{"id":"876e342b-8a9f-4e0f-819e-450bcb385727","arxiv_id":"2501.12661","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Scale cuts break the equivalence between the galaxy angular power spectrum and the angular correlation function for primordial non-Gaussianity, with the correlation function retaining fNL information on surprisingly small angular scales.","lead":"This paper shows that for galaxy surveys searching for primordial non-Gaussianity, the angular power spectrum and the angular correlation function are mathematically equivalent only when every angular scale is used. Once analysts cut scales to avoid systematics, the two statistics diverge, and the correlation function can recover non-Gaussianity information from small angles or partial-sky surveys.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The partial-sky claim in the abstract is not supported by the Fisher calculation: fsky=1 throughout and the integral constraint is the full-sky monopole, so 'partial area coverage' is only an extrapolation from discarding large separations in an all-sky survey.","rationale":"The strongest claim that could change practice is that small-angle or partial-sky w(theta) measurements can deliver sigma_fNL ~ 1. The small-angle part is a legitimate consequence of the Legendre transform: low-ell modes contribute to all theta, so cutting theta below theta_NL does not remove them, and for high-redshift shot-noise-dominated bins the constant PNG offset can be measured. The partial-sky part, however, is where the argument breaks. The calculations never vary fsky; Eq. (16) is written with fsky but the text sets fsky=1. Fig. 4's theta_max=10 deg result uses all-sky covariance and the full-sky integral constraint. This is equivalent to asking how much information is lost by not using pairs with theta > 10 deg on a full sky, not to observing a 10-degree patch. In a patch, the window couples multipoles and the integral constraint suppresses the very low multipoles where the PNG signal lives; this is a different estimator with a different covariance. The expected degradation, roughly 1/sqrt(fsky), is large enough to change the conclusion from sigma_fNL ~ 1.5 to > 15. The paper's final remark that window effects are 'straightforward' does not establish the claim; a calculation or a clearly stated qualifier is needed. I therefore regard the full-sky mathematical result and the scale-cut asymmetry as solid and independently supported by the derivation, but the abstract's partial-sky implication is not. The proposed test, a masked Fisher forecast with fsky=(1-cos(theta_max))/2 and the masked integral constraint, would settle it. If the authors prefer not to add such a calculation, the abstract and conclusion should be revised to say 'small-angle pairs from a wide-area survey' rather than 'partial area coverage.' This moves the reader's ACCEPT to CONDITIONAL: the core result is accepted, but the advertised practical implication needs either support or qualification.","tokens_in":13968,"tokens_out":9437,"duration_ms":103448,"concrete_test":"Replace the full-sky Fisher forecast for the z=[4,7] (and 'All') row in Table I with a masked survey of angular radius theta_max=10 deg: set fsky=(1-cos(theta_max))/2 in Eq. (16), and model the observed correlation function as tilde-w_mask(theta) = w(theta) - (1/Omega) int_Omega w(theta') dOmega', or equivalently use a windowed/pseudo-C_ell covariance. Recompute sigma_fNL with the same multipole and separation binning. If sigma_fNL degrades by roughly sqrt(1/fsky) ~ 11 as expected from Eq. (16), the partial-area claim in the abstract fails; if it remains near 1.5, the claim is supported. A simpler check: take the Fisher information from w(theta <= 10 deg) with the full-sky covariance and multiply the covariance by 1/fsky; if the projected sigma_fNL exceeds current constraints, the abstract needs revision.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The full-sky equivalence in Eqs. (9)-(13) is correct, and the scale-cut asymmetry is real: a low-ell PNG signal necessarily appears at all theta in w(theta). The load-bearing problem is the paper's quantitative extension to partial sky. The Fisher analysis sets fsky=1 (text after Eq. 16) and uses tilde-w from Eq. (12), which subtracts the full-sky monopole C_00/(4*pi). Fig. 4's variation of theta_max therefore discards large-separation pairs in an all-sky survey; it does not model a survey that only covers those separations. A real partial survey imposes a different integral constraint: the mean density is estimated within the footprint, which in harmonic space removes or