Recognition: 2 theorem links
· Lean TheoremThe contribution from small scales on two-point shear analysis: comparison between power spectrum and correlation function
Pith reviewed 2026-05-16 20:49 UTC · model grok-4.3
The pith
Harmonic-space cosmic shear power spectra produce smaller S8 biases from uncertain small-scale modeling than real-space correlation functions.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
When the analysis is extended to smaller scales, the harmonic-space power spectrum yields cosmological constraints on S8 that remain more stable across choices of baryonic feedback model and intrinsic alignment model than the real-space correlation function. Model-driven shifts in S8 are two to three times smaller in harmonic space. The BACCO emulator combined with TATT produces the most consistent constraints between the two spaces, whereas HMCode-2016 generates an approximately 1.1 sigma tension between them.
What carries the argument
The Bessel-function mapping between harmonic and real space, which converts hard scale cuts in one representation into soft cuts in the other and causes small-scale astrophysical effects to imprint differently on the two statistics.
If this is right
- Flexible simulation-based emulators such as BACCO paired with TATT yield consistent cosmological constraints across analysis spaces even when small scales are included.
- Standard models such as HMCode-2016 can produce apparent tensions between harmonic-space and real-space results.
- Real-space statistics separate baryonic feedback effects more clearly and will help distinguish feedback models in future surveys.
Where Pith is reading between the lines
- Analyses could prioritize harmonic space for primary cosmological constraints to minimize model-dependent shifts while reserving real space for cross-checks and model discrimination.
- The differing robustness could guide calibration of small-scale models so that they produce consistent results in both spaces by construction.
- In higher-precision data from upcoming surveys, the clearer separation of baryonic effects in real space may become the practical route to testing feedback scenarios.
Load-bearing premise
The chosen baryonic feedback models, intrinsic alignment models, and the Bessel-function correspondence of scale cuts together capture the true small-scale astrophysics without leaving systematic residual biases that differ between the two statistics.
What would settle it
A new baryonic feedback model validated against high-resolution hydrodynamical simulations that removes the 1.1 sigma tension between spaces when HMCode-2016 is replaced, or that reverses the relative size of S8 shifts between harmonic and real space.
read the original abstract
A known problem in cosmic shear two-point statistics is the apparent inconsistency between analyses performed in harmonic space (power spectrum) and real space (angular correlation). This arises mainly from two factors: first, scale cuts in one space correspond to soft cuts in the other, as the relationship between the two spaces is mediated by Bessel functions. For the same reason, astrophysical effects that are compact in one space may not be in the other, which can lead to biased parameter estimates. In this paper, we argue that these two statistics are complementary: we expect a robust theory to provide consistent constraints regardless of the chosen scale cuts. We present the consequences of pushing our analysis to smaller scales in both spaces, accounting for different models of Intrinsic Alignment and Baryonic Feedback in HSC Y3 data: we find that the harmonic-space analysis is significantly less sensitive to the specific modeling of small-scale physics, with model-choice-driven biases in $S_8$ being 2-3 times smaller than in real space. We show that using a flexible, simulation-based emulator for baryonic feedback (BACCO) in combination with the TATT model for intrinsic alignments provides the most consistent cosmological constraints between the two spaces when pushing to the smallest scales. In contrast, the standard HMCode-2016 model results in a $\sim 1.1\sigma$ tension between the two statistics. While harmonic space appears more robust for cosmological inference given current model uncertainties, real-space analyses offer a clearer separation of baryonic effects and will play a crucial role in distinguishing between baryonic feedback models in upcoming surveys.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper compares cosmic shear two-point statistics in harmonic space (power spectrum) versus real space (angular correlation function) on HSC Y3 data. It examines the impact of pushing to smaller scales while varying models for baryonic feedback (BACCO emulator vs. HMCode-2016) and intrinsic alignments (TATT), finding that harmonic-space constraints on S8 exhibit 2-3 times smaller model-driven biases than real-space analyses. BACCO+TATT yields the most consistent cosmological parameters between the two spaces, whereas HMCode-2016 produces ~1.1σ tension; the work concludes that harmonic space is more robust under current small-scale uncertainties while real space aids model discrimination.
Significance. If the central comparison holds after verification of scale-cut correspondence and model residuals, the result would usefully demonstrate the complementarity of the two statistics for cosmic shear. It would support preferring harmonic-space analyses for robustness in the presence of baryonic and IA uncertainties, while highlighting real-space utility for distinguishing feedback models in Stage-IV surveys. The explicit use of a flexible simulation-based emulator (BACCO) strengthens the analysis relative to fixed parametric models.
major comments (3)
- [§3] §3 (scale cuts and correspondence): the claim that harmonic-space analysis is intrinsically less sensitive rests on the Bessel-function-mediated correspondence between scale cuts; however, the manuscript does not quantify the residual mismatch in effective k-sensitivity after the soft cuts are applied, leaving open whether the reported 2-3× factor arises from the statistic itself or from how HMCode-2016’s specific suppression couples to the real-space kernel.
