Pith. sign in

REVIEW 3 major objections 4 minor 3 cited by

CSST Cosmological Emulator II: Generalized Accurate Halo Mass Function Emulation

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read A generalized emulator predicts halo mass functions to 2 percent at low masses for CSST cosmology.

desk verdict A useful and mostly clean HMF emulator paper whose headline accuracy numbers are probably optimistic because the leave-one-out test shares the fitting-formula denominator with the held-out cosmology. read the letter →

arxiv 2506.09688 v2 pith:UYRORYW2 submitted 2025-06-11 astro-ph.CO

Zhao Chen , Yu Yu This is my paper
classification astro-ph.CO PACS 95.75.-z98.65.Dx98.80.-k
keywords halomassfunctioncosmologicalemulatorcumulativebinningeffectGaussianprocessregressionN-bodysimulationsCSSTclustercosmology
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 claims that the cumulative halo mass function can be emulated at percent-level accuracy over a wide, eight-dimensional cosmological parameter space, for three different halo mass definitions at once, by fitting the ratio of a B-spline smoothed measurement to a calibrated fitting formula. The authors first quantify how mass binning biases the differential halo mass function by up to 5% at high masses and redshifts, and demonstrate that the cumulative form is binning-independent, making it the correct observable for emulation. Their leave-one-out validation shows the emulator predicting cumulative HMFs within 2% for $M\le10^{13}\,h^{-1}M_\odot$, 5% for $M\le10^{14}\,h^{-1}M_\odot$, and 10% for $M\le10^{15}\,h^{-1}M_\odot$ at $z\le1$, comparable to the statistical errors of the training simulations. If correct, this provides cluster abundance analyses with a fast, unbiased theoretical predictor for the forthcoming CSST survey.

What carries the argument

The load-bearing object is the ratio $R_j(\ge M_j,z)=N_j(\ge M_j,z)/N_{j,\mathrm{C23}}(\ge M_j,z)$ between the B-spline smoothed cumulative halo mass function and the integrated, recalibrated Castro23 fitting formula. This ratio strips away the steep mass, redshift, and cosmology dependence of the raw abundance, leaving a slowly varying field that ten principal components capture and Gaussian process regression interpolates over the eight cosmological parameters. The cumulative form itself is the second mechanism: because it is defined at fixed mass thresholds rather than over mass bins, it has no binning bias, and it is smoother with smaller statistical errors at the high-mass end than the differential form.

What would settle it

Run a resolution-convergence test at a low-$\sigma_8$ cosmology with massive neutrinos inside the training volume (for instance a 250 $h^{-1}$Mpc box with $768^3$ and $1536^3$ particles); if the cumulative HMF ratio at $M=10^{12}\,h^{-1}M_\odot$ deviates by more than 2% at any redshift up to $z=3$, the low-mass accuracy bound fails for that region of parameter space. An independent check would compare the emulator's leave-one-out predictions at the extreme corners of the Sobol-sampled design against a separate high-resolution simulation suite.

Watch

Extended reading notes

Core claim

On its own terms, the central claim is that a generalized framework—B-spline smoothing of the measured cumulative HMF, recalibration of the Castro23 fitting formula as a physical prior, and emulation of the resulting ratio via PCA-compressed Gaussian process regression—reproduces the simulated halo abundance to statistical precision for three distinct halo mass definitions (Rockstar $M_{200m}$, Rockstar $M_{vir}$, and FoF $M_{200c}$) across the $w_0w_a$CDM + $\sum m_\nu$ parameter space, for $M\ge10^{12}\,h^{-1}M_\odot$ up to $z=3$. The paper also establishes that binning choices cause percent-level systematic shifts in the differential HMF, while the cumulative HMF is immune to them, and that the same accuracy holds across halo finders and mass definitions.

Load-bearing premise

The accuracy claims rest on the assumption that the 2% mass-resolution convergence measured at the Planck 2018 cosmology without massive neutrinos holds unchanged across all 129 cosmologies in the training space, including models with massive neutrinos and extreme values of $\sigma_8$, $w_0$, and $w_a$; the resolution test is performed at the fiducial cosmology only.

Editorial extensions

If this is right

  • The emulator predicts cumulative and differential HMFs for $M_{200m}$, $M_{vir}$, and $M_{200c}$ in about 0.3 seconds per cosmology, fast enough for Monte Carlo parameter estimation.
  • At $z\le1$, the 2%/5%/10% accuracy bands mean cluster number counts can avoid the multi-percent non-universality biases of universal fitting formulas.
  • A CSST-like cluster count forecast using the emulator recovers the fiducial $\Omega_m$ and $\sigma_8$, whereas the Tinker08 and Castro23 formulas produce several-sigma shifts, implying that unbiased Stage-IV cluster cosmology requires emulated predictions.
  • The same generalized framework can be extended to other physical halo boundaries, such as splashback or depletion radii, once corresponding halo catalogs exist.