attenuates modes with ell less than about pi/theta_survey through a window convolution, not only ell=0. For theta_max=10 deg, fsky=(1-cos(theta_max))/2 ~ 0.0076, so the Gaussian covariance in Eq. (16) degrades sigma_fNL by roughly sqrt(1/fsky) ~ 11 relative to fsky=1; the z=[4,7] forecast of sigma_fNL ~ 1.5 would become sigma_fNL > 15, which is not competitive. Thus the abstract's 'partial area coverage' implication is an extrapolation, not a derived result. The paper acknowledges window effects at the end ('rather straightforward... not challenging'), but no calculation is shown, and the direction of the effect is not benign: it suppresses exactly the low-ell modes that carry the PNG signal. The mathematical equivalence and the full-sky scale-cut violation remain solid; only the partial-sky quantitative claim is affected.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies whether the galaxy angular power spectrum C_ℓ and the angular correlation function w(θ) are equivalent for constraining local-type primordial non-Gaussianity (f_NL). It defines the integral-constrained power spectrum ~C_ℓ by subtracting the monopole (Eq. 9) and the corresponding correlation function ~w(θ) by subtracting the full-sky average (Eq. 12), and shows that the two are Legendre transforms of each other (Eq. 13). The authors argue that the f_NL signal in C_ℓ is confined to low multipoles, whereas in w(θ) the integral constraint spreads a constant offset across all angular scales, so that scale cuts break the equivalence. Using Gaussian Fisher forecasts for a six-bin LSST-like sample plus a Lyman-break-galaxy sample at 4<z<7, they report σ_f_NL ≈ 1.5 from w(θ) up to θ_max ≈ 10 deg for the high-redshift sample, and σ_f_NL ≈ 1 for all bins combined. The abstract and conclusion further claim that f_NL can be extracted from w(θ) measured in a survey with partial area coverage.","tokens_in":14313,"tokens_out":5783,"duration_ms":56316,"significance":"The exact derivation of the integral constraint in Eqs. (9)–(13) is clean, self-contained, and mathematically solid; it makes a useful conceptual point about how a low-multipole signal is redistributed in configuration space. The observation that a scale cut in ℓ versus a scale cut in θ breaks the equivalence differently is an original and practically relevant insight, particularly for surveys with limited sky coverage. The Fisher setup is transparent and the forecasts are reproducible from the text. However, the paper's headline claim about partial-sky surveys is not backed by the calculation, which assumes f_sky=1 and a full-sky integral constraint. Since that claim is load-bearing in the abstract and conclusion, the manuscript needs revision.","major_comments":[{"comment":"The abstract and Section V state that PNG information can be extracted from w(θ) measured in a survey with partial area coverage, but the Fisher analysis never models a partial survey. The covariance in Eq. (16) assumes f_sky = 1, and the integral constraint in Eq. (12) subtracts the full-sky monopole C_{ℓ=0}/4π. The variation of θ_max in Fig. 4 and Table I corresponds to discarding pairs at large separations in an all-sky survey, not to restricting the survey footprint. A real partial survey covers a fraction f_sky of the sky; its window function removes or attenuates modes with ℓ ≲ π/θ_survey, not only ℓ = 0, and the Gaussian covariance in Eq. (16) scales roughly as 1/f_sky. For θ_max = 10 deg, f_sky = (1−cos 10°)/2 ≈ 0.0076, so the quoted σ_{f_NL} ≈ 1.5 for the z = [4,7] sample would degrade by about a factor of 11, making the forecast noncompetitive. The closing remark in Section V that window effects are 'rather straightforward' is not sufficient, because the effect is not benign: it suppresses the low-ℓ modes that carry the PNG signal.","section":"Abstract, Section IV.A.2, Section V"},{"comment":"Fig. 3 presents forecasts for w(θ) as θ_min is decreased below the nonlinear scale θ_NL, which is the region to the left of the vertical dashed line in each panel, and the abstract highlights that the PNG signature in w(θ) extends below the nonlinear scale. However, the model in Eqs. (3)–(5) and the covariance in Eq. (16) are linear-theory and Gaussian quantities, and the text states that linear theory is assumed throughout. No nonlinear bias model or nonlinear covariance is provided for the small-angle bins that drive the improved σ_{f_NL} in that region. The forecast numbers in that part of Fig. 3 are therefore not a robust prediction. This does not affect the exact equivalence result in Eqs. (9)–(13), but it does affect the quantitative conclusion that the