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Dark Quest. I. Fast and Accurate Emulation of Halo Clustering Statistics and Its Application to Galaxy Clustering

Takahiro Nishimichi, Masahiro Takada, Ryuichi Takahashi, Ken Osato, Masato Shirasaki, Taira Oogi, Hironao Miyatake, Masamune Oguri, Ryoma Murata, Yosuke Kobayashi, Naoki Yoshida

arXiv:1811.09504v2astro-ph.CO

TL;DR

Galaxy-bias uncertainty complicates extracting cosmological information from galaxy-survey observables. Dark Emulator emulates halo clustering statistics from N-body simulations, achieving ∼2% rms error for halo-matter cross-correlation and ∼3–4% for halo auto-correlation in typical CMASS-like host halos.

  • Problem

    A central challenge is extracting cosmological information from galaxy-survey observables while accounting for uncertainty in the relation between dark matter and large-scale structure tracers.

    Source: §1 · p. 1

  • Method

    Dark Emulator uses N-body realizations and multi-redshift halo catalogs to compute halo mass functions and halo-matter and halo auto-correlations across halo mass, separation, redshift, and cosmology.

    Source: §1 · p. 2

  • Results

    ∼2% rms error for halo-matter cross-correlation and ∼3–4% for halo auto-correlation were achieved for halos with number density 10^-4(h^-1Mpc)^-3.

    Source: §6 · p. 30

  • Takeaways & Limitations

    Dark Emulator can be used to study halo mass-density profiles, including concentration-mass relations and splashback features across redshift, halo mass, and cosmology.

    Source: §6 · p. 31

    It remains unclear how much cosmological information halo-clustering dependences provide after nuisance marginalization; assessing this requires realistic survey-like mocks and parameter inference.

    Source: §6 · p. 31

Abstract

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We perform an ensemble of $N$-body simulations with $2048^3$ particles for 101 flat $w$CDM cosmological models sampled based on a maximin-distance Sliced Latin Hypercube Design. By using the halo catalogs extracted at multiple redshifts in the range of $z=[0,1.48]$, we develop Dark Emulator, which enables fast and accurate computations of the halo mass function, halo-matter cross-correlation, and halo auto-correlation as a function of halo masses, redshift, separations and cosmological models, based on the Principal Component Analysis and the Gaussian Process Regression for the large-dimensional input and output data vector. We assess the performance of the emulator using a validation set of $N$-body simulations that are not used in training the emulator. We show that, for typical halos hosting CMASS galaxies in the Sloan Digital Sky Survey, the emulator predicts the halo-matter cross correlation, relevant for galaxy-galaxy weak lensing, with an accuracy better than $2\%$ and the halo auto-correlation, relevant for galaxy clustering correlation, with an accuracy better than $4\%$. We give several demonstrations of the emulator. It can be used to study properties of halo mass density profiles such as the mass-concentration relation and splashback radius for different cosmologies. The emulator outputs can be combined with an analytical prescription of halo-galaxy connection such as the halo occupation distribution at the equation level, instead of using the mock catalogs, to make accurate predictions of galaxy clustering statistics such as the galaxy-galaxy weak lensing and the projected correlation function for any model within the $w$CDM cosmologies, in a few CPU seconds.

1. INTRODUCTION

Dark Emulator is designed to provide fast, accurate predictions of halo clustering statistics across flat wCDM cosmologies, giving galaxy-survey analyses a flexible basis for modeling observables while accounting for galaxy-bias uncertainty.

Introduction

  • Galaxy-bias uncertainty complicates cosmological inference because galaxies are observed tracers of dark matter, while halo bias varies with mass, cosmology, and scale in the nonlinear regime.Halos provide a theory-motivated link between galaxies and dark matter, but their small-scale bias is complex and scale-dependent.Source: §1 · pp. 1–2
  • Galaxy-galaxy lensing and galaxy auto-correlation offer complementary information, and combining clustering observables can tighten cosmological constraints while calibrating systematic bias uncertainty.Lensing probes projected matter around tracers, while small-scale galaxy correlations have high signal-to-noise but are complicated by the one-halo contribution.Source: §1 · p. 2
  • Dark Emulator predicts the halo mass function, halo-matter cross-correlation, and halo auto-correlation across halo mass, separation, redshift, and flat wCDM cosmologies using PCA and Gaussian Process regression.It uses halo catalogs from multiple N-body simulation realizations and propagates statistical uncertainties in its predictions.Source: §1 · p. 2
  • Unlike direct cosmology-to-galaxy-statistic emulators, Dark Emulator models halo clustering without prescribing a galaxy-halo connection, leaving users to combine its outputs with HOD or other physics models analytically.The paper demonstrates equation-level combination with an HOD and satellite-galaxy distributions to predict galaxy-galaxy weak lensing and projected galaxy correlation functions.Source: §1 · pp. 2–3