Reading between the lines

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

  • If the fiducial-only resolution test is representative, the emulator's low-mass reach means galaxy–halo connection models and cluster counts down to $10^{12}\,h^{-1}M_\odot$ can inherit the same systematic floor, a reach most previous emulators do not offer.
  • The binning-independence of the cumulative HMF suggests a practical route for comparing cluster catalogs built with different mass estimators: convert each catalog to cumulative counts above common thresholds rather than attempting to match differential bins.
  • The same ratio-plus-PCA/GPR recipe could likely be transferred to other mass-dependent statistics, such as halo bias or concentration–mass relations, wherever a fitting formula provides an imperfect prior.
  • A decisive stress test would extend the convergence analysis to a couple of extreme Kun cosmologies, since the current 2% floor is empirically established at a single fiducial point.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper presents a generalized emulator for the cumulative halo mass function (HMF) covering three halo mass definitions (M200m, Mvir, M200c) over a broad w0waCDM + m_nu parameter space, based on the 129 Kun simulations. The authors first quantify the effect of mass binning on the differential HMF and argue that the cumulative HMF is robust to binning choices. They then build the emulator by B-spline smoothing the simulated cumulative HMF, calibrating a Castro23-style fitting formula as a cosmology-dependent denominator, and using PCA plus Gaussian process regression on the ratio. The headline claim is 2% accuracy for M <= 1e13 h^-1 M_sun, 5% for M <= 1e14 h^-1 M_sun, and 10% for M <= 1e15 h^-1 M_sun at z <= 1.0, supported by leave-one-out cross-validation. The paper also includes comparisons with existing emulators and fitting formulae and a forecast for CSST cluster cosmology.

Significance. If the accuracy claims hold, this would be a valuable public tool for CSST cluster cosmology, extending HMF emulation to multiple halo mass definitions and to a broad cosmological parameter space including massive neutrinos and dynamical dark energy. The paper's strengths are its careful treatment of binning effects, the use of a cosmology-dependent fitting-formula prior to reduce extrapolation risk, the consistency of performance across different halo definitions and finders, and the public release of the code. The central accuracy claim, however, rests on a cross-validation procedure whose denominator is calibrated on all 129 simulations including the held-out cosmology, so the reported LOO residuals are not a clean measure of predictive accuracy on unseen cosmologies.

major comments (3)
  1. [Sec. 4.2 and Sec. 4.4, Eqs. (13) and (14)] The Castro23 parameters in Table 1 are obtained by maximizing the likelihood in Eq. (13) summed over all 129 Kun cosmologies, including the cosmology that is later held out in each LOO fold. This calibrated Castro23 model is then used to construct the ratio R_j in Eq. (14) for every cosmology. Consequently, for each LOO test the held-out cosmology contributes to the denominator of the very quantity the emulator is asked to predict, so the residuals in Figs. 8 and 9 are not true prediction errors for an unseen cosmology. Because the headline 2%/5%/10% accuracy claim is based entirely on this LOO validation, this leakage is load-bearing. I recommend re-calibrating the Castro23 parameters separately for each training fold (excluding the held-out cosmology) and reporting the scatter in the fitted parameters, or alternatively validating on a small number of simulations not used in any calibration step. The authors should also state explicitly whether the quoted accuracy refers to predictions with a fixed Castro23 prior or to a fully re-calibrated prior.
  2. [Sec. 3.2, Fig. 4] The resolution convergence test establishes a 2% convergence for M >= 1e12 h^-1 M_sun only at the Planck 2018 fiducial cosmology without massive neutrinos, in boxes of side 250 h^-1 Mpc. The paper then assumes this convergence holds across the full Kun parameter space of Eq. (6), including models with massive neutrinos and widely varying sigma8, w0, and wa. Since the headline accuracy claim extends down to M = 1e12 h^-1 M_sun, a resolution systematic above 2% in any part of the parameter space would directly invalidate the low-mass end of the claim. I ask the authors either to run the resolution test at least one high-sigma8 and one massive-neutrino cosmology, or to explicitly limit the stated accuracy range to a regime where the resolution convergence has been verified.
  3. [Sec. 4.4, Fig. 8] The paper states that the difference between LOO predictions and simulations is 'mainly caused by statistical fluctuations' and that residuals are within the median Poisson noise for sigma8, cb > 0.7. This conclusion is based on visual comparison of residuals with a Poisson-noise band, but the B-spline smoothing and the Castro23 denominator both introduce cosmology-dependent systematics that are not included in that noise estimate. A quantitative decomposition of the LOO residual into Poisson noise, B-spline smoothing error, and emulator interpolation error would make the validation more convincing and would clarify whether the claimed percent-level accuracy is limited by statistics or by methodology.
minor comments (4)
  1. [Throughout] There are several typographical errors: 'Marten-5/2' should be 'Matérn-5/2', 'Cartro23' should be 'Castro23', 'Serval' should be 'Several', and 'K un' should be 'Kun'.
  2. [Sec. 2.2 and 2.3] The statement that the cumulative HMF is independent of the binning scheme should be qualified: this is true at mass thresholds that coincide with bin edges, whereas thresholds lying inside a bin can still suffer a small binning effect. Fig. 3 demonstrates the former case; the text currently overstates the generality.
  3. [Sec. 4.1, Eq. (9)] The choice of alpha = 0.5 for the maximum-mass cut is stated to be 'a good choice' for all halo definitions and redshifts, but no supporting figure or quantitative criterion is shown. A brief validation of this choice, or a reference to an analogous test, would improve transparency.
  4. [Sec. 5.2] The forecast in Sec. 5.2 uses a very simplified covariance that includes only shot noise and a 30% systematic contribution, and the paper acknowledges this is optimistic. The conclusion that Tinker08 and Castro23 introduce 'significant systematic bias' should be interpreted with this caveat in mind; the quantitative sigma_Omega_m and sigma_sigma8 values depend on the assumed covariance and would change under a more realistic mass-calibration model.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the LOO denominator is a global fit, so the held-out cosmology's influence is a minor validation caveat, not a construction-level circularity.

