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REVIEW 4 major objections 4 minor 46 references

The final WaZP galaxy cluster catalog of the Dark Energy Survey and comparison with SZE data

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read WaZP, a red-sequence-free cluster finder applied to six years of Dark Energy Survey data, yields over 400,000 clusters, and its cross-match with SPT and ACT shows essentially all Sunyaev-Zel'dovich clusters in the overlap have a WaZP…

desk verdict A carefully validated final DES-Y6 WaZP catalog with a large, honest SZE cross-match; the redshift-based validation has a partial circularity worth fixing, but the catalog itself is a solid resource. read the letter →

arxiv 2507.05360 v2 pith:QK6A45JP submitted 2025-07-07 astro-ph.CO

C. Benoist , M. Aguena , L. da Costa , J. Gschwend , S. Allam , O. Alves , F. Andrade-Oliveira , D. Bacon
show 38 more authors
L. Bleem D. Brooks A. Carnero Rosell J. Carretero F.J. Castander M. Costanzi J. De Vicente S. Desai S. Dodelson P. Doel S. Everett B. Flaugher J. Frieman J. Garcia-Bellido G. Giannini P. Giles R. Gruendl G. Gutierrez S. Hinton D.L. Hollowood K. Honscheid D. James K. Kuehn S. Lee J. Marshall J. Mena-Fernández R. Miquel A. Plazas Malagón K. Romer E. Sanchez B. Santiago I. Sevilla M. Smith E. Suchyta G. Tarle N. Weaverdyck J. Weller M.E. da Silva Pereira
This is my paper · ORCID
classification astro-ph.CO
keywords galaxyclustersclusterfindingphotometricredshiftsSunyaev-Zel'dovicheffectDarkEnergySurveyWaZPrichnessmulti-wavelengthcross-match
topics 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

The paper presents the final galaxy cluster catalog produced by the WaZP cluster finder on the full six-year Dark Energy Survey data, and it argues that the catalog is reliable enough for cosmology. WaZP detects clusters as projected overdensities in photometric redshift slices rather than by looking for a red sequence, so the sample covers a different and complementary part of cluster-selection space than red-sequence finders. The main numbers are a full catalog of more than 400,000 clusters with richness $N_{\mathrm{gals}} \ge 5$ reaching $z\simeq1.3$, and a homogenized cosmology subsample of about 33,000 rich clusters with $N_{\mathrm{gals}} > 25$. The paper's key external validation is a cross-match with Sunyaev-Zel'dovich (SZE) clusters from SPT and ACT: essentially all SZE clusters in the overlapping footprint have a WaZP counterpart, and 90% of WaZP clusters with richness above about 150 are recovered by the deeper SZE regions. A reliable, red-sequence-independent cluster sample of this size, checked against an independent millimeter-wave tracer, is what makes the catalog useful for cluster abundance and mass calibration studies.

What carries the argument

The carrying object is the WaZP (Wavelet Z-Photometric) cluster finder. It builds pixelized galaxy density maps in slices of photometric redshift, weights each galaxy by its redshift probability, extracts wavelet-based overdensity peaks, and merges peaks across slices; no red-sequence assumption enters. Membership probabilities are assigned from cluster-centric distance, magnitude, and photometric redshift following the Castignani & Benoist prescription, and richness $N_{\mathrm{gals}}$ is the sum of those probabilities for members brighter than $m^*(z)+1.5$. This machinery does the work because it produces the independent optical detections whose angular and redshift agreement with SPT and ACT validates the catalog, and because the same member assignments feed the redshift estimates, the centering analysis, and the multiplicity measurement.

What would settle it

Run the same redshift and multiplicity analysis using only SZE clusters with genuine spectroscopic redshifts, roughly one third of the sample. If the consistent-pair fraction stays near 93% and the multiplicity fraction stays at 15-20%, the paper's redshift-based validation survives; if the numbers drop substantially, the apparent agreement is partly inherited from the shared optical photo-z assignment used for the SZE redshifts.

