Baryonification Techniques in Cosmology & QCD

• Baryonification techniques are a set of methods that incorporate baryonic processes — including gas cooling, star formation, and feedback — into dark-matter simulations to refine cosmic structure predictions.
• They employ diverse approaches such as holographic duality, six-quark dispersion, and map-level corrections to reproduce observables like matter power spectra, bispectra, and halo properties.
• Emulator and symbolic regression methods further optimize these techniques, achieving percent-level accuracy in modeling baryonic feedback across multiple scales.

Baryonification techniques are a class of postprocessing and modeling methods developed to incorporate the effects of baryonic physics—such as gas cooling, star formation, and feedback from supernovae and active galactic nuclei (AGN)—into dark-matter-only simulations and effective field-theoretic or holographic models of hadronic structure. These approaches enable the efficient prediction of mass clustering, matter power spectra, bispectra, and halo properties in contexts where direct hydrodynamical simulation or first-principles nonperturbative calculation would be computationally prohibitive or conceptually inaccessible. Baryonification further allows for a physically motivated calibration of galaxy and cluster observables with minimal computational overhead, while systematically incorporating phenomenology from QCD, string theory, and astrophysical datasets.

1. Holographic Baryonification in QCD

Holographic baryonification leverages gauge/gravity duality, exemplified by the D4–D8 string theory model, to provide a dual description of baryons and their interactions within strong coupling QCD (Yi, 2011). In this framework, a stack of $N_c$ D4-branes and flavor D8-branes in type IIA string theory generates a five-dimensional background, with the warped geometry determined by the function $u(w)$, setting the mass scale $M_{KK} \sim 0.94$ GeV. Baryons are realized as instanton solitons—coherent states with unit Pontryagin index—in the five-dimensional $U(N_f)$ flavor gauge theory localized in the holographic direction $w$.

These solitons serve as holographic duals of Skyrmions but are substantially modified by the coupled infinite tower of vector and axial-vector mesons arising from dimensional reduction. The resulting baryon is not a simple chiral soliton but incorporates higher-spin and nontrivial profile corrections, allowing calculation of observables (size, couplings) without adjustable parameters. The effective low-energy action for baryons features both minimal couplings (via covariant derivatives with correct $U(1)_v$ and $SU(2)$ charges) and nonminimal terms related to the instanton’s field strength ($\bar{B}\gamma^{MN} F_{MN} B$). Overlap integrals between the sharply localized baryon wavefunctions and meson mode profiles yield predictive expressions for nucleon–meson couplings.

Empirically, this approach produces nucleon–meson couplings in good agreement with phenomenological values, such as the tensor-to-vector coupling ratio for $\rho$ mesons—a dominant effect in nucleon potentials—while predicting vanishing tensor couplings for both all axial-vectors and isosinglet vectors. The leading order tensor coupling for isotriplet vectors is given by: $2 M_{KK}\, g_{dV}^{(k)} \sim 1.3 N_c\, g_V^{(k)},$ which matches empirical trends upon extrapolation to $N_c=3$ and inclusion of subleading effects.

2. Six-Quark Dispersion Techniques and Multiquark Baryonification

Heavy baryonification at the hadron level, including charmed baryonia, has been pursued via relativistic six-quark dispersion relation techniques (Gerasyuta et al., 2012). These models construct explicit six-body ($BB$) amplitude equations, decomposed into subamplitudes per pairwise subsystem, without quark–antiquark mixing. Leading singularities—dominant contributions from two-body and triangular thresholds—are extracted to reduce the coupled integral equations to a system of algebraic equations at a fixed Dalitz plot point $s_0$. The pole positions in the reduced amplitudes, often parameterized as

$$a_j(s, s_0) \propto \frac{I_j(s, s_0)}{1 - g I_j(s, s_0)},$$

determine the baryonia mass spectrum. This reduces the complex six-body dynamics to tractable algebraic constraints, facilitating the prediction of bound states and masses accurate to within $\sim$1 MeV. This systematic reduction and the explicit inclusion of color-matched quark configurations constitute a concrete baryonification technique in nonperturbative QCD, particularly relevant for exotic hadrons.

