Quantifying the accuracy of the Alcock-Paczynski scaling of baryon acoustic oscillation measurements

Asta Heinesen; Chris Blake; David L. Wiltshire

REVIEW 3 major objections 4 minor 2 cited by

Quantifying the accuracy of the Alcock-Paczynski scaling of baryon acoustic oscillation measurements

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that the standard constant AP scaling of BAO measurements is accurate only when fiducial and true metric gradients are comparable, and that replacing the scaling parameters by their survey averages is more accurate; it…

desk verdict A careful, genuinely useful BAO systematics paper: the new survey-averaged AP scaling and the ~1% residual deserve referee time, though the residual's origin is not fully pinned down. read the letter →

arxiv 1908.11508 v2 pith:AMEGI2HN submitted 2019-08-30 astro-ph.CO gr-qc

Asta Heinesen , Chris Blake , David L. Wiltshire This is my paper · ORCID

classification astro-ph.COgr-qc

keywords baryonacousticoscillationsAlcock–PaczyńskiscalinggalaxyclusteringfiducialcosmologyBAOdistancescaletimescapemodelcorrelationfunctioncosmologicaldependence

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

This paper asks whether the standard Alcock–Paczyński scaling used to convert baryon acoustic oscillation measurements between cosmologies is trustworthy. It argues that the usual practice of evaluating the two AP scaling parameters at the survey's mean redshift is only valid when the fiducial and true metrics have similar gradients. It proposes replacing those values with redshift-averaged parameters, which it shows recovers the BAO scale accurately even for toy models with large metric oscillations. It also finds a residual systematic error of about 1% in the recovered BAO scale when the true model is far from the fiducial, independent of the fitting method, meaning BAO distance errors in the standard literature may be underestimated for models outside the concordance cosmology.

What carries the argument

The engine of the argument is the small-separation geodesic distance approximation $D_T^2 \approx g_{zz}(\delta z)^2 + g_{\theta\theta}(\delta\Theta)^2$, which lets any spherically-symmetric template metric be parametrised relative to a fiducial one by the redshift-dependent AP functions $\alpha(z)$ and $\epsilon(z)$. From these, the paper constructs the exact redshift-dependent map of the 2-point correlation function and its wedge integrals, then expands the redshift average to first order; this expansion identifies the survey-averaged parameters $\bar\alpha$ and $\bar\epsilon$ as the correct constant parameters. The empirical Gaussian-plus-polynomial model of the BAO feature, fitted to the mean correlation function of mock catalogues, converts the analytic predictions into measured peak shifts and warping parameters.

What would settle it

Recompute the 'exact' redshift-dependent AP predictions for the toy models $D(z)[1 + A\cos(f z + \Phi)]$ using full geodesic distances rather than the local-metric approximation. If the resulting BAO peak shifts differ from the paper's benchmark by more than about $0.1\%$ at separations near $100\,\mathrm{Mpc}/h$ for the $f = 30$ and $f = 50$ cases, then the benchmark itself is approximate and the stated accuracy of the modified AP scaling for all test models is not established.

Watch

Extended reading notes

Core claim

The paper's central claim is that the conventional constant Alcock–Paczyński approximation, which takes the two scaling functions at the survey's mean redshift, is only a special case of a more accurate constant choice: the survey-averaged values. In the spherically-symmetric template framework, a first-order expansion of the redshift-integrated correlation function identifies the survey-averaged parameters as the ones that enter the effective BAO template. On mock catalogues the two choices are nearly indistinguishable for smooth large-scale cosmologies, but the modified choice remains accurate for toy models with large metric oscillations where the conventional choice fails. The paper also finds a residual systematic error of about 1% in the recovered BAO scale when the true model differs from the fiducial by roughly ten percent in its distance measures, present even when the exact redshift-dependent AP functions are used, and similar for both the empirical Gaussian fit and the standard template fit; it therefore presents this as a systematic that must be added to the error budget when BAO distances are extrapolated beyond the fiducial cosmology.

Load-bearing premise

The load-bearing premise is the small-separation approximation that treats the distance between galaxy pairs as if the metric components were constant across the pair; the paper verifies it for FLRW and timescape models but applies it without independent verification to the high-frequency toy models where the metric oscillates on scales comparable to the pair separation.

Editorial extensions

If this is right

Published constant AP parameters from BAO analyses should be read as survey-averaged distance ratios, not as values at a single effective redshift.
For pairs of smooth large-scale cosmologies the conventional constant AP approximation is adequate, so existing concordance-model results do not need to be redone.
When a BAO measurement is used to constrain a model whose distance measures differ from the fiducial by about ten percent, a systematic error of roughly one percent in the acoustic scale should be added to the error budget.
The modified AP scaling can be applied to existing analyses by reinterpreting their fitted parameters, which matters for models with significant curvature gradients.
The standard approach breaks down when the true metric has large gradients relative to a smooth fiducial metric; in such cases evaluating AP parameters at the mean redshift misstates the effective distance ratios.

