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REVIEW 3 major objections 4 minor 61 references

Light path averages in spacetimes with non-vanishing average spatial curvature

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

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Pith's one-line read Nonzero spatial curvature does not change how light-path averages compare with volume averages in Swiss-cheese models, tests with 1000 light rays show.

desk verdict A careful numerical study whose central claim about 1D and volume averages is not directly tested; still a useful data point. read the letter →

arxiv 1909.00610 v2 pith:XDHWNDME submitted 2019-09-02 astro-ph.CO

S. M. Koksbang This is my paper · ORCID
classification astro-ph.CO
keywords spatialcurvatureSwisscheesemodelsLTBvoidslightpathaveragesvolumeredshift-distancerelationHubblediagraminhomogeneouscosmology
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 tests a worry specific to curved cosmologies: light rays average the universe along one-dimensional lines, and in non-Euclidean space such one-dimensional averages might not converge to the three-dimensional volume averages that background models assume. The test uses Swiss cheese models, exact LTB voids embedded in FLRW backgrounds, with flat, open, and closed cases ($\Omega_{K,0} = -0.2, -0.1, 0, 0.1, 0.2$), tracing 1000 light rays per model out to $z=0.35$. Along each ray the paper samples the accumulated density contrast, the redshift fluctuations from integrated expansion rate and shear, and the fluctuations in angular diameter distance. The mean and dispersion of these quantities are similar across the models, and the small differences track the backgrounds' expansion rates and density contrasts rather than their curvature. The paper concludes that there is no indication that the relationship between one-dimensional light-path averages and volume averages depends significantly on background curvature, leaving flat-background results for observables such as the Hubble diagram applicable to mildly curved universes.

What carries the argument

The machinery is a Swiss cheese model built on the fly: spherically symmetric Lemaitre-Tolman-Bondi (LTB) structures of comoving radius $r_b=40\,\mathrm{Mpc}$ are embedded in FLRW (Friedmann-Lemaitre-Robertson-Walker) backgrounds, with each light ray turned around just outside a structure ($r=r_b+1\,\mathrm{Mpc}$) and sent through it with a random impact parameter; the same sequence of impact parameters is reused for every background so that sample variance is shared between models. The central identity is the decomposition of the observed redshift into background and fluctuation parts, $$1+z=(1+z_{\mathrm{bg}})\exp\left[\int_{t(\$\lambda$)}^{t_0}dt\left(\tfrac{1}{3}\$\Delta$\Theta+$c^{2}$\$\sigma$^\$\alpha${}_\$\beta$ e_\$\alpha$ e^\$\beta$\right)\right],$$ where $\Theta$ is the local expansion rate and $\sigma^\alpha{}_\beta e_\alpha e^\beta$ the shear projected onto the light ray's spatial direction. Along with the accumulated density contrast $\int\delta\,d\lambda/(\lambda_e-\lambda_0)$, these integrals are what connect what a light ray samples to what a volume average would give. Their near-cancellation between shear and expansion-rate terms across all backgrounds is what carries the argument that curvature does not change the statistical relationship.

What would settle it

Recompute the mean accumulated density contrast and the mean $\Delta D_A/D_A$ at $z=0.35$ using identical density-contrast profiles in every background, for example the same $k_{\max}$ or a profile scaled to the background, with the same 1000 impact parameters; if the curved models separate from the flat models by more than the sample variance seen across realizations, the null conclusion fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the finding is a null result with a practical consequence: in Swiss cheese spacetimes with LTB inhomogeneities placed in FLRW backgrounds of curvature $\Omega_{K,0}=-0.2,-0.1,0,0.1,0.2$, the mean accumulated density contrast along light rays converges to approximately $-0.08$ in every model, and the mean redshift and angular-diameter-distance fluctuations differ between models only at levels attributable to the background expansion rate and density contrast. The integrated shear and expansion-rate contributions to the redshift are found to cancel each other almost exactly in all the models, as previously seen in flat-background Swiss cheese studies. The author reads this as evidence that the relation between one-dimensional spatial averages, which approximate light-path averages when structures evolve slowly, and volume averages does not depend significantly on background curvature. Consequently, flat-background Swiss cheese results for the mean and dispersion of distance indicators and local Hubble-parameter estimates should remain valid if the real universe has a small non-zero curvature, including curvature that emerges only at late times through cosmic backreaction.

Load-bearing premise

The conclusion rests on the assumption that the small differences between the models come from background curvature and not from the fact that the inhomogeneities were given slightly different strengths in each model.