full rationale

The claimed derivation is self-contained. The HMF data come from the Kun N-body suite; the B-spline smoothing (Sec. 4.1) and the PCA+GPR ratio emulator (Sec. 4.3) are trained on the simulations, and the LOO validation (Sec. 4.4) genuinely removes one cosmology from the ratio-emulator training. The Castro23 denominator is a global eight-parameter fitting formula calibrated once over all 129 cosmologies (Eqs. 10-13); it is used only to compress the dynamic range of the emulated ratio (Eq. 14). Because the parameters are shared globally rather than re-fit per cosmology, the held-out cosmology's inclusion in the Castro23 calibration is a minor validation contamination, not a construction-level circularity: the final prediction is not statistically forced by the held-out data, and removing one cosmology from the global fit would change the denominator only slightly. No load-bearing self-citation chain or imported uniqueness theorem appears; the comparison against external emulators (DarkQuest, Aemulus-ν) and fitting formulae in Sec. 5.1 provides independent support. The resolution-convergence caveat is a robustness concern, not circularity.

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

The central claim rests on the Castro23 fitting formula parameters, which are fitted to the Kun simulations, and on several modeling assumptions about the simulations, resolution convergence, and noise model. No new physical entities are introduced.

free parameters (5)
  • Castro23 parameters for M200m = a1=0.7728, a2=0.3686, az=-0.0321, p1=-0.4608, p2=-0.5342, q1=0.3734, q2=-0.3232, qz=-0.1920
    Eight parameters of the Castro23 multiplicity function fitted to the Kun simulation HMF measurements for the Rockstar M200m catalog (Sec. 4.2, Table 1). Used as the denominator in the emulated ratio.
  • Castro23 parameters for Mvir = a1=0.7645, a2=0.4171, az=-0.0618, p1=-0.4686, p2=-0.6511, q1=0.3883, q2=-0.3696, qz=-0.0069
    Same fitting for Rockstar Mvir catalog (Sec. 4.2, Table 1).
  • Castro23 parameters for M200c = a1=0.7660, a2=0.3854, az=-0.1423, p1=-0.5236, p2=-0.7343, q1=0.3208, q2=-0.4046, qz=0.1527
    Same fitting for FoF M200c catalog (Sec. 4.2, Table 1).
  • Maximum halo mass coefficient alpha = 0.5
    Chosen by hand after checking, used in Eq. 9 to set the high-mass cut for B-spline fitting. Controls the number of high-mass data points used.
  • Hyperparameters of the pipeline = B-spline degree k=2, knot spacing 0.25 dex, min B-spline mass 1e11, min halo count 20, NPCA=10, sigma_sys=0.005N_sim
    Chosen by hand or by standard practice; affect the smoothness and the emulator's dynamic range. These are not fitted to data but are ad hoc choices for the central pipeline.
assumptions (5)
  • domain assumption N-body simulations with the fixed-amplitude method and 2LPT initial conditions accurately produce halo populations under the w0waCDM+Σmν model.
    The entire training set is generated by Gadget-4 with these settings (Sec. 3.1); any bias in the simulations propagates to the emulator.
  • domain assumption The Castro23 multiplicity function (Eq. 10-12) is flexible enough to describe the HMF across all 129 cosmologies and redshifts within a few percent.
    Used as the denominator in the emulated ratio (Sec. 4.2); if the fitting formula cannot track the true HMF in some corner of parameter space, the ratio becomes difficult to emulate.
  • domain assumption The resolution convergence results from the CT0/CT3/CT4 test simulations, run at the Planck 2018 cosmology without massive neutrinos, hold for all Kun cosmologies.
    Sec. 3.2 uses these three simulations to conclude 2% convergence at M>=10^12 for all mass definitions; generalization to the full parameter space is assumed.
  • domain assumption The binning-effect bias predicted using Tinker08 matches the true bias in simulations.
    Used in Sec. 2.2 to argue cumulative HMF is binning-independent; validated only against the Quijote suite at fiducial-like cosmology.
  • domain assumption The Poisson likelihood with an added 0.5% systematic noise (Eq. 13) is an appropriate noise model for the HMF measurements.
    Adopted from Castro et al. 2023; governs the Castro23 parameter fits.

how reviews work

0 comments
Cite this review

Pith. "Pith review of CSST Cosmological Emulator II: Generalized Accurate Halo Mass Function Emulation." pith.science (2026). https://pith.science/paper/UYRORYW2

@misc{pith2026250609688,
  author       = {Pith},
  title        = {Pith review of: CSST Cosmological Emulator II: Generalized Accurate Halo Mass Function Emulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UYRORYW2}},
  note         = {Machine review of arXiv:2506.09688}
}
abstract

