{"id":"c7baf870-3a97-408c-a232-8bbdccc6a20f","arxiv_id":"1909.00791","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Including braiding and in-survey covariance increases forecast error bars on the dark energy parameter w by about 120% for a Euclid-like galaxy survey and is a necessary ingredient for a mathematically valid covariance matrix.","lead":"This paper works out how non-Gaussian clustering terms, called braiding and in-survey covariance, change the expected error bars for next-generation galaxy surveys like Euclid. It finds that ignoring them can understate the uncertainty on the dark energy parameter w by more than a factor of two.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"All forecast and positive-definiteness numbers depend on the unvalidated Bij approximation (Eq. 16) for braiding covariance; if exact Eq. 13 differs at the 10–20% level, the reported 120% w-impact and 17–85% HOD ranges could shift.","rationale":"The reader's weakest assumption identifies the load-bearing numerical input: the Bij approximation is used without direct validation against Eq. (13). My read agrees and treats this as the single most important concern. The analytical regulator argument in Sec. 3.2 is largely independent of the approximation, since it is based on diagrammatic counting and the rank-deficiency of SSC and 1h terms, so the qualitative necessity of braiding is not in serious doubt. However, the quantitative forecasts—the 120% increase on sigma_w, the 17-85% HOD increases, and the 9.4% S/N impact on top of SSC—all pass through the approximate braiding covariance. The paper's Limber-based argument is suggestive but does not provide a numerical error bound, and the public code makes a direct check inexpensive. This is precisely the kind of addressable, concrete uncertainty that supports a CONDITIONAL rather than ACCEPT verdict; since the reader already issued CONDITIONAL with moderate confidence, no verdict change is needed. If the exact-vs-approximate test passes at the few-percent level, the paper would be substantially strengthened and could be accepted on this point.","tokens_in":16222,"tokens_out":7496,"duration_ms":96137,"concrete_test":"Modify the released notebook at https://github.com/fabienlacasa/BraidingArticle to compute the exact braiding covariance of Eq. (13) for a representative subset: the nine multipoles of Sec. 3.1 and the ten redshift bins of Sec. 4.1, covering all ell-ell' pairs plus a sample of bin pairs with the largest Fisher weight. Compare elementwise with the Bij approximation of Eq. (16), reporting fractional differences for diagonal and off-diagonal entries. Then rerun the Fisher forecast replacing the Bij braiding block with the exact block (and verify that Sigma_ONG remains positive definite). If the forecast changes by more than ~10% relative to the published numbers for w or for the HOD parameters, the headline impact percentages require revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The article's quantitative claims—the positive definiteness of the ONG block, the S/N reduction, and every Fisher error-bar percentage—are computed with the Bij approximation of Eq. (16), not with the exact braiding expression of Eq. (13). The validation offered in Sec. 2.2 is indirect: the analogy with the Sij approximation for SSC, the identity B_{0,0}=sigma^2, and the Limber argument that B becomes sharper at high multipoles. These arguments indicate that the approximation should improve with multipole, but they do not quantify the error for the specific survey configuration used here: ten redshift bins, nine input multipoles, and 29 interpolated bins up to ell~2290. The approximation also enters the cross-redshift bins through the n_g(z)^2-weighted integrals in Eqs. (18)-(20), where the slow-variation assumption on Psi^alt is least justified. Because the Fisher forecast inverts the full covariance matrix in Eq. (30), errors in individual braiding elements are not simply averaged away; their effect on marginalised parameters is nonlinear and potentially amplified after inversion. The same approximate braiding term is also part of the Sigma_ONG matrix whose positive definiteness is the numerical basis for the 'necessary condition' conclusion in Sec. 3.2, so a failure of the approximation would affect not only the impact percentages but also the demonstration that braiding regulates the 2h1+3, 3h-base0, and 4h-3 terms. This is a correctness risk, not an internal inconsistency, and it is directly checkable because the code and data are public.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops and applies an analytical halo-model framework for non-Gaussian covariance terms of the galaxy angular power spectrum, focusing on braiding covariance. It introduces the Bij approximation (Eq. 16) for the braiding term, claims that braiding is a necessary condition for including the in-survey 2h1+3, 3h-base0, and 4h-3 covariance terms because those terms alone produce correlation coefficients greater than unity and negative eigenvalues, and uses Fisher forecasts for a Euclid-like survey to quantify the impact on cosmological and HOD parameter error bars. The reported impacts are: ONG increases the marginalized error bar on w by about 50% alone and by about 120% in total non-Gaussianity; HOD error bars increase by 17% to 85% in total; and including the 1-halo trispectrum on top of SSC is insufficient. Public code and a Python notebook are provided.","tokens_in":16590,"tokens_out":5923,"duration_ms":61551,"significance":"If