Reading between the lines

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

The same Taylor-expansion argument suggests the survey-averaging correction applies to any redshift-dependent distance scale fitted through a correlation-function wedge, so the reinterpretation of AP parameters may carry over to other clustering statistics.
Because the ~1% residual appears even with exact redshift-dependent scaling, BAO-only constraints on models far from the fiducial cosmology are limited by a systematic of the same order as current statistical errors; combining probes reduces statistics, not this model-dependence.
A natural testable extension is to reverse the roles of true and fiducial models in the toy-model mock analysis; the paper expects its conclusions to hold under reversal but does not demonstrate it.
If the local-metric approximation fails for the high-frequency toy models, both the conventional and modified constant AP methods inherit the same benchmark error, so the modified method's reported advantage in those cases would need to be re-evaluated.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper does something concrete: it proposes replacing the usual constant AP scaling parameters evaluated at the mean survey redshift with survey-averaged parameters (ᾱ, ε̄), derives analytic bounds on the difference between the two choices, and tests both on ΛCDM mocks and on toy models with large metric gradients. The mock work uses up to 1000 QPM catalogues with known true cosmology, and the authors cross-check their empirical fitting model against a ΛCDM template fitter. The analytic bounds in Eqs. (3.9) and (3.14) are real results, not placeholders, and the finding that the standard constant AP approximation breaks down for large metric gradients while the modified version stays accurate is well supported for the tested toy models.

The main substantive finding is the ~1% systematic in BAO scale recovery when the true model is far from the fiducial, which persists even when using the exact redshift-dependent AP scaling. The authors are careful to say this is not attributable to any constant AP approximation. I think that claim is as well supported as it can be given their setup, because the exact calculation shows the same shift. But the stress-test worry is legitimate: the exact calculation still shares the standard-ruler assumption and the empirical Gaussian shape (or the fixed template), so the residual could in principle come from those modeling choices rather than from a real failure of the AP framework. The authors do not identify the source of the 1%, and they acknowledge the standard-ruler approximation itself can shift the BAO scale by up to ~0.5% in ΛCDM. That is not a fatal flaw, but it means the headline quantitative claim is not yet as clean as the abstract suggests.

A smaller soft spot: the local-metric approximation in Eq. (2.4) is verified for FLRW and timescape but not for the high-frequency toy models where the metric oscillates on scales comparable to pair separations. This matters for the toy-model demonstration of the modified AP scaling's superiority, but not for the main ~1% residual, which appears in smooth large-scale models.

The paper is honest about its limitations: the modified AP scaling's superiority is argued heuristically in Appendix B, the toy models are unphysical, and the analysis is restricted to spherically symmetric template metrics. None of these are hidden.

Overall, this is a solid methodological study. The new AP scaling variant and the bounds are worth having, and the ~1% warning is important even if its ultimate origin remains unclear. A serious referee should engage with it. I would send it to review, and I would probably cite it in future BAO systematics work.

Referee Report

3 major / 4 minor

Summary. The paper investigates the validity of the conventional constant Alcock-Paczyński (AP) scaling used in baryon acoustic oscillation (BAO) analyses. It derives analytic bounds on the difference between the standard approximation, in which the AP parameters α and ϵ are evaluated at the survey mean redshift, and a modified approximation in which they are replaced by survey-averaged values ᾱ and ϵ̄. The authors propose that published AP parameters should be reinterpreted as survey-averaged distance ratios, and they test this proposal on QPM mock catalogues for a range of ΛCDM, timescape, and toy-model cosmologies. They report systematic errors of about 1% in the recovery of the BAO scale when the fiducial and true cosmologies are distant, and show that these errors persist when the full redshift-dependent AP functions are used instead of constant parameters and that a similar trend appears with a ΛCDM template fitting procedure.

Significance. If correct, the paper has two significant consequences. First, the constant AP parameters reported in the literature are more accurately interpreted as survey-averaged ratios rather than values at a single effective redshift; this matters for models with large metric gradients. Second, BAO-derived distance measurements carry a model-dependent systematic at the percent level when extrapolated beyond the fiducial cosmology, a systematic that is not captured by the constant-AP approximation itself. The analytic bounds in Eqs. (3.9) and (3.14) are parameter-free and useful, the mock tests use 200–1000 QPM catalogues with estimated covariance, and the ΛCDM template cross-check provides a nontrivial robustness test. The main limitations are that the residual systematics are not traced to a specific source and that some claims rest on assumptions shared by the two fitting procedures.

major comments (3)