Accurate theoretical prediction for halo mass function across a broad cosmological space is crucial for the forthcoming China Space Station Telescope (CSST) observations, which will capture cosmological information from multiple probes, e.g., cluster abundance, and weak lensing. In this work, we quantify the percent-level impact of different mass binning schemes when measuring the differential halo mass function from simulations, and demonstrate that the cumulative form of the halo mass function is independent of the binning scheme. Through the recently finished Kun simulation suite, we propose a generalized framework to construct multiple accurate halo mass function emulators for different halo mass definitions, including $M_{200m}$, $M_{vir}$, and $M_{200c}$. This extends our CSST Emulator to provide fast and accurate halo mass function predictions for halo mass $M\geq 10^{12}\,h^{-1}M_{\odot}$ up to $z=3.0$. For redshifts $z\leq 1.0$, the accuracy is within $2\%$ for $M\leq 10^{13}\,h^{-1}M_{\odot}$, $5\%$ for $M\leq 10^{14}\,h^{-1}M_{\odot}$, and $10\%$ for $M\leq 10^{15}\,h^{-1}M_{\odot}$, which is comparable with the statistical errors of training simulations. This tool is integrated in CSST Emulator and publicly available at https://github.com/czymh/csstemu, providing a fast and accurate theoretical tool to obtain unbiased cosmological constraints of the upcoming CSST survey.

Discussion (0). Sign in to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Hermes - Towards an Optimal High-Performance Algorithm for Cosmic Statistics of Large Data Sets

    astro-ph.CO 2026-07 conditional novelty 6.0 of 10

    Hermes/PyHermes reconstructs catalogues in a scaling-function basis and unifies CIC, 2PCF, 3PCF, marked, and operator-based cosmic statistics as reusable window operations with FFT/MPI/GPU scaling.

  2. Artifacts in Halo Shapes: Imprints of the Initial Condition

    astro-ph.CO 2025-07 conditional novelty 6.0 of 10

    Grid-based initial conditions imprint roughly one-percent alignment artifacts on simulated halo shapes, with a redshift-dependent sign flip whose origin is not yet pinned down.

  3. Extending CSST Emulator to post-DESI era

    astro-ph.CO 2025-10 conditional novelty 5.0 of 10

    A tuned 'spectral equivalence' mapping lets the CSST emulator predict nonlinear matter power spectra at ~1% accuracy across the DESI DR2+CMB dynamic-dark-energy posterior.

Reference graph

Works this paper leans on

116 extracted references · 21 canonical work pages · cited by 3 Pith papers

  1. [1]

    Borgani , P

    S. Borgani , P. Rosati , P. Tozzi , S. A. Stanford , P. R. Eisenhardt , C. Lidman et al., 561, 13 (2001), arXiv: astro-ph/0106428

    arXiv 2001

  2. [2]

    Schuecker , H

    P. Schuecker , H. B \"o hringer , C. A. Collins , and L. Guzzo , 398, 867 (2003), arXiv: astro-ph/0208251

    arXiv 2003

  3. [3]

    Vikhlinin , A

    A. Vikhlinin , A. V. Kravtsov , R. A. Burenin , H. Ebeling , W. R. Forman , A. Hornstrup et al., 692, 1060 (2009), arXiv: 0812.2720

    arXiv 2009

  4. [4]

    Y. Zu , R. Mandelbaum , M. Simet , E. Rozo , and E. S. Rykoff , 470, 551 (2017), arXiv: 1611.00366

    arXiv 2017

  5. [5]

    Marulli , A

    F. Marulli , A. Veropalumbo , M. Sereno , L. Moscardini , F. Pacaud , M. Pierre et al., 620, A1 (2018), arXiv: 1807.04760

    arXiv 2018

  6. [6]

    McClintock , T

    T. McClintock , T. N. Varga , D. Gruen , E. Rozo , E. S. Rykoff , T. Shin et al., 482, 1352 (2019), arXiv: 1805.00039

    arXiv 2019

  7. [7]

    Bocquet , J

    S. Bocquet , J. P. Dietrich , T. Schrabback , L. E. Bleem , M. Klein , S. W. Allen et al., 878, 55 (2019), arXiv: 1812.01679

    arXiv 2019

  8. [8]

    Fumagalli , M

    A. Fumagalli , M. Costanzi , A. Saro , T. Castro , and S. Borgani , 682, A148 (2024), arXiv: 2310.09146

    arXiv 2024

Show all 116 references
  1. [9]

    S. W. Allen , A. E. Evrard , and A. B. Mantz , 49, 409 (2011), arXiv: 1103.4829

    2011 arXiv

  2. [10]

    A. V. Kravtsov and S. Borgani , 50, 353 (2012), arXiv: 1205.5556

    2012 arXiv

  3. [11]

    D. H. Weinberg , M. J. Mortonson , D. J. Eisenstein , C. Hirata , A. G. Riess , and E. Rozo , 530, 87 (2013), arXiv: 1201.2434

    2013 arXiv

  4. [12]

    Miyatake , arXiv e-prints arXiv:2505.07697 (2025), arXiv: 2505.07697

    H. Miyatake , arXiv e-prints arXiv:2505.07697 (2025), arXiv: 2505.07697

    2025 arXiv

  5. [13]

    E. S. Rykoff , E. Rozo , M. T. Busha , C. E. Cunha , A. Finoguenov , A. Evrard et al., 785, 104 (2014), arXiv: 1303.3562

    2014 arXiv

  6. [14]

    T. M. C. Abbott , M. Aguena , A. Alarcon , S. Allam , O. Alves , A. Amon et al., 105, 023520 (2022), arXiv: 2105.13549

    2022 arXiv

  7. [15]

    G. F. Lesci , F. Marulli , L. Moscardini , M. Sereno , A. Veropalumbo , M. Maturi et al., 659, A88 (2022), arXiv: 2012.12273

