CSSTEmu: Fast Cosmological Emulator Suite

• CSSTEmu is a modular cosmological emulator suite that provides fast, simulation-calibrated predictions for nonlinear matter spectra and generalized halo mass functions in an eight-dimensional parameter space.
• The methodology employs PCA compression, Gaussian process regression, and B-spline smoothing with variance reduction techniques to achieve percent-level accuracy and mitigate systematic biases.
• These capabilities support high-precision CSST analyses including weak lensing, galaxy clustering, and cluster cosmology, addressing gaps in traditional analytic fitting approaches.

CSSTEmu, the CSST Cosmological Emulator, is a public emulator suite built to provide fast, accurate theoretical predictions for key large-scale-structure observables needed by the China Space Station Telescope cosmology program. It was introduced first as a simulation-based surrogate model for the nonlinear matter power spectrum in an eight-dimensional flat $w_0w_a\mathrm{CDM}+\sum m_\nu$ parameter space, and was then extended in the second CSSTEmu paper to a generalized halo mass function emulator supporting multiple halo definitions, including $M_{200m}$, $M_{vir}$, and $M_{200c}$ (Chen et al., 16 Feb 2025, Chen et al., 11 Jun 2025). The suite is publicly released at https://github.com/czymh/csstemu, with documentation at https://csst-emulator.readthedocs.io/.

1. Definition, scope, and series development

The CSSTEmu program is framed as a broader “CSST Cosmological Emulator” effort whose purpose is to supply fast theoretical predictions at the accuracy required by future CSST imaging and spectroscopic analyses. In its first major realization, the suite targeted the nonlinear matter power spectrum. In its second, it targeted the halo mass function, explicitly extending the framework beyond a single halo-mass definition and beyond a single fitting choice (Chen et al., 16 Feb 2025, Chen et al., 11 Jun 2025).

The matter-power-spectrum component is designed to predict nonlinear clustering of both the total matter field and the cold component $P_\mathrm{cb}$. The halo-mass-function component is designed to provide fast and accurate predictions for halo masses $M\geq 10^{12}\,h^{-1}M_{\odot}$ up to $z=3.0$, with a generalized workflow that can be applied once the relevant halo catalog is available. Taken together, these two components position CSSTEmu as a CSST-focused prediction engine for nonlinear structure observables that enter weak lensing, galaxy clustering, and cluster cosmology analyses.

A plausible implication is that CSSTEmu is best understood not as a single emulator of one statistic, but as a modular suite of simulation-calibrated emulators constructed on a shared simulation backbone and a shared cosmological design space.

2. Scientific motivation in CSST cosmology

The scientific motivation is the mismatch between the precision expected from CSST observations and the accuracy of older theoretical approximations. CSST will measure very large samples of galaxy clusters, weak lensing, and clustering with few-percent statistical precision. To extract unbiased cosmological constraints, the theory side must match that precision. The power-spectrum paper states that traditional fitting formulas such as HaloFit and HMCODE are useful, but their accuracy is not always adequate across the broad parameter space needed for CSST, especially once one includes dynamical dark energy and massive neutrinos (Chen et al., 16 Feb 2025).

For halo statistics, the motivation is more specific. The halo mass function is central because it controls cluster abundance / number counts, connections between halos and galaxies, the nonlinear structure growth history, and weak-lensing and clustering modeling through halo-based frameworks. Older analytic HMF fits are limited by non-universality: the HMF depends at the 10–20% level on redshift, cosmology, halo definition, and halo finder. That level of variation is too large for CSST-era analyses, and therefore motivates a simulation-calibrated emulator spanning a broad cosmological space (Chen et al., 11 Jun 2025).

This scientific framing makes CSSTEmu a response to a concrete inference problem. Direct $N$-body recalibration at every likelihood evaluation is impractical, while analytic prescriptions are insufficiently accurate across the parameter volume of interest. The emulator formalism supplies a surrogate between those two limits.

3. Kun simulation suite and cosmological design space

CSSTEmu is trained on the Kun simulation suite, which serves as the simulation backbone of the project. The suite contains 129 high-resolution simulations with box size $L=1\,h^{-1}{\rm Gpc}$ and $3072^3$ particles. The simulations use fixed-amplitude initial conditions, initial redshift $M_{200m}$0, and a modified Gadget-4 evolution. Outputs are stored at

$M_{200m}$1

The first CSSTEmu paper further states that the full suite required about $M_{200m}$2 CPU hours and about $M_{200m}$3 PB of storage (Chen et al., 16 Feb 2025).