full f_NL information can be recovered from w(θ) on scales below θ_NL.","section":"Section IV.A.3 and Fig. 3"}],"minor_comments":[{"comment":"The columns labeled σ_f_NL (θ_max = [10,180] deg) and σ_f_NL (ℓ_min = [1,30]) are ambiguous: they contain two numbers per redshift bin without a clear indication of which number corresponds to which scale cut. The authors should split these into separate columns or label the pairs explicitly.","section":"Table I"},{"comment":"The notation for the angular power spectrum and correlation function alternates between C^i_i_{gg,ℓ} and C_{gg,ℓ}, and similarly for w, which makes the text slightly harder to follow. A consistent notation, including the tilde for integral-constrained quantities, would improve readability.","section":"Notation throughout"},{"comment":"The shot-noise covariance formula in Eq. (18) is derived for an all-sky survey with N_pair ∝ 4π; for partial sky the pair count should scale with the survey area. This is consistent with the paper's all-sky assumption, but it should be stated explicitly near Eq. (18) so that readers do not apply it to partial-sky cases.","section":"Section IV.A.2, Eq. (18)"}],"recommendation":"major_revision","confidential_remarks":"The core derivation of the integral constraint is sound and the full-sky scale-cut asymmetry is a valuable result. The main obstacle is the unsupported extension to partial sky in the abstract and conclusion, which is repeated in the title's promise of 'equivalence' in a broader context. If the authors remove the partial-sky claim or supply a genuine windowed calculation, the paper would be suitable for publication. The nonlinear-scale part of Fig. 3 also needs either a disclaimer or a nonlinear model. I would be comfortable with acceptance after these points are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The first half of this paper is the useful half. The authors show explicitly that in an all-sky survey the integral constraint removes only the monopole in harmonic space, while in configuration space it acts as an additive, scale-dependent correction. That is why the local-PNG signal in w(theta) appears at all angles, including below the nonlinear scale, even though its C_l signature is confined to low multipoles. They then point out that any practical scale cut breaks the formal C_l/w(theta) equivalence. That is a clean, correct, and genuinely clarifying result, and they deserve credit for stating it without overclaiming the math. The relation in Eq. (13) is just the Legendre transform, but the emphasis on the scale-cut violation and the Fisher comparison is a useful contribution for survey design.\n\nThe soft spot is the partial-sky extension. The Fisher analysis sets fsky=1 throughout and varies theta_max by discarding large-separation pairs in an all-sky survey. That does not model a survey that only covers those separations. In a real partial survey, the mean density is estimated within the footprint and the window suppresses the low-ell modes that carry the PNG signal; the direction of the effect is not benign. The paper acknowledges the window effect and calls it 'straightforward,' but it does not compute it. For theta_max = 10 deg, the effective fsky is about 0.008, which would degrade the Gaussian covariance by a factor of roughly 11 in sigma, turning the z=4-7 forecast of sigma_fNL ~ 1.5 into something like 15. So the abstract's implication about partial area coverage is an extrapolation, not a derived result. The qualitative full-sky point survives, but the quantitative forecasts are optimistic.\n\nA minor caveat: the forecasts assume the linear-theory scale-dependent bias model holds below the nonlinear scale and use Gaussian covariance with fixed cosmology. The authors are upfront about this, but the sigma_fNL numbers should be read as idealized upper bounds.\n\nThis paper is for people planning fNL analyses with photometric surveys. The first half is a genuinely useful conceptual clarification. The second half needs either removal of the partial-sky implication or a proper window-function treatment. I would send it to peer review, with the recommendation that the authors either moderate the abstract or do the partial-sky calculation properly.","headline":"The all-sky scale-cut equivalence argument is clean and worth publishing, but the abstract's partial-sky claim is an extrapolation from fsky=1 forecasts, not a computed result.","tokens_in":14823,"tokens_out":1647,"would_cite":true,"duration_ms":18790,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["98.80.