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Extended reading notes

Core claim

The central claim is that the DES-Y6 WaZP catalog is both large and independently confirmed. Using only angular proximity for the cross-match, the paper recovers a WaZP counterpart for essentially every SPT and ACT SZE cluster that lies within the DES-Y6 footprint, has adequate local coverage, and sits below the local depth limit; 801 of 816 SPT clusters and 1842 of 1873 ACT clusters are matched. Going the other way, the fraction of WaZP clusters with an SZE counterpart rises with richness and exceeds 90% above $N_{\mathrm{gals}} \simeq 150$ in the deeper SZE regions. The paper further claims that WaZP cluster redshifts are accurate to a scatter of 1.4% and an offset of 0.2%, with no bias against SZE spectroscopic redshifts, and that 15-20% of SZE-matched systems contain more than one WaZP counterpart at the same redshift within the SZE $R_{500c}$, which it reads as evidence of interacting or unrelaxed systems.

Load-bearing premise

The redshift-consistency and multiplicity claims lean on SZE redshifts that, for the majority of SZE clusters, were assigned by cross-matching the millimeter detection to optical cluster catalogs; if those redshifts inherit the same optical selection or richness biases as WaZP, then the 93% consistency and the 15-20% multiplicity rate are not independent validations, although the angular matching rates do not depend on this assumption.

Editorial extensions

If this is right

  • The full 416,947-cluster catalog and the 33,000-cluster cosmology subsample provide a red-sequence-independent sample large enough for cluster abundance measurements over 4,545 deg^2.
  • The near-complete recovery of SZE clusters implies WaZP selection is essentially complete for massive systems within the DES-Y6 depth limits, so the catalog can be used to calibrate richness-mass relations against SZE and X-ray mass proxies.
  • The 15-20% multiplicity among SZE-matched systems means any cosmology analysis using this catalog must account for a substantial population of clusters that appear as two optical concentrations inside one SZE radius.
  • The measured redshift scatter of 1.4% and offset of 0.2% support using WaZP photometric redshifts as cluster redshifts out to $z\sim1$ for statistical studies.

Reading between the lines

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

  • Inference: comparing WaZP member lists with a red-sequence-based finder on the same data could quantify how much the red-sequence assumption biases richness and photo-z estimates at high redshift.
  • Inference: the 15-20% of SZE systems with multiple WaZP counterparts are natural targets for X-ray or dynamical follow-up, and optical multiplicity may provide a statistical tracer of merging systems.
  • Inference: the wavelet-in-photo-z detection strategy should transfer to upcoming wide-field surveys, where dropping the red-sequence prior may help recover high-redshift clusters with poorly sampled red sequences.
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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

4 major / 4 minor

Summary. This paper presents the final DES-Y6 WaZP galaxy cluster catalog: 416,947 detections with richness Ngals>=5 over 4,545 deg^2 out to z~1.32, together with a homogenized cosmology sample of 33k clusters with Ngals>25. The catalog is validated internally with 24,447 spectroscopic redshifts, by comparison with the DES-Y1 WaZP catalog, and against external optical catalogs (CLuMPR, WH24) and SZE catalogs from SPT and ACT. The main reported results are high Y1-to-Y6 recovery rates (95-99%), nearly complete matching of SZE clusters in well-covered footprint regions (801/816 SPT, 1842/1873 ACT), 90% recovery of the richest optical clusters in the deeper SZE regions, ~93% redshift consistency between WaZP and SZE counterparts, a 15-20% multiplicity rate interpreted as possible interacting/unrelaxed systems, and a two-component Rayleigh model for optical-SZE centering offsets.

Significance. If the results hold, this is a valuable red-sequence-independent cluster sample for cosmology and cluster astrophysics, and the public release of the catalog is a useful community resource. The paper is careful in several important respects: it uses a large spectroscopic sample for redshift validation, traces unmatched SZE systems to masking, coverage, or zmax limits, provides a cosmology-ready homogenized subsample, and makes the data available online. The angular matching rates and the richness-dependent recovery fractions are robust and do not depend on SZE redshift assumptions. The main caveat is that the redshift-based SZE validation is not fully independent: the text states that many SZE photometric redshifts were assigned by cross-matching with other optical cluster finders, so the ~93% consistency claim, the <0.6% random-association estimate, and the multiplicity interpretation are partly circular with optical selection. The spectroscopic SZE subsample (160 SPT, 456 ACT) is large enough to provide a clean test, and the authors should present it as the primary validation.