3. Cosmological Baryonification: Hydrodynamics, Simulation, and Postprocessing

3.1. Algorithmic Structure and Profile-Based Corrections

In cosmological large-scale structure studies, baryonification refers to a family of postprocessing algorithms and baryonic correction models (BCM) that displace mass elements in $N$-body simulations to reproduce the net effect of baryonic feedback processes (Aricò et al., 2019, Aricò et al., 2020, Zhou et al., 12 May 2025, Schneider et al., 10 Jul 2025, Kovač et al., 10 Jul 2025). The standard workflow is as follows:

1. Halo identification: Generate a halo catalog from a dark-matter-only simulation, fitting each halo with an NFW or more complex profile.
2. Analytical profiles for baryonic components: Assign analytic forms (with parameters calibrated to simulations/observations) to the central galaxy ($\rho_{\mathrm{CG}}$), hot gas ($\rho_{\mathrm{BG}}$), ejected gas ($\rho_{\mathrm{EG}}$), and corrected dark matter ($\rho_{\mathrm{RDM}}$). E.g.,

$$\rho_{\mathrm{GrO}}(r) = \frac{\rho_0}{(r/r_s)(1+r/r_s)^2}; \quad \rho_{\mathrm{CG}}(r) = f_{\mathrm{CG}} \frac{M_{200}}{4\pi^{3/2} R_h r^2} \exp\left[-\left(\frac{r}{2R_h}\right)^2\right];$$

1. Construction of the baryon-corrected mass profile: Sum the individual components to yield $M_{\mathrm{BC}}(r)$.
2. Particle (shell/pixel) displacement: Compute the mapping $M_{\mathrm{GrO}}(r) = M_{\mathrm{BC}}(r')$ and obtain the displacement field $\Phi(r) = r' - r$; displace particles or shells accordingly to reproduce the corrected mass profile and spatial statistics.
3. Dark matter back-reaction: Include adiabatic relaxation/expansion to account for baryonic redistribution's impact on the DM profile (e.g., via quasi-adiabatic contraction formulae).

This pipeline can efficiently reproduce the $P(k)$ suppression/enhancement due to AGN- or star-formation-driven gas ejection and central condensation, matching hydrodynamical simulation outputs to percent-level accuracy over $k \lesssim 5$--$10\,h\,\mathrm{Mpc}^{-1}$ and $z \lesssim 2$. Ejecta distributions (parameterized by $\eta$) and mass thresholds (e.g., $M_{\mathrm{c}}$ for gas retention) are central parameters.

3.2. Extensions: Thermodynamics and Feedback Constraints

Recent extensions assign pressure and temperature profiles—using polytropic equations of state and hydrostatic equilibrium—to the gas components, enabling self-consistent prediction of observable fields such as the thermal and kinetic Sunyaev-Zel'dovich (tSZ, kSZ) effects (Aricò et al., 3 Jun 2024, Kovač et al., 10 Jul 2025). For example,

$$P_{\mathrm{BG}}(r) = P_0 [\rho_{\mathrm{BG}}(r)]^\Gamma,$$

with non-thermal corrections,

$$P_{\mathrm{BG,th}}(r) = f_{\mathrm{th}}(x) P_{\mathrm{BG}}(r),$$

and gas temperature from the ideal gas law,

$$T_{\mathrm{BG}}(r) = \frac{\mu m_p}{k_B} \frac{P_{\mathrm{BG,th}}(r)}{\rho_{\mathrm{BG}}(r)}.$$

Calibration to simulation pressure and density profiles is achieved either by direct fitting (e.g., to FLAMINGO or TNG profiles) or by parameterizing the hot gas profile in terms of a mass-dependent slope and truncation (Schneider et al., 10 Jul 2025).

The component-wise BFC approach (Baryonification with Full Components) further decomposes gas and stellar matter into central, hot, and satellite components; for each, empirically determined profiles (core, truncation, and mass-dependent slopes) are specified, and particle splitting with component-wise displacement is performed prior to assigning thermodynamical quantities (Schneider et al., 10 Jul 2025, Kovač et al., 10 Jul 2025). This ensures mass conservation per component and allows direct comparison with multiwavelength datasets (e.g., X-ray, SZ, eROSITA, ACT).

4. Map-Level and Higher-Order Baryonification

Map-level baryonification applies the displacement prescription not to simulation particles but directly to pixelized shells or 2D projected density/grids, enabling rapid construction of baryon-corrected full-sky maps (Anbajagane et al., 5 Sep 2024, Zhou et al., 12 May 2025). This method supports fast emulation of observables relevant for weak lensing ($\kappa$), tSZ ($y$), and their higher-order statistics. The framework is especially powerful for forward modeling higher-order (non-Gaussian) statistics—moments, scattering coefficients, wavelet phase harmonics—of the lensing and tSZ fields, with summary statistics such as: $$m_3(\theta) = \langle (\kappa \star G(\theta))^3 \rangle;\quad Q(k) = B(k)/[3P(k)^2],$$ matching the baryonic suppression in $P(k)$ and $B(k)$ seen in hydrodynamical runs within $1$--$2\%$ across a broad parameter space and up to $\ell \approx 2000$ for widely used survey configurations.

