Baryonic Emulation Frameworks: Efficient Modeling

• Baryonic emulation frameworks are computational techniques that surrogate high-resolution hydrodynamic simulations to predict baryonic effects on cosmological observables.
• They leverage methods like PCA, machine learning, and symbolic regression to efficiently capture baryonic phenomena while maintaining percent-level accuracy.
• These frameworks enable rapid calibration of cosmic surveys by modeling key processes such as star formation, feedback, and gas dynamics.

Baryonic Emulation Frameworks refer to a family of computational techniques designed to replace or augment high-resolution hydrodynamical simulations with surrogate models that rapidly predict the impact of baryonic physics—such as cooling, star formation, and feedback—on cosmological observables. These frameworks span symbolic regression, principal component analysis, machine learning, and hybrid analytic/ML surrogates, collectively pushing towards accurate, efficient, and flexible modeling of baryons in both large-scale cosmological contexts and mesoscopic (e.g., halo, galaxy, soliton) regimes.

1. Motivation and Scope of Baryonic Emulation

Hydrodynamical simulations, though physically comprehensive, carry prohibitive computational cost, especially when exploring high-dimensional parameter spaces or generating large ensembles of mock universes. Baryonic emulation frameworks seek to bridge this gap by learning from suites of hydro runs or analytic prescriptions, efficiently mapping from dark-matter-only or coarsened simulations to full baryon-informed outputs required by weak lensing, galaxy clustering, Lyα statistics, and other precision probes. The systematic suppression of power in the nonlinear matter spectrum (and bispectrum), scale-dependent redistribution of gas/stars, and cluster-to-galaxy-scale feedback are central targets for emulation frameworks, given their status as dominant astrophysical systematics in Stage IV surveys (LSST, Euclid, SKA).

2. Principal-Component and Basis-Reduction Approaches

A foundational approach for baryonic emulation is the principal component analysis (PCA) parametrization as introduced for cosmic shear observables (Mohammed et al., 2017). Given a suite of hydrodynamic simulations (fixed cosmology, varied subgrid baryonic physics), one computes the baryonic contrast vector:

$$\delta C_{ij}(\ell) \equiv \frac{C_{ij}^{\rm BAR}(\ell)}{C_{ij}^{\rm DMO}(\ell)} - 1$$

Data vectors $D^{(m)}$ of dimension $n$ (e.g., $n=180$ for 15 tomographic bins $\times$ 12 multipoles) are mean centered, and the sample covariance matrix is decomposed:

${\mathbf C} e_a = \lambda_a e_a$

The baryonic impact is then approximated as the mean plus a truncated set of $k$-largest eigenmodes:

$$D^{(k)}_i = \bar D_i + \sum_{a=1}^k \alpha_a e_{a,i}$$

For the LSST-like test case, $k=4$ captures $>99.99\%$ of the variance, with RMS errors $D^{(m)}$0 ($D^{(m)}$1), provided the training set includes physically plausible "outliers" (e.g., ART14-CX with strong cooling). Excluding such outliers degrades accuracy by a factor $D^{(m)}$2, emphasizing the need for a training set that spans the true systematics envelope. In practical applications, the coefficients $D^{(m)}$3 become nuisance parameters, marginalized over (with appropriate priors) during cosmological inference (Mohammed et al., 2017).

3. Machine Learning and Field-Level Emulation

A class of frameworks leverages machine-learning architectures—convolutional neural networks, stochastic interpolants, or symbolic regression—for field-level or summary-statistic emulation:

• EMBER-2 (Bernardini et al., 21 Feb 2025) employs a context-modulated U-Net–GAN, mapping projected dark-matter density/velocity to field-level baryonic quantities (gas, HI, velocity, temperature) for $D^{(m)}$4. The architecture integrates a redshift-style MLP, modulated convolutions, and multi-scale discriminator, reaching percent-level gas-mass conservation and cross-correlation error $D^{(m)}$5 at $D^{(m)}$6 (up to $D^{(m)}$7), enabling direct integration into mock-catalog/statistical analyses.
• BaryonBridge (Horowitz et al., 22 Oct 2025) introduces a stochastic interpolant (SI) U-Net conditioned on cosmology/feedback parameters. The SI paradigm interpolates between input DM fields and baryonic targets via a neural SDE, trained to minimize a score-matching loss. Field-level HI density, temperature, and Lyα flux PDF are recovered to better than $D^{(m)}$8 ($D^{(m)}$9 for $n$0), and map-level residuals remain $n$1 when applied to new TNG50-class volumes.
• Symbolic Regression frameworks (e.g., syren-baryon (Kammerer et al., 10 Jun 2025)) provide closed-form analytic expressions for power-spectrum suppression $n$2. Operon-fueled Pareto optimization ensures boundary conditions ($n$3 as $n$4, $n$5), with model-dependent uncertainties consistently $n$6 for $n$7. Each analytic model is tailored to a specific hydro or baryonification dataset, enabling rapid evaluation and physical interpretability.

These ML-based emulators see use in painting baryonic fields onto DM-only simulations, forward-modeling weak-lensing observables, and enabling cosmology–astrophysics joint inference.

