BACCO Emulator for Cosmological Analysis

• The BACCO emulator is a neural-network framework that accurately predicts two-point clustering statistics of matter and galaxies using hybrid Lagrangian bias expansions.
• It combines high-resolution N-body simulations with cosmology rescaling and machine learning to achieve sub-percent precision across an 8-dimensional cosmological parameter space.
• The emulator integrates theoretical models with rapid neural predictions to support full-shape power spectrum analyses in modern galaxy surveys, including baryonic and redshift-space effects.

The BACCO emulator is a neural-network-based framework for rapidly predicting two-point clustering statistics—most notably the power spectrum—of matter and biased tracers (e.g., galaxies) in both real and redshift space, as a function of cosmology, redshift, galaxy bias, and astrophysical parameters. Designed by the BACCO Simulation Project, it combines the flexibility of perturbative Lagrangian bias expansions with the accuracy of high-resolution N-body dynamics, utilizing cosmology-rescaling, machine learning, and hybridization of theory and simulation. BACCO achieves high-precision predictions across an extended cosmological parameter space, including massive neutrinos and evolving dark energy, and is validated for scales extending well into the quasi-linear and mildly nonlinear regime relevant to modern galaxy surveys (Ibáñez et al., 2024, Pellejero-Ibanez et al., 2022, Zennaro et al., 2021).

1. Theoretical Foundations: Hybrid Lagrangian Bias Expansion

The BACCO emulator models the galaxy-matter connection via a second-order Lagrangian bias expansion, including a higher-derivative term, up to five basis fields:

• $\delta_L$ (linear overdensity)
• $\delta_L^2$ (local quadratic)
• $s^2$ (tidal shear, $$s_{ij} = \partial_i\partial_j\phi_L - \tfrac13\delta_{ij}\delta_L$$)
• $\nabla^2\delta_L$ (higher-derivative)
• $1$ (trivial)

The tracer overdensity in Lagrangian space is given by

$$\delta_g(q) = b_1\,\delta_L(q) + b_2\,(\delta_L^2(q) - \langle\delta_L^2\rangle) + b_{s^2}(s^2(q) - \langle s^2\rangle) + b_{\nabla^2}\nabla^2\delta_L(q)$$

This expansion is then advected to Eulerian space using the non-linear displacement field $\psi(q)$ measured in N-body simulations:

$$1+\delta_g(x) = \int d^3q\,F[\delta_L(q)]\,\delta_D[x - q - \psi(q)]$$

Redshift-space distortions (RSD), including the effects of peculiar velocities, are incorporated by shifting the line-of-sight coordinate of each element by its velocity component:

$$\psi(q) \to \psi(q) + \frac{v(x)\cdot\hat{z}}{aH}\hat{z}$$

A phenomenological Fingers-of-God (FoG) model convolves the redshift-space density along the line of sight with a mixture of a Dirac delta (centrals) and an exponential (satellites), parameterized by the satellite fraction $\delta_L^2$0 and velocity dispersion scale $\delta_L^2$1 (Ibáñez et al., 2024, Pellejero-Ibanez et al., 2022, Collaboration et al., 28 Jan 2026).

2. Simulation Suite and Cosmology-Rescaling

The emulator is trained on a simulation suite consisting of high-resolution, large-volume gravity-only N-body runs (e.g., $\delta_L^2$2 particles in $\delta_L^2$3Mpc boxes, $\delta_L^2$4), employing fixed-and-paired initial conditions to minimize sample variance (Angulo et al., 2020, Pellejero-Ibanez et al., 2022). Rather than rerunning simulations for every cosmological model, BACCO utilizes the Angulo & White cosmology-rescaling algorithm, extended to massive neutrinos and time-varying dark energy, to remap particle positions, velocities, and halo properties to thousands of target cosmologies. This results in coverage of an 8-dimensional cosmological parameter space:

$\delta_L^2$5

spanning $\delta_L^2$6 the 68% Planck uncertainties (Ibáñez et al., 2024, Angulo et al., 2020). Cross-spectra of Lagrangian basis fields—including all relevant bias and RSD terms—are measured in real and redshift space at each rescaled cosmology.