    2022 arXiv

  8. [16]

    R. A. Burenin , A. Vikhlinin , A. Hornstrup , H. Ebeling , H. Quintana , and A. Mescheryakov , 172, 561 (2007), arXiv: astro-ph/0610739

    2007 arXiv

  9. [17]

    Pacaud , N

    F. Pacaud , N. Clerc , P. A. Giles , C. Adami , T. Sadibekova , M. Pierre et al., 592, A2 (2016), arXiv: 1512.04264

    2016 arXiv

  10. [18]

    Y. E. Bahar , E. Bulbul , N. Clerc , V. Ghirardini , A. Liu , K. Nandra et al., 661, A7 (2022), arXiv: 2110.09534

    2022 arXiv

  11. [19]

    Planck Collaboration , P. A. R. Ade , N. Aghanim , M. Arnaud , M. Ashdown , J. Aumont et al., 594, A27 (2016), arXiv: 1502.01598

    2016 arXiv

  12. [20]

    Hilton , M

    M. Hilton , M. Hasselfield , C. Sif \'o n , N. Battaglia , S. Aiola , V. Bharadwaj et al., 235, 20 (2018), arXiv: 1709.05600

    2018 arXiv

  13. [21]

    Huang , L

    N. Huang , L. E. Bleem , B. Stalder , P. A. R. Ade , S. W. Allen , A. J. Anderson et al., 159, 110 (2020), arXiv: 1907.09621

    2020 arXiv

  14. [22]

    A. B. Mantz , A. von der Linden , S. W. Allen , D. E. Applegate , P. L. Kelly , R. G. Morris et al., 446, 2205 (2015), arXiv: 1407.4516

    2015 arXiv

  15. [23]

    Zubeldia and A

    \'I . Zubeldia and A. Challinor , 489, 401 (2019), arXiv: 1904.07887

    2019 arXiv

  16. [24]

    Costanzi , A

    M. Costanzi , A. Saro , S. Bocquet , T. M. C. Abbott , M. Aguena , S. Allam et al., 103, 043522 (2021), arXiv: 2010.13800

    2021 arXiv

  17. [25]

    LSST Science Collaboration , P. A. Abell , J. Allison , S. F. Anderson , J. R. Andrew , J. R. P. Angel et al., arXiv e-prints arXiv:0912.0201 (2009), arXiv: 0912.0201

    2009 arXiv

  18. [26]

    Laureijs , J

    R. Laureijs , J. Amiaux , S. Arduini , J. L. Augu \`e res , J. Brinchmann , R. Cole et al., arXiv e-prints arXiv:1110.3193 (2011), arXiv: 1110.3193

    2011 arXiv

  19. [27]

    Mellier , Abdurro'uf , J

    Euclid Collaboration , Y. Mellier , Abdurro'uf , J. A. Acevedo Barroso , A. Ach \'u carro , J. Adamek et al., arXiv e-prints arXiv:2405.13491 (2024), arXiv: 2405.13491

    2024

  20. [28]

    Dore , C

    O. Dore , C. Hirata , Y. Wang , D. Weinberg , T. Eifler , R. J. Foley et al., 51, 341 (2019), arXiv: 1904.01174

    2019 arXiv

  21. [29]

    Gong , X

    Y. Gong , X. Liu , Y. Cao , X. Chen , Z. Fan , R. Li et al., 883, 203 (2019), arXiv: 1901.04634

    2019 arXiv

  22. [30]

    Miao , Y

    H. Miao , Y. Gong , X. Chen , Z. Huang , X.-D. Li , and H. Zhan , 519, 1132 (2023), arXiv: 2206.09822

    2023 arXiv

  23. [31]

    Zhang , M

    Y. Zhang , M. Chen , Z. Wen , and W. Fang , Research in Astronomy and Astrophysics 23, 045011 (2023), arXiv: 2302.05010

    2023 arXiv

  24. [32]

    Gong , H

    Y. Gong , H. Miao , X. Zhou , Q. Xiong , Y. Song , Y. Jiang et al., Science China Physics, Mechanics, and Astronomy 68, 280402 (2025), arXiv: 2501.15023

    2025 arXiv

  25. [33]

    Gong , H

    CSST Collaboration , Y. Gong , H. Miao , H. Zhan , Z.-Y. Li , J. Shangguan et al., arXiv e-prints arXiv:2507.04618 (2025), arXiv: 2507.04618

    2025

  26. [34]

    JUST Team , C. Liu , Y. Zu , F. Feng , Z. Li , Y. Yu et al., Astronomical Techniques and Instruments 1, 16 (2024), arXiv: 2402.14312

    2024 arXiv

  27. [35]

    W. H. Press and P. Schechter , 187, 425 (1974)

    1974

  28. [36]

    J. R. Bond , S. Cole , G. Efstathiou , and N. Kaiser , 379, 440 (1991)

    1991

  29. [37]

    Jenkins , C

    A. Jenkins , C. S. Frenk , S. D. M. White , J. M. Colberg , S. Cole , A. E. Evrard et al., 321, 372 (2001), arXiv: astro-ph/0005260

    2001 arXiv

  30. [38]

    Reed , J

    D. Reed , J. Gardner , T. Quinn , J. Stadel , M. Fardal , G. Lake et al., 346, 565 (2003), arXiv: astro-ph/0301270

    2003 arXiv

  31. [39]