2. HALO COSMOLOGY

The section connects projected galaxy lensing and clustering observables to three-dimensional galaxy correlations, then motivates modeling halo statistics as the basis for predicting those observables. Dark Emulator predicts key halo statistics that can be combined analytically with an HOD prescription to compute galaxy clustering statistics.

2.1. Galaxy observables

  • Galaxy-galaxy weak lensing probes projected ξgm, while projected galaxy clustering traces ξgg; combining lensing and clustering can break the linear-bias–σ8 normalization degeneracy.The cross coefficient rgm quantifies departures from the linear deterministic galaxy–matter relation due to nonlinear corrections and stochasticity.Source: §2.1 · p. 3

2.2. Halo model approach to galaxy bias

  • At ∼100 milliseconds, Dark Emulator computes halo mass functions and halo-matter and halo-halo correlations across Ωm; the figure varies Ωm while fixing other parameters to fiducial values.The mass functions are shown at z = 0, 0.5, and 1; the correlation functions use three halo masses at z = 0.5. Halo masses use the M200 definition.Source: §2.2 · p. 4, Figure 1 · p. 4
  • On mildly nonlinear scales, scale-dependent rgm and halo bias matter; on strongly nonlinear scales, one-halo lensing constrains host halo mass and HOD parameters, supporting quasi bias-free use of large-scale clustering.For a delta-function HOD at mass M, the lensing-derived halo mass informs large-scale bias and breaks the bias–clustering degeneracy.Source: §2.2 · p. 5
  • The emulator models dn/dM, ξhh, and ξhm across halo mass, redshift, separation, and cosmology, enabling fast HOD-based galaxy predictions suited to high-dimensional parameter inference.The galaxy predictions use the HOD form proposed by Zheng et al. (2005) and generalized by More et al. (2015b).Source: §2.2 · pp. 3–5
  • The method projects predicted three-dimensional clustering by line-of-sight integration, allowing any projection direction and selectable width; its wgg module models redshift-space distortions when the width is small.Source: §2.2 · p. 5

3. SIMULATION ENSEMBLE

The Dark Quest suite uses a sliced design of wCDM simulations and specified numerical and analysis procedures to calibrate halo clustering statistics across cosmologies and redshifts.

3.1. Simulation design

  • The ensemble samples 100 six-dimensional wCDM models using maximin-distance sliced Latin hypercube design, with 80 simulations for training and 20 for cross-validation.The five slices each contain 20 samples, allowing training and validation sets to cover the parameter space while remaining reasonably separated.Source: §3.1 · pp. 5–6, §3.2 · p. 7
  • The sampled parameter ranges are centered on Planck best-fit values and are broad, but predictions outside the emulator’s supported region require additional simulations or extrapolation.The paper notes that a survey-inferred posterior may lie outside the supported region if its best-fit model differs from the Planck fiducial cosmology.Source: §3.1 · p. 6

3.2. Box size and resolution

  • HR and LR simulations use 2048^3 particles in 1 and 2 h^-1Gpc boxes, respectively; HR resolution supports HMF and halo-matter cross-correlation estimates down to about 10^12 h^-1M⊙.LR runs primarily calibrate the noisier halo auto-correlation, while additional small-box runs test numerical convergence.Source: §3.2 · pp. 6–8

3.3. Initial conditions

  • Initial conditions use second-order Lagrangian perturbation theory, and varied random seeds are used for the reported results except in Appendix C.Twenty HR simulations in Slice 1 share a seed; the paper varies seeds elsewhere to reduce propagated sample variance.Source: §3.3 · p. 8

3.4. Time integration

  • Each realization is stored at 21 redshifts from z = 0 to 1.48, enabling interpolation to arbitrary intermediate redshifts because the relevant statistics evolve slowly and monotonically.Source: §3.4 · p. 8

3.5. Halo catalogs

  • The halo catalog uses Rockstar host halos, defines mass as M200m within the comoving R200m boundary, and retains halos above 10^12 h^-1M⊙.Subhalos are excluded from the primary catalog; a less massive halo is classified as a satellite if its center lies within the more massive halo’s R200m.Source: §3.5 · pp. 8–9

