The impact of braiding covariance and in-survey covariance on next-generation galaxy surveys

Fabien Lacasa

REVIEW 2 major objections 3 minor 23 references

The impact of braiding covariance and in-survey covariance on next-generation galaxy surveys

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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%.

desk verdict 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. read the letter →

arxiv 1909.00791 v2 pith:QTI7FVWM submitted 2019-09-02 astro-ph.CO

Fabien Lacasa This is my paper · ORCID

classification astro-ph.CO

keywords galaxyclusteringangularpowerspectrumcovariancematrixnon-Gaussianbraidingsuper-samplehalomodelFisherforecast

open problems Dark Energy

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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

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.

What 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.

The 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.

Citations 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.

Who 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.

Referee Report

2 major / 3 minor

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.

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 (2)

[Sec. 2.2, Eq. (16)] 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.
[Sec. 3.2] 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.

minor comments (3)

[Sec. 3.3] The heading contains a typo: 'Alhough' should be 'Although'.
[Sec. 3.2, bottom of page 4] 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'.
[Sec. 4.2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the positive-definiteness and Fisher-forecast claims are computed from imported covariance expressions, not equivalent to those inputs by construction.

full rationale

The load-bearing quantitative claims are not equivalent to their inputs by construction. The non-Gaussian covariance terms in Eqs. (7)-(15) are imported from the author's earlier Lacasa (2018) derivation, and the Bij approximation in Eq. (16) is motivated by analogy with the Sij approximation of Lacasa & Grain (2019); these are self-citations, but they supply the input equations, not the conclusions. The positive-definiteness result in Sec. 3.2 is obtained by computing term-by-term correlation matrices and the summed ONG matrix, and each Fisher impact in Secs. 3.3 and 4 is obtained by inverting the corresponding covariance matrix; no parameter is fitted to the reported error-bar increases. The HOD fit in Appendix A only sets the fiducial model for the forecasts and does not determine the NG-induced fractional impacts. The genuine caveats are correctness risks rather than circularity: the Bij approximation is not directly validated against Eq. (13), and the word 'necessary' in the abstract overstates what is demonstrated (a demonstrated sufficient construction plus arguments that Gaussian, SSC, and 1-halo terms alone cannot regulate). Because the numerical results are code-released and are compared with external analyses, the central derivation is self-contained against external benchmarks.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central numerical results depend on standard halo-model ingredients taken from the literature, on a polynomial HOD fit to the Euclid n(z), and on the new Bij approximation. No new physical entities are postulated. The key untested input is the accuracy of the Bij approximation.

free parameters (2)

Polynomial HOD coefficients (Ma_min, Mb_min, Mc_min, Md_min) = 11.020, -0.143, 0.549, -0.105
Fitted in Appendix A to reproduce the Euclid photometric galaxy redshift distribution n(z) to 2.5% precision. The fiducial power spectrum and all covariance terms depend on these values.
Fixed HOD parameters (sigma_logM, alpha_sat, Mratio) = 0.5, 1, 10
Chosen by hand in Sec. 3.1 and Appendix A rather than fitted. The covariance forecasts depend on these values.

assumptions (3)

domain assumption The halo model with Tinker et al. (2008) mass function, Tinker et al. (2010) bias, and the Zehavi et al. (2011) HOD prescriptions describes galaxy clustering and its covariance at the needed precision.
Invoked in Sec. 3.1 to compute C_l and all covariance terms. If the halo model is inaccurate at relevant scales, the numerical impact estimates change.
domain assumption The halo model at tree level predicts the galaxy angular power spectrum to about 10% precision, sufficient for covariance forecasting.
Stated in Sec. 3.3. The Fisher forecasts use this modeling for the observable and its derivatives.
ad hoc to paper The Bij approximation (Eq. 16) is valid because the separable element Psi^alt varies slowly with redshift and B(l,l') varies quickly enough, by analogy with the Sij approximation.
All numerical results use this approximation. Its accuracy is argued from analogy with Lacasa & Grain (2019) and Limber behavior, but no direct numerical validation is provided in the paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The impact of braiding covariance and in-survey covariance on next-generation galaxy surveys." pith.science (2026). https://pith.science/paper/QTI7FVWM

