Baryonification (BFC) Model

• Baryonification (BFC) model is a framework that remaps dark matter halos by incorporating baryonic effects like gas cooling, star formation, and feedback.
• It decomposes halos into components (gas, stars, ejected matter) to reconstruct mass profiles, thereby mimicking full hydrodynamical simulations.
• Calibration against simulations and observations enables the model to accurately predict key observables such as the matter power spectrum and weak lensing signals.

Searching arXiv for recent and foundational baryonification/BFC papers to ground the article. arXiv search query: all:baryonification OR all:"baryon correction model" OR all:BCEmu Baryonification, often denoted BFC and closely related to the baryon correction model (BCM), is a post-processing framework that transforms a gravity-only $N$-body realization into a baryon-corrected matter field by displacing matter around halos according to physically motivated prescriptions for gas, stars, and the baryon-induced response of collisionless matter. Its purpose is to reproduce the redistribution of mass caused by gas cooling, star formation, supernova feedback, AGN feedback, gas ejection, and dark-matter back-reaction without rerunning full hydrodynamical simulations. In contemporary large-scale-structure analyses, baryonification is used to model the nonlinear matter power spectrum, bispectrum, weak lensing, galaxy-galaxy lensing, kinematic and thermal Sunyaev–Zel’dovich signals, and, more recently, fast-radio-burst dispersion-measure statistics (Aricò et al., 2020, Aricò et al., 2024, Schneider et al., 10 Jul 2025, Torkamani et al., 26 Jan 2026).

1. Physical definition and modeling objective

The central premise of baryonification is that dark-matter-only simulations capture the collisionless gravitational backbone of structure formation, but omit the redistribution of matter caused by baryonic physics. On sufficiently large scales this omission can be subdominant, whereas on nonlinear scales it directly alters halo density profiles and therefore the observables derived from them. The method is therefore not a replacement for gravity-only evolution; it is a physically motivated correction layer applied after the $N$-body calculation (Aricò et al., 2024, Anbajagane et al., 2024).

A standard formulation begins from a dark-matter-only halo plus its environment and replaces that profile by a baryon-corrected decomposition. In the map-level formulation this is written as

$$\rho_{\rm DMO}=\rho_{\rm NFW}+\rho_{\rm 2h}, \qquad \rho_{\rm DMB}=\rho_{\rm CLM}+\rho_{\rm gas}+\rho_{\rm star}+\rho_{\rm 2h},$$

where $\rho_{\rm CLM}$ denotes collisionless matter after baryonic response, $\rho_{\rm gas}$ the redistributed gas, $\rho_{\rm star}$ the stellar component, and $\rho_{\rm 2h}$ the two-halo contribution (Anbajagane et al., 2024). Other implementations use a more granular partition into dark matter, central galaxy, satellite galaxies, bound gas, ejected gas, and reaccreted gas (Aricò et al., 2020), or into hot gas, inner/cold gas, central-galaxy stars, and satellite-galaxy stars (Schneider et al., 10 Jul 2025, Kovač et al., 10 Jul 2025).

This construction is designed to preserve the baryon budget while changing the radial distribution of matter. In BACCO, for example, the baryonified field is built so that

$$\text{(stellar mass + bound gas + ejected gas)}=\frac{\Omega_b}{\Omega_m},$$

which makes the method a field-level prescription rather than a pure fitting formula for a single statistic (Aricò et al., 2020).

2. Halo decomposition and displacement formalism

The defining algorithmic operation is a radial displacement field that remaps matter from the gravity-only enclosed-mass profile to the target baryonified enclosed-mass profile. In the standard BCM formulation, the gravity-only halo is modeled as an NFW profile, and particles are moved spherically around halo centers so that cumulative mass ordering is preserved (Aricò et al., 2020). The model thereby converts a DMO halo into a sum of parametric halo components, each with its own profile and mass fraction.

