{"id":"11b0de8f-f947-4f5a-9ca1-a09b3e555ab2","arxiv_id":"2506.06902","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A spherical Fourier-Bessel analysis of galaxy clustering lets survey analysts cut only the angular and radial modes contaminated by systematics, preserving large-scale modes that standard multipole analyses would discard.","lead":"This paper shows how a spherical Fourier-Bessel basis separates angular and radial structure in galaxy survey data, placing smooth observational errors mostly in just one large-scale radial mode. That would let future surveys remove only contaminated modes instead of discarding all large-scale clustering data, sharpening constraints on early-universe physics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Mode-retention claim depends on real systematics having broad, smooth, separable radial profiles; Fig. 7 shows narrow profiles contaminate many n modes, and the stellar example's uniform radial assumption is unrealistic—so the n=0 cut may not retain modes for actual surveys.","rationale":"The paper is a strong methods contribution: it gives a clear interpretation of SFB radial indices, derives the plane-parallel limit (Eq. 20) with percent-level checks at high ell, and uses exact SFB computations for systematics. The central claim is conditional on the radial broadness of systematics, and the authors explicitly acknowledge the narrow-Gaussian failure mode in Fig. 7 and call for realistic templates in Sec. VII. However, the abstract's advocacy is not conditioned on a survey-specific demonstration, and the stellar contamination example—the paper's flagship realistic case—uses an unjustified uniform radial assumption. I do not regard the velocity boundary condition choice as load-bearing, since it is an analysis choice the authors recommend with clear motivation (better plane-parallel accuracy and faster convergence in Appendix B), and any adopted basis with matching estimator/theory is internally consistent. The narrow-profile issue is more serious because it is a property of the physics, not the analysis convention; if real systematics are narrow, no choice of BC can make them localize in n=0. The proposed concrete test—recomputing Fig. 8 with a realistic stellar radial distribution—directly settles whether the claimed mode-retention advantage survives for the paper's own example. Pending that, CONDITIONAL acceptance is the right verdict, so I recommend no change to the reader's verdict.","tokens_in":29264,"tokens_out":12337,"duration_ms":143009,"concrete_test":"Recompute the stellar contamination example of Fig. 8 using a realistic radial distribution for stars in the survey sample—e.g., a delta-function at z=0 convolved with the survey's photometric redshift error distribution—instead of a uniform profile; with the velocity boundary condition, determine the smallest n_cut(ell) for ell≤50 at which the stellar SFB power spectrum falls below the fNL=1 PNG signal. If n_cut≥1 for any ell where PNG constraining power resides, the n=0-only cut advocated in Sec. VI A fails for stellar contamination, and the PSM-contrast claim of Sec. V D weakens.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract's central claim that SFB enables retaining large-scale radial modes by cutting only n=0 requires that dominant survey systematics have broad, smooth radial profiles. The paper's demonstration uses a separable model S(hat_n)R(x) (Eq. 27), with R(x) uniform for the stellar contamination example in Sec. V D. This uniform assumption is physically unmotivated: stars misclassified as galaxies are at z≈0 and enter a z=1–1.5 bin only via photometric-redshift outliers, whose radial distribution can be narrow. Figure 7 shows that a narrow Gaussian R(x) contaminates many radial modes and falls at a rate comparable to the PNG k^-2 signal, so the n=0 cut would not remove the contamination while retaining n≥1 modes. The paper acknowledges this limitation qualitatively but does not establish that real survey systematics (DESI, Euclid, SPHEREx) are broad; it merely asserts they are 'reasonable.' Because the quantitative demonstration (Fig. 8, only n=0 cut for 2.5% stellar contamination) depends on this profile, the practical mode-retention advantage over a PSM k_min cut is not yet demonstrated for actual surveys. The framework's mathematical capability is solid; the missing piece is empirical support for the profile assumption.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":29490,"tokens_out":8324,"duration_ms":95878,"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":[{"comment":"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.","section":"Sec. V B, Eq. (27), Fig. 7"},{"comment":"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.","section":"Sec. V D, Fig. 8"},{"comment":"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.","section":"Appendix B, Fig. 10; Sec. VI A"}],"minor_comments":[{"comment":"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.","section":"Fig. 7 caption"},{"comment":"Typo: \"cosmologiucal\" should be \"cosmological.\"","section":"Fig. 9 caption"},{"comment":"Typo: \"Idenitities\" should be \"Identities.\"","section":"Appendix D heading"},{"comment":"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.","section":"Sec. V D, Fig. 9"},{"comment":"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.","section":"Sec. V B, after Fig. 7"},{"comment":"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.","section":"Sec. IV, Eq. (20) and Fig. 4"},{"comment":"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).","section":"Sec. IV, Eq. (21)"}],"recommendation":"major_revision","confidential_remarks":"The core mathematics of the paper is sound and the analytical derivations are a genuine strength. The main reservation is the gap between the generic profile assumption and the survey-level advocacy: the stellar contamination example uses a uniform radial profile that is unlikely to hold for real photo-z outliers, and the paper's own Fig. 7 shows that a narrow profile invalidates the n=0-only cut. I would ask the authors either to add a realistic 3D systematic template or to narrow the abstract and conclusions accordingly. Also, please request a published or detailed reference for the SFB-to-PSM mapping currently cited as Ref. [78] 'in preparation,' since it is used in the quantitative comparison of Fig. 9."