3.6. Hybrid Fourier-direct method to measure the correlation signal

  • The hybrid method switches from direct pair counting to FFT at 5 h^-1Mpc for HR runs; its smoothed result stays within target accuracy despite a small residual near the switch.The FFT uses 1024^3 grids, with direct pair counting below 5 h^-1Mpc for HR runs and 10 h^-1Mpc for LR runs.Source: Figure 3 · p. 9, §3.6 · p. 9

4. EMULATION

Dark Emulator is built from the Dark Quest simulation suite and the analysis pipeline described in the preceding section. This section explains how its halo-statistic prediction modules are constructed and combined.

4. EMULATION

  • Dark Emulator combines modules that predict different halo statistical quantities, using the Dark Quest simulations and the preceding section’s analysis pipeline.The section describes construction of the individual modules and their combination into the emulator.Source: §4 · p. 9

4.1. Overall design

The emulator predicts basic halo-clustering quantities rather than galaxy observables from mock catalogs built under a fixed halo-galaxy prescription. Utility modules combine its outputs analytically with a changeable halo-galaxy connection to predict galaxy-clustering observables.

4.1. Overall design

  • Emulating galaxy quantities from mocks generated under a restricted halo-galaxy prescription risks inaccuracies and makes later changes to that prescription difficult.The passage identifies uncertain satellite radial profiles and potentially off-centered central galaxies as sources of variation that a restricted prescription may not capture.Source: §4.1 · p. 10
  • The emulator predicts basic halo-clustering quantities, which can be combined analytically with a changeable halo-galaxy prescription to predict galaxy-clustering observables without retraining on mock catalogs.The core halo predictions depend on cosmological parameters, halo mass, separation scale, and redshift.Source: §4.1 · p. 10
  • The design has Linear Modules for matter perturbations, Halo Modules for dark-matter halo statistics, and Utility Modules that combine them to compute observables.Utility Modules can project three-dimensional correlations along the line of sight and offer options for baryonic corrections near halo centers and redshift-space distortions.Source: §4.1 · p. 10

4.2. Resolution study and matching scheme

The study tests how simulation resolution affects halo statistics and develops matching and large-scale modeling strategies to combine results across simulations. Resolution limits matter for halo mass functions and small-scale correlations, while finite-volume sample variance complicates large-scale predictions.

4.2.1. Halo mass function

  • At the 100-particle mass scale, the halo mass function is underestimated by about 10% across redshifts, so several hundred particles are needed for percent-level accuracy.The four simulations agree at the high-mass end; the 100-particle result motivates a more conservative 200-particle threshold for calibration.Source: §4.2.1 · p. 10, Figure 5 · p. 11
  • Correcting spherical-overdensity halo masses improves agreement among resolution levels down to the resolution limit, but overcorrects the mass function below it.The correction addresses underestimated SO masses in low-resolution simulations, where the surrounding matter density is itself underestimated; the emulator uses HR simulations and halos with at least 200 particles.Source: §4.2.1 · pp. 11–12, Figure 6 · p. 12

4.2.2. Correlation functions

  • Matching halo samples by number density improves cross-resolution agreement over fixed mass thresholds, but small-scale halo-matter cross-correlations still require HR simulations; LR simulations suffice for halo auto-correlations.Resolution deviations begin below about 1 h−1Mpc and are larger for lower mass thresholds; the auto-correlation is relatively robust, with small-scale scatter related to halo exclusion.Source: §4.2.2 · p. 12, Figure 7 · p. 13, Figure 8 · p. 13

4.2.3. Large-scale limit

  • Emulating the propagator and combining it with the linear power spectrum yields large-scale correlation predictions that closely match simulations while reducing sample variance.The propagator’s low-k limit gives linear bias, while its damping captures BAO smearing; residuals are consistent with zero within realization scatter, including for low-density halo auto-correlations.Source: §4.2.3 · pp. 14–15, Figure 10 · p. 15

4.3. Implementation detail and performance

This section describes Dark Emulator’s PCA- and Gaussian-process-based modeling of halo statistics and evaluates its performance for the halo mass function and halo correlations, including auto-correlation resampling and propagator validation. It also examines how halo exclusion, Poisson noise, and cosmology affect the auto-correlation modeling.