Editorial extensions

If this is right

  • Flat-background Swiss cheese results for the mean and dispersion of Hubble-diagram observables remain statistically applicable if the universe has a small non-zero spatial curvature.
  • Small differences between curved and flat models can be attributed to differences in background expansion rate and density contrast, so those factors, not curvature, should be adjusted when comparing models.
  • Local Hubble-parameter estimates and Hubble-diagram scatter computed in flat Swiss cheese models do not need to be recomputed for mildly curved backgrounds.
  • The near-exact cancellation between integrated shear and expansion-rate fluctuations holds across backgrounds of different curvature, making it a robust feature of these models.

Reading between the lines

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

  • Editorial inference: because the paper uses a single void scale and an on-the-fly packing scheme, the null result may not survive in a multi-scale cosmic web; a curved-background Swiss cheese with several void sizes, or a curved N-body simulation with matched density contrasts, would test this.
  • Editorial inference: the small positive tails in the accumulated density contrast seen for the negative-curvature models are the most promising place to look for a genuine curvature effect; tracing many more than 1000 rays per model would show whether they are only sample variance.
  • Editorial inference: the paper does not compute volume averages directly in the curved backgrounds, so the strongest form of the conclusion, that one-dimensional averages converge to volume averages in curved space, remains indirect and could be settled by a direct volume-average calculation.
  • Editorial inference: because $k_{\max}$ was tuned separately for each model to keep structures non-linear and free of shell crossings, a curvature effect could be masked; running the same light-ray experiment with a fixed physical density-contrast profile across all backgrounds would make the curvature comparison cleaner.
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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

3 major / 4 minor

Summary. The paper studies whether a non-vanishing spatial background curvature affects the relation between light-path averages and volume averages, motivated by the concern that one-dimensional spatial averages may not converge to volume averages in non-Euclidean space. This is investigated by constructing Swiss cheese models with LTB structures embedded in FLRW backgrounds with different spatial curvature, including two flat models for comparison, and tracing 1000 light rays per model. The author computes the accumulated density contrast, the integrated expansion rate and shear, redshift fluctuations, and angular-diameter-distance fluctuations along the rays, and finds that the mean values and dispersions of these quantities are similar across models. The conclusion is that the study does not indicate a significant dependence of the 1D/volume-average relation on background curvature.

Significance. If the central claim were fully supported, the paper would provide an important justification for applying flat-background Swiss cheese results to universes with small non-zero curvature, and it would address a known theoretical concern about the convergence of one-dimensional spatial averages in curved space. The paper has several strengths: it uses a well-defined numerical setup for light propagation in LTB Swiss cheese models, it controls sample variance by reusing impact-parameter series across models, and Appendix A provides a useful explicit sample-variance study based on five realizations of 1000 light rays. The numerical integrations of the geodesic and transport equations are standard. However, the main conclusion is currently overreaching, because the paper does not actually compute the one-dimensional spatial averages or the volume averages that the abstract's claim is about, and the per-model tuning of kmax introduces a confounding factor.