4. Baryonification and Analytic/Hybrid Surrogates

Baryonification frameworks efficiently modify DM N-body outputs at the particle or halo level, calibrated on hydro simulations or observations:

• Component-wise Baryonification (BFC) (Schneider et al., 10 Jul 2025) splits original N-body particles into DM and baryonic children, then displaces each using halo mass–dependent analytic profiles. Gas pressure and temperature are assigned via hydrostatic equilibrium (plus empirical non-thermal fractions), and the complete model is calibrated on gas/stellar mass ratios from reference hydros (TNG, FLAMINGO). A reduced (2+1)-parameter version, incorporating feedback/ redshift evolution, achieves $n$8 accuracy in power-spectrum suppression for $n$9.
• BACCO baryonification emulator (Aricò et al., 2020) applies displacements derived from multi-component analytic profiles (stellar, bound/ejected gas) and cosmology scaling, using a neural network to regress power-spectrum ratios $n=180$0 on 15 input parameters (8 cosmology, 7 baryon). RMS interpolation error is $n=180$1 across validation suites.
• Post-processing frameworks (CGMBrush) (Williams et al., 2022) "paint" spherically symmetric, calibrated gas profiles onto halo catalogs from large N-body boxes, constructing secondary observables such as DM statistics, tSZ/kSZ maps, and baryon-modified density fields. These frameworks are modular, supporting profile families motivated by zoom-in hydrodynamics, X-ray/SZ observations, or semi-analytical precipitation models.

Hybrid analytic/ML schemes (e.g., equilibrium + ExtraTrees (Moews et al., 2020)) merge physically motivated halo/gas cycling models with non-linear ML regression, yielding sub-percent accuracies for principal stellar and gas properties and $n=180$2 speedup over full hydrodynamics.

5. Statistical Emulation for Model Calibration and Inference

Gaussian-process regression (GPR) and principal-component reduction are routinely used for high-fidelity emulation of simulation outputs and Bayesian parameter inference:

• In (Ramachandra et al., 12 Jan 2026), a GPR–PCA emulator is trained on a Latin-hypercube grid of cosmological hydrodynamic runs (sampling star-formation winds, AGN feedback, etc.). The GP maps subgrid parameter vectors $n=180$3 to flattened observables (e.g., stellar mass function, gas profiles), capturing $n=180$4 variance with $n=180$5 mean error (large-box phase). Bayesian inference against synthetic and real data reveals multimodal likelihoods, and diagnose degeneracies between AGN kinetic parameters and halo-scale gas observables.
• For field-level mode corrections, GP regression is used to model the transfer function $n=180$6 across cosmology/feedback parameter space (Sharma et al., 2024). Median emulator errors are $n=180$7–$n=180$8 for $n=180$9 and $\times$0–$\times$1 at $\times$2 over diverse hydrodynamics.

A critical requirement across all emulation frameworks is to accurately represent the full range of plausible baryonic behaviors, especially "outlier" feedback or cooling mechanisms, as omission of such models severely biases uncertainty coverage and model reliability (Mohammed et al., 2017).

6. Hierarchical Multi-Fidelity and Specialized Emulators

Hierarchical/multi-fidelity emulation expands the scope of surrogate modeling to nuclear theory and fuzzy dark matter:

• BANNANE (Belley et al., 27 Feb 2025) employs a hierarchical Bayesian neural network for nuclear observables (binding energies, radii) trained across a sequence of many-body model truncations. Embedding nuclear structure (Z, N, fidelity), the emulator delivers percent-level RMSE and automated global sensitivity analysis, supporting zero-shot extrapolation to new isotopes/fidelities.
• In specialized galaxy/soliton contexts, symbolic regression and genetic algorithms are used to emulate baryon equations of motion, e.g., $\times$3 for FDM soliton cores (Wang et al., 1 Dec 2025), achieving $\times$4 reproduction errors for equilibrium profiles.

Practical frameworks often combine analytic "backbones" (angular momentum conservation, hydrostatic equilibrium, merger-tree averaging) with ML-based residual-learning or symbolic fitting for additional baryonic channels, thermodynamic phases, or generalization to unseen parameter regimes.

7. Applications, Performance, and Limitations

Baryonic emulation frameworks have found broad application in weak lensing forecasts, BAO/RSD analysis, FRB dispersion modeling, mock data synthesis, and hydro-observable parameter calibration. Quantitative error budgets typically remain at or below $\times$5–$\times$6 for power spectrum ratios up to $\times$7 and $\times$8, with some degradation at smaller scales or higher redshift. Table-driven parameter reduction studies identify minimal subsets (e.g., four-parameter baryonification) sufficient for sub-percent recovery over relevant regimes (Giri et al., 2021, Schneider et al., 10 Jul 2025).

The dominant limitations of existing frameworks stem from: incomplete sampling of feedback/cooling extremes, breakdowns in hydrostatic or spherical symmetry assumptions at small scales or high redshift, lack of detailed halo-to-halo scatter modeling, and memory/speed constraints for 3D field-level emulation at high resolution (Schneider et al., 10 Jul 2025, Horowitz et al., 22 Oct 2025). Extension to multi-fidelity, cross-code emulators, and the explicit propagation of uncertainty through cosmological parameter inference, are active areas of development (Ramachandra et al., 12 Jan 2026, Belley et al., 27 Feb 2025).

Baryonic emulation frameworks are now essential for enabling the full scientific return of Stage-IV and next-generation cosmological surveys, providing a rigorous statistical link between the microphysics of galaxy formation and macroscopic cosmological probes across inaccessible parameter domains.