One widely used generation of BCM includes the following ingredients: dark matter, a central galaxy, satellite galaxies, bound halo gas, ejected gas, and late-time reaccreted gas. The bound-gas component is described by a double-power-law profile, the central galaxy by a compact profile with fiducial inner slope $\alpha_g=2$, ejected gas is set by mass conservation, and reaccreted gas is assigned a Gaussian-like profile. In that formulation, ejected gas is identified as the dominant driver of power-spectrum suppression, while bound gas and dark-matter back-reaction are responsible for the small-scale bispectrum bump seen especially in Illustris TNG-300 and EAGLE (Aricò et al., 2020).

A later component-wise BFC refines the displacement itself by first splitting each simulation particle into separate dark-matter and baryonic components and then displacing the two species independently. The displacement is written as

$$d_{\rm A}(r_{\rm i}|M_{200},c_{200},\theta_{\rm BFC}) = r_{{\rm A},f}(M_{\rm A})-r_{{\rm A},i}(M_{\rm A}), \qquad {\rm A}=\{{\rm dm,bar}\},$$

with cumulative masses

$N$0

In this formulation, the initial dark-matter and baryon profiles are both constructed from a truncated NFW halo plus a two-halo term, while the final baryonic sector is decomposed into hot gas, inner gas, and stellar components (Schneider et al., 10 Jul 2025).

The physical profile choices differ across implementations, but recurring patterns are clear. Hot gas is commonly represented by a cored or double-power-law profile with a mass-dependent slope and an outer truncation; central stars are modeled as compact $N$1-type profiles with an exponential cutoff; and collisionless matter undergoes contraction or expansion in response to the rearranged baryonic potential (Aricò et al., 2020, Bigwood et al., 2024, Schneider et al., 10 Jul 2025). Some map-level frameworks replace 3D particle displacements by 2D projected-shell displacements,

$N$2

to construct full-sky weak-lensing and tSZ maps efficiently (Anbajagane et al., 2024).

3. Parameterization and reduced model families

Baryonification is not a single parameterization but a family of related models. A recurrent parameter is $N$3, although its operational definition depends on the implementation. In BACCO, $N$4 is the parameter that sets the gas fraction retained per halo mass and is identified as the main control on baryonic suppression in the power-spectrum ratio $N$5 (Aricò et al., 2020). In BCEmu, $N$6 is the characteristic mass scale where the gas-profile slope transitions (Bigwood et al., 2024). In the FRB-dispersion application, $N$7, $N$8, and $N$9 are the primary drivers of the PDF shape (Torkamani et al., 26 Jan 2026).

The 2020 BCM fit to matter power spectrum and bispectrum uses seven free parameters,

$$\rho_{\rm DMO}=\rho_{\rm NFW}+\rho_{\rm 2h}, \qquad \rho_{\rm DMB}=\rho_{\rm CLM}+\rho_{\rm gas}+\rho_{\rm star}+\rho_{\rm 2h},$$0

with $$\rho_{\rm DMO}=\rho_{\rm NFW}+\rho_{\rm 2h}, \qquad \rho_{\rm DMB}=\rho_{\rm CLM}+\rho_{\rm gas}+\rho_{\rm star}+\rho_{\rm 2h},$$1 and $$\rho_{\rm DMO}=\rho_{\rm NFW}+\rho_{\rm 2h}, \qquad \rho_{\rm DMB}=\rho_{\rm CLM}+\rho_{\rm gas}+\rho_{\rm star}+\rho_{\rm 2h},$$2 controlling retained halo gas fractions, $$\rho_{\rm DMO}=\rho_{\rm NFW}+\rho_{\rm 2h}, \qquad \rho_{\rm DMB}=\rho_{\rm CLM}+\rho_{\rm gas}+\rho_{\rm star}+\rho_{\rm 2h},$$3 the extent of ejected gas, and $$\rho_{\rm DMO}=\rho_{\rm NFW}+\rho_{\rm 2h}, \qquad \rho_{\rm DMB}=\rho_{\rm CLM}+\rho_{\rm gas}+\rho_{\rm star}+\rho_{\rm 2h},$$4 the shape of the hot-gas profile (Aricò et al., 2020). BACCO embeds this baryonic sector in a 15-dimensional emulator space composed of 8 cosmological parameters and 7 baryonic parameters, while emulating

$$\rho_{\rm DMO}=\rho_{\rm NFW}+\rho_{\rm 2h}, \qquad \rho_{\rm DMB}=\rho_{\rm CLM}+\rho_{\rm gas}+\rho_{\rm star}+\rho_{\rm 2h},$$5

rather than the full power spectrum directly (Aricò et al., 2020).