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Robin — worth your time. This is a genuinely useful methods paper: it gives the discrete spherical-Fourier-Bessel power spectrum a clean plane-parallel interpretation (Eq. 20) that maps each (ell,n) mode to a clustering-wedge piece P(k, mu) with an explicit line-of-sight mode k_||,nell = n pi / (xmax - x_t). The approximation is checked against exact SFB computations and hits percent level at large ell; the authors are upfront that it fails at low ell where wide-angle effects matter. That is honest. The clarification that n counts radial oscillations and that n=0 modes are purely angular is also well done and should settle a recurring confusion. The systematics localization results — angular systematics with broad radial profiles fall as k^-8 along n, under velocity boundary conditions — are analytic, not fits, and the comparison with the k^-2 PNG scaling in SFB space is a nice demonstration.\n\nThe soft spots are real but not fatal. The paper's headline practical claim is that you can cut n=0 and keep the large-scale radial modes that a PSM k_min cut would discard. That only works if the systematics in the actual survey are radially broad and smooth. The stellar-contamination example simply assumes a uniform radial distribution. In a real z=1–1.5 bin, misclassified stars are at z≈0 and enter the bin through photometric-redshift outliers; that radial distribution can be narrow. Fig. 7 shows a narrow Gaussian contaminates many n modes and falls at a rate comparable to the PNG signal, so the n=0 cut would not do what the abstract advertises. The authors acknowledge this limitation qualitatively but do not establish that DESI, Euclid, or SPHEREx systematics are actually broad. They may well be; the paper just doesn't show it. The k^-8 result also depends on the velocity boundary condition; the potential condition yields k^-4. The authors justify the choice, but it is a choice. Finally, Fig. 9 relies on an unpublished mapping (Ref. [78]); that needs to be public before the PSM comparison can be fully evaluated. These are all fixable or qualifyable in revision.\n\nNet: the math is solid, the framework is a real advance for SFB-based analyses, and the systematics-localization idea is worth taking seriously. The paper has not yet demonstrated the mode-retention advantage for real surveys. That's a limitation of the evidence, not a flaw in the derivation. Send it to a serious referee; ask for the radial-profile assumption to be sharpened and the unpublished mapping released. I'd probably cite it for the plane-parallel bridge even while remaining skeptical of the survey-level payoff.","headline":"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.","tokens_in":30060,"tokens_out":3486,"would_cite":true,"duration_ms":34545,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["spherical Fourier-Bessel power spectrum","angular-radial mode separation","systematics mitigation","clustering wedge","primordial non-Gaussianity","large-scale structure","radial modes"],"falsifier":"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.","tokens_in":29040,"feed_emoji":"🌌","tokens_out":10640,"duration_ms":108965,"temperature":0.7,"pith_summary":"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.","feed_headline":"A spherical-Fourier basis confines systematics to one radial mode","feed_subtitle":"Broad, smooth systematics stay in the n=0 modes, so large-scale radial modes survive without a k_min cut.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the discrete SFB radial basis to cosmology; the basis used in Eq. (10) comes from here.","marker":"[38]"},{"why":"Defines the discrete SFB power spectrum estimator and boundary-condition choices; supplies the estimator the paper's case rests on.","marker":"[40]"},{"why":"Provides the exact SFB power spectrum computation with linear RSD used as the numerical benchmark for the plane-parallel approximation and systematics comparisons.","marker":"[41]"},{"why":"Earlier SFB study of angular foregrounds with the continuous basis; the point of departure the discrete-basis localization claim is built against.","marker":"[55]"},{"why":"Earlier plane-parallelization of the continuous SFB power spectrum; the discrete version here is compared with and simplified relative to it.","marker":"[65]"},{"why":"Clustering-wedge systematics results and the required multipole order; the toolbox the SFB basis inherits at quasi-linear scales.","marker":"[49]"},{"why":"Derives the $k^{-2}$ scaling of PNG and GR signals in SFB space; the contrast that makes the steeper systematics fall-off meaningful.","marker":"[28]"},{"why":"Radial integral constraint formalism; supports the claim that purely radial systematics sit in the $\\ell=0$ SFB mode.","marker":"[75]"},{"why":"Demonstrates PSM large-scale systematics excess and motivates $k_{\\rm min}$ removal; the baseline being improved on.","marker":"[23]"}],"fun_headline_variants":["Systematics confined to single radial mode with new power spectrum","Spherical-Fourier spectrum zeros out systematics except one mode","New basis isolates systematics, preserving large-scale modes","SFB power spectrum localizes systematics, avoids k_min cuts","One radial mode traps systematics in full-sky 3D clustering"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Systematics confined to single radial mode with new power spectrum","Spherical-Fourier spectrum zeros out systematics except one mode","New basis isolates systematics, preserving large-scale modes","SFB power spectrum localizes systematics, avoids k_min cuts","One radial mode traps systematics in full-sky 3D clustering"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000777,"raw_usage":{"total_tokens":3576,"prompt_tokens":1225,"completion_tokens":2351,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":841,"completion_tokens_details":{"reasoning_tokens":2264}},"tokens_in":841,"tokens_out":2351,"duration_ms":16989,"temperature":1.0,"reasoning_tokens":2264,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:48:04.867954+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the discrete SFB radial basis to cosmology; the basis used in Eq. (10) comes from here."},{"cited_title":"3DEX: a code for fast spherical Fourier-Bessel decomposition of 3D surveys","cited_arxiv_id":"1111.3591","evidence_quote":"Defines the discrete SFB power spectrum estimator and boundary-condition choices; supplies the estimator the paper's case rests on."},{"cited_title":"Spectral Line De-confusion in an Intensity Mapping Survey","cited_arxiv_id":"1604.07833","evidence_quote":"Earlier SFB study of angular foregrounds with the continuous basis; the point of departure the discrete-basis localization claim is built against."}],"review_version":1}