major comments (4)
  1. [4.3, Fig. 18, Table 3] The claim that 93% of WaZP-SZE matches have consistent redshifts is not an independent multi-wavelength validation because, as stated in Section 4.3, the SZE photometric redshifts for most of the sample (583 SPT and 1,255 ACT systems) were 'assigned by cross-matching with the results of several optical cluster finders (e.g. redMaPPer, Camira, AMICO, CLuMPR, WH24)'. Since WaZP is itself an optical cluster finder, these redshifts correlate with optical cluster redshifts by construction. Please recompute the consistency fraction, bias, and scatter using only the SZE spectroscopic subsample (160 SPT and 456 ACT in Table 3) and present it as the primary validation; the photometric-SZE comparison should be labeled explicitly as a consistency check that inherits optical redshift information.
  2. [4.2, random-association estimate] The estimate that random associations are less than 0.6% is obtained by imposing the redshift-consistency condition from Section 4.3. Because that condition relies partly on non-independent SZE photo-zs, the quoted probability is likely underestimated. Please recompute the random-association rate using the angular-only criterion (~6% per catalog) as the headline number, or provide a version restricted to SZE systems with independent spectroscopic redshifts.
  3. [4.4] The 15-20% multiplicity fraction is defined using a redshift window derived from the scatter in Table 3, whose photometric rows inherit the same optical-correlation issue described above. As a result, the conclusion that these systems indicate 'possible interacting or unrelaxed systems' is not yet independently supported by the data as presented. Please recompute the multiplicity fraction for SZE clusters with spectroscopic redshifts only and report how the fraction changes; if the subsample is too small for a robust statement, the claim should be explicitly downgraded to a preliminary result.
  4. [4.5, Table 4] The centering model is fit only to multiplicity-1 systems, and the secondary Rayleigh component is poorly constrained: for example, sigma1 = 0.57(+0.68,-0.35) for WaZP-SPT with SNR>5 and sigma1 = 0.90(+0.44,-0.33) for WaZP-ACT with SNR>5. The abstract's statement that 'WaZP and SZE centerings are found to be consistent' should be qualified to reflect that the data mainly constrain the dominant well-centered component; the tail population is essentially unconstrained, and the robustness check shows that sigma1 depends on how multiple matches are resolved.
minor comments (4)
  1. [3.1 and Fig. 8] The number of zCG spectroscopic clusters is given as 23,570 in Section 3.1 and Table 2, but Figure 8's caption says 23,850; please correct the inconsistency.
  2. [4.2] The sentence 'The very few SZE systems without counterparts (lesssim4%) are cross-matched using a 5.3 arcmin aperture' is confusing because the final unmatched fractions are 1.8% for SPT and 1.7% for ACT; please clarify that this is the fraction before the second, larger-aperture pass.
  3. [Data Availability] The Data Availability sentence contains a grammatical error: 'can be found are available' should be 'can be found' or 'are available'.
  4. [3.2 and Conclusions] The paper explicitly defers the quantitative impact of photometric-redshift quality on detection completeness to a separate paper; given that, the conclusions should state more prominently that the Y1-Y6 recovery comparison is not corrected for this effect, so the word 'robustness' should be accompanied by this caveat.

Circularity Check

1 steps flagged · score 3.0 of 10

Redshift-consistency and multiplicity claims are partially built on SZE photometric redshifts that were assigned via other optical cluster finders; angular cross-matches and SZE spec-z comparisons remain independent.

  1. fitted input called prediction [Section 4.3 (redshift comparison, Fig. 18 and Table 3), propagated to Section 4.2 (random-association estimate) and Section 4.4 (multiplicity)]
    "The two SZE catalogs used to compare with our optical detection provide a spectroscopic redshift for one-third of the sample and a photometric redshift for the rest. The latter were assigned by cross-matching with the results of several optical cluster finders (e.g. redMaPPer, Camira, AMICO, CLuMPR, WH24) and/or from targeted follow-up observations."