@misc{pith2026190900791,
  author       = {Pith},
  title        = {Pith review of: The impact of braiding covariance and in-survey covariance on next-generation galaxy surveys},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QTI7FVWM}},
  note         = {Machine review of arXiv:1909.00791}
}

read the original abstract

As galaxy surveys become more precise and push to smaller scales, the need for accurate covariances beyond the classical Gaussian formula becomes more acute. Here, I investigate the analytical implementation and impact of non-Gaussian covariance terms that I previously derived for galaxy clustering. Braiding covariance is such a class of terms and it gets contribution both from in-survey and super-survey modes. I present an approximation for braiding covariance which speeds up the numerical computation. I show that including braiding covariance is a necessary condition for including other non-Gaussian terms: the in-survey 2-, 3- and 4-halo covariance, which yield covariance matrices with negative eigenvalues if considered on their own. I then quantify the impact on parameter constraints, with forecasts for a Euclid-like survey. Compared to the Gaussian case, braiding and in-survey covariances significantly increase the error bars on cosmological parameters, in particular by 50% for w. The Halo Occupation Distribution (HOD) error bars are also affected between 12% and 39%. Accounting for super-sample covariance (SSC) also increases parameter errors, by 90% for w and between 7% and 64% for HOD. In total, non-Gaussianity increases the error bar on w by 120% (between 15% and 80% for other cosmological parameters), and the error bars on HOD parameters between 17% and 85%. Accounting for the 1-halo trispectrum term on top of SSC is not sufficient for capturing the full non-Gaussian impact: braiding and the rest of in-survey covariance have to be accounted for. Finally, I discuss why the inclusion of non-Gaussianity generally eases up parameter degeneracies, making cosmological constraints more robust to astrophysical uncertainties. The data and a Python notebook reproducing the results and plots of the article are available at \url{https://github.com/fabienlacasa/BraidingArticle}. [Abridged]

Figures

Figures reproduced from arXiv: 1909.00791 by the authors.

**Figure 2.** Correlation matrices for the different non-Gaussian covariance terms, normalised by its own diagonal. Top: SSC, 1h, Braiding. Bottom: 2h1+3, 3h-base0, 4h-3; the color bar is clipped at 7. In the top row we see well-behaved terms which yield matrices with all eigenvalues ≥ 0 : SSC, 1-halo and braiding. The correlation coefficients are all in [-1,1]. In the bottom row we see the 2h1+3, 3h-base0 and 4h-3 terms for whi… view at source ↗

**Figure 1.** Different non-Gaussian contributions to the variance of the angular power spectrum per multipole in the redshift bin 0.9 < z < 1.019. We first see that the 3h-base0 and 4h-3 terms are negligible compared to all other terms. This means that the perturbative contributions to variances are excellently encapsulated inside super-sample covariance and braiding covariance. We can then focus on the other covariance term… view at source ↗

**Figure 4.** Diagrams for some of the trispectrum terms involved in the covariance of the galaxy angular power spectrum. Left: 2-halo part of Braiding, right: 1-halo term. We can find the regulator via the diagram discussion. Since the 2h1+3 term quantifies how the 2-halo part of the spectrum is correlated with the 1-halo part due to halo coincidence, it has to be regulated by a first term which quantifies how the 2-halo part o… view at source ↗

Figures from the paper (8 more)

**Figure 3.** Figure 3: Diagrams for some of the trispectrum terms involved in the covariance of the galaxy angular power spectrum Cov C gal ` ,C gal ` 0 . From left to right: 4-halo, 3-halo, and 2-halo 1+3 term. Galaxies 1 and 2 are the source of the first power spectrum C gal ` , while…