-k","98.80.Es"],"model":"deepseek-v4-flash","headline":"Scale cuts break the equivalence of galaxy angular power spectrum and correlation function, making small-angle clustering a viable fNL probe.","keywords":["primordial non-Gaussianity","fNL","angular correlation function","angular power spectrum","integral constraint","scale-dependent bias","galaxy clustering","Lyman break galaxies"],"falsifier":"Run a large-volume N-body simulation seeded with a known nonzero $f_{\\rm NL}$ (or with halos assigned the scale-dependent bias of Eq. 4), populate it with a high-redshift galaxy-like sample, measure $w(\\theta)$ on a partial-sky footprint, and compare the recovered $f_{\\rm NL}$ constraint with the paper's Fisher forecast; if the small-angle bins below the nonlinear scale do not retain the predicted $f_{\\rm NL}$ signal, or the realized covariance is larger, the central claim would be refuted.","tokens_in":13770,"feed_emoji":"🔭","tokens_out":7470,"duration_ms":69497,"temperature":0.7,"pith_summary":"The paper argues that the galaxy angular power spectrum and the angular correlation function are mathematically equivalent descriptions of the same clustering, but only when every angular scale is included. Once practical scale cuts are applied, as surveys routinely do to avoid nonlinear and systematic effects, the equivalence breaks: the primordial non-Gaussianity (PNG) signature in $C_\\ell$ is confined to low multipoles, whereas in $w(\\theta)$ the integral constraint spreads the same signal across all angular separations, including small nonlinear scales. This means real-space measurements can retain most of the $f_{\\rm NL}$ information even when restricted to angles below about 10 degrees or to a partial sky. The authors' Fisher forecasts for photometric samples over $0<z<7$, including a Lyman-break-galaxy sample at $4<z<7$, show $\\sigma_{f_{\\rm NL}}$ of order unity for high-redshift samples, suggesting a new practical route to constraining PNG.","feed_headline":"Small-scale galaxy correlation can reveal primordial non-Gaussianity","feed_subtitle":"With the integral constraint, fNL leaks into all angular separations, so partial-sky surveys may hit sigma_fNL ~ O(1).","key_machinery":"The load-bearing mechanism is the integral constraint: in an all-sky survey, the observed two-point statistics are defined relative to the mean galaxy density estimated from the data itself, which removes the monopole $\\ell=0$ (equivalently subtracts $C_{\\ell=0}/4\\pi$ from the correlation function). This subtraction is a single local operation in harmonic space but a non-local, angle-dependent correction in real space, so the PNG-induced low-multipole power is redistributed across all angular separations of $w(\\theta)$. Combined with the scale-dependent bias $\\Delta b(k)\\propto f_{\\rm NL}/k^2$ of Eq. (4), this makes small-angle and partial-sky correlation measurements carry $f_{\\rm NL}$ information that harmonic-space scale cuts would discard.","core_discovery":"The central claim is that the observed correlation function and observed power spectrum are equivalent, $\\widetilde w(\\theta) = \\sum_{\\ell\\ge0} \\frac{2\\ell+1}{4\\pi} \\widetilde C_\\ell P_\\ell(\\cos\\theta)$ (Eq. 13), once the integral constraint is imposed by dropping the monopole. Yet this equivalence is unstable in practice: the local-type PNG signal in $C_\\ell$ is a low-multipole phenomenon, while the same signal enters $w(\\theta)$ as an additive, angle-dependent correction from the integral constraint, so it appears at all separations, even below the nonlinear scale. Consequently, when scale cuts remove the largest scales, $C_\\ell$ loses most of its $f_{\\rm NL}$ sensitivity while $w(\\theta)$ keeps it; the paper forecasts that $w(\\theta)$ measured up to about 10 degrees for a high-redshift sample ($4<z<7$) yields $\\sigma_{f_{\\rm NL}}\\sim O(1)$ and that combining all bins can reach $\\sigma_{f_{\\rm NL}}\\sim 1$. The paper further notes that the same mechanism should apply to three-dimensional clustering, since any finite survey must estimate its mean density from itself.","pith_inferences":["By the same mechanism, any configuration-space statistic that estimates the mean from the same data will spread a low-wavenumber signal across all scales; this could be exploited for other scale-dependent signatures (e.g., modified gravity or massive neutrinos), not just local PNG.","The paper's logic implies a testable strategy: instead of discarding small-angle correlation data to avoid nonlinear bias, one could keep it and marginalize over nonlinear nuisance parameters, since exactly those scales carry the PNG signal.","A natural extension is