The emulator domain is the eight-dimensional $M_{200m}$4 model: $M_{200m}$5

$M_{200m}$6

$M_{200m}$7

The fiducial cosmology is Planck 2018, and the 128 training cosmologies are generated using Sobol sampling (Chen et al., 16 Feb 2025).

The power-spectrum construction adds further variance-reduction stages. For each cosmology, two matched and paired FastPM low-resolution runs are generated, one with the same initial phases and one with inverted phases. A paired estimate is then constructed and smoothed with a third-order Savitzky–Golay filter using the 1-loop LPT prediction from velocileptors as a reference. This design is intended to suppress cosmic variance and residual fluctuations without introducing measurable bias.

4. Nonlinear matter power spectrum emulator

The first CSSTEmu paper presents a nonlinear matter power spectrum emulator intended to provide percent-level theoretical accuracy for cosmological inference from small-scale large-scale-structure data. Rather than predicting $M_{200m}$8 directly, the emulator predicts a ratio,

$M_{200m}$9

where $M_{vir}$0 is one of linear theory, HaloFit, or HMCODE-2020. The paper reports that the HMCODE-2020 ratio gives the best overall performance. The ratio field is compressed with PCA, using $M_{vir}$1, and the cosmology dependence of the PCA coefficients is interpolated with Gaussian process regression using an RBF kernel multiplied by a constant amplitude factor (Chen et al., 16 Feb 2025).

The emulator provides predictions for both the cold-component power spectrum $M_{vir}$2 and the total matter power spectrum $M_{vir}$3. For neutrino cosmologies, it works naturally with the cold-component spectrum from Newtonian-motion-gauge simulations and then converts to total matter using the stated $M_{vir}$4 relation involving $M_{vir}$5, $M_{vir}$6, and $M_{vir}$7.

Its quoted validity range is $M_{vir}$8 and $M_{vir}$9. The headline claim is $M_{200c}$0 accuracy for $M_{200c}$1 and $M_{200c}$2. At $M_{200c}$3, the quoted performance remains good but degrades at the smallest scales, with about $M_{200c}$4 accuracy for $M_{200c}$5–$M_{200c}$6 depending on redshift and cosmology. The leave-one-out error for the best-performing HMCODE-based strategy is described as comparable to the simulation error and below $M_{200c}$7 for $M_{200c}$8 (Chen et al., 16 Feb 2025).

The same paper compares CSSTEmu with BACCO, EuclidEmulator2, and Mira-Titan IV. BACCO is described as typically showing 2–4% errors at $M_{200c}$9, EuclidEmulator2 as showing a $P_\mathrm{cb}$0 discrepancy at low redshift in the particular validation cases used and becoming less accurate at higher redshift and smaller scales, and Mira-Titan IV as often performing at the $P_\mathrm{cb}$1 level for its quoted range. The comparison is used to place CSSTEmu within the existing emulator landscape rather than to redefine the underlying methodology.

5. Generalized halo mass function emulation

The second CSSTEmu paper addresses halo mass function emulation and begins from a methodological distinction between the differential and cumulative forms. The cumulative HMF is defined as

$P_\mathrm{cb}$2

Its central result is that the differential HMF is sensitive to the mass binning scheme, whereas the cumulative HMF is effectively binning-independent because the variable is a mass threshold rather than a bin-averaged mass (Chen et al., 11 Jun 2025).

The differential-HMF analysis tests bin widths of 0.05 dex ($P_\mathrm{cb}$3), 0.1 dex ($P_\mathrm{cb}$4, fiducial), and 0.2 dex ($P_\mathrm{cb}$5). The reported binning-induced bias can reach about 5% at $P_\mathrm{cb}$6 and $P_\mathrm{cb}$7 for the fiducial 0.1 dex binning, becomes larger at higher redshift and higher mass, and worsens for coarser bins. The paper validates this behavior with averaged measurements from 250 Quijote-HR simulations and reports excellent agreement with the predicted binning bias. By contrast, cumulative HMF curves from different bin widths lie on top of each other, aside from tiny sample-variance fluctuations at the highest masses. The emulator is therefore built on the cumulative HMF because it is smoother, less noisy, and free from binning systematics (Chen et al., 11 Jun 2025).