Sensitivity analysis of these moments with respect to parameter variations (e.g., $M_c$, $\theta_{\mathrm{ej}}$) allows joint cosmology–baryon inference and supports robust marginalization over baryonic uncertainties. Public pipelines such as BaryonForge enable community application of these tools (Anbajagane et al., 5 Sep 2024).

5. Emulator and Symbolic Regression Approaches

Given the high dimensionality of the problem (spanning multiple cosmological and baryonic feedback parameters), neural network and symbolic regression-based emulators are increasingly used to interpolate baryonification predictions (Aricò et al., 2020, Kammerer et al., 10 Jun 2025, Burger et al., 23 Jun 2025). Emulators are trained on $>50{,}000$ baryonified power spectrum measurements and can predict the baryon-to-gravity-only suppression ratio

$$S(k, z, \vec{\theta}) = \frac{P_{\mathrm{baryon}}(k, z, \vec{\theta})}{P_{\mathrm{DM\text{-}only}}(k, z, \vec{\theta})}$$

across a 15-dimensional parameter space with $1$--$2\%$ accuracy. Symbolic regression provides compact analytic parametrizations, e.g.,

$$\log S(k, z, \theta) = \frac{\alpha_1 k (\alpha_2 k)^{\alpha_3(\alpha_4 - z)}}{(\alpha_5 z - \alpha_6^{-2z})(\alpha_7 + \alpha_8 k + k^2)} + (\alpha_9 k - 1 + e^{-\alpha_{10} k}) e^{-\alpha_{11} z},$$

with $\alpha_i$ tied to cosmology or feedback parameters. These analytic forms are enforced to recover $S(k) \to 1$ as $k \to 0$ or $z \to \infty$ and support rapid Bayesian parameter inference.

For higher-order and non-Gaussian statistics (e.g., bispectrum), dedicated neural network emulators trained on baryonified BACCO outputs can predict baryonic suppression in $B(\mathbf{k})$ to better than $2\%$ for the $68\%$ percentile across triangle configurations and scales $k \in [0.01, 20]\,h\,\text{Mpc}^{-1}$ (Burger et al., 23 Jun 2025).

6. Observational and Cosmological Implications

Baryonification techniques, especially when jointly constrained by multiwavelength observations (e.g., kSZ from ACT, gas fractions from eROSITA, X-COP cluster profiles), yield feedback models consistent with strong-feedback hydrodynamical simulations, often predicting suppressions at the $1$--$8\%$ level for $k = 0.3$--$1\,h\,\mathrm{Mpc}^{-1}$ and exceeding $20\%$ at $k=5\,h\,\mathrm{Mpc}^{-1}$ (Kovač et al., 10 Jul 2025). Models with increased flexibility (more free feedback parameters) can accommodate a wider range of feedback strengths and suppressions, but at the cost of increased degeneracy with cosmological parameters (e.g., shifting $S_8$ in weak lensing analyses) (Bigwood et al., 9 Apr 2024).

Component-wise baryonification calibrated to multi-tracer data produces models that self-consistently describe gas and pressure profiles across halos, providing agreement with cluster-scale X-ray and SZ profiles and supporting joint cosmological–astrophysical inference. Such techniques enable robust marginalization over baryonic uncertainties, critical for precision weak lensing, galaxy clustering, and CMB lensing analyses in current and next-generation cosmological surveys.

7. Limitations, Challenges, and Future Directions

Principal limitations include (i) the empirical calibration of analytic profiles (tied to simulations or selected observations); (ii) uncertainty in the universality and redshift evolution of feedback parameters; (iii) possible breakdown near or within halos with complex dynamical histories or extreme AGN feedback not well captured by analytic forms; and (iv) subtleties in map-level corrections on highly non-linear/small scales, where hydrodynamical simulation resolution is itself limited.

Baryonification techniques, especially when combined with cosmology rescaling, provide a computationally efficient path to robust large-scale structure modeling, enabling joint parameter estimation for cosmology and physics of feedback. Continued refinement of thermodynamic modeling, higher-order statistics, and calibration against expanding simulation and observational datasets is expected. The development of public pipelines (e.g., BaryonForge) and analytic/symbolic regression emulators further democratizes the application and cross-comparison of baryonification approaches, placing them at the core of contemporary and future cosmological analyses.