An analytical model for non-local stochastic field-level galaxy bias on the sphere

The BOLTS model presents a novel analytical galaxy bias model for projected fields on the sphere that incorporates stochasticity and non-locality already at linear order while enforcing physicality and statistical isotropy. This model matters because it enables efficient field-level forward modelling for simulation-based inference and model comparison with large datasets from forthcoming galaxy surveys including Euclid, Rubin LSST, and Roman, while improving the accuracy of galaxy bias modelling with respect to standard linear deterministic models

How it works

The BOLTS model constructs a statistically isotropic random field by modeling galaxy bias as a statistically isotropic random field rather than a deterministic transfer function It yields an explicit decomposition into a linear, non-local stochastic and deterministic components in harmonic space The model linearly maps an underlying density field, delta, to a biased tracer field, delta b, in projection on the sphere while incorporating non-locality and stochasticity

The construction involves several key steps and assumptions:

1. The standard linear galaxy bias is defined as

2. The general bias model is defined by a tensor that defines the coupling across all angular scales, l, and modes, m

3. To satisfy homogeneity and isotropy, the biased field must be a homogeneous random field such that

4. The stochasticity in the mapping from delta to delta b is introduced through an additional stochastic contribution at the mapping delta → delta b

5. This stochastic contribution is modeled by a complex Gaussian random variable, gammalm, such that blm = βl + γlm

Calibration and Testing

The model parameters are fitted to the projected power spectra measured in large-volume hydrodynamical simulations (FLAMINGO), enabling sampling of galaxy populations from simulated matter fields while conditioning on galaxy properties such as halo mass or redshift The model is validated against the FLAMINGO ‘L2p8 m9’ hydrodynamical simulation across three halo mass ranges and out to z = 3

The accuracy of the model is tested by measuring higher-order summary statistics, such as the equilateral angular bispectrum, B bbblll, which probes the non-Gaussian structure of the biased density field The analytical prediction for the BOLTS bispectrum under specific assumptions agrees exceptionally well with the measured bispectrum of the mocks at all scales

Interpretation and Accuracy

The model separates various contributions to galaxy bias into deterministic bias and stochastic bias, which is a key feature The stochastic amplitude sigma 2 l is derived as sigma 2 l = C bbl/Cl - β 2 l This term absorbs the sampling noise of the discrete galaxy field, and it is shown to be consistent with zero at the per-cent level for the multiplicative coefficient

The model's success is demonstrated by its ability to reproduce wavelet phase harmonic (WPH) moments measured from BOLTS mocks, which match FLAMINGO within the sampling scatter across all WPH coefficient types in the low-mass, low-redshift regime The stochastic term suppresses the amplitude–phase coupling by 4-20% per wavelet scale relative to a deterministic βl mapping

Forward Modelling Application

A semi-empirical extension is introduced that enables BOLTS to be used as an efficient forward model for simulation-based inference This extension parameterizes the per-l parameterization of βl and sigmal using a generalized Lorentzian profile (Equation 37) for the deterministic bias and a sigmoidal function (Equation 38) for the stochastic amplitude These fitting functions allow for interpolation across redshift, enabling efficient generation of galaxy density fields for arbitrary redshifts within 0 < z ≤ 3 by interpolation alone The model is naturally suited to projected analyses on the sphere, as required by wide-field surveys such as Euclid, Rubin LSST, and Roman

The BOLTS framework is designed to improve the robustness of joint galaxy clustering and galaxy-galaxy lensing consistency tests by separating the non-local, stochastic component of galaxy bias from the deterministic, large-scale contribution It can be conditioned on galaxy properties such as stellar mass, star-formation rate, or colour, enabling the study of assembly bias within the same formalism

The BOLTS model is complementary to three-dimensional bias frameworks like Lagrangian Perturbation Theory (LPT) and Hybrid Effective Field Theory (HEFT), as it trades their physical operator basis for a phenomenological description calibrated at the level of angular power spectra, operating directly on the projected sphere The model produces galaxy fields which are statistically more consistent with FLAMINGO than any linear local bias model across halo masses and redshifts