3. Neural-Network Emulator Design and Workflow

Emulation is performed by feed-forward neural networks. The general structure is:

• Input: 8 cosmological parameters (optionally 7 baryonic parameters for baryonification).
• Architecture: 2 hidden layers (200–400 neurons with ReLU activation), 1 output layer (principal-component amplitudes or cross-spectra).
• Output: Principal component (PC) amplitudes that reconstruct the desired power spectra (monopole $\delta_L^2$7, quadrupole $\delta_L^2$8, hexadecapole $\delta_L^2$9, and cross-spectra $s^2$0).

The emulator predicts the nonlinear galaxy power spectrum multipoles in redshift space as:

1. Neural network predicts the set of 15 cross-spectra $s^2$1 as functions of cosmology.
2. These templates are linearly combined with Lagrangian bias weights to yield $s^2$2.
3. Legendre expansion obtains multipoles $s^2$3 ($s^2$4).
4. FoG convolution and shot-noise corrections are applied.
5. Alcock–Paczynski scaling is performed, and the model is evaluated in likelihood analyses (e.g., MultiNest, emcee) with typical evaluation times $s^2$5 s (BOSS) or $s^2$610 ms (Euclid-like applications) (Ibáñez et al., 2024, Pellejero-Ibanez et al., 2022, Collaboration et al., 28 Jan 2026).

Emulator accuracy is at the sub-percent level for the monopole ($s^2$70.5%), percent for the quadrupole, and at the 10% level for the hexadecapole, all below current BOSS data errors (Ibáñez et al., 2024).

4. Extensions: Baryonic Effects, Real-Space Tracers, Linear/Nonlinear Matter, and Modular Stacking

BACCO provides emulators for a variety of related observables:

• Baryonic suppression: A dedicated "baryonification" emulator, trained on gravity-only N-body runs augmented with a baryonic algorithm, delivers $s^2$8–$s^2$9 precision on the ratio $$s_{ij} = \partial_i\partial_j\phi_L - \tfrac13\delta_{ij}\delta_L$$0 across $$s_{ij} = \partial_i\partial_j\phi_L - \tfrac13\delta_{ij}\delta_L$$1–$$s_{ij} = \partial_i\partial_j\phi_L - \tfrac13\delta_{ij}\delta_L$$2 and $$s_{ij} = \partial_i\partial_j\phi_L - \tfrac13\delta_{ij}\delta_L$$3–$$s_{ij} = \partial_i\partial_j\phi_L - \tfrac13\delta_{ij}\delta_L$$4, with a 15-dimensional parameter input (8 cosmological, 7 baryonic) (Aricò et al., 2020).
• Real-space biased tracers: A similar emulator predicts $$s_{ij} = \partial_i\partial_j\phi_L - \tfrac13\delta_{ij}\delta_L$$5 and $$s_{ij} = \partial_i\partial_j\phi_L - \tfrac13\delta_{ij}\delta_L$$6 for galaxies and matter using 15 cross-spectra and Lagrangian bias parameters, achieving $$s_{ij} = \partial_i\partial_j\phi_L - \tfrac13\delta_{ij}\delta_L$$7 accuracy up to $$s_{ij} = \partial_i\partial_j\phi_L - \tfrac13\delta_{ij}\delta_L$$8 at $$s_{ij} = \partial_i\partial_j\phi_L - \tfrac13\delta_{ij}\delta_L$$9 (Zennaro et al., 2021).
• Linear/nonlinear matter spectra: Additional neural network emulators for $\nabla^2\delta_L$0 and nonlinear boost factors are included, covering identical parameter volumes, with $\nabla^2\delta_L$1 accuracy for $\nabla^2\delta_L$2 and sub-100 ms evaluation times (Angulo et al., 2020, Aricò et al., 2021).
• Modular stacking: All BACCO emulators share conventions, interface structure, and k-grid, permitting seamless stacking (e.g., $\nabla^2\delta_L$3 add nonlinear $\nabla^2\delta_L$4 boost $\nabla^2\delta_L$5 add baryon correction $\nabla^2\delta_L$6 model galaxy bias) (Aricò et al., 2021).