    Crocce , P

    M. Crocce , P. Fosalba , F. J. Castander , and E. Gazta \ n aga , 403, 1353 (2010), arXiv: 0907.0019

    2010 arXiv

  32. [40]

    R. E. Angulo , V. Springel , S. D. M. White , A. Jenkins , C. M. Baugh , and C. S. Frenk , 426, 2046 (2012), arXiv: 1203.3216

    2012 arXiv

  33. [41]

    W. A. Watson , I. T. Iliev , A. D'Aloisio , A. Knebe , P. R. Shapiro , and G. Yepes , 433, 1230 (2013), arXiv: 1212.0095

    2013 arXiv

  34. [42]

    Bocquet , A

    S. Bocquet , A. Saro , K. Dolag , and J. J. Mohr , 456, 2361 (2016), arXiv: 1502.07357

    2016 arXiv

  35. [43]

    Tinker , A

    J. Tinker , A. V. Kravtsov , A. Klypin , K. Abazajian , M. Warren , G. Yepes et al., 688, 709 (2008), arXiv: 0803.2706

    2008 arXiv

  36. [44]

    Despali , C

    G. Despali , C. Giocoli , R. E. Angulo , G. Tormen , R. K. Sheth , G. Baso et al., 456, 2486 (2016), arXiv: 1507.05627

    2016 arXiv

  37. [45]

    Diemer , 903, 87 (2020), arXiv: 2007.10346

    B. Diemer , 903, 87 (2020), arXiv: 2007.10346

    2020 arXiv

  38. [46]

    Ondaro-Mallea , R

    L. Ondaro-Mallea , R. E. Angulo , M. Zennaro , S. Contreras , and G. Aric \`o , 509, 6077 (2022), arXiv: 2102.08958

    2022 arXiv

  39. [47]

    Salvati , M

    L. Salvati , M. Douspis , and N. Aghanim , 643, A20 (2020), arXiv: 2005.10204

    2020 arXiv

  40. [48]

    Artis , J.-B

    E. Artis , J.-B. Melin , J. G. Bartlett , and C. Murray , 649, A47 (2021), arXiv: 2101.02501

    2021 arXiv

  41. [49]

    Castro , A

    Euclid Collaboration , T. Castro , A. Fumagalli , R. E. Angulo , S. Bocquet , S. Borgani et al., 671, A100 (2023), arXiv: 2208.02174

    2023 arXiv

  42. [50]

    Bhattacharya , K

    S. Bhattacharya , K. Heitmann , M. White , Z. Luki \'c , C. Wagner , and S. Habib , 732, 122 (2011), arXiv: 1005.2239

    2011 arXiv

  43. [51]

    A. G. Adame , J. Aguilar , S. Ahlen , S. Alam , D. M. Alexander , M. Alvarez et al., 2025, 021 (2025), arXiv: 2404.03002

    2025 arXiv

  44. [52]

    Abdul-Karim , J

    DESI Collaboration , M. Abdul-Karim , J. Aguilar , S. Ahlen , S. Alam , L. Allen et al., arXiv e-prints arXiv:2503.14738 (2025), arXiv: 2503.14738

    2025 arXiv

  45. [53]

    R. E. Angulo and O. Hahn , Living Reviews in Computational Astrophysics 8, 1 (2022), arXiv: 2112.05165

    2022 arXiv

  46. [54]

    Moriwaki , T

    K. Moriwaki , T. Nishimichi , and N. Yoshida , Reports on Progress in Physics 86, 076901 (2023), arXiv: 2303.15794

    2023 arXiv

  47. [55]

    McClintock , E

    T. McClintock , E. Rozo , M. R. Becker , J. DeRose , Y.-Y. Mao , S. McLaughlin et al., 872, 53 (2019), arXiv: 1804.05866

    2019 arXiv

  48. [56]

    Nishimichi , M

    T. Nishimichi , M. Takada , R. Takahashi , K. Osato , M. Shirasaki , T. Oogi et al., 884, 29 (2019), arXiv: 1811.09504

    2019 arXiv

  49. [57]

    Bocquet , K

    S. Bocquet , K. Heitmann , S. Habib , E. Lawrence , T. Uram , N. Frontiere et al., 901, 5 (2020), arXiv: 2003.12116

    2020 arXiv

  50. [58]

    Shen , N

    D. Shen , N. Kokron , J. DeRose , J. Tinker , R. H. Wechsler , A. Banerjee et al., 2025, 056 (2025), arXiv: 2410.00913

    2025 arXiv

  51. [59]

    Heitmann , M

    K. Heitmann , M. White , C. Wagner , S. Habib , and D. Higdon , 715, 104 (2010), arXiv: 0812.1052

    2010 arXiv

  52. [60]

    Heitmann , E

    K. Heitmann , E. Lawrence , J. Kwan , S. Habib , and D. Higdon , 780, 111 (2014), arXiv: 1304.7849

    2014 arXiv

  53. [61]

    Knabenhans , J

    Euclid Collaboration , M. Knabenhans , J. Stadel , S. Marelli , D. Potter , R. Teyssier et al., 484, 5509 (2019), arXiv: 1809.04695

    2019 arXiv

  54. [62]

    Knabenhans , J

    Euclid Collaboration , M. Knabenhans , J. Stadel , D. Potter , J. Dakin , S. Hannestad et al., 505, 2840 (2021), arXiv: 2010.11288