For weak-lensing and kSZ inference, BCEmu introduces reduced-complexity variants: BCEmu1, where only $$\rho_{\rm DMO}=\rho_{\rm NFW}+\rho_{\rm 2h}, \qquad \rho_{\rm DMB}=\rho_{\rm CLM}+\rho_{\rm gas}+\rho_{\rm star}+\rho_{\rm 2h},$$6 varies; BCEmu3, where $$\rho_{\rm DMO}=\rho_{\rm NFW}+\rho_{\rm 2h}, \qquad \rho_{\rm DMB}=\rho_{\rm CLM}+\rho_{\rm gas}+\rho_{\rm star}+\rho_{\rm 2h},$$7, $$\rho_{\rm DMO}=\rho_{\rm NFW}+\rho_{\rm 2h}, \qquad \rho_{\rm DMB}=\rho_{\rm CLM}+\rho_{\rm gas}+\rho_{\rm star}+\rho_{\rm 2h},$$8, and $$\rho_{\rm DMO}=\rho_{\rm NFW}+\rho_{\rm 2h}, \qquad \rho_{\rm DMB}=\rho_{\rm CLM}+\rho_{\rm gas}+\rho_{\rm star}+\rho_{\rm 2h},$$9 vary; and BCEmu7, where all seven baryonification parameters vary (Bigwood et al., 2024). The paper states that BCEmu7 and BCEmu3 can reproduce simulated power suppression to better than $\rho_{\rm CLM}$0, whereas BCEmu1 is only accurate at the few-percent level and is generally too restrictive (Bigwood et al., 2024).

The component-wise BFC of 2025 organizes its parameter space into an 8-parameter full BFC and a reduced model. In the full version the free parameters are $\rho_{\rm CLM}$1, $\rho_{\rm CLM}$2, $\rho_{\rm CLM}$3, $\rho_{\rm CLM}$4, $\rho_{\rm CLM}$5, $\rho_{\rm CLM}$6, $\rho_{\rm CLM}$7, and $\rho_{\rm CLM}$8; in the reduced version only $\rho_{\rm CLM}$9 and $\rho_{\rm gas}$0 remain free; and a later extension adds one redshift-evolution parameter to form a 2+1 parameter model (Schneider et al., 10 Jul 2025).

4. Calibration strategies, emulators, and numerical performance

A major attraction of baryonification is that it can be calibrated either to hydrodynamical simulations or, in some applications, to observations. The 2020 BCM study calibrated its seven-parameter model jointly to the matter power spectrum and equilateral reduced bispectrum from six hydrodynamical simulations—BAHAMAS standard AGN, BAHAMAS low-AGN, BAHAMAS high-AGN, EAGLE, Illustris, and Illustris TNG-300—and reported typical accuracies of $\rho_{\rm gas}$1 for the power spectrum and $\rho_{\rm gas}$2 for the bispectrum over $\rho_{\rm gas}$3 and $\rho_{\rm gas}$4 (Aricò et al., 2020). The same work showed that calibrating only on the power spectrum can leave the bispectrum wrong by as much as $\rho_{\rm gas}$5, which established higher-order statistics as a nontrivial constraint on the mass-redistribution model (Aricò et al., 2020).