major comments (3)
  1. [Section IV and Section V] The central claim in the abstract and in Section V is that the study does not yield an indication that the relationship between one-dimensional spatial averages and volume averages depends significantly on background curvature. This claim is not directly tested by the computations presented. The paper computes light-path averages of the accumulated density contrast, integrated expansion/shear, and distance fluctuations along null geodesics, but it never computes either a 1D spatial average at fixed time or a volume average over the curved background. Indeed, in Section IV the author states that 'it is not entirely clear how this quantity should be related to neither 1 dimensional spatial averages nor volume averages.' That statement applies to the accumulated density contrast, the very quantity used to draw the conclusion about 1D versus volume averages. The observed similarity of light-path means across models shows only that these particular ray-tracing statistics are insensitive to background curvature in these models; it does not constrain the convergence of 1D spatial averages to volume averages. To support the abstract, the author should either compute the relevant 1D and volume averages from the model data, or substantially weaken the conclusion to a statement about light-path averages only. This is a load-bearing issue because the paper's stated purpose and title concern the 1D/volume relation.
  2. [Section II, Table I] The comparison across models is partly confounded by the per-model choice of kmax. As shown in Table I, kmax is 5.4 Mpc^-2 for the flat ΛCDM model, 4 Mpc^-2 for ΛCDM2, 5.3 Mpc^-2 for Ω_K,0 = -0.1, and 5.1 Mpc^-2 for Ω_K,0 = -0.2. The author explains that these values are chosen to make the structures nonlinear while avoiding shell crossings, but this means that the present-day density contrasts differ between models (as visible in Fig. 1). Since the accumulated density contrast and the fluctuations in the distance-redshift relation depend directly on the structure amplitude, the small differences between models can partly be absorbed by the kmax tuning. The text acknowledges this trade-off, but no quantitative robustness check is provided. A more convincing test would be to vary kmax in the flat models over a range comparable to the values used in the curved models, or to match nonlinear density contrasts across models by construction, and then verify that the conclusions are unchanged. Without such a check, the interpretation that the observed similarities are due to curvature being unimportant is not fully supported.
  3. [Section II A] The on-the-fly Swiss cheese construction uses a single structure scale with rb = 40 Mpc and a single LTB profile shape. The author argues that a single structure size should be sufficient to determine statistical effects of background curvature, but this is an assumption rather than a demonstrated fact. The geometric effect that governs the relation between 1D spatial averages and volume averages in curved space could plausibly depend on the distribution of structure sizes and on the packing fraction. The paper does not test whether the conclusion changes with rb or with the profile parameters p1 and p2. Since the central claim is a null result, the robustness of that null result to the model choices should be addressed, at least by a short discussion of the expected dependence or by an additional model variant.
minor comments (4)
  1. [Fig. 1 caption] The caption contains a typo: 'are therefor not shown' should be 'are therefore not shown'.
  2. [Eq. (8)] The symbol z_bg is used in Eq. (8) but its explicit definition as the background redshift is only given in the subsequent sentence; the definition should be stated at first use.
  3. [Section IV, Figure 9] The histograms in Figures 9-12 are informative, but the text does not state whether the bin counts are normalized or whether the same number of light rays is used in every histogram. Since each model uses 1000 rays, the bin counts are directly comparable, but a brief statement would help the reader.
  4. [Section V] The final paragraph of the summary says that the results indicate that flat-background results 'regarding e.g. mean and dispersion in H0' remain valid even with small curvature. The paper, however, does not compute the Hubble constant or the H0-problem directly; it computes distance fluctuations and redshift fluctuations. The connection between the computed quantities and H0 estimates should be made explicit, or the sentence should be rephrased to avoid overstating the direct applicability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the ray-tracing results are computed independently of the conclusion; the 1D/volume-average gap is an external-validity limitation, not a definitional reduction.

full rationale

The derivation chain here is a self-contained numerical experiment rather than a formal derivation. The paper fixes the Swiss-cheese construction parameters (rb = 40 Mpc, p1 = p2 = 6, and the per-model kmax values in Table I) on feasibility and non-linearity grounds, not by fitting to the null result; it then ray-traces 1000 light rays per model and compares means and dispersions of the accumulated density contrast, integrated expansion/shear contributions, and Delta D_A (Figs. 5-12). Nothing in Eqs. (1)-(8) defines the target 1D-spatial-average/volume-average relation in terms of the simulated light-path quantities, so the abstract's conclusion is an inductive inference from a proxy rather than a quantity equal to its input by construction. The paper itself flags the proxy gap in Section IV: 'Mainly due to metric measures in spatial averages in curved space, it is not entirely clear how this quantity should be related to neither 1 dimensional spatial averages nor volume averages.' This admission is a limitation on external validity, not circularity, because the compared items are computed independently of the claimed conclusion. Self-citations to the author's earlier work ([25], [53], [56]) are used for context, such as the cancellation of shear and expansion fluctuations and the insensitivity of light propagation to profile details; they are not load-bearing in the sense of making the conclusion true by cited authority. No uniqueness theorem is imported, and no fitted parameter is renamed as a prediction. The per-model kmax variation is a possible confound that could mask a curvature effect, but those values are chosen before the comparison and are justified by shell-crossing and non-linearity constraints, so this is a correctness risk rather than a circular step. No quoted equation or constructed quantity reduces to the paper's own inputs, so the circularity score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central comparison depends on several chosen model parameters, especially kmax, and on assumptions that the Swiss cheese construction is representative. The paper introduces no new particles, fields, forces, or dimensions. The main burden is that model differences are not solely due to curvature.