    2021 arXiv

  55. [63]

    K. R. Moran , K. Heitmann , E. Lawrence , S. Habib , D. Bingham , A. Upadhye et al., 520, 3443 (2023), arXiv: 2207.12345

    2023 arXiv

  56. [64]

    Chen , Y

    Z. Chen , Y. Yu , J. Han , and Y. Jing , Science China Physics, Mechanics, and Astronomy 68, 289512 (2025), arXiv: 2502.11160

    2025 arXiv

  57. [65]

    McClintock , E

    T. McClintock , E. Rozo , A. Banerjee , M. R. Becker , J. DeRose , S. McLaughlin et al., arXiv e-prints arXiv:1907.13167 (2019), arXiv: 1907.13167

    2019 arXiv

  58. [66]

    Kwan , K

    J. Kwan , K. Heitmann , S. Habib , N. Padmanabhan , E. Lawrence , H. Finkel et al., 810, 35 (2015), arXiv: 1311.6444

    2015 arXiv

  59. [67]

    Zhai , J

    Z. Zhai , J. L. Tinker , M. R. Becker , J. DeRose , Y.-Y. Mao , T. McClintock et al., 874, 95 (2019), arXiv: 1804.05867

    2019 arXiv

  60. [68]

    B. D. Wibking , D. H. Weinberg , A. N. Salcedo , H.-Y. Wu , S. Singh , S. Rodr \' guez-Torres et al., 492, 2872 (2020), arXiv: 1907.06293

    2020 arXiv

  61. [69]

    Yuan , L

    S. Yuan , L. H. Garrison , D. J. Eisenstein , and R. H. Wechsler , 515, 871 (2022), arXiv: 2203.11963

    2022 arXiv

  62. [70]

    Zhai , J

    Z. Zhai , J. L. Tinker , A. Banerjee , J. DeRose , H. Guo , Y.-Y. Mao et al., 948, 99 (2023), arXiv: 2203.08999

    2023 arXiv

  63. [71]

    Storey-Fisher , J

    K. Storey-Fisher , J. L. Tinker , Z. Zhai , J. DeRose , R. H. Wechsler , and A. Banerjee , 961, 208 (2024), arXiv: 2210.03203

    2024 arXiv

  64. [72]

    Hadzhiyska , C

    B. Hadzhiyska , C. Garc \' a-Garc \' a , D. Alonso , A. Nicola , and A. Slosar , 2021, 020 (2021), arXiv: 2103.09820

    2021 arXiv

  65. [73]

    Zennaro , R

    M. Zennaro , R. E. Angulo , M. Pellejero-Ib \'a \ n ez , J. St \"u cker , S. Contreras , and G. Aric \`o , 524, 2407 (2023), arXiv: 2101.12187

    2023 arXiv

  66. [74]

    Pellejero Iba \ n ez , R

    M. Pellejero Iba \ n ez , R. E. Angulo , M. Zennaro , J. St \"u cker , S. Contreras , G. Aric \`o et al., 520, 3725 (2023), arXiv: 2207.06437

    2023 arXiv

  67. [75]

    DeRose , N

    J. DeRose , N. Kokron , A. Banerjee , S.-F. Chen , M. White , R. Wechsler et al., 2023, 054 (2023), arXiv: 2303.09762

    2023 arXiv

  68. [76]

    Zhou , Z

    S. Zhou , Z. Chen , and Y. Yu , arXiv e-prints arXiv:2506.04671 (2025), arXiv: 2506.04671

    2025

  69. [77]

    J. Liu , S. Bird , J. M. Zorrilla Matilla , J. C. Hill , Z. Haiman , M. S. Madhavacheril et al., 2018, 049 (2018), arXiv: 1711.10524

    2018 arXiv

  70. [78]

    Liu and M

    J. Liu and M. S. Madhavacheril , 99, 083508 (2019), arXiv: 1809.10747

    2019 arXiv

  71. [79]

    Z. Li , J. Liu , J. M. Zorrilla Matilla , and W. R. Coulton , 99, 063527 (2019), arXiv: 1810.01781

    2019 arXiv

  72. [80]

    W. R. Coulton , J. Liu , M. S. Madhavacheril , V. B \"o hm , and D. N. Spergel , 2019, 043 (2019), arXiv: 1810.02374

    2019 arXiv

  73. [81]

    Wang and P

    Y. Wang and P. He , 111, L041302 (2025), arXiv: 2408.13876

    2025 arXiv

  74. [82]

    G. L. Bryan and M. L. Norman , 495, 80 (1998), arXiv: astro-ph/9710107

    1998 arXiv

  75. [83]

    Castorina , E

    E. Castorina , E. Sefusatti , R. K. Sheth , F. Villaescusa-Navarro , and M. Viel , 2014, 049 (2014), arXiv: 1311.1212

    2014 arXiv

  76. [84]

    Kitayama and Y

    T. Kitayama and Y. Suto , 469, 480 (1996), arXiv: astro-ph/9604141

    1996 arXiv

  77. [85]

    Courtin , Y

    J. Courtin , Y. Rasera , J. M. Alimi , P. S. Corasaniti , V. Boucher , and A. F \"u zfa , 410, 1911 (2011), arXiv: 1001.3425

    2011 arXiv

  78. [86]

