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On the equivalence between galaxy angular correlation function and power spectrum in constraining primordial non-Gaussianity

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Scale cuts break the equivalence of galaxy angular power spectrum and correlation function, making small-angle clustering a viable fNL probe.

desk verdict 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. read the letter →

arxiv 2501.12661 v1 pith:Z4IORIQV submitted 2025-01-22 astro-ph.CO

Ryo Terasawa , Yue Nan , Masahiro Takada This is my paper · ORCID
classification astro-ph.CO PACS 98.80.-k98.80.Es
keywords primordialnon-GaussianityfNLangularcorrelationfunctionpowerspectrumintegralconstraintscale-dependentbiasgalaxyclusteringLymanbreakgalaxies
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

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

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

2 major / 3 minor

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.

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 (2)
  1. [Abstract, Section IV.A.2, Section V] 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.
  2. [Section IV.A.3 and Fig. 3] 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.
minor comments (3)
  1. [Table I] 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.
  2. [Notation throughout] 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.
  3. [Section IV.A.2, Eq. (18)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central equivalence is an algebraic identity and the fNL forecasts are parameter-free Fisher projections from an assumed fiducial model.

full rationale

The paper's main derivation, Eqs. (9)-(13), defines the integral-constrained power spectrum and correlation function by subtracting the monopole and then notes that the two are Legendre transforms of each other. This is an identity, and the paper does not dress it up as an empirically fitted prediction. The substantive claim—that scale cuts break the equivalence because the PNG contribution to C_l is confined to low multipoles while the integral-constraint subtraction spreads that signal across all angular separations in w(theta)—follows from the Legendre transform and the k^{-2} behavior of the scale-dependent bias, not from any fitted parameter. The Fisher forecasts use a stated fiducial cosmology, assumed bias and number densities, and Gaussian covariance with fsky=1; no observational data are fitted and then renamed as a prediction. The suggestion that w(theta) from partial sky coverage could be useful is an extrapolation from the full-sky theta_max variation and is explicitly qualified by the remark that survey window effects are not modeled; that is a scope/correctness concern, not circularity. Self-citations (e.g., Kurita & Takada 2023 for window effects, Takada & Jain 2003 for pair counting) appear only as background methodology and are not load-bearing for the paper's core result. The paper is therefore self-contained against the benchmark of its own claims, and no derivation step reduces to its inputs.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new particles, forces, or physical entities. Its central result rests on standard cosmology inputs and several clearly stated modeling choices: the local-type PNG bias model, the all-sky integral constraint, and a Gaussian covariance. The qualitative claim that scale cuts break the C_l / w(theta) equivalence is independent of these choices, while the quantitative sigma_fNL forecasts are sensitive to them.

free parameters (3)
  • Linear galaxy bias b1(z)=1+z = 1+z
    Fiducial bias assumed for all tomographic bins; the Fisher analysis marginalizes over b1 in each bin, so the projected fNL errors depend on this choice and on the neglect of bias evolution within bins.
  • Nonlinear-scale threshold epsilon = 0.3
    Chosen following Senatore and Zaldarriaga to define k_NL; this sets which multipoles and angles count as linear and therefore directly affects the scale-cut forecasts.
  • PNG bias parameter p = 1
    Adopted in Eq. (4) following the review by Biagetti; a convention that shifts the amplitude of the scale-dependent bias but does not change the qualitative conclusion.
assumptions (6)
  • standard math Legendre transform relation w(theta) = sum (2l+1)/(4pi) C_l P_l(cos theta) and orthogonality of Legendre polynomials
    Used in Eqs. (10)-(13) to derive the integral constraint in real space and the equivalence relation between C_l and w(theta).
  • domain assumption Local-type PNG model Phi = phi + fNL(phi^2 - <phi^2>)
    Defines the target signal; the scale-dependent bias formula in Eq. (4) follows from this model and from Dalal et al.
  • domain assumption Scale-dependent bias formula Delta b = 2(b1-p) fNL delta_c / alpha(k)
    Adopted from Dalal et al. via Eq. (4); the value p=1 is a convention from the literature, and the forecast precision depends on this model.
  • domain assumption All-sky survey with fsky=1 and exact integral constraint as monopole subtraction
    The central derivation applies to an idealized full-sky survey; partial-sky window effects are deferred to future work.
  • domain assumption Gaussian covariance with shot noise, no non-Gaussian or cross-bin covariance
    Eqs. (16)-(17) define the error model; this is optimistic but standard for Fisher forecasts.
  • domain assumption Linear theory plus variance threshold epsilon=0.3 defines the linear regime
    Used to set k_NL and the scale cuts; nonlinear corrections up to about 30 percent are neglected.

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Pith. "Pith review of On the equivalence between galaxy angular correlation function and power spectrum in constraining primordial non-Gaussianity." pith.science (2026). https://pith.science/paper/Z4IORIQV

@misc{pith2026250112661,
  author       = {Pith},
  title        = {Pith review of: On the equivalence between galaxy angular correlation function and power spectrum in constraining primordial non-Gaussianity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z4IORIQV}},
  note         = {Machine review of arXiv:2501.12661}
}
abstract

We investigate the angular power spectrum ($C_\ell)$ and angular correlation function ($w(\theta)$) of galaxy number density field in the presence of the local-type primordial non-Gaussianity (PNG), explicitly accounting for the integral constraint in an all-sky survey. We show that the PNG signature in $C_{\ell}$ is confined to low multipoles in the linear regime, whereas its signature in $w(\theta)$ extends across a wide range of angular scales, including those below the nonlinear scale. Therefore, the equivalence between $C_\ell$ and $w(\theta)$ can be violated when scale cuts of multipoles or angular scales -- for example, to mitigate systematic effects -- are applied in the analysis. Assuming samples of photometric galaxies divided into multiple redshift bins in the range $0<z<7$, we forecast the precision of constraining the PNG parameter ($f_{\rm NL}$) from the hypothetical measurements of $C_\ell$ or $w(\theta)$ assuming different scale cuts in the multipoles or angular scales, respectively. Our results imply that the PNG information can be extracted from $w(\theta)$ on relatively small angular scales such as $\lesssim 10$ degree for a high-redshift galaxy sample or from $w(\theta)$ measured in a survey with partial area coverage.

Figures

Figures reproduced from arXiv: 2501.12661 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Sensitivity of the angular power spectra, [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The marginalized [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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