    The SZE photometric redshifts used in the validation are not independent millimeter-wavelength redshifts: the paper states they were assigned by matching SZE signals to optical cluster finders. WaZP is itself an optical cluster finder operating on the same DES galaxies, so agreement between WaZP redshifts and these SZE photo-z is partly agreement between two optical selections rather than a SZE confirmation. The reported 93% consistent-redshift fraction, the <0.6% random-association estimate that conditions on redshift consistency, and the 15-20% multiplicity rate (defined in Section 4.4 using a redshift window derived from Table 3, which includes these photo-z rows) all inherit that correlation.

full rationale

The main deliverable, the DES-Y6 WaZP catalog, and the angular cross-matching claims are self-contained: richnesses, the 801/816 SPT and 1842/1873 ACT angular matches, the >90% recovery for richness above ~150, and the cluster redshift scatter of 1.4% measured against public spec-z are all derived from DES data plus independent SZE positions and published spectroscopy. The only material circularity is in the redshift-consistency and multiplicity claims, which partly use SZE photo-z that were assigned by cross-matching SZE detections to other optical cluster finders. Because WaZP is also an optical cluster finder, those comparisons are not fully independent multi-wavelength validations; however, the SZE spec-z subsample provides genuine independent support, and the angular matching and catalog claims do not depend on the photo-z input. Overall circularity is therefore modest rather than structural.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

No new physical entities are introduced. The ledger instead captures the fixed detection thresholds, matching radii, coverage cuts, and the fitted Rayleigh offset model that the paper's validation claims rest on. The most consequential entries are the selection of the richest WaZP counterpart and the reliance on SZE photometric redshifts that were themselves derived from optical cluster catalogs.

free parameters (6)
  • delta_mag detection = 2 mag
    Limiting magnitude below m*(z) used to define zmax maps and detection depth; chosen as a fixed value in Section 2.3.
  • delta_mag richness = 1.5 mag
    Cut for member counts in richness Ngals; chosen in Section 3.
  • wavelet pixel scale = 1/16 Mpc
    Resolution of density maps; sets the centering beam in Section 3.
  • SZE-WaZP match aperture = 2.6 arcmin and 5.3 arcmin
    Angular windows matching 1 Mpc at z=0.5 and 1 Mpc at z=0.2; chosen for cross-match in Section 4.2.
  • coverage fraction thresholds = 0.5 and 0.8
    Thresholds for footprint coverage in cross-match and centering analyses in Sections 4.2 and 4.5.
  • centering model parameters rho0, sigma0, sigma1 = rho0 ~0.8-0.91, sigma0 ~0.03-0.07 R500c, sigma1 ~0.3-0.9 R500c
    Fitted by MCMC to the WaZP-SZE offset distribution (Table 4); these are calibrated parameters, not derived from first principles.
assumptions (6)
  • domain assumption Validity of the WaZP cluster finder as described in Paper-1: wavelet-based overdensities in photo-z slices correspond to galaxy clusters.
    The paper reuses the finder and code from Aguena et al. 2021 and only summarizes it in Section 3.
  • domain assumption Galaxy weights in each photo-z slice follow a Gaussian redshift PDF.
    Section 3 states galaxies are weighted based on their redshift PDF, here Gaussian.
  • ad hoc to paper The richest WaZP cluster inside the SZE aperture is the optical counterpart sourcing the SZE signal.
    Section 4.2: 'Multiple candidates are solved by selecting the richest WaZP cluster.' This drives the pairing and affects downstream multiplicity claims.
  • domain assumption SZE redshift assignments from SPT/ACT are reliable for comparison, including photometric redshifts obtained from other optical cluster catalogs.
    Section 4.3 states SZE photometric redshifts were 'assigned by cross-matching with the results of several optical cluster finders', so they are not fully independent.
  • domain assumption NFW-weighted coverage fraction correctly quantifies DES footprint overlap for SZE clusters.
    Used to define c_f and exclude poorly covered systems in Section 4.2, following Rykoff et al. 2012.
  • domain assumption The zmax map derived from m_lim = m*(zmax)+2 correctly defines redshift completeness of richness.
    Section 2.3 assumes the luminosity-function knee m*(z) and a fixed delta mag; the quantitative completeness impact is deferred to future simulations.