BACCO converts this idea into a neural-network emulator trained on roughly $\rho_{\rm gas}$6 baryonified power spectra, with smoothing by a Savitzky–Golay filter, PCA compression to 6 principal components, and a feed-forward network with 2 hidden layers of 400 neurons each, ReLU activations, Adam optimization, and mean-squared-error loss (Aricò et al., 2020). The reported emulator precision is unbiased at the $\rho_{\rm gas}$7 level, with $\rho_{\rm gas}$8 precision for $\rho_{\rm gas}$9, $\rho_{\rm star}$0 up to $\rho_{\rm star}$1, and an overall quoted accuracy of $\rho_{\rm star}$2 over $\rho_{\rm star}$3 and $\rho_{\rm star}$4 (Aricò et al., 2020).

Thermodynamic extensions generalize baryonification from mass redistribution to gas pressure and temperature. One such extension adds only two new free parameters and reproduces the suppression in the matter power spectrum induced by baryons at the percent level while fitting the electron-pressure auto- and cross-power spectra at $\rho_{\rm star}$5 for the BAHAMAS suite. The same model reproduces BAHAMAS convergence power spectra within $\rho_{\rm star}$6 and thermal Sunyaev–Zel’dovich angular power spectra within $\rho_{\rm star}$7 down to $\rho_{\rm star}$8 (Aricò et al., 2024).

The improved component-wise BFC of 2025 calibrates directly to gas and stellar mass-ratio profiles from FLAMINGO and TNG. When fit to these profiles and then used to baryonify $\rho_{\rm star}$9-body boxes, it reaches $\rho_{\rm 2h}$0 agreement in total-matter power-spectrum suppression up to $\rho_{\rm 2h}$1 across all tested feedback scenarios and redshifts relevant for cosmological surveys, while its reduced 2+1 parameter model remains better than $\rho_{\rm 2h}$2 over $\rho_{\rm 2h}$3 to $\rho_{\rm 2h}$4 (Schneider et al., 10 Jul 2025).

5. Observables and inference pipelines

Baryonification has moved from two-point matter statistics to multi-probe inference. In the SHAMe framework for galaxy clustering and galaxy-galaxy lensing, baryonification is applied not to the galaxy-population model itself but to the underlying dark-matter simulation used for the lensing calculation. The stated workflow is: populate the gravity-only simulation with galaxies using SHAMe, apply a baryonification correction to the same gravity-only simulation, and compute lensing around the SHAMe galaxies in that baryonified mass field. In this use case the correction improves lensing, not clustering, and extends the validity of simultaneous clustering+lensing modeling from $\rho_{\rm 2h}$5 to $\rho_{\rm 2h}$6 (Contreras et al., 2022).

For weak lensing combined with kSZ, BCEmu is used as a flexible analytic mapping from dark-matter-only halos to baryon-loaded halos with gas ejection and redistribution. In the DES Y3 + ACT DR5 analysis, WL-only with BCEmu7 gives

$\rho_{\rm 2h}$7

while the joint WL+kSZ analysis gives

$\rho_{\rm 2h}$8

The same paper reports that WL+kSZ prefers a suppression of the nonlinear matter power spectrum that is more extreme than most hydrodynamical simulations, and that the kSZ data materially improve constraints on baryonic parameters, especially $\rho_{\rm 2h}$9 (Bigwood et al., 2024).

A subsequent observational application constrains the component-wise BFC using ACT kSZ profiles and eROSITA halo gas fractions. The paper reports that these data sets are jointly consistent, with $$\text{(stellar mass + bound gas + ejected gas)}=\frac{\Omega_b}{\Omega_m},$$0 and $$\text{(stellar mass + bound gas + ejected gas)}=\frac{\Omega_b}{\Omega_m},$$1, and imply matter-power-spectrum suppression exceeding the percent level above $$\text{(stellar mass + bound gas + ejected gas)}=\frac{\Omega_b}{\Omega_m},$$2, growing to $$\text{(stellar mass + bound gas + ejected gas)}=\frac{\Omega_b}{\Omega_m},$$3 at $$\text{(stellar mass + bound gas + ejected gas)}=\frac{\Omega_b}{\Omega_m},$$4 and $$\text{(stellar mass + bound gas + ejected gas)}=\frac{\Omega_b}{\Omega_m},$$5 at $$\text{(stellar mass + bound gas + ejected gas)}=\frac{\Omega_b}{\Omega_m},$$6 (Kovač et al., 10 Jul 2025). That analysis also finds excellent agreement with X-COP gas density and pressure profiles, which functions as a consistency check across mass scales (Kovač et al., 10 Jul 2025).

