{"id":"64a19882-c752-4630-a7dd-f45f8e1d42c8","arxiv_id":"1909.00610","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Light-path averaged observables in Swiss cheese cosmologies show nearly the same mean and scatter in curved and flat FLRW backgrounds, indicating curvature does not significantly alter the relationship between line-of-sight and volume averages.","lead":"This paper tests whether the spatial curvature of the universe changes how light rays average over cosmic structures, using simplified Swiss cheese models. It finds no sign that curvature affects these averages, so flat-universe results for distances and Hubble measurements may still hold if curvature is small.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central claim not directly tested: the paper computes light-path averages but never computes 1D or volume averages, so the curvature dependence of the 1D/volume relation remains untested.","rationale":"The reader's verdict is CONDITIONAL, and this stress-test agrees with that assessment. The paper is a careful numerical study with a transparent parameter table and a sample-variance appendix, and it does not overstate its internal numerical results. However, the abstract and Section V interpret the similarity of light-path averages as bearing on the relationship between 1D spatial averages and volume averages. That connection is asserted, not demonstrated: the accumulated density contrast is a path integral along null geodesics, not a fixed-time spatial average, and no volume average is computed. The author's own caveat in Section IV makes this limitation explicit. The per-model kmax tuning is a real confound, but it is secondary to the missing direct computation; even with perfectly matched density contrasts, the paper would still not have measured the 1D/volume relation. Thus the central claim is plausible but under-supported. The appropriate response is to keep the verdict CONDITIONAL, requiring the explicit 1D and volume average comparison before the abstract's conclusion can be accepted as established.","tokens_in":20361,"tokens_out":4195,"duration_ms":145313,"concrete_test":"Run the same model specifications but explicitly compute (a) the volume average of the density contrast over the full curved spatial hypersurface (or a large comoving region) and (b) the 1D spatial average along many random fixed-time straight lines, for each Omega_K. Compare the ratio (1D average)/(volume average) between flat and curved models. Also repeat with kmax fixed to the flat-model value to test whether per-model kmax tuning masks a curvature effect. If the ratio varies significantly with Omega_K, the paper's null conclusion fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim concerns the relation between 1D spatial averages and volume averages in curved space (abstract; Section V). The actual numerical comparison is between light-path averages of the accumulated density contrast, integrated expansion/shear, and DA fluctuations in different Swiss cheese models (Section IV). No 1D spatial average at fixed time and no volume average over the curved background is computed. The author even notes that \"it is not entirely clear how this quantity should be related to neither 1 dimensional spatial averages nor volume averages\" (Section IV, paragraph on accumulated density contrast). Consequently, the observed similarity of light-path means across models does not directly constrain the convergence of 1D spatial averages to volume averages in non-Euclidean space; it only shows that this particular ray-tracing statistic is insensitive to background curvature in these models. The comparison is further complicated by per-model kmax choices (Table I and Section II): matching non-linear density contrasts across models may partially absorb a curvature effect. For the abstract's conclusion to be supported, the 1D and volume averages themselves must be computed and compared.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20588,"tokens_out":4143,"duration_ms":39645,"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":[{"comment":"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.","section":"Section IV and Section V"},{"comment":"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.","section":"Section II, Table I"},{"comment":"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.","section":"Section II A"}],"minor_comments":[{"comment":"The caption contains a typo: 'are therefor not shown' should be 'are therefore not shown'.","section":"Fig. 1 caption"},{"comment":"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.","section":"Eq. (8)"},{"comment":"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.","section":"Section IV, Figure 9"},{"comment":"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.","section":"Section V"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a useful numerical study of light propagation in curved Swiss cheese models and the sample-variance appendix is a genuine strength. The main issue is that the abstract and summary claim a conclusion about 1D versus volume averages, while the actual computations concern light-path averages. This can be fixed either by adding the missing average calculations or by reframing the claim. The kmax confound also needs a robustness check. I therefore recommend major revision rather than rejection, as the underlying numerical results appear sound and the paper could be publishable after appropriate changes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThis is the first systematic ray-tracing comparison of light-path averages in Swiss cheese models with LTB structures placed in positively, flat, and negatively curved FLRW backgrounds, aimed at a specific worry from Lavinto, Rasanen, and Szybka about 1D versus volume averages in curved space. The numerical work is careful: the same random impact parameters are reused across models, there is a sample-variance appendix with five independent realizations, and the parameter table is transparent. The observed cancellation between integrated shear and expansion-rate fluctuations along the light rays in all backgrounds is a nice confirmation of earlier findings.