4.3. Implementation detail and performance

  • For each halo statistic, the workflow builds simulation data vectors, compresses them with weighted PCA, and uses GP regression to interpolate the retained coefficients across cosmologies.The stated setup trains on 80 simulations and reserves 20 models plus the fiducial Planck model for cross-validation; PCA weighting can assign zero weight to missing data.Source: §4.3 · p. 16

4.3.1. Halo mass function

  • The HMF emulator predicts held-out cosmologies to better than a few percent up to several times 10^14 h−1M⊙; larger-mass errors are comparable to statistical scatter.A larger-volume validation found corrected cluster-mass measurements close to predictions, though their mean ratio was 1–2% above unity.Source: Figure 13 · p. 17, §4.3.1 · p. 18, Figure 15 · p. 18
  • They recalibrate Tinker HMF parameters A and a for each of 100 cosmological models, retaining b and c; fits are within ±5%, with typical accuracy of ∼1% (3%) at 10^13 (10^14) h−1M⊙.Source: §4.3.1 · pp. 17–18

4.3.2. Halo-matter cross-correlation function

  • Keeping five PCA coefficients reproduces halo-matter correlations within 5% for x ≲30 h−1Mpc, with cross-model rms typically below 2% to ∼20 h−1Mpc.GP predictions for held-out cosmologies match the correlations about as well as simulation sample variance; PCA is inadequate at very large separations, where the propagator method is used instead.Source: §4.3.2 · p. 19, Figure 16 · p. 20

4.3.3. Halo auto-correlation function

  • Resampling halo-auto-correlation separations gives 1–2% rms error across models on scales of roughly 1–40 h−1Mpc, with worst cases around 3%.The larger small-scale errors arise from halo exclusion; the analysis uses eight high-number-density bins and 36 sample-pair combinations.Source: §4.3.3 · pp. 19–20, Figure 17 · p. 20
  • For halo auto-correlation, GP validation reaches 2–3% rms over 1 ≲ x/[h−1Mpc] ≲ 20; training accuracy is 1–5% rms depending on scale.Eight principal components reduce the data vector from 1.8×10^6 points to 8 while retaining a few-percent error, with slight degradation at scales ≳10 h−1Mpc.Source: §4.3.3 · p. 21
  • Halo auto-correlation predictions are less accurate than cross-correlation predictions because Poisson noise increases measurement noise, while halo exclusion suppresses small-scale values and makes ratios near zero-crossing noisy.Source: §4.3.3 · p. 21

4.3.4. Propagator

  • The propagator fit residual has a typical amplitude below a few percent relative to its low-k limit, and its amplitude and decay wavenumber vary strongly with cosmology.The result is shown for halo samples with nh = 10−4 (h−1Mpc)−3 at z = 0.55.Source: §4.3.4 · p. 21
  • For the halo-propagator GP regression, predictions on the validation set—the remaining 20 models in Slice 5 plus fiducial Planck cosmology—stay within ±5%, with accuracy not degraded versus training.Source: Figure 18 · p. 22, §4.3.4 · p. 22

5. USAGE OF THE EMULATOR

This section describes how Dark Emulator is used to predict halo properties and combine halo statistics with HOD models to calculate galaxy-clustering observables, including halo density profiles, concentration and splashback trends, lensing, and projected clustering. It also covers the stated accuracy requirements for Subaru HSC and SDSS measurements, cross-correlation analyses, and the code’s inputs, outputs, configurable galaxy models, and limitations.

5.1.1. Implementation detail

  • Even a pessimistic 4% cumulative mass-function error induces correlation-function shifts within ±5% and mostly within ±3%; realistic errors yield shifts below typical emulator accuracy.The test shifts number density for two mass thresholds at z = 0.55; halo auto-correlation responds more strongly than halo-matter cross-correlation, while exclusion features are specific to halo clustering.Source: §5.1.1 · pp. 22–23, Figure 19 · p. 23
  • Halo bias is consistent with Tinker et al. (2010) to no worse than 10% across the examined ranges and is rather universal with little cosmology or redshift dependence.The comparison spans different Ωm cosmologies and three redshifts.Source: §5.1.1 · p. 24, Figure 22 · p. 24
  • The emulator predicts boosted BAO peak structure for low-number-density halo samples, enabled by propagator parameters g2 and g4.The reported scale-dependent bias feature is consistent with peak-model expectations and prior simulation findings.Source: §5.1.1 · p. 24, Figure 23 · p. 24
  • Cubic-spline interpolation and stitching at xswitch = 60 h−1Mpc combine halo predictions across number density, redshift, and scale; low-density halo auto-correlation uses bias scaling that cannot reproduce exclusion features.The authors report no sizable interpolation error; the low-density approximation is not a severe error for CMASS galaxies because their small-scale correlation is mainly one-halo dominated.Source: §5.1.1 · pp. 24–25, Figure 26 · p. 25