[Sec. 5.3; Sec. 2.3] The paper's central claim that the ~1% systematic 'cannot be attributed to any constant AP approximation' is supported by comparing fits with constant AP parameters to fits using the redshift-dependent functions (5.8). However, both fitting models assume a redshift-independent BAO scale r and both use the same approximate redshift integral (3.3). Section 2.3 acknowledges that the standard-ruler assumption has shifts below ~0.5% in ΛCDM at z>0.3, which is comparable to the claimed 1% effect at |ᾱ−1|~0.1. The ΛCDM template cross-check fixes the template shape and therefore shares the same standard-ruler assumption. Please quantify the impact of a redshift-dependent BAO scale, for example by fitting with a redshift-dependent r or by using mocks where the BAO scale variation is known, or explicitly scope the abstract and Section 6 claims to the standard-ruler ansatz.
[Sec. 5.4; Eq. (2.4)] The local-metric approximation (2.4) is verified in Section 2.1 only for FLRW and timescape models. It is then applied without verification to the toy models of Section 5.4, whose distance-redshift relations oscillate as A cos(fz+Φ) with f up to 50. For a radial pair at z≈0.5 with separation 100 Mpc/h, δz≈0.03, so f δz≈1.5 and the metric varies substantially across the pair. In this regime the 'exact' redshift-dependent AP predictions (5.8) used as the benchmark in Figure 7b are themselves approximate, and the statement that the modified constant AP scaling is efficient for all tested models rests on an unvalidated premise. Please verify (2.4) for the toy-model parameters, for instance by direct numerical geodesic integration, or restrict the toy-model conclusions to the regime where (2.4) is accurate.
[Abstract; Sec. 6] The statement that the level of systematic uncertainty is 'robust to the exact fitting method employed' is stronger than the evidence presented: only two fitting methods are compared, and both use a fixed template or a fixed standard ruler. The evidence supports robustness to the two fitting methods considered. Please rephrase the claim accordingly, or add a third, less parametric BAO extraction method.

minor comments (4)

[Sec. 5.4] The paragraph following Figure 7a refers to 'figure 3b' when comparing the modified constant AP scaling to the exact AP scaling; this should be 'figure 7b'.
[Sec. 2.1] The claimed verification that corrections to Eq. (2.4) are below 10^-3 for FLRW and timescape models is stated without quantitative details; please provide the numerical values or a reference to a figure or table.
[Sec. 5.1] The text says that 1000 QPM mock catalogues are used for the reference model and 200 for the remaining trial cosmologies; please ensure the sample sizes are stated consistently in the figure captions and in the text describing the fits.
[Introduction] There is a typo in the first paragraph of Section 1: 'use of of N-body mock catalogues' should read 'use of N-body mock catalogues'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: claims are benchmarked against mock catalogues with known true cosmology, and the modified AP scaling follows from an explicit Taylor expansion rather than from the data it explains.

full rationale

The paper's central claims are tested against QPM mock catalogues with a known reference cosmology (Omega_M = 0.29), providing an external benchmark for comparing the standard constant AP, modified constant AP, and redshift-dependent 'exact' AP expressions. The modified AP approximation is introduced via the first-order expansion (3.4), where the survey-averaged parameters (bar-alpha, bar-epsilon) are defined by (3.5); the claim that using these averages removes the first-order error is an analytic consequence of Taylor's theorem, not a parameter fitted to the final answer. The bounds (3.9) and (3.14) are derived directly from Taylor remainders and triangle inequalities, without invoking the target result. The residual ~1% systematics are measured relative to the calibrated reference scale r = 102.47 Mpc/h, which is determined from the same mocks; this calibration is a standard device to remove an overall offset, and it does not pre-determine the residual trend as a function of (bar-alpha). The robustness claim is supported by the independent Lambda-CDM-template cross-check (figure 5). Although reference [26] (with overlapping authorship) supplies the empirical fitting model, the model is an explicit ansatz rather than a hidden input, and the key conclusions are confirmed with a different fitting procedure. The unvalidated local-metric approximation for the high-frequency toy models is a correctness concern, not a circular reduction: eq. (5.8) does not reduce to the fitting model by construction, and the toy-model results are not used to justify the theoretical bounds. No fitted parameter is renamed as a prediction, and no load-bearing self-citation chain is present.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claims rest on four domain assumptions: the spherically-symmetric template metric (2.1), the local geodesic-distance approximation (2.4), the near-multiplicative separability of pair counts (A.13), and the redshift-independent standard-ruler ansatz for the BAO peak. The first two are weakest for the high-frequency toy models. Free parameters are mostly nuisance parameters of the empirical fitting model plus hand-chosen toy-model and redshift-distribution inputs; no ad hoc physical constants are introduced. No invented entities are postulated.

free parameters (4)