Map-level baryonification extends the formalism to direct full-sky modeling of weak-lensing and tSZ fields and their higher-order moments. Using auto- and cross-moments of order $$\text{(stellar mass + bound gas + ejected gas)}=\frac{\Omega_b}{\Omega_m},$$7, the model is reported to fit IllustrisTNG within measurement uncertainties for scales above roughly $$\text{(stellar mass + bound gas + ejected gas)}=\frac{\Omega_b}{\Omega_m},$$8 and across multiple redshifts, while simulation-based forecasts indicate that the combination of lensing and tSZ moments can jointly constrain cosmology and baryons because their degeneracy directions are often orthogonal (Anbajagane et al., 2024).

The formalism has also been pushed to one-point statistics of line-of-sight observables. In the FRB application, BFC is combined with the halo mass function and halo bias to derive an analytic PDF for dispersion measures up to $$\text{(stellar mass + bound gas + ejected gas)}=\frac{\Omega_b}{\Omega_m},$$9. The paper reports excellent agreement with IllustrisTNG and states that $\alpha_g=2$0 FRBs can constrain the gas-profile parameters to precision comparable to kSZ and X-ray observations (Torkamani et al., 26 Jan 2026).

6. Limitations, calibration dependence, and scope

The principal limitations of baryonification arise from its deliberately reduced description of galaxy formation. Spherical symmetry, halo-centered radial displacements, hydrostatic assumptions for pressure modeling, approximate two-halo terms, and simplified redshift dependence are all explicit ingredients of current BFC implementations (Aricò et al., 2024, Kovač et al., 10 Jul 2025, Torkamani et al., 26 Jan 2026). These assumptions are often sufficient for percent-level matter suppression, but they can be stressed by extreme feedback prescriptions, non-monotonic halo gas fractions, or observables dominated by rare massive systems (Aricò et al., 2020, Anbajagane et al., 2024).

Calibration dependence is a second major issue. The SHAMe analysis explicitly shows that a TNG300-calibrated baryonification and a BAHAMAS-tuned baryonification produce significantly different galaxy-galaxy lensing below $\alpha_g=2$1, underscoring that small-scale lensing is sensitive to the chosen baryonic model (Contreras et al., 2022). The DES Y3 + ACT study likewise emphasizes a flexibility-versus-precision trade-off: more flexible baryonic modeling can shift $\alpha_g=2$2 upward and enlarge uncertainties, while restrictive models such as BCEmu1 can bias $\alpha_g=2$3 low in strong-feedback scenarios (Bigwood et al., 2024).

Redshift evolution remains an active difficulty. The 2020 BCM study finds that best-fit baryonic parameters are not redshift independent; if $\alpha_g=2$4 parameters are applied unchanged to $\alpha_g=2$5 and $\alpha_g=2$6, the reduced bispectrum errors can reach $\alpha_g=2$7 (Aricò et al., 2020). Later reduced BFC models address this by adding an explicit redshift-evolution parameter, but those are still compact phenomenological descriptions rather than first-principles hydrodynamics (Schneider et al., 10 Jul 2025).

A final point is terminological. In the astrophysical literature surveyed here, BFC denotes a baryonification or baryon-correction framework. In unrelated graph-learning work, however, BFC can denote Balanced Forman Curvature; that usage is explicitly not the astrophysical baryonification model (Hardiman-Mostow et al., 29 Oct 2025). Within cosmology, by contrast, baryonification consistently refers to the construction of a baryon-modified matter field from a gravity-only baseline through interpretable halo-scale mass redistribution (Aricò et al., 2020, Schneider et al., 10 Jul 2025).