5.1.2. Demonstrations

  • NFW profiles fit emulator density profiles to better than about 5% up to R200m, but fail beyond it; a DK fit captures the splashback feature.The DK fit’s logarithmic slope has a sharp dip marking the splashback radius.Source: §5.1.2 · pp. 26–27, Figure 27 · p. 26
  • The concentration–peak-height relation has little Ωm dependence at high redshift but clear dependence at lower redshifts, which the emulator incorporates.The reported trend also includes increasing concentration toward lower redshift and positive Ωm dependence.Source: Figure 28 · p. 26, §5.1.2 · p. 27
  • The splashback radius is similar to R200m and decreases slightly with peak height, while Rsp/R200m is higher for models with larger Ωm.These trends are qualitatively consistent with fitting formulae that depend on Ωm(z) and peak height.Source: §5.1.2 · p. 27, Figure 29 · p. 27

5.2. Projected galaxy clustering statistics

  • Dark Emulator’s HOD-based predictions agree remarkably with mock measurements of galaxy-galaxy lensing and projected clustering, and take a few CPU seconds.The comparison uses 72 mock realizations for fiducial Planck cosmology; the figure reports that predictions meet the stated uncertainty requirements across the separation range examined.Source: §5.2 · pp. 27–28, Figure 30 · p. 28
  • The emulator’s accuracy meets the requirements for Subaru HSC and SDSS: m ≲0.04 for weak lensing and m ≲0.029 for the projected correlation function.These limits correspond to overall signal-to-noise ratios of about 25 and 35, respectively.Source: §5.2 · p. 28

5.3. Cross-correlation coefficient

  • For the HOD galaxy sample, emulator predictions agree with mock-galaxy measurements; above several h−1Mpc, bias scale dependence is weak and the cross-correlation coefficient is within ∼10% of unity.Reconstructing underlying matter clustering from biased tracers is supported for the simple HOD model considered, while broader galaxy populations require further study; the matter auto-correlation module used here was a development version.Source: §5.3 · pp. 28–29, Figure 32 · p. 29

5.4. Summary of the current code

  • Code inputs cover wCDM cosmology, redshift, halo samples, HOD occupation, and galaxy profiles, while outputs include halo abundance and clustering, propagators, and derived galaxy statistics.Halo samples can be specified by mass, mass threshold, or number density; linear quantities are evaluated at z = 0 and scaled by the linear growth factor.Source: §5.4 · pp. 29–30
  • HOD and galaxy-profile modules are modifiable, with options for satellite NFW or matter-profile distributions and central-galaxy off-centering; assembly bias is not implemented.The code also allows parametric baryonic changes to halo mass profiles and linear-theory residual redshift-space distortion effects for finite projection widths.Source: §5.4 · pp. 29–30

6. SUMMARY

Dark Quest simulations underpin Dark Emulator, which rapidly predicts halo clustering statistics and enables galaxy-clustering calculations through halo-galaxy prescriptions. Validation shows percent-level accuracy for key halo statistics, while applications include halo profiles and galaxy observables.

