshold as a function of profile width and amplitude), or an explicit statement that the n=0-only cut has been demonstrated only for the uniform profile used in Fig. 8.
  2. [Sec. V D, Fig. 8] The stellar contamination example assumes that misclassified stars are uniformly distributed in comoving distance over z=1-1.5. This assumption is load-bearing for the central comparison with the PSM k_min cut, because Fig. 8 shows the n=0-only cut working. In a real z~1 galaxy sample, stars enter through photometric-redshift outliers whose radial PDF can be narrow, and the narrow-Gaussian curve in Fig. 7 shows that such a profile contaminates many n modes at a rate comparable to the PNG k^-2 signal. Please replace the uniform radial assumption with a realistic photo-z outlier distribution for at least one survey, or, if that is outside the present scope, revise the abstract and conclusion so that the mode-retention advantage is claimed only for systematics whose radial profile is known to be broad.
  3. [Appendix B, Fig. 10; Sec. VI A] The strong k^-8 localization is specific to the velocity (Neumann) boundary condition; under the potential boundary condition the leakage is only k^-4. The paper recommends the velocity BC and gives good reasons, but the abstract's statement that systematics "primarily concentrate in the n=0 modes" is made without this caveat. Since both boundary conditions yield consistent cosmological SFB spectra, an estimator built with the potential BC would not enjoy the advertised strong localization. Please qualify the abstract and the Sec. VI A claim "one can cut the n=0 modes from the analysis to be robust against any systematics with broad radial distributions" by adding "under the velocity boundary condition," or show that the n=0 cut remains valid under the potential BC for a realistic systematic amplitude.
minor comments (7)
  1. [Fig. 7 caption] Please give the functional forms and widths of the exponential and Gaussian radial profiles; without these the "broad" versus "narrow" distinction cannot be reproduced by the reader.
  2. [Fig. 9 caption] Typo: "cosmologiucal" should be "cosmological."
  3. [Appendix D heading] Typo: "Idenitities" should be "Identities."
  4. [Sec. V D, Fig. 9] The SFB-to-PSM mapping used for Fig. 9 cites Ref. [78] as "in preparation." Since this mapping underpins the PSM comparison, please include the necessary equations in an appendix or cite a published derivation.
  5. [Sec. V B, after Fig. 7] The suggestion that localized systematics "can be directly mitigated in real space" is not developed; a short explanation of the proposed real-space mitigation and how it would complement the SFB mode cut would strengthen the argument.
  6. [Sec. IV, Eq. (20) and Fig. 4] For low ℓ the fractional error of Eq. (20) reaches order unity. The text already notes this qualitatively, but a quantitative validity condition (for example, in terms of k_⊥/k_|| or the ratio of angular to radial scales) would help the reader know when the approximation can be used.
  7. [Sec. IV, Eq. (21)] The effective distance x_eff in Eq. (21) is defined with a general radial selection R(x), but the numerical validation in Sec. IV assumes a uniform R(x). Please state explicitly which figures assume uniform selection and which use a non-uniform R(x).

Circularity Check

0 steps flagged · score 2.0 of 10

No substantive circularity: the SFB-to-wedge mapping and the k^-8 localization scalings are analytic derivations checked against exact computations; only minor self-citations, including the in-preparation Ref. [78], support auxiliary mapping steps without encoding the central result.

full rationale

The paper's central derivation chain is self-contained. Equation (20) is obtained analytically in Appendix A by combining flat-sky angular harmonics (Eqs. A5-A10) with the plane-wave approximation of the SFB radial basis (Eqs. A13-A18); the result is validated against the exact SFB computation in Fig. 4 rather than fitted. The n=0 localization claim follows from the explicit coefficient relation C^S_ell n1 n2 = d_n1 ell d_n2 ell C^S_ell (Eq. 26) with d_n ell ~ k^-4 under the velocity boundary condition (Eq. 25 and Fig. 10), a mathematical property of the chosen basis. The paper transparently shows that the alternative potential boundary condition gives only a k^-4 drop-off (Fig. 10), so the choice is not smuggled in as a forced uniqueness result. The stellar-contamination example is a forward model using a Gaia angular template plus an explicitly stated uniform radial profile, not a fit of the predicted modes. The main self-citations are to the authors' SFB formalism (Refs. [25,28,40,41]) and to the in-preparation Ref. [78] for the auxiliary SFB-to-PSM mapping used only for the monopole comparison in Sec. V D; these do not encode or presuppose the systematics-localization result, so they are not load-bearing. The paper also explicitly acknowledges the narrow-Gaussian limitation (Fig. 7), which is a scope caveat rather than circular reasoning. Overall, no derivation reduces by construction to its own inputs; the score reflects only the presence of minor, non-load-bearing self-citations.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The new derivations introduce no fitted parameters; the illustrative comparison uses assumed stellar contamination parameters. The main assumptions are physical and statistical (Kaiser RSD, additive systematics), technical (velocity boundary condition, uniform radial selection for the default demo), and representativeness (broad smooth radial profiles). No new physical entities are introduced.