BAO peak calibration scale r (reference model) = 102.47 Mpc/h
Fitted to the mean isotropic correlation function of 1000 QPM mock catalogues in the flat ΛCDM reference model (section 5.2) and used as the benchmark r_th for all AP error estimates. Its interpretation as the true BAO scale is an ansatz with <1% precision in ΛCDM (section 2.3), so the reported systematic floor is measured relative to a calibration with its own model dependence.
Empirical model shape parameters (A, σ, C0, C1, C2) = not quoted in paper
Nuisance parameters of the empirical fitting function (2.12), fitted to each mock correlation function. A and σ are assumed constant in redshift; the polynomial coefficients are allowed redshift dependence. The adequacy of this empirical model for the toy cosmologies is an ansatz.
Toy-model distortion parameters (A, f, Φ) = A in {0.001, 0.003, 0.005, 0.01}; f in {10, 12, 15, 20, 30, 40, 50}; Φ in {-0.6, -0.3, 0}
Chosen by hand to construct the artificial metric-gradient cosmologies (3.18)-(3.20) used to test where the standard AP scaling breaks down. They are test-design inputs, not fitted to data, and the selected values are not justified by any physical model.
Truncated Gaussian redshift distribution parameters (μ, σ) = CMASS: μ=0.54, σ=0.075; LOWZ: μ=0.31, σ=0.10
Used to approximate the survey redshift distributions in computing ᾱ, ε̄ and the bounds in section 3.3. The paper notes that using the exact redshift distributions produces nearly identical results.

assumptions (4)

domain assumption The large-scale geometry and galaxy distribution are described by an observer-adapted, spherically-symmetric template metric (2.1)-(2.3), with monotone z(r).
The AP scaling framework, the reduced 2-point correlation function, and all model tests assume this metric ansatz. The authors acknowledge in section 6 that realistic lumpy space-times break spherical symmetry and would require six generalized AP functions instead of two.
domain assumption Local geodesic distance approximation (2.4), neglecting second-order variations of the spatial metric over galaxy-pair separations.
Verified in section 2.1 for FLRW and timescape models (corrections below 10^-3 at 100 Mpc/h) but assumed valid for the high-frequency toy models with f up to 50 without separate verification.
domain assumption Almost multiplicative separability of the pair-count density, f(D,μ,z) ≈ f(D,μ)P(z)(1+δ) with δ and δ_Poisson much less than 1 (A.13), allowing ξ(D,μ) ≈ ∫ dz P(z) ξ(D,μ,z) to first order.
This integral relation (3.3) underlies the fitting models (4.2), (5.6), and (5.8); only first order in the non-separable parts is retained, per Appendix A.
domain assumption The BAO feature peak is a redshift-independent statistical standard ruler of scale r.
Section 2.3 states: 'We assume the peak of the BAO feature r to be a standard ruler independent of redshift.' The paper notes this is good to below 0.5% in ΛCDM and 'less obviously good in non-ΛCDM cosmology,' where it is an ansatz.

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Pith. "Pith review of Quantifying the accuracy of the Alcock-Paczynski scaling of baryon acoustic oscillation measurements." pith.science (2026). https://pith.science/paper/AMEGI2HN

@misc{pith2026190811508,
  author       = {Pith},
  title        = {Pith review of: Quantifying the accuracy of the Alcock-Paczynski scaling of baryon acoustic oscillation measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AMEGI2HN}},
  note         = {Machine review of arXiv:1908.11508}
}

abstract

We investigate - in a generic setting - the regime of applicability of the Alcock-Paczynski (AP) scaling conventionally applied to test different cosmological models, given a fiducial measurement of the baryon acoustic oscillation (BAO) characteristic scale in the galaxy 2-point correlation function. We quantify the error in conventional AP scaling methods, for which our ignorance about the true cosmology is parameterised in terms of two constant AP scaling parameters. We propose a new, and as it turns out, improved version of the constant AP scaling, also consisting of two scaling parameters. The two constant AP scaling methods are almost indistinguishable when the fiducial model used in data reduction and the "true" underlying cosmology are not differing substantially in terms of metric gradients, but are otherwise expected to differ. Our new methods can be applied to existing analyses through a reinterpretation of the results of the conventional AP scaling. This reinterpretation might be important in model universes where curvature gradients above the scale of galaxies are significant. We test our theoretical findings on $\Lambda$CDM mock catalogues. The conventional constant AP scaling methods are surprisingly successful for pairs of large-scale metrics, but eventually break down when toy models allowing for large metric gradients are tested. The new constant AP scaling methods proposed in this paper are efficient for all test models examined. We find systematic errors of ~1% in the recovery of the BAO scale when the true model is distant from the fiducial, which cannot be attributed to any constant AP approximation. The level of systematic uncertainty is robust to the exact fitting method employed. This indicates that caution must be taken with the error budget when extrapolating the BAO acoustic scale measurements obtained in the standard literature.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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