The generalized emulator workflow has three stages. First, the measured cumulative HMF is smoothed with B-splines,

$P_\mathrm{cb}$8

using spline degree $P_\mathrm{cb}$9, knot spacing $M\geq 10^{12}\,h^{-1}M_{\odot}$0, minimum mass for spline fitting $M\geq 10^{12}\,h^{-1}M_{\odot}$1, only points with at least 20 halos, and Poisson noise as weights. Second, a theoretical baseline is fitted using the Euclid preparation formula from C23, with eight free parameters recalibrated for each halo definition. Third, residuals are emulated through

$M\geq 10^{12}\,h^{-1}M_{\odot}$2

followed by PCA compression, retention of the first 10 principal components, and GPR interpolation over the 8D cosmological space with a combination of a constant kernel and a Matérn-$M\geq 10^{12}\,h^{-1}M_{\odot}$3 kernel.

A major extension of the framework is explicit support for multiple halo mass definitions:

Halo definition	Catalog source	Overdensity convention
$M\geq 10^{12}\,h^{-1}M_{\odot}$4	Rockstar	$M\geq 10^{12}\,h^{-1}M_{\odot}$5 mean matter density
$M\geq 10^{12}\,h^{-1}M_{\odot}$6	Rockstar	virial overdensity
$M\geq 10^{12}\,h^{-1}M_{\odot}$7	FoF-based	$M\geq 10^{12}\,h^{-1}M_{\odot}$8 critical density

The paper emphasizes that once a halo catalog is available, the same pipeline can produce an emulator for that mass definition. This generalization directly addresses the fact that previous emulators often targeted only one halo definition, typically $M\geq 10^{12}\,h^{-1}M_{\odot}$9 or $z=3.0$0.

6. Accuracy, numerical limits, public release, and related frameworks

Before emulation, the halo-mass-function paper tests numerical resolution using additional convergence simulations and concludes that, for the fiducial setup, the cumulative HMF is converged to within about 2% for $z=3.0$1 across all redshifts and all three halo definitions. The final emulator range is therefore mass $z=3.0$2 and redshift up to $z=3.0$3. For redshifts $z=3.0$4, the quoted accuracy is within 2% for $z=3.0$5, 5% for $z=3.0$6, and 10% for $z=3.0$7. The conclusion further states that for low-mass halos $z=3.0$8 at $z=3.0$9, errors are roughly 1–2%, while for cluster masses $N$0 errors are within 5% (10%). The accuracy is reported to be similar for $N$1, $N$2, and $N$3, and the emulator errors are generally comparable to the Poisson noise in the simulations (Chen et al., 11 Jun 2025).

The suite is publicly released at https://github.com/czymh/csstemu. The power-spectrum paper states that the released package can run using only NumPy and SciPy, because it internally contains the needed ingredients and does not require external cosmological solvers or machine-learning libraries. Its intended applications include nonlinear matter clustering, weak lensing and galaxy clustering analyses, constraints on dark energy evolution, and neutrino-mass measurements (Chen et al., 16 Feb 2025).

The broader CSST literature also contains adjacent frameworks that have been discussed under similar naming. One example is a field-level neural network emulator for cosmological $N$4-body simulations, built from two convolutional neural networks that emulate nonlinear displacements and velocities from linear Zel’dovich-approximation inputs, with cosmology dependence encoded as style parameters and accuracy down to scales of $N$5 (Jamieson et al., 2022). Another is a mock-observation framework for the CSST mission that generates high-fidelity, pixel-level synthetic imaging and slitless spectroscopic data, built upon GalSim and CSST-specific modules (Wei et al., 10 Nov 2025). A plausible implication is that the acronym has acquired a wider ecosystem meaning in CSST preparation work, while its most explicit published usage denotes the CSST Cosmological Emulator itself.

In the narrow sense established by the 2025 CSSTEmu papers, the suite is a simulation-calibrated emulator infrastructure for nonlinear structure prediction in the CSST era: first for the matter power spectrum with one percent accuracy to $N$6 over most of the redshift range, and then for a generalized, accurate halo mass function emulator that replaces binning-sensitive differential modeling with cumulative-HMF emulation across multiple halo definitions (Chen et al., 16 Feb 2025, Chen et al., 11 Jun 2025).