free parameters (2)
  • Stellar contamination fraction = 2.5% (assumed for illustration)
    Chosen by hand for the stellar contamination example; the specific claim that only n=0 modes need removal depends on this amplitude relative to the fNL=1 PNG signal (Sec. V D).
  • Stellar angular power-law index = -2.3
    Fit to Gaia DR3 point-source angular power spectrum and used as a smooth template in Fig. 9; the structural localization result does not depend on this index, but the PSM comparison figure does (Sec. V D, Eq. 30).
assumptions (7)
  • domain assumption Statistical isotropy and translational invariance for the matter field at fixed redshift (Eq. 3).
    Used throughout to write the matter power spectrum P_m(k,z) and to justify the plane-parallel approximation; breaks down for observed light-cone fields, which the SFB basis is meant to handle.
  • domain assumption Kaiser formula Eq. 7 for the local power spectrum under linear Newtonian RSD.
    The plane-parallel approximation in Sec. IV evaluates P(k,mu) with this formula; nonlinear and wide-angle corrections are not included in the default computation.
  • domain assumption Systematics enter as additive, uncorrelated contamination in Eq. 22, and the additive part dominates in the density contrast.
    The localization analysis in Sec. V treats additive systematics only; multiplicative and window effects are deferred.
  • domain assumption Uniform radial selection function for the default numerical demonstrations (Secs. IV and V B).
    The k^-8 localization is derived for uniform R(x); exponential and broad Gaussian profiles give k^-4, and narrow Gaussian profiles contaminate many more modes (Fig. 7).
  • domain assumption Velocity (Neumann) boundary condition Eq. 12 is the appropriate choice for the SFB basis.
    The paper recommends it because it matches cosine phases and makes angular systematics decay faster (Appendix B); the potential boundary condition yields k^-4 instead of k^-8, so the localization strength is tied to this choice.
  • domain assumption Representative systematics have broad, smooth radial distributions.
    The central localization claim is demonstrated for uniform, exponential, and broad Gaussian profiles; narrow radial systematics are shown to contaminate higher n modes, so the practical recommendation depends on this assumption about real surveys.
  • domain assumption SFB-to-PSM mapping of Refs. [25,78] is valid, with Ref. [78] unpublished.
    Fig. 9 converts SFB power spectra to the PSM monopole using this mapping; without the unpublished equations the comparison cannot be independently checked.

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Cite this review

Pith. "Pith review of Separating Angular and Radial Modes with Spherical-Fourier Bessel Power Spectrum on All Scales and Implications for Systematics Mitigation." pith.science (2026). https://pith.science/paper/HBWSTTZK

@misc{pith2026250606902,
  author       = {Pith},
  title        = {Pith review of: Separating Angular and Radial Modes with Spherical-Fourier Bessel Power Spectrum on All Scales and Implications for Systematics Mitigation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HBWSTTZK}},
  note         = {Machine review of arXiv:2506.06902}
}
abstract

Current and upcoming large-scale structure surveys place stringent requirements on the mitigation of observational systematics in order to achieve their unprecedented constraining power. In this work, we investigate the potential use of the spherical Fourier-Bessel (SFB) power spectrum in controlling systematics, leveraging its capability of disentangling angular and radial scales. We first clarify how the discrete SFB basis describes radial scales via the index $n$ and demonstrate that the SFB power spectrum reduces to the clustering wedge $P(k,\mu)$ in the plane-parallel limit, enabling it to inherit results from past literature based on the clustering wedge. While the parallel and perpendicular Fourier mode $(k_{||}, k_\perp)$ decomposition underlying the wedge is only valid for surveys of small angular coverage with a well-defined global line-of-sight, the SFB basis provides a natural generalization that can be applied to the full sky. Crucially, the separation of angular and radial scales allows systematics to be localized in SFB space. In particular, systematics with broad and smooth radial distributions primarily concentrate in the $n=0$ modes corresponding to the largest radial scales. This localization behavior enables one to selectively remove only particular angular and radial modes contaminated by systematics. This is in contrast to standard 3D clustering analyses of wide-field surveys based on power spectrum multipoles, where systematic effects necessitate the removal of all modes below a given $k_{\rm min}$. Our findings advocate for adopting the SFB basis in 3D clustering analyses where systematics have become a limiting factor.

Figures

Figures reproduced from arXiv: 2506.06902 by the authors.

Figure 1
Figure 1. FIG. 1. The values of the total Fourier [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The SFB radial basis functions at [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Comparison between the exact SFB PS and the corresponding plane-parallel [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figures from the paper (8 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison between the cosmology (abbreviated as “cosmo.”, calculated assuming Planck 2018 ΛCDM cosmology [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison between the cosmology and angular systematic signals (with uniform radial distributions) for the diagonal [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison between the different drop-off behaviors of angular systematic under different radial distributions. The left [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Comparison between stellar contamination (with con [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Comparison of SFB radial basis functions under the velocity and potential boundary conditions. The left panel [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Reconstruction of the unit function [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Comparison between the exact SFB PS and its plane-parallel approximation with [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. The approximation of [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]

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