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Cite this review

Pith. "Pith review of The final WaZP galaxy cluster catalog of the Dark Energy Survey and comparison with SZE data." pith.science (2026). https://pith.science/paper/QK6A45JP

@misc{pith2026250705360,
  author       = {Pith},
  title        = {Pith review of: The final WaZP galaxy cluster catalog of the Dark Energy Survey and comparison with SZE data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QK6A45JP}},
  note         = {Machine review of arXiv:2507.05360}
}
read the original abstract

In this work, we present and characterize the galaxy cluster catalog detected by the WaZP cluster finder, which is not based on red-sequence identification, on the full six years of observations of the Dark Energy Survey (DES-Y6). The full catalog contains over 400k detected clusters with richnesses, Ngals, above 5 and that reach redshifts up to 1.3. We also provide a version of the catalog where the observation depth and richness computation are homogenized to be used for cosmology, containing 33k rich (Ngals >25) clusters. We compare our results with the previous WaZP catalog obtained from the DES first-year data release (DES-Y1). We find that essentially all clusters within the common footprint and depth limit are recovered. The deeper observations on DES-Y6 and the more complete available spectroscopic redshift sample lead to improvements in the redshifts of the clusters, resulting in an average scatter of 1.4% and offset of 0.2%. The optical clusters are also cross-matched with Sunyaev Zel'dovich Effect (SZE) cluster samples detected by the South Pole Telescope (SPT) and the Atacama Cosmology Telescope (ACT). We find that essentially all SZE clusters with reasonable overlapping footprint have a corresponding WaZP cluster. Conversely, 90% of the optical detections with richness greater than 150 have a counterpart in the deeper regions of the SZE surveys. Based on cross-match with the SZE catalogs, we also find that 15-20% of the SZE matched systems have more than one possible WaZP counterpart at the same redshift and within the SZE R500c, indicating possible interacting or unrelaxed systems. Finally, given the optical and SZE beams, WaZP and SZE centerings are found to be consistent. A more detailed study of the SZE-WaZP mass-richness relation will be presented in a separate paper.

Figures

Figures reproduced from arXiv: 2507.05360 by the authors.