6. SUMMARY

  • Dark Quest used 2048^3 particles in 1 or 2 Gpc/h boxes for 100 six-parameter wCDM models, extracting halos above roughly 10^12 h−1M⊙ at 21 redshifts from z = 0 to 1.48.The cosmological models were sampled around a fiducial ΛCDM cosmology, and the parameter range is consistent with existing cosmological datasets.Source: §6 · p. 30
  • Dark Emulator uses PCA for dimension reduction and Gaussian Process regression to predict the halo mass function, halo-matter cross-correlation, and halo auto-correlation, which can be combined with an HOD prescription to compute galaxy statistics.Validation used separate N-body simulations sampled to cover the parameter space, including points distant from the nearest training data.Source: §6 · p. 30
  • The emulator achieves 1–2% halo mass-function accuracy, ∼2% halo-matter cross-correlation accuracy, and ∼3–4% halo auto-correlation accuracy for CMASS-like halos.These are rms errors; the cross-correlation result applies over 0.1–30 h−1Mpc for halos with number density 10−4(h−1Mpc)−3. The massive-end mass function is limited by Poisson error, accuracy worsens at x ≲1 h−1Mpc, and the largest discrepancy is no worse than 5%.Source: §6 · p. 30
  • A propagator module models large-scale clustering by encoding large-scale bias and BAO damping, using an estimator in which sample variance mostly cancels.The module is trained and validated to support accurate correlation-function predictions at large separations.Source: §6 · p. 31
  • Dark Emulator outputs support studies of halo mass-density profiles, including the concentration-mass relation and splashback feature, across redshift, halo mass, and cosmology.Source: §6 · p. 31
  • Combined with an HOD prescription, the emulator predicts projected galaxy correlation and galaxy-galaxy weak-lensing profiles, including small-scale effects such as off-centering and satellite distributions.Its FFTLog implementation rapidly converts three-dimensional correlation functions to projected statistics.Source: §6 · p. 31
  • Halo and galaxy statistics typically evaluate in ∼100 milliseconds and a few seconds, respectively, while halo-matter and HOD-populated galaxy-matter cross-correlation coefficients remain close to unity on large scales.Galaxy-statistic calculations are slower because they usually involve integrals over halo masses.Source: §6 · p. 31
  • The current emulator is adequate for ongoing wide-area surveys such as Subaru HSC but not future LSST, Euclid, or WFIRST, and its approximations include independent PCA-coefficient modeling and omission of assembly bias.It also does not support halos below roughly 10^12 h−1M⊙; improving precision may require more simulations, while survey-specific inference and realistic mock-catalog validation remain future work.Source: §6 · pp. 31–32

APPENDIX LINEAR MODULES

The Linear Modules use PCA and Gaussian-process regression to emulate three linear-theory quantities, achieving sub-percent cross-validation errors and millisecond evaluation.

APPENDIX LINEAR MODULES

  • PCA/GPR emulate Plin(k), σM, and σd; 360 of 400 SLHD cosmologies train the model, 40 validate it, and σd is modeled directly.The emulator retains 13 principal components for Plin(k) and 4 for σM.Source: APPENDIX LINEAR MODULES · pp. 33–34
  • Cross-validation errors are always below 1%: Plin(k) is ~0.3–0.4%, while σM and σd are typically 0.01%.The larger Plin(k) error is attributed to baryon acoustic oscillations; σM varies smoothly and monotonically with M.Source: Figure 33 · p. 34, APPENDIX LINEAR MODULES · p. 34
  • The Linear Modules evaluate all three quantities in a few milliseconds per cosmological model, a negligible share of Dark Emulator's total computation time.The inputs include cosmological parameters, halo mass, and redshift; σM and σd require wavenumber integrals.Source: APPENDIX LINEAR MODULES · pp. 33–34

INITIAL CONDITIONS OF N-BODY SIMULATIONS · Optimal initial redshift

The appendix identifies an initial-redshift choice that balances particle-discreteness suppression at high starting redshift against insufficient nonlinear evolution at low starting redshift. For the tested resolutions, the late-time power-spectrum peak occurs when the rms initial displacement is about 25% of the mean particle separation.

Optimal initial redshift

  • The test varied five initial redshifts, ZA and 2LPT, and HR/LR resolutions using identical realizations, then compared evolving power spectra with same-realization GridSPT predictions.The simulations used a 250 h−1Mpc box with 512^3 or 256^3 particles.Source: Optimal initial redshift · p. 34, Figure 34 · p. 35
  • At k=0.03 h Mpc−1, early power suppression is about 0.2%, rising to about 1% at k=0.53 h Mpc−1 for 512^3 particles.The sudden drop is about twice as large for 256^3 particles and occurs with both ZA and 2LPT initial conditions.Source: Optimal initial redshift · p. 35, Figure 34 · p. 35
  • Transient effects are stronger for smaller zin, with a substantial zin dependence for ZA but only a subtle dependence for 2LPT.At k ≳0.1 h Mpc−1, perturbation theory also over-predicts power at later epochs, contributing to the declining simulation-to-theory ratio.Source: Optimal initial redshift · p. 35, Figure 34 · p. 35
  • The optimal starting redshift balances high-z particle-discreteness suppression against insufficient nonlinear evolution at low zin.The high-z suppression is attributed to particle discreteness, while low-z inaccuracy reflects insufficient nonlinear evolution.Source: Optimal initial redshift · pp. 34–35
  • At z=0, the 512^3 runs peak at zin=59, while the 256^3 peak shifts lower; both peak near σd/Δpart=0.25, adopted as zin=59 (HR) and 29 (LR).The peak locations for the two resolutions are almost identical when plotted against the displacement-to-separation ratio.Source: Optimal initial redshift · pp. 35–36, Figure 35 · p. 36
  • The displacement-based initial-redshift prescription holds for cosmologies neighboring the fiducial Planck model, although zin may vary significantly across cosmologies.The authors adopt σd(zin)/Δpart=0.25 for the paper's main HR and LR simulations.Source: Optimal initial redshift · p. 36