the results hold, the paper fills an important gap in analytic covariance modeling for next-generation photometric surveys: it provides a computationally tractable braiding term and, for the first time, a quantitative argument that ignoring it invalidates the usual in-survey non-Gaussian covariance terms. The analytical demonstration of correlation coefficients greater than unity for the 2h1+3 term is clean and instructive, and the numerical eigenvalue checks support the main structural claim. The paper also gives an explicit comparison to the Euclid 10% precision requirement, making the practical relevance concrete. The availability of reproducible code and data strengthens the reliability and utility of the work.","major_comments":[{"comment":"The Bij approximation is used to compute every numerical covariance matrix and every Fisher forecast in the paper, yet its accuracy is not directly validated against the exact expression of Eq. (13) for any configuration. The arguments presented (the analogy to the Sij approximation, the B_{0,0}=sigma^2 identity, and the qualitative Limber limiting behavior) do not bound the error for the specific survey setup: ten redshift bins, 29 interpolated multipoles up to ell~2290, and the cross-redshift integrals in Eqs. (18)-(20). Since the Fisher forecast inverts the full covariance in Eq. (30), errors in braiding entries are not simply averaged away and can be amplified after inversion. The reported impact percentages (e.g., 120% on w, 17-85% on HOD, 9.4% S/N reduction) therefore rest on an unvalidated approximation. A direct comparison of the exact and approximate braiding terms for representative multipole and redshift pairs is needed, along with a sensitivity test of the forecasts to the approximating assumptions.","section":"Sec. 2.2, Eq. (16)"},{"comment":"The statement that braiding is 'necessary' for including 2h1+3, 3h-base0, and 4h-3 is supported by showing that the Gaussian term, SSC, and the 1h term individually cannot regulate the negative eigenvalues, and by a numerical check that the ONG group (with braiding computed under the Bij approximation) is positive definite. This excludes only three candidate regulators and does not prove that no other combination of terms from the same halo-model decomposition could yield a positive-definite matrix. The numerical eigenvalue check is also performed for one cosmology, one redshift binning, and the approximate braiding term. I recommend either arguing necessity more generally within the halo-model term set or rephrasing the conclusion as a demonstrated property of the considered term set and configuration.","section":"Sec. 3.2"}],"minor_comments":[{"comment":"The heading contains a typo: 'Alhough' should be 'Although'.","section":"Sec. 3.3"},{"comment":"The phrase 'or in other term the matrix restricted to these two points has a negative eigenvalue' should read 'or in other words the matrix restricted to these two points has a negative eigenvalue'.","section":"Sec. 3.2, bottom of page 4"},{"comment":"The attribution of the ONG impact to 'braiding and 2h1+3' is based on separating the 1h term from the rest of ONG, but the paper does not separately quantify the off-diagonal contributions of 3h-base0 and 4h-3 after covariance inversion; a brief justification or caveat would make the attribution more precise.","section":"Sec. 4.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is well structured and the central claim is interesting, but the quantitative results depend on the unvalidated Bij approximation. I would be willing to accept after the author adds a numerical validation of Eq. (16) against Eq. (13) for representative redshift and multipole pairs, and shows that the forecast percentages and the positive-definiteness conclusion are stable under the approximation. The paper relies heavily on the author's own previous work, but the new results are computed and the debt to prior work is explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plainly: this paper does a useful thing. It takes the non-Gaussian covariance terms from the author's earlier derivation, introduces the Bij approximation to make braiding covariance tractable, proves analytically that the 2h1+3 term yields correlation coefficients above unity (hence negative eigenvalues), and then shows numerically that braiding is needed to regulate that behavior. The Fisher forecasts are the payoff: relative to a Gaussian covariance, the full non-Gaussian treatment inflates the w error bar by ~120% for an Euclid-like survey, and HOD parameter errors by 17–85%. Those numbers are above Euclid's 10% precision requirement, which makes this a practical paper, not a purely formal one.\n\nWhat I like: the paper is honest about provenance—the expressions come from Lacasa (2018)—and it releases code and data so every number can be reproduced. The analytic argument for why 2h1+3 alone is pathological is clean. The discussion of parameter degeneracies being eased (or not) by NG is thoughtful and goes beyond a simple error-bar report.\n\nThe soft spot, as flagged by the stress-test note, is the Bij approximation itself. Every quantitative result—the positive-definiteness check, the S/N curves, and the Fisher percentages—uses Eq. 16 rather than the exact Eq. 13. The support given is indirect: an analogy with Sij in Lacasa & Grain (2019) and a Limber-based argument that the kernel sharpens at high multipoles. That's plausible but not a validation. Because the Fisher forecast inverts the full covariance, element-wise errors can be amplified. This does not look like a fatal flaw; the code is public and the check is straightforward. But without it, the headline percentages carry an unknown systematic. I'd also like the multipole interpolation (9 to 29 bins) quantified, though that is a minor issue.