\n\nThe main soft spot is that the abstract overreaches. The paper computes light-path means and dispersions of accumulated density contrast, integrated expansion/shear, and DA fluctuations. It does not compute 1D spatial averages or volume averages, and it does not directly test their convergence. The author acknowledges in Section IV that for the accumulated density contrast it is 'not entirely clear how this quantity should be related to neither 1 dimensional spatial averages nor volume averages.' The step from 'these ray-averages look similar across curvatures' to 'the 1D/volume relation does not depend on curvature' is logically unsupported. What is established is that these particular light-path statistics are insensitive to background curvature in this family of model setups, which is useful but weaker.\n\nSecond, model control is imperfect. The parameter kmax is adjusted per background (Table I) to keep structures non-linear and avoid shell crossings, so the density contrasts are not identical across models. The author attempts to control for expansion-rate differences with two flat models, but curvature is not fully isolated. The histogram comparisons are visual rather than backed by significance tests, though the sample-variance appendix helps. Finally, no code or data are released, which is a real limitation for a purely numerical study.\n\nOverall this is a solid, honest numerical contribution to a real question. The null result is plausible. But the central claim should be scaled back to match the actual computation, or supplemented by explicitly computing volume averages and 1D spatial averages. It deserves a serious referee, and the report should ask for that.\n\nRecommendation: send to peer review; the likely outcome is a major revision that either softens the abstract or adds the missing averages.","headline":"A careful numerical study whose central claim about 1D and volume averages is not directly tested; still a useful data point.","tokens_in":21034,"tokens_out":3752,"would_cite":true,"duration_ms":32074,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Nonzero spatial curvature does not change how light-path averages compare with volume averages in Swiss-cheese models, tests with 1000 light rays show.","keywords":["spatial curvature","Swiss cheese models","LTB voids","light path averages","volume averages","redshift-distance relation","Hubble diagram","inhomogeneous cosmology"],"falsifier":"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.","tokens_in":20147,"feed_emoji":"🌌","tokens_out":12563,"duration_ms":115852,"temperature":0.7,"pith_summary":"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.","feed_headline":"Curved space barely shifts light-ray averages in Swiss-cheese models","feed_subtitle":"If right, flat-universe Hubble-diagram results remain valid for mildly curved space.","key_machinery":"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,\n$$1+z=(1+z_{\\mathrm{bg}})\\exp\\left[\\int_{t(\\$\\lambda$)}^{t_0}dt\\left(\\tfrac{1}{3}\\$\\Delta$\\Theta+$c^{2}$\\$\\sigma$^\\$\\alpha${}_\\$\\beta$ e_\\$\\alpha$ e^\\$\\beta$\\right)\\right],$$\nwhere $\\Theta$ 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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Raises the concern that one-dimensional averages may not converge to volume averages in curved space, which this paper is designed to test.","marker":"[29]"},{"why":"Argues that small non-vanishing spatial curvature is not ruled out by observations and should be included in inhomogeneous cosmology studies.","marker":"[28]"},{"why":"Earlier Swiss cheese study that established the cancellation between shear and expansion-rate contributions and provided statistics of distance fluctuations that the present models extend to curved backgrounds.","marker":"[25]"},{"why":"Prior Swiss cheese computation of CMB distance fluctuations and mean angular-diameter-distance shifts used as comparison for the mean values found here.","marker":"[19]"},{"why":"Swiss cheese study with 1000 light rays per model that showed similar statistical significance and is cited when assessing sample variance and packing fraction.","marker":"[21]"},{"why":"Perturbative results that mean values of observables are unbiased under ensemble and directional averages, giving the flat-space expectation this study checks against curvature.","marker":"[12]"},{"why":"Demonstrates the cancellation of expansion-rate and shear contributions in near-FRW models, supporting the same cancellation observed in all the present backgrounds.","marker":"[55]"},{"why":"Provides the general formula for redshift fluctuations along light paths in statistically homogeneous and isotropic dust universes used in the analysis.","marker":"[30]"}],"fun_headline_variants":["Curved space fails to skew light-ray averages","Light averages unaffected by curved Swiss cheese","Swiss cheese with curvature: light averages hold steady","Curved backgrounds don't change light-ray averages","No curvature effect on light averages in Swiss cheese"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Curved space fails to skew light-ray averages","Light averages unaffected by curved Swiss cheese","Swiss cheese with curvature: light averages hold steady","Curved backgrounds don't change light-ray averages","No curvature effect on light averages in Swiss cheese"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000766,"raw_usage":{"total_tokens":3409,"prompt_tokens":970,"completion_tokens":2439,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":586,"completion_tokens_details":{"reasoning_tokens":2369}},"tokens_in":586,"tokens_out":2439,"duration_ms":18064,"temperature":1.0,"reasoning_tokens":2369,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:41:34.527907+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier Swiss cheese study that established the cancellation between shear and expansion-rate contributions and provided statistics of distance fluctuations that the present models extend to curved backgrounds."},{"cited_title":"close-ups","cited_arxiv_id":null,"evidence_quote":"Prior Swiss cheese computation of CMB distance fluctuations and mean angular-diameter-distance shifts used as comparison for the mean values found here."},{"cited_title":"Effect of inhomogeneities on high precision measurements of cosmological distances","cited_arxiv_id":"1408.4390","evidence_quote":"Swiss cheese study with 1000 light rays per model that showed similar statistical significance and is cited when assessing sample variance and packing fraction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Demonstrates the cancellation of expansion-rate and shear contributions in near-FRW models, supporting the same cancellation observed in all the present backgrounds."}],"review_version":1}