Impact on halo statistics

The supplied figure examines how the halo mass function depends on initial redshift and identifies zin = 59 as the fiducial setting for the HR simulation.

Impact on halo statistics

  • The figure examines the halo mass function’s dependence on initial redshift and gives zin = 59 as the fiducial setting for the HR simulation.The caption describes this setting as equivalent to “5123” and ends before clarifying that notation.Source: Figure 36 · p. 37

RANDOM VS FIXED PHASE SIMULATIONS

The emulator’s accuracy was not strongly affected by using fixed rather than varied random seeds, so the authors used different seeds across cosmological models for their main results.

RANDOM VS FIXED PHASE SIMULATIONS

  • Common seeds may reduce sample-variance contamination for neighboring cosmologies, but nonlinear mode coupling can ruin that cancellation for sufficiently distant models.Thus, the authors note that common seeds do not guarantee converged statistical estimates from a limited number of simulation realizations.Source: RANDOM VS FIXED PHASE SIMULATIONS · p. 37
  • The fixed- and varied-seed GP models differed mostly below 1%, except at very large scales (~70 h^-1 Mpc), and had similar cross-validation ratios to Slice 5 simulations.The comparison used halo-matter cross-correlation functions and reused the hyperparameters optimized for the full sample.Source: RANDOM VS FIXED PHASE SIMULATIONS · p. 37
  • Seed choice did not largely affect emulator accuracy, so the main results use different random seeds for each cosmological model as the more conservative choice.The comparison used 20 fixed-seed and 20 varied-seed simulations on Slice 1.Source: RANDOM VS FIXED PHASE SIMULATIONS · p. 37

EFFECT OF MASSIVE NEUTRINOS

The simulations treat massive neutrinos only through the linear initial transfer function, and the resulting linear-theory deviation stays below 1% across the target redshifts z ≲ 1.5.

EFFECT OF MASSIVE NEUTRINOS

  • The treatment fixes the neutrino mass sum at 0.06 eV and includes its effects only at the linear-transfer-function level.The paper notes that a fuller treatment of massive-neutrino effects on nonlinear structure formation would be important for cosmological tests.Source: EFFECT OF MASSIVE NEUTRINOS · p. 38
  • The initial distribution uses the z = 0 CAMB total-matter transfer function, scaled to the initial redshift, but the N-body evolution omits massive neutrinos and radiation.The growth factor assumes wCDM with neutrino density included in the matter content and ignores scale-dependent neutrino growth.Source: EFFECT OF MASSIVE NEUTRINOS · p. 38
  • Below 1% is the linear-theory deviation at target redshifts z ≲ 1.5; it grows to ∼3% by z ∼ 30.The comparison is between CAMB’s matter power spectrum and the simulation’s underlying linear power spectrum, scaled using a growth factor computed without massive neutrinos.Source: EFFECT OF MASSIVE NEUTRINOS · p. 38

DEPENDENCE ON THE HALO FINDER

Halo clustering predictions are robust to the halo finder overall, but depend on the criterion separating central halos from substructures. Changes in that criterion affect correlation functions, while differences between Rockstar and Subfind can be absorbed by adjusting the criterion.