\n\nCitations are not a problem: the heavy self-citation reflects a genuine research program, and the new results are computed, not restated. This is a serious, workmanlike paper. It deserves peer review. I would send it to referees with a request for a direct Bij-versus-exact comparison as the main revision.\n\nWho for: survey teams building covariance models, and anyone doing Fisher forecasts for photometric galaxy clustering. I'd take it to reading group and would cite it once the approximation is validated.","headline":"A careful, transparent paper showing that braiding and in-survey covariance terms materially change Euclid-like Fisher forecasts, with the main caveat being an unvalidated but checkable approximation.","tokens_in":17096,"tokens_out":3554,"would_cite":true,"duration_ms":36992,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Braiding covariance is the term that makes full non-Gaussian galaxy-survey covariances valid, and it raises the dark-energy error bar by about 120%.","keywords":["galaxy clustering","angular power spectrum","covariance matrix","non-Gaussian covariance","braiding covariance","super-sample covariance","halo model","Fisher forecast"],"falsifier":"Evaluate the exact braiding covariance formula, Equation (13), for the same multipole and redshift-bin pairs used in the paper and compare it with the Bij approximation, Equation (16); if the differences exceed a few percent at the multipoles that drive the dark-energy constraint, the quoted 120% error-bar increase would need revision.","tokens_in":15989,"feed_emoji":"🔭","tokens_out":10447,"duration_ms":90797,"temperature":0.7,"pith_summary":"This paper argues that galaxy clustering analyses of next-generation surveys cannot stop at Gaussian or Gaussian-plus-super-sample covariances. It implements a full set of halo-model non-Gaussian covariance terms centred on 'braiding covariance', a class of trispectrum terms coupling in-survey and super-survey modes, and shows that braiding is the term that makes the other in-survey terms usable: the in-survey 2-, 3-, and 4-halo terms separately produce covariance matrices with negative eigenvalues, but combined with braiding they yield a positive-definite matrix. To make the calculation feasible, the paper introduces a fast approximation for braiding covariance, the Bij approximation. In Fisher forecasts for a survey with Euclid-like galaxy density and angular coverage, the full non-Gaussian covariance increases the marginalised error on the dark-energy equation-of-state parameter $w$ by about 120%, with HOD parameter errors rising by 17% to 85%. The conclusion is that braiding and in-survey covariance must be included to meet the roughly 10% precision target of next-generation surveys.","feed_headline":"Overlooked covariance terms can double dark-energy error bars","feed_subtitle":"Forecasts for a Euclid-like survey show Gaussian-only error bars are too small once braiding and in-survey covariance are included.","key_machinery":"The load-bearing object is the braiding covariance, a class of halo-model trispectrum terms that get contributions from both in-survey and super-survey modes. Its exact form is a double redshift integral of a response function $\\Psi^{\\rm alt}$ with a braiding kernel $B_{\\ell,\\ell'}$, itself a weighted sum of matter angular power spectra (Eqs. 13-15); the numerical shortcut, the Bij approximation, factors this integral by separately integrating the response and the kernel (Eqs. 16-20), analogous to the earlier Sij approximation for super-sample covariance. The second mechanism is the regulator argument: the $2h_{1+3}$, $3h$-base0, and $4h$-$3$ terms individually over-correlate off-diagonal multipoles because they link the 2-halo and 1-halo parts of the spectrum, and braiding supplies the matching same-pairing counterpart needed for a positive-definite total covariance matrix.","core_discovery":"The central claim is that braiding covariance is a necessary component of a valid non-Gaussian covariance for the galaxy angular power spectrum, not an optional refinement. Without it, the in-survey $2h_{1+3}$, $3h$-base0, and $4h$-$3$ terms, taken alone, give correlation coefficients larger than unity and hence negative eigenvalues; the paper proves this analytically for $2h_{1+3}$ in a limiting regime and shows numerically that adding braiding (with the 1-halo term) restores positive definiteness. With the total covariance, the Fisher forecast for a Euclid-like survey increases the marginalised error on $w$ by about 120% relative to Gaussian, with braiding and in-survey covariance alone contributing 50% and super-sample covariance 90%; HOD errors rise by 17% to 85%. The paper also argues that super-sample covariance plus the 1-halo trispectrum is insufficient: braiding and the rest of in-survey covariance are required to capture the full non-Gaussian impact.","pith_inferences":["A practical diagnostic falls out of the regulator argument: any galaxy-clustering covariance pipeline that adds the 2-, 3-, or 4-halo in-survey terms without braiding is mathematically guaranteed to be invalid, so checking for negative eigenvalues is a quick way to catch a missing braiding term.","The pattern of impacts, largest for $w$ and $n_s$ and smaller for amplitude after marginalising, suggests that extensions changing the shape of the matter power spectrum, such as neutrino mass or a running spectral index, may inherit especially large braiding-driven error increases in future forecasts.","The Bij approximation could be validated cheaply by evaluating the exact braiding integral at a handful of representative multipole and redshift pairs; this would test whether the 120% figure is robust before it is baked into survey pipelines.","Because the paper's real-space to harmonic-space mapping is linear, configuration-space clustering analyses inherit the same braiding requirement; simulations-based covariance estimates that capture only super-sample variance will miss it."],"forward_implications":["Gaussian-only covariance forecasts are optimistic: with the full non-Gaussian covariance, the marginalised error on $w$ for a Euclid-like survey is about 2.2 times the Gaussian value.","Super-sample covariance plus the 1-halo trispectrum is not enough: braiding and the remaining in-survey terms add about 15% to the $w$ error on top of SSC and push several parameter errors past the 10% precision target.","The in-survey $2h_{1+3}$, $3h$-base0, and $4h$-$3$ terms cannot be used alone; they must be combined with braiding (and the 1-halo term) to form a positive-definite covariance matrix.","Braiding and in-survey covariance matter for HOD constraints too, increasing marginalised HOD parameter errors by 17% to 85%, with four parameters affected beyond 10%.","Non-Gaussianity generally reduces parameter degeneracies, especially between dark-energy and HOD parameters, since accounting for it distributes constraining power more evenly across scales rather than concentrating it in low-noise small-scale measurements."],"supporting_citations":[{"why":"Supplies the full analytical derivation of the non-Gaussian covariance terms, including braiding, that this paper implements and approximates.","marker":"Lacasa (2018)"},{"why":"Provides the Sij approximation for super-sample covariance whose success motivates the analogous Bij approximation for braiding.","marker":"Lacasa & Grain (2019)"},{"why":"Defines super-sample covariance and supplies the response and kernel formalism used in the halo-model covariance.","marker":"Takada & Hu (2013)"},{"why":"Supplies the survey specifications, sky fraction, and redshift binning used in the Fisher forecasts.","marker":"Euclid Collaboration et al. (2019)"},{"why":"Provides the halo mass function used in all halo-model computations.","marker":"Tinker et al. (2008)"},{"why":"Provides the halo bias used in the halo-model power spectra and covariances.","marker":"Tinker et al. (2010)"},{"why":"Supplies the Halo Occupation Distribution parametrization used to model galaxy occupation.","marker":"Zehavi et al. (2011)"},{"why":"Independent weak-lensing forecast whose connected non-Gaussian covariance impact the paper cites as agreeing with its ONG impact.","marker":"Barreira et al. (2018a)"},{"why":"Independent result that in-survey non-Gaussian terms can dominate SSC, cited as agreement with the paper's finding.","marker":"Wadekar & Scoccimarro (2019)"}],"fun_headline_variants":["Braiding covariance doubles dark-energy error bars in forecasts","Missing braiding covariance makes Euclid error bars too small","Non-Gaussian covariance boosts w error by 120% in galaxy surveys","Without braiding, galaxy covariance breaks – forecasts too optimistic","Braiding and in-survey covariance dominate galaxy covariance errors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"All numerical results use the fast 'Bij' approximation for braiding covariance, whose accuracy is argued from analogy and from a large-multipole limiting behaviour rather than tested directly against the exact formula; if the approximation is inaccurate at the multipoles and redshifts used, the quoted error-bar increases would shift.","fun_headline_variants_meta":{"raw":{"variants":["Braiding covariance doubles dark-energy error bars in forecasts","Missing braiding covariance makes Euclid error bars too small","Non-Gaussian covariance boosts w error by 120% in galaxy surveys","Without braiding, galaxy covariance breaks – forecasts too optimistic","Braiding and in-survey covariance dominate galaxy covariance errors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001011,"raw_usage":{"total_tokens":4371,"prompt_tokens":1144,"completion_tokens":3227,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":760,"completion_tokens_details":{"reasoning_tokens":3144}},"tokens_in":760,"tokens_out":3227,"duration_ms":123143,"temperature":1.0,"reasoning_tokens":3144,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:35:34.568926+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the exact braiding covariance formula, Equation (13), for the same multipole and redshift-bin pairs used in the paper and compare it with the Bij approximation, Equation (16); if the differences exceed a few percent at the multipoles that drive the dark-energy constraint, the quoted 120% error-bar increase would need revision.","supporting_citations":[{"cited_title":"2018, A&A, 611, A83","cited_arxiv_id":null,"evidence_quote":"Supplies the full analytical derivation of the non-Gaussian covariance terms, including braiding, that this paper implements and approximates."},{"cited_title":"H., et al","cited_arxiv_id":null,"evidence_quote":"Supplies the Halo Occupation Distribution parametrization used to model galaxy occupation."}],"review_version":1}