free parameters (4)
  • kmax (Mpc^-2) = 5.4 (LambdaCDM), 4 (LambdaCDM2), 5.4 (OmegaK=0.1), 5.4 (OmegaK=0.2), 5.3 (OmegaK=-0.1), 5.1 (OmegaK=-0.2)
    Chosen per background to ensure non-linear density contrasts and avoid shell crossings (Table I). This tuning means the models differ in structure amplitude and density contrast in addition to curvature, confounding the comparison.
  • rb (Mpc) = 40
    Sets the comoving size of the LTB inhomogeneity and the homogeneity scale of the Swiss cheese model, chosen as a compromise with observations and computational cost.
  • p1, p2 = 6, 6
    Shape parameters in the curvature profile k(r) chosen to give bucket-shaped voids with non-linear overdensities while keeping computation time manageable.
  • Turnaround radius = rb + 1 Mpc
    Chosen to be close to the structure boundary, increasing the effective packing fraction and thus the inhomogeneity effects sampled by light rays.
assumptions (5)
  • standard math LTB metric and dust plus Lambda field equations describe the structures and background.
    The paper models inhomogeneities with Lemaitre-Tolman-Bondi exact solutions to Einstein's equations (Section II), an accepted general relativity background.
  • domain assumption Under statistical homogeneity and isotropy with slowly evolving structures, light-path averages approximate 1D spatial averages.
    This interpretation, taken from [29,30], is what connects the computed light-path quantities to the 1D versus volume average question.
  • ad hoc to paper A single-scale Swiss cheese construction with on-the-fly placement and rb = 40 Mpc is sufficient to reveal curvature effects at a statistical level.
    The author states that using a single structure size should be sufficient for the statistical question (Section II), but this is not independently established.
  • domain assumption Neglecting radiation when setting initial conditions does not affect the late-time results.
    Initial conditions are set at scale factor 1/1200 without radiation, and the argument is that this is acceptable since the models are only considered at late times (Section II).
  • ad hoc to paper Reusing the same impact parameter series across models keeps sample variance similar enough for model comparisons.
    The paper uses this technique to protect against false positives from sample variance (Section II A), plus a five-realization check for the flat model in Appendix A.

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Cite this review

Pith. "Pith review of Light path averages in spacetimes with non-vanishing average spatial curvature." pith.science (2026). https://pith.science/paper/XDHWNDME

@misc{pith2026190900610,
  author       = {Pith},
  title        = {Pith review of: Light path averages in spacetimes with non-vanishing average spatial curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XDHWNDME}},
  note         = {Machine review of arXiv:1909.00610}
}
read the original abstract

Effects of inhomogeneities on observations have been vastly studied using both perturbative methods, N-body simulations and Swiss cheese solutions to the Einstein equations. In nearly all cases, such studied setups assume vanishing spatial background curvature. While a spatially flat Friedmann-Lemaitre-Robertson-Walker model is in accordance with observations, a non-vanishing curvature is not ruled out. It is therefore important to note that, as has been pointed out in the literature, 1 dimensional averages might not converge to volume averages in non-Euclidean space. If this is indeed the case, it will affect the interpretation of observations in spacetimes with non-vanishing average spatial curvature. This possibility is therefore studied here by computing the integrated expansion rate and shear, the accumulated density contrast, and fluctuations in the redshift-distance relation in Swiss cheese models with different background curvatures. It is found that differences in mean and dispersion of these quantities in the different models are small and naturally attributable to differences in background expansion rate and density contrasts. Thus, the study does not yield an indication that the relationship between 1 dimensional spatial averages and volume averages depends significantly on background curvature.

Figures

Figures reproduced from arXiv: 1909.00610 by the authors.

Figure 1
Figure 1. FIG. 1. Present time 1D density profiles of LTB structures. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Density contrast along fiducial light ray in the Swiss [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 5
Figure 5. FIG. 5. Mean and dispersion of accumulated density contrast [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figures from the paper (14 more)
Figure 6
Figure 6. Figure 6: FIG. 6. Mean and dispersion of fluctuations of the red [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 4
Figure 4. Figure 4: FIG. 4. Mean and dispersion of accumulated density contrast, [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Mean and dispersion of fluctuations in the redshift split into contributions from the projected shear and fluctuations [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Mean and dispersion of ∆ [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Histograms showing the accumulated density contrast at [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Histogram showing the accumulated density con [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Histograms showing the redshift fluctuations at [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Histograms showing fluctuations of [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Mean and dispersion of the accumulated density [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Mean and dispersion of fluctuations in the redshift [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Mean and dispersion of ∆ [PITH_FULL_IMAGE:figures/full_fig_p014_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Histograms showing the accumulated density contrast at [PITH_FULL_IMAGE:figures/full_fig_p015_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Histograms showing the redshift fluctuations at [PITH_FULL_IMAGE:figures/full_fig_p016_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Histograms showing fluctuations in [PITH_FULL_IMAGE:figures/full_fig_p017_18.png]

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