**Figure 5.** Figure 5: Correlation matrix for ONG, the sum of non-Gaussian covariance terms other than SSC: 1h, 2h1+3, 3h-base0, 4h-3, and Braiding. 3.3. Impact on the signal-to-noise ratio Alhough Braiding is necessary to include ONG covariance, the question remains of whether ONG has a si…

**Figure 6.** Figure 6: Cumulative signal to noise ratio for the measurement of C gal ` in the bin 0.9 < z < 1.019 as a function of maximum multipole of analysis. Left, from top to bottom: Gaussian covariance only, Gaussian + ‘other NG’, Gaussian + SSC, Gaussian + SSC + 1h, total covariance. …

**Figure 8.** Figure 8: Marginalised error bars on each cosmological parameters using all redshift bins as a function of the maximum multipole of analysis. When using the full multipole range, non-Gaussian covariance terms have a large impact on the information content for all cosmological p…

**Figure 7.** Figure 7: (Square root of the) cumulative Fisher elements for the cosmological parameters in the considered redshift bin, as a function of the maximum multipole of the analysis. If the analysis is carried out on the full range of multipoles then non-Gaussian covariance terms wo…

**Figure 9.** Figure 9: Fisher ellipses on cosmological parameters, using all redshift bins and the full multipole range. The color coding is identical to the other figures: blue=Gaussian, orange=Gaussian+ONG, green=Gaussian+SSC, violet=total covariance. 4.8×107 in the Gaussian case to 1.4×…

**Figure 10.** Figure 10: (Square root of the) cumulative Fisher elements for the HOD parameters in the considered redshift bin as a function of the maximum multipole of analysis. If the analysis is carried out on the full range of multipoles, then non-Gaussian covariance terms have a large i…

**Figure 12.** Figure 12: Fisher ellipses on HOD parameters, using all redshift bins and the full multipole range. The color coding is identical to the other figures: blue=Gaussian, orange=Gaussian+ONG, green=Gaussian+SSC, violet=total covariance. 5. Discussion By way of a summary of previous…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 11 canonical work pages

[1]

R., Balmès, I., Lacasa, F., & Lima, M

Abramo, L. R., Balmès, I., Lacasa, F., & Lima, M. 2015, MNRAS, 454, 2844

work page 2015
[2]

& Takada, M

Akitsu, K. & Takada, M. 2017, ArXiv e-prints [arXiv:1711.00012]

arXiv 2017
[3]

2018b, J

Barreira, A., Krause, E., & Schmidt, F. 2018b, J. Cosmology Astropart. Phys., 6, 015 Euclid Collaboration, Blanchard, A., Camera, S., et al. 2019, arXiv e-prints, arXiv:1910.09273 Harnois-Déraps, J. & Pen, U.-L. 2013, MNRAS, 431, 3349

arXiv 2019
[4]

2017, MNRAS, 465, 1454

Hildebrandt, H., Viola, M., Heymans, C., et al. 2017, MNRAS, 465, 1454

2017
[5]

& Eiﬂer, T

Krause, E. & Eiﬂer, T. 2017, MNRAS, 470, 2100

work page 2017
[6]

F., Zuntz, J., et al

Krause, E., Eiﬂer, T. F., Zuntz, J., et al. 2017, ArXiv e-prints [arXiv:1706.09359]

arXiv 2017
[7]

& Grain, J

Lacasa, F. & Grain, J. 2019, A&A, 624, A61

2019
[8]

& Kunz, M

Lacasa, F. & Kunz, M. 2017, A&A, 604, A104

2017

Show all 23 references

[9]

2018, A&A, 611, A83

Lacasa, F., Lima, M., & Aguena, M. 2018, A&A, 611, A83

2018
[10]

2014, MNRAS, 439, 123

Lacasa, F., Pénin, A., & Aghanim, N. 2014, MNRAS, 439, 123

2014
[11]