    Li and R

    Y. Li and R. E. Smith , arXiv e-prints arXiv:2411.18722 (2024), arXiv: 2411.18722

    2024 arXiv

  79. [87]

    S. Yue , L. Feng , W. Ju , J. Pan , Z. Huang , F. Fang et al., 535, 3500 (2024), arXiv: 2408.16398

    2024 arXiv

  80. [88]

    Villaescusa-Navarro , C

    F. Villaescusa-Navarro , C. Hahn , E. Massara , A. Banerjee , A. M. Delgado , D. K. Ramanah et al., 250, 2 (2020), arXiv: 1909.05273

    2020 arXiv

  81. [89]

    J. Han , M. Li , W. Jiang , Z. Chen , H. Wang , C. Wei et al., arXiv e-prints arXiv:2503.21368 (2025), arXiv: 2503.21368

    2025 arXiv

  82. [90]

    Chevallier and D

    M. Chevallier and D. Polarski , International Journal of Modern Physics D 10, 213 (2001), arXiv: gr-qc/0009008

    2001 arXiv

  83. [91]

    E. V. Linder , 90, 091301 (2003), arXiv: astro-ph/0208512

    2003 arXiv

  84. [92]

    Partmann , C

    C. Partmann , C. Fidler , C. Rampf , and O. Hahn , 2020, 018 (2020), arXiv: 2003.07387

    2020 arXiv

  85. [93]

    Heuschling , C

    P. Heuschling , C. Partmann , and C. Fidler , 2022, 068 (2022), arXiv: 2201.13186

    2022 arXiv

  86. [94]

    Sobol, Vychisl

    I. Sobol, Vychisl. Mat. i Mater. Phys 7, 784 (1967)

    1967

  87. [95]

    M. D. McKay, R. J. Beckman, and W. J. Conover, Technometrics 21, 239 (1979)

    1979

  88. [96]

    Tang, Journal of the American Statistical Association 88, 1392 (1993)

    B. Tang, Journal of the American Statistical Association 88, 1392 (1993)

    1993

  89. [97]

    R. E. Angulo and A. Pontzen , 462, L1 (2016), arXiv: 1603.05253

    2016 arXiv

  90. [98]

    P. S. Behroozi , R. H. Wechsler , and H.-Y. Wu , 762, 109 (2013), arXiv: 1110.4372

    2013 arXiv

  91. [99]

    Springel , S

    V. Springel , S. D. M. White , G. Tormen , and G. Kauffmann , 328, 726 (2001), arXiv: astro-ph/0012055

    2001 arXiv

  92. [100]

    S \'a ez-Casares , Y

    I. S \'a ez-Casares , Y. Rasera , T. R. G. Richardson , and P. S. Corasaniti , 691, A323 (2024), arXiv: 2410.05226

    2024 arXiv

  93. [101]

    Virtanen, R

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau et al., Nature Methods 17, 261 (2020)

    2020

  94. [102]

    Castro , S

    T. Castro , S. Borgani , and J. Dakin , arXiv e-prints arXiv:2504.07608 (2025), arXiv: 2504.07608

    2025 arXiv

  95. [103]

    Pedregosa, G

    F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel et al., Journal of Machine Learning Research 12, 2825 (2011)

    2011

  96. [104]

    J. A. Fillmore and P. Goldreich , 281, 1 (1984)

    1984

  97. [105]

    Bertschinger , 58, 39 (1985)

    E. Bertschinger , 58, 39 (1985)

    1985

  98. [106]

    Adhikari , N

    S. Adhikari , N. Dalal , and R. T. Chamberlain , 2014, 019 (2014), arXiv: 1409.4482

    2014 arXiv

  99. [107]

    Diemer and A

    B. Diemer and A. V. Kravtsov , 789, 1 (2014), arXiv: 1401.1216

    2014 arXiv

  100. [108]

    Fong and J

    M. Fong and J. Han , 503, 4250 (2021), arXiv: 2008.03477

    2021 arXiv

  101. [109]

    Fong , J

    M. Fong , J. Han , J. Zhang , X. Yang , H. Gao , J. Wang et al., 513, 4754 (2022), arXiv: 2205.01816

    2022 arXiv

  102. [110]

    H. Gao , J. Han , M. Fong , Y. P. Jing , and Z. Li , 953, 37 (2023), arXiv: 2303.10887

    2023 arXiv

  103. [111]

    Zhou and J

    Y. Zhou and J. Han , 525, 2489 (2023), arXiv: 2303.10886

    2023 arXiv

  104. [112]

    Zhou and J

    Y. Zhou and J. Han , 979, 55 (2025), arXiv: 2407.08381

    2025 arXiv

  105. [113]

    Costanzi , E

    M. Costanzi , E. Rozo , M. Simet , Y. Zhang , A. E. Evrard , A. Mantz et al., 488, 4779 (2019), arXiv: 1810.09456

    2019 arXiv

  106. [114]

    C. R. Harris , K. J. Millman , S. J. van der Walt , R. Gommers , P. Virtanen , D. Cournapeau et al., 585, 357 (2020), arXiv: 2006.10256

    2020 arXiv

  107. [115]

    , " * write output.state after.block = add.period write newline

    ENTRY address archive author booktitle chapter collaboration edition editor eid eprint howpublished institution journal key month note number numpages organization pages publisher school series title type url volume year label INTEGERS output.state before.all mid.sentence afte...

  108. [116]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.