5. Validation, Performance Metrics, and Applications

Validation employs mock catalogs (e.g., Nseries, BOSS-like HOD, SHAMe mocks, Euclid Flagship) and comparisons to analytic or other emulator-based predictions:

• Accuracy: Guaranteed for $\nabla^2\delta_L$7 in BOSS analyses, $\nabla^2\delta_L$8 for high-density and mildly nonlinear tests, errors below the data statistical errors in all relevant regimes (Ibáñez et al., 2024, Pellejero-Ibanez et al., 2022).
• Parameter recovery: Posteriors for $\nabla^2\delta_L$9 and nuisance parameters are unbiased, with stable errors across tested $1$0; $1$1 constraints from BOSS alone exhibit a $1$2–$1$3 tension with Planck, consistent with lower $1$4 from lensing analyses, but within overall compatibility (Ibáñez et al., 2024).
• Comparison to PT models: Outperforms Effective Field Theory (EFT) at small scales ($1$5) and matches or surpasses alternatives such as VDG$1$6 up to $1$7 for unbiasedness, although VDG$1$8 provides marginally tighter constraints at higher $1$9 in some configurations (Collaboration et al., 28 Jan 2026).

6. Limitations and Future Development

Several limitations are noted in the literature:

• Parameter volume: Performance degrades near the edges of training coverage; extension to new physics (e.g., non-flat cosmologies, exotic dark sectors) requires retraining with new simulation anchors (Pellejero-Ibanez et al., 2022).
• Small-scale reliability: The Lagrangian hybrid expansion may break at $$\delta_g(q) = b_1\,\delta_L(q) + b_2\,(\delta_L^2(q) - \langle\delta_L^2\rangle) + b_{s^2}(s^2(q) - \langle s^2\rangle) + b_{\nabla^2}\nabla^2\delta_L(q)$$0 or for highly sparse samples where shot-noise or unmodeled physics dominate (Pellejero-Ibanez et al., 2022).
• Residual systematics: At high redshift and in the highest-precision hexadecapole, interpolation artifacts may appear; active work focuses on increasing k-bin resolution, anchor simulation density, or modifying neural architectures (Collaboration et al., 28 Jan 2026).
• Stochastic terms: Shot noise is treated phenomenologically; further modeling of stochasticity or higher-order bias fields may be needed for next-generation data (Pellejero-Ibanez et al., 2022).

7. Software Distribution and Community Usage

The BACCO emulator suite is public and documented, with Python modules, Jupyter notebook examples, command-line interfaces, and support for embedding in standard inference workflows (emcee, MultiNest). Installation is via pip (pip install baccoemu or pip install bacco_emulator for the original version), with code repositories at http://www.dipc.org/bacco (Angulo et al., 2020, Aricò et al., 2021, Aricò et al., 2020, Pellejero-Ibanez et al., 2022). All necessary neural network weights, principal component bases, and reference cosmologies are included. Users are advised to respect the emulator's validated parameter bounds and to combine modules as needed for nonlinear, baryonic, and tracer-specific effects.

The BACCO emulator is used as a workhorse in full-shape power spectrum analyses of state-of-the-art galaxy surveys (e.g., BOSS, Euclid), providing a robust, rapid, and extensible solution for extracting precision cosmology from large-scale structure data (Ibáñez et al., 2024, Collaboration et al., 28 Jan 2026).