- [§5] §5 (model comparison results): the factor of 2-3 smaller S8 biases in harmonic space is shown only for the BACCO vs. HMCode-2016 swap; without an additional baryon prescription whose small-scale shape differs qualitatively from both (e.g., a model with stronger scale-dependent suppression at k>5 h Mpc⁻¹), it remains unclear whether the relative robustness is general or specific to the chosen models’ residuals.
- [§4–5] §4–5 (tension calculation): the reported ~1.1σ tension between spaces under HMCode-2016 is presented without explicit details on whether the covariance matrix includes cross-space correlations or accounts for the shared data vector; this affects whether the tension value is load-bearing for the robustness conclusion.
minor comments (2)
- [Figure 3] Figure 3 (or equivalent results figure): axis labels and legend entries should explicitly state which curve corresponds to each model combination (BACCO+TATT, HMCode+TATT, etc.) to avoid ambiguity when comparing S8 shifts.
- [§2] The abstract and §2 cite the correspondence via Bessel functions but omit a reference to the specific prior work that first quantified the soft-cut effect for cosmic shear; adding this citation would improve traceability.
Simulated Author's Rebuttal
We thank the referee for the constructive and detailed comments, which have helped us improve the clarity and robustness of our analysis. We address each major comment below and have revised the manuscript accordingly where possible.
read point-by-point responses
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Referee: [§3] §3 (scale cuts and correspondence): the claim that harmonic-space analysis is intrinsically less sensitive rests on the Bessel-function-mediated correspondence between scale cuts; however, the manuscript does not quantify the residual mismatch in effective k-sensitivity after the soft cuts are applied, leaving open whether the reported 2-3× factor arises from the statistic itself or from how HMCode-2016’s specific suppression couples to the real-space kernel.
Authors: We agree that explicitly quantifying the residual mismatch strengthens the interpretation. In the revised manuscript we have added a new paragraph and accompanying figure in §3 that computes the effective k-window functions for both statistics after the chosen scale cuts. These show that the mismatch in k-sensitivity is small (<10% in the relevant range) and cannot account for the factor of 2–3 difference in S8 biases. This confirms that the reduced sensitivity is intrinsic to the harmonic-space statistic rather than an artifact of the specific model coupling. revision: yes
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Referee: [§5] §5 (model comparison results): the factor of 2-3 smaller S8 biases in harmonic space is shown only for the BACCO vs. HMCode-2016 swap; without an additional baryon prescription whose small-scale shape differs qualitatively from both (e.g., a model with stronger scale-dependent suppression at k>5 h Mpc⁻¹), it remains unclear whether the relative robustness is general or specific to the chosen models’ residuals.
Authors: We acknowledge that testing an additional qualitatively different baryonic model would further generalise the result. BACCO is a flexible simulation-based emulator spanning a range of feedback strengths, while HMCode-2016 is a widely used parametric prescription; their small-scale residuals are already distinct in both amplitude and scale dependence. We have expanded the discussion in §5 to explain why these two models are representative of current uncertainties and to note the limitation. Adding a third model would require new high-resolution simulations outside the scope of the present work. revision: partial
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Referee: [§4–5] §4–5 (tension calculation): the reported ~1.1σ tension between spaces under HMCode-2016 is presented without explicit details on whether the covariance matrix includes cross-space correlations or accounts for the shared data vector; this affects whether the tension value is load-bearing for the robustness conclusion.
Authors: We thank the referee for this important clarification. The ~1.1σ tension is computed from the joint covariance matrix of the combined harmonic- and real-space data vectors, which explicitly includes the cross-covariance terms arising from the shared HSC Y3 data. We have now added the relevant equations and a short paragraph in §4 describing the covariance construction and tension formula. revision: yes
Circularity Check
No significant circularity in derivation chain
full rationale
The paper conducts an empirical comparison of cosmic shear two-point statistics on HSC Y3 data by applying external baryon models (BACCO emulator and HMCode-2016) and IA models (TATT) while varying scale cuts in both harmonic and real space. The central result—that harmonic-space S8 biases are 2-3 times smaller—follows directly from the computed parameter shifts under these model swaps and is not obtained by defining any quantity in terms of itself, fitting a parameter to a subset and relabeling it a prediction, or invoking a self-citation chain as the sole justification. The scale-cut correspondence via Bessel functions is stated as a known property and used to set comparable cuts, without circular reduction. The analysis remains self-contained against external public data and models.
Axiom & Free-Parameter Ledger
free parameters (2)
- baryonic feedback parameters
- intrinsic alignment parameters
axioms (1)
- standard math Scale cuts in harmonic space map to soft cuts in real space via Bessel functions.
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/AlexanderDuality.leanalexander_duality_circle_linking unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
the relationship between the two spaces is mediated by Bessel functions... astrophysical effects that are compact in one space may not be in the other
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
HMCode-2016... BACCO emulator... TATT model for intrinsic alignments
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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2PCF results for HSC-Y3 ( θmin = 0 .3′) shown in shaded orange. Right: suppressions inferred from harmonic-space analyses, highlighting this paper’s HMCode+TA TTand BACCO+T A TT results (ℓmax = 14200) alongside the recent constraints from García-García et al.[ 19] (green), obtained by combining HSC DR1, DES Y3, and KiDS-1000 up to ℓmax = 8192. – 30 –
discussion (0)
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