DEPENDENCE ON THE HALO FINDER

  • Changing the substructure-exclusion criterion most affects halo-matter cross-correlation near 1 h^-1Mpc and can alter the overall bias of halo auto-correlation.The criterion changes which outskirts structures count as substructures, altering the halo population and especially its recent-merger-history mix.Source: DEPENDENCE ON THE HALO FINDER · p. 41, Figure 42 · p. 41, Figure 43 · p. 41
  • Subfind differs from Rockstar in the innermost cross-correlation at x ≲ 0.05 h^-1Mpc because the finders define halo centers differently.This scale is close to the 0.024 h^-1Mpc softening length and below the emulator’s main target scale of ≳0.1 h^-1Mpc.Source: DEPENDENCE ON THE HALO FINDER · pp. 40–41
  • Halo-finder dependence is subdominant: Subfind correlation functions can be matched to Rockstar by adjusting the central–substructure separation parameter.The analysis concludes that halo clustering properties are robust to finder choice, although the central–satellite separation parameter can affect results.Source: DEPENDENCE ON THE HALO FINDER · pp. 41–42, Figure 42 · p. 41, Figure 43 · p. 41
  • The authors speculate that halo-definition dependence in galaxy clustering may be absorbed by HOD parameters, with reclassified structures populated as satellites.They defer explicit tests of this possibility to future work.Source: DEPENDENCE ON THE HALO FINDER · p. 42
  • Using R200c, R500c, or R2500c instead of R200m generally changes correlation functions by a few percent or less, except for the extreme R2500c choice.The boundary choice produces trends similar to varying the maximum allowed mass increase, but with smaller variation.Source: Figure 42 · p. 41, Figure 43 · p. 41, DEPENDENCE ON THE HALO FINDER · p. 42

DEPENDENCE OF THE PERFORMANCE ON REDSHIFT AND HALO NUMBER DENSITY

Validation shows that halo mass function modeling is reasonable relative to simulation uncertainties; for correlation functions, accuracy worsens at lower halo number densities, with no clear redshift dependence for halo-matter cross-correlation.

DEPENDENCE OF THE PERFORMANCE ON REDSHIFT AND HALO NUMBER DENSITY

  • Across redshifts, halo mass function emulator scatter is comparable to the scatter among realizations of the fiducial Planck cosmology, indicating modeling accuracy is reasonable given simulation uncertainties.The cross-validation uses 20 cosmologies in Slice 5, which were not used in Gaussian Process regression.Source: DEPENDENCE OF THE PERFORMANCE ON… · p. 42, Figure 44 · p. 42
  • Halo-matter cross-correlation accuracy degrades as halo number density decreases, reflecting increased simulation-data noise, but shows no clear dependence on redshift.The cross-validation covers multiple number densities and redshifts.Source: DEPENDENCE OF THE PERFORMANCE ON… · pp. 42–43, Figure 45 · p. 43
  • Halo-halo correlation tests likewise show greater scatter for low-density samples, while the propagators provide a practical guide to code accuracy by halo number density or mass.The tests include auto-correlations of the same halo samples and cross-correlations of samples with different number densities.Source: DEPENDENCE OF THE PERFORMANCE ON… · p. 43

HALO OCCUPATION DISTRIBUTION MODEL

This section presents a default HOD module for connecting halos to galaxies and using Dark Emulator to predict galaxy abundance, clustering, and galaxy-galaxy weak lensing. The predictions combine halo statistics with assumptions about central and satellite occupations and their spatial distributions.

HALO OCCUPATION DISTRIBUTION MODEL

  • The example HOD, based on More et al. (2015b), predicts galaxy abundance using Dark Emulator’s halo mass function.The prescription is provided as a default package; galaxy number density is computed from the HOD and dn/dM for the chosen cosmology.Source: HALO OCCUPATION DISTRIBUTION MODEL · pp. 43–44
  • The seven-parameter HOD includes central-galaxy incompleteness and Bernoulli central and conditional Poisson satellite occupations.The incompleteness fraction is log-linear in halo mass, while satellites are populated only when a central galaxy exists.Source: HALO OCCUPATION DISTRIBUTION MODEL · pp. 43–44
  • Galaxy-galaxy weak lensing is computed from the galaxy-matter power spectrum, with Gaussian central off-centering and satellite profiles modeled using halo concentration.The default uses the Diemer & Kravtsov (2015) concentration relation, which is not a free parameter; an alternative distributes satellites in proportion to 1 + ξhm(x). Seven HOD parameters plus foff and Roff specify the halo-galaxy connection.Source: HALO OCCUPATION DISTRIBUTION MODEL · pp. 45–46
  • The projected galaxy correlation function wgg(R) is computed from the galaxy auto-power spectrum using the same nine halo-galaxy parameters as the lensing prediction.The auto-power spectrum follows from the model’s assumptions for central and satellite occupations.Source: HALO OCCUPATION DISTRIBUTION MODEL · p. 46
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