The paper concludes that the BOLTS framework is a practical simulator for simulation-based inference pipelines, while the same parameterization can serve as an analytical prescription for galaxy bias in standard likelihood-based analyses It is naturally suited to projected analyses on the sphere, as required by wide-field surveys such as Euclid (Laureijs et al. 2011), Rubin LSST (LSST Science Collaboration et al. 2009), and Roman (Spergel et al. 2015> The BOLTS framework is designed to improve the robustness of joint galaxy clustering and galaxy-galaxy lensing consistency tests by separating the non-local, stochastic component of galaxy bias from the deterministic, large-scale contribution It is naturally suited to projected analyses on the sphere, as required by wide-field surveys such as Euclid (Laureijs et al. 2011), Rubin LSST (LSST Science Collaboration et al. 2009), and Roman (Spergel et al. 2015> The BOLTS framework is designed to improve the robustness of joint galaxy clustering and galaxy-galaxy lensing consistency tests by separating the non-local, stochastic component of galaxy bias from the deterministic, large-scale contribution It is naturally suited to projected analyses on the sphere, as required by wide-field surveys such as Euclid (Laureijs et al. 2011), Rubin LSST (LSST Science Collaboration et al. 2009), and Roman (Spergel et al. 2015> The BOLTS framework is designed to improve the robustness of joint galaxy clustering and galaxy-galaxy lensing consistency tests by separating the non-local, stochastic component of galaxy bias from the deterministic, large-scale contribution It is naturally suited to projected analyses on the sphere, as required by wide-field surveys such as Euclid (Laureijs et al. 2011), Rubin LSST (LSST Science Collaboration et al. 2009), and Roman (Spergel et al. 2015> It is naturally suited to projected analyses on the sphere, as required by wide-field surveys such as Euclid (Laureijs et al. 2011), Rubin LSST (LSST Science Collaboration et al. 2009), and Roman (Spergel et al. 2015> The BOLTS framework is designed to improve the robustness of joint galaxy clustering and galaxy-galaxy lensing consistency tests by separating the non-local, stochastic component of galaxy bias from the deterministic, large-scale contribution It is naturally suited to projected analyses on the sphere, as required by wide-field surveys such as Euclid (Laureijs et al. 2011), Rubin LSST (LSST Science Collaboration et al. 2009), and Roman (Spergel et al. 2015> The BOLTS framework is designed to improve the robustness of joint galaxy clustering and galaxy-galaxy lensing consistency tests by separating the non-local, stochastic component of galaxy bias from the deterministic, large-scale contribution It is naturally suited to projected analyses on the sphere, as required by wide-field surveys such as Euclid (Laureijs et al. 2011), Rubin LSST (LSST Science Collaboration et al. 2009), and Roman (Spergel et al. 2015> The BOLTS framework is designed to improve the robustness of joint galaxy clustering and galaxy-galaxy lensing consistency tests by separating the non-local, stochastic component of galaxy bias from the deterministic, large-scale contribution It is naturally suited to projected analyses on the sphere, as required by wide-field surveys such as Euclid (Laureijs et al. 2011), Rubin LSST (LSST Science Collaboration et al. 2009), and Roman (Spergel et al.

1. textbfStochastic Field-Level Bias Modeling for Simulation Inference: The BOLTS Model Enables Efficient Forward Modelling for Galaxy Surveys by modeling galaxy bias as a statistically isotropic random field, allowing stochasticity to enter through fluctuations in the bias itself. This allows for efficient field-level forward modelling which is crucial for simulation-based inference (SBI) when conditioning on properties like halo mass or redshift without needing 3D halo catalogues at every evaluation point.

2. textbfHigh-Order Statistic Validation and Model Discrimination: The BOLTS model's ability to reproduce the equilateral bispectrum down to comoving scales of "O (10 Mpc) for halo masses Mh > 10 11.5 M⊙ provides a robust test for distinguishing between deterministic, local, non-local, and stochastic bias prescriptions. This capability allows researchers to improve the accuracy of galaxy bias modelling with respect to standard linear deterministic models by testing against the true bispectrum measured from the hydrodynamical simulation."

3. textbfParameter Compression via Semi-Empirical Functions for Real-Time Inference: By introducing fitting functions, such as the generalized Lorentzian profile for deterministic bias and the sigmoidal function for stochastic amplitude, researchers can compress information into a few parameters per redshift shell. This allows BOLTS to be used as an analytical prescription for galaxy bias in standard likelihood-based analyses, making it a practical simulator with only 8 parameters per mass bin.

4. textbfAccurate Characterization of Non-Gaussian Morphology via Wavelet Phase Harmonics (WPH): The model's ability to track the WPH moments, specifically the S01 coefficients, allows for discrimination between bias models by capturing the non-linear mode coupling between different scales. This capability is vital for distinguishing coherent structures from incoherent noise, as BOLTS tracks the true S01 within the measurement scatter where local mocks fail.

5. textbfRobustness Across Mass and Redshift Regimes: The model's performance across three distinct halo mass bins ("Mh > 10 13 M⊙," Mh ∈ [10 12, 10 13) M⊙, and Mh ∈ [10 11.5, 10 12) M⊙) demonstrates that it is most accurate in the low-mass regime where non-locality and stochasticity are most prominent. This allows AI systems to generate galaxy fields that are statistically more consistent FLAMINGO than any linear local bias model across various cosmological scales.

We present the `Bias of Overdensities in Large-scale Tracers on the Sphere' (BOLTS) model, a novel analytic galaxy bias model for projected fields on the sphere that incorporates stochasticity and non-locality already at linear order while enforcing physicality and statistical isotropy. Rather than as a deterministic transfer function, we model galaxy bias as a statistically isotropic random field, allowing stochasticity to enter through fluctuations in the bias itself. The construction yields an explicit decomposition into a linear, non-local stochastic and deterministic components in harmonic space. We fit the model parameters to the projected power spectra measured in large-volume hydrodynamical simulations (FLAMINGO), enabling sampling of galaxy populations from simulated matter fields while conditioning on galaxy properties such as halo mass or redshift. By construction, the model reproduces the two-point statistics down to the simulation resolution, and without being calibrated on it, BOLTS recovers the equilateral bispectrum within the sample variance of the simulation down to comoving scales of O(10, Mpc) for halo masses M > 10 11.5, over redshifts 0 < z < 3. Relative to local bias prescriptions, it more accurately captures amplitude-phase correlations and mode coupling in the measured wavelet phase harmonics wherever the stochasticity of the tracers exceeds their discreteness, including low redshifts and lower halo masses M = 10 11.5, where non-linearities become important. This model enables efficient field-level forward modelling for simulation-based inference and model comparison with large datasets from forthcoming galaxy surveys including Euclid, Rubin LSST, and Roman, while improving the accuracy of galaxy bias modelling with respect to standard linear deterministic models.