to replace the $f_{\\rm sky}=1$ Gaussian covariance with the true survey window function; the paper approximates partial-sky effects, so a full window-function treatment may shift which angular bins dominate the $f_{\\rm NL}$ signal.","If validated, these forecasts suggest that wide but shallow surveys optimized for photometric galaxies at $z\\gtrsim4$ could be competitive with spectroscopic surveys for $f_{\\rm NL}$, because the needed angular scales are modest."],"forward_implications":["A high-redshift galaxy sample ($4<z<7$) can reach $\\sigma_{f_{\\rm NL}}\\sim O(1)$ from $w(\\theta)$ measured only up to about 10 degrees, nearly matching the full-sky harmonic-space result.","Combining all six photometric redshift bins, small-angle $w(\\theta)$ gives $\\sigma_{f_{\\rm NL}}\\sim 1$, comparable to the $C_\\ell$ analysis, when the nonlinear scale is included.","For partial-sky surveys, real-space analysis is more efficient than harmonic-space analysis, because the PNG signal is not concentrated on the largest scales that are hardest to recover from a cut sky.","The equivalence between $C_\\ell$ and $w(\\theta)$ Fisher forecasts is recovered only when $w(\\theta)$ reaches scales below the nonlinear scale; cutting $w(\\theta)$ at the nonlinear scale loses most of the $f_{\\rm NL}$ information.","The same non-local integral-constraint redistribution should appear in 3D correlation functions of spectroscopic surveys, suggesting small-scale 3D clustering also carries PNG information, with radial modes adding further constraining power."],"supporting_citations":[{"why":"Introduces the scale-dependent bias $\\Delta b(k)\\propto f_{\\rm NL}/k^2$ that produces the PNG signal in galaxy clustering.","marker":"[5]"},{"why":"Provides the Gaussian covariance matrix for $C_\\ell$ used to compute the Fisher errors on $f_{\\rm NL}$.","marker":"[30]"},{"why":"Defines the nonlinear scale through the variance threshold $\\epsilon=0.3$, setting the scale cuts in the forecasts.","marker":"[33]"},{"why":"Supplies the observed number densities of high-redshift Lyman break galaxies that define the $4<z<7$ sample.","marker":"[29]"},{"why":"Supplies the survey-like galaxy redshift distribution used for the lower-redshift tomographic bins.","marker":"[27]"},{"why":"Provides the FFTLog algorithm used to evaluate the $k$-integrals in the angular power spectrum.","marker":"[21]"}],"fun_headline_variants":["Small-scale galaxy correlation captures fNL when power spectrum fails","Integral constraint lets w(θ) reveal fNL on small angular scales","Partial-sky surveys can constrain fNL via small-angle galaxy correlations","For high-z galaxies, w(θ) up to 10° measures fNL to ~1"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The forecast assumes the linear scale-dependent bias model of Eqs. (3)-(5) stays valid down to the small angular scales included, and that the Gaussian, $f_{\\rm sky}=1$ covariance is an accurate error model; if nonlinear galaxy bias or unmodeled small-scale systematics contaminate those scales, the projected precision on $f_{\\rm NL}$ would not be reached.","fun_headline_variants_meta":{"raw":{"variants":["Small-scale galaxy correlation captures fNL when power spectrum fails","Integral constraint lets w(θ) reveal fNL on small angular scales","Partial-sky surveys can constrain fNL via small-angle galaxy correlations","For high-z galaxies, w(θ) up to 10° measures fNL to ~1"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00035,"raw_usage":{"total_tokens":1965,"prompt_tokens":1051,"completion_tokens":914,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":667,"completion_tokens_details":{"reasoning_tokens":832}},"tokens_in":667,"tokens_out":914,"duration_ms":10308,"temperature":1.0,"reasoning_tokens":832,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T16:56:36.371558+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a large-volume N-body simulation seeded with a known nonzero $f_{\\rm NL}$ (or with halos assigned the scale-dependent bias of Eq. 4), populate it with a high-redshift galaxy-like sample, measure $w(\\theta)$ on a partial-sky footprint, and compare the recovered $f_{\\rm NL}$ constraint with the paper's Fisher forecast; if the small-angle bins below the nonlinear scale do not retain the predicted $f_{\\rm NL}$ signal, or the realized covariance is larger, the central claim would be refuted.","supporting_citations":[{"cited_title":"Dodelson and F","cited_arxiv_id":null,"evidence_quote":"Introduces the scale-dependent bias $\\Delta b(k)\\propto f_{\\rm NL}/k^2$ that produces the PNG signal in galaxy clustering."}],"review_version":1}