& Rosenfeld, R

Lacasa, F. & Rosenfeld, R. 2016, J. Cosmology Astropart. Phys., 8, 005

2016
[12]

Laureijs, R. et al. 2011, ArXiv e-prints [arXiv:1110.3193]

2011 arXiv
[13]

Li, Y ., Schmittfull, M., & Seljak, U. 2018, J. Cosmology Astropart. Phys., 2, 022

2018
[14]

M., Gaztañaga, E., & Croton, D

Norberg, P., Baugh, C. M., Gaztañaga, E., & Croton, D. J. 2009, MNRAS, 396, 19 Planck Collaboration, Aghanim, N., Akrami, Y ., et al. 2018, ArXiv e-prints [arXiv:1807.06209]

2009 arXiv
[15]

2019, MNRAS, 490, 4688

Rizzato, M., Benabed, K., Bernardeau, F., & Lacasa, F. 2019, MNRAS, 490, 4688

2019
[16]

& Nishimichi, T

Sato, M. & Nishimichi, T. 2013, Phys. Rev. D, 87, 123538

2013
[17]

& Starck, J.-L

Sellentin, E. & Starck, J.-L. 2019, J. Cosmology Astropart. Phys., 2019, 021

2019
[18]

Takada, M. & Hu, W. 2013, Phys. Rev. D, 87, 123504

2013
[19]

V ., Klypin, A., et al

Tinker, J., Kravtsov, A. V ., Klypin, A., et al. 2008, ApJ, 688, 709

2008
[20]

L., Robertson, B

Tinker, J. L., Robertson, B. E., Kravtsov, A. V ., et al. 2010, ApJ, 724, 878

2010
[21]

A., Krause, E., Chang, C., et al

Troxel, M. A., Krause, E., Chang, C., et al. 2018, MNRAS, 479, 4998

2018
[22]

& Scoccimarro, R

Wadekar, D. & Scoccimarro, R. 2019, arXiv e-prints, arXiv:1910.02914

2019 arXiv
[23]

H., et al

Zehavi, I., Zheng, Z., Weinberg, D. H., et al. 2011, ApJ, 736, 59 Article number, page 10 of 11 Fabien Lacasa: Braiding and in-survey covariance Appendix A: Redshift dependent Halo Occupation Distribution The speciﬁcation Eq. 27 for the galaxy redshift distribution n(z) corres...

2011

Pith tools

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

[1] [1]

R., Balmès, I., Lacasa, F., & Lima, M

Abramo, L. R., Balmès, I., Lacasa, F., & Lima, M. 2015, MNRAS, 454, 2844

work page 2015

[2] [2]

& Takada, M

Akitsu, K. & Takada, M. 2017, ArXiv e-prints [arXiv:1711.00012]

arXiv 2017

[3] [3]

2018b, J

Barreira, A., Krause, E., & Schmidt, F. 2018b, J. Cosmology Astropart. Phys., 6, 015 Euclid Collaboration, Blanchard, A., Camera, S., et al. 2019, arXiv e-prints, arXiv:1910.09273 Harnois-Déraps, J. & Pen, U.-L. 2013, MNRAS, 431, 3349

arXiv 2019

[4] [4]

2017, MNRAS, 465, 1454

Hildebrandt, H., Viola, M., Heymans, C., et al. 2017, MNRAS, 465, 1454

2017

[5] [5]

& Eiﬂer, T

Krause, E. & Eiﬂer, T. 2017, MNRAS, 470, 2100

work page 2017

[6] [6]

F., Zuntz, J., et al

Krause, E., Eiﬂer, T. F., Zuntz, J., et al. 2017, ArXiv e-prints [arXiv:1706.09359]

arXiv 2017

[7] [7]

& Grain, J

Lacasa, F. & Grain, J. 2019, A&A, 624, A61

2019

[8] [8]

& Kunz, M

Lacasa, F. & Kunz, M. 2017, A&A, 604, A104

2017