Figure 1
Figure 1. Magnitude distribution (top panel) of the 10σ DES-Y6 depth map on z-band, and the total area of the survey below said magnitude thresholds. latter, we adopted the classification scheme provided by the DES data release team, based on the so-called EXTENDED_FITVD parameter available within the internal release of DES-Y6 data (Bechtol et al. 2025; Hartley et al. 2022). This parameter is a small variation from the EXTEN… view at source ↗
Figure 2
Figure 2. Distribution of spectroscopic redshifts of galaxies used for train￾ing and validating DNF photo-zs for this work (DES-Y6) and for Paper￾1. The complete sample used in the training and the m ∗ (z) + 2 limited sample used by WaZP for cluster detection (solid areas) are shown as transparent and opaque histograms, respectively [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Comparison of photo-z bias, scatter, and fraction of catastrophic failures as a function of spec-z obtained with DES-Y6 (this work) and DES-Y1 (Paper-1) data. Metrics were computed using the complete samples (transparent lines) and samples brighter than m ∗ (z) + 2 (full lines). of the DES-Y6 i− and z− bands are similar, covering approxi￾mately the same area at a given redshift, with a slight advantage in the case o… view at source ↗
Figures from the paper (16 more)
Figure 5
Figure 5. Figure 5: Distribution of the zphot error computed in cells of ∼ 3 deg2 across the DES footprint and its relation to the local z-band depth. In green and orange are the cells deviating by more than 2- and 3-σ, respectively, from the median value of the survey. The left panel sho…
Figure 6
Figure 6. Figure 6: Distribution of the offset between the nominal WaZP center derived from smooth density maps and the associated brightest clus￾ter member when it is closer than 150 kpc from the WaZP center, i.e. ∼ 3 pixels in the WaZP density maps [PITH_FULL_IMAGE:figures/full_fig_p00…
Figure 7
Figure 7. Figure 7: Cluster density per arcmin2 of rich (Ngals ≥ 25) WaZP detections per area detected of WaZP detections on DES-Y1 and DES-Y6 data, considering the full and the cosmology (cluster z < zmax map) samples. ∆θ < 1 arcmin; 2) Clusters with ∆r⊥ < 300 kpc and ∆z < σz , where ∆r⊥…
Figure 10
Figure 10. Figure 10 [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]
Figure 11
Figure 11. Figure 11: Richness comparison of WaZP-Y6 and WaZP-Y1 matched clusters with ∆θ < 100 kpc and ∆z < 0.023. The smaller panel shows the distribution of the difference between richnesses, with the colors (blue, orange, green) corresponding to the N Y6 gals bins: [10, 25, 50, 300]. l…
Figure 12
Figure 12. Figure 12: The recovery rates between WaZP clusters and other optical cluster catalogs as a function of WaZP richness [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]
Figure 13
Figure 13. Figure 13: DES-Y6 footprint (red) compared to ACT (yellow) and the different SPT (blue) footprints. where cf is the Navarro-Frenk-White (NFW) weighted area (see Rykoff et al. 2012) fraction covered by the DES footprint in a 1 Mpc aperture around the SZE cluster. The visual inspe…
Figure 14
Figure 14. Figure 14: SZE clusters inside WaZP DES-Y6 footprint. The colors of the points correspond the redshift type of the SZE counterparts: spec-z shown in black and photo-z in green. We also flagged clusters according to their coverage on the DES footprint: clusters with low coverage …
Figure 15
Figure 15. Figure 15: Diagram of the number of clusters matched considering the WaZP, SPT, and ACT catalogs inside the DES footprint. optical detection provide a spectroscopic redshift for one-third of the sample and a photometric redshift for the rest. The latter were assigned by cross-ma…
Figure 16
Figure 16. Figure 16: Normalized distributions of WaZP-SZE matched clusters. The angular separation between WaZP-SZE cluster centers of pairs with c SZ f , c WaZP f > 0.8 is shown on the left panel. On the right panel, the distribution of the WaZP richnesses for the matched pairs is displa…
Figure 17
Figure 17. Figure 17: The recovery rates of WaZP clusters by ACT or SPT as a func￾tion of WaZP richness, estimated as the fraction of WaZP clusters of a given richness for which there is an ACT or SPT cluster within a ra￾dius of 5.3 arcmin. Only WaZP clusters with a 100% coverage of the SZ…
Figure 18
Figure 18. Figure 18: Redshift relation of WaZP-SPT (left) and WaZP-ACT (right) matched clusters. The sample is split into clusters with spectroscopic SZE redshifts (top) and photometric redshifts (bottom). The blue-shaded regions correspond to 3 times the scatter of the sample with the ou…
Figure 19
Figure 19. Figure 19: Four examples of SZE detections for which our detection and matching procedure assigned a primary redshift different from that of the SZE consortia. However, in these cases, a secondary counterpart on almost the same line of sight has a consistent redshift. These exam…
Figure 20
Figure 20. Figure 20: Case for which the WaZP counterpart to the ACT detection is found at a very different redshift (z = 0.71) from the ACT redshift (z = 0.38). This is an ambiguous case where two clear cD galaxies at redshifts 0.38 and 0.71 are located next to each other. However, only t…
Figure 21
Figure 21. Figure 21: Number of WaZP clusters within a cylinder composed of an aperture of R500c and height of 3σz from [PITH_FULL_IMAGE:figures/full_fig_p015_21.png]
Figure 23
Figure 23. Figure 23: Left: Projected angular separation of matched clusters with SNR>5 and c WaZP f , c SZE f > 0.8. Histograms correspond to WaZP-SPT and WaZP-ACT pairings, and the dashed lines represent the convolution of the offset models for each cluster catalog pairing. Right: Constr…

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    Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching, Germany

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    2.4, we evaluate the spatial homogeneity of the photo- metric redshift errors used in this work (Fig

    Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany Appendix A: Photometric redshifts quality and dust absorption In Sec. 2.4, we evaluate the spatial homogeneity of the photo- metric redshift errors used in this work (Fig. 5). In Fig. A.1, the...

    2020

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