Galaxy Bias Expansion Framework

• Galaxy bias expansion is a systematic, perturbative approach that models galaxy overdensity as a sum of local and nonlocal operators derived from density and tidal fields.
• It employs renormalization and effective field theory techniques to absorb small-scale complexities and capture nonlinear gravitational evolution in clustering statistics.
• Applications of this framework enable robust predictions for galaxy clustering observables in ongoing and future large-scale structure surveys.

Galaxy bias expansion is the systematic, perturbative framework that describes the statistical relationship between the observed spatial distribution of galaxies (or other luminous tracers) and the underlying dark matter density field. The expansion is organized in terms of local and non-local operators constructed from the large-scale gravitational and dynamical fields (e.g., the matter density, tidal tensor, higher derivatives), with coefficients ("bias parameters") encapsulating how galaxy formation processes respond to the long-wavelength environment. This formalism absorbs the complexities of baryonic and feedback physics at a given epoch into a finite set of parameters, enabling robust predictions for galaxy clustering statistics essential for cosmological inference. Significant generalizations have been developed to account for non-local gravitational evolution, radiative-transfer effects, non-Gaussian initial conditions, relativistic corrections, and practical systematics. The current paradigm employs renormalization and effective field theory (EFT) techniques to control small-scale sensitivity and loop corrections, thereby systematically connecting the bias model to the observable galaxy correlation functions and power spectra relevant for ongoing and future large-scale structure surveys.

1. Theoretical Foundations of the Bias Expansion

The bias expansion is motivated by the observation that galaxies, as tracers of matter, are not unbiased samples of the underlying mass field but rather form preferentially in specific environments. Under the assumption that, on sufficiently large (quasi-linear) scales, all relevant physics can be parametrized by gravitational observables at a given point, the galaxy overdensity is written as a sum over operators $O_i$: $$\delta_g(x,\tau) = \sum_{O} b_O(\tau)\, [O](x,\tau) + \epsilon(x,\tau)$$ where $[O](x,\tau)$ represents a set of renormalized operators constructed from the density field, tidal field, and their derivatives, $b_O$ are deterministic bias parameters, and $\epsilon(x,\tau)$ encapsulates stochastic effects arising from small-scale physics (Desjacques et al., 2016).

The expansion proceeds in perturbative order: at linear order, $\delta_g = b_1\delta$; at quadratic, terms like $\delta^2$ and the quadratic tidal field $(K_{ij})^2$ enter; cubic and higher orders follow similarly. Operators involving spatial derivatives, such as $\nabla^2\delta$, represent nonlocality arising from the finite scale of galaxy formation—typically the Lagrangian radius or associated astrophysical scales.

The perturbative construction carefully respects the equivalence principle and general relativity on large scales, ensuring that no operator depending solely on the gravitational potential (without derivatives) appears; only quantities built from second derivatives (i.e., tidal fields and density) are allowed (Desjacques et al., 2016). The expansion is closed under renormalization, so any ultraviolet (UV) divergences arising in loop corrections are absorbed into the redefinition of the bias parameters (Assassi et al., 2015).

2. Nonlocality, Nonlinearity, and Gravitational Evolution

While galaxies may be formed with a local-in-density bias at their formation time—i.e., $$\delta_g^\ast = b_1^\ast \delta^\ast + \frac{b_2^\ast}{2}(\delta^\ast)^2 + ...$$—gravitational evolution necessarily induces nonlocal and nonlinear contributions. Even pure local formation bias is rendered nonlocal by nonlinear gravitational clustering, which generates terms involving the tidal field and other invariants constructed from the deformation tensor: $$\delta_g(x) = \sum_k b_k O^{\text{local}}_k(\delta(x)) + \sum_j \gamma_j \mathcal{G}_j(\Phi_v(x)) + ...$$ where the $\mathcal{G}_j$ are Galileon invariants (e.g., $$\mathcal{G}_2(\Phi_v) = (\nabla_i\nabla_j\Phi_v)^2 - (\nabla^2\Phi_v)^2$$) associated with the tidal and higher multipole fields (Chan et al., 2012). Nonlocal contributions are particularly important for highly biased tracers (such as luminous galaxies or massive halos), where simulations show strong departures from purely local bias models.

The inclusion of velocity bias (i.e., differences between galaxy and matter velocities) further introduces new contributions, including dipole terms at second order that break Galilean invariance, manifesting as additional nonlocal operators (Chan et al., 2012).

3. Renormalization, Operator Completeness, and Effective Field Theory

Ultraviolet divergences arise when evaluating composite operators (e.g., $\delta^2$) at the same spatial position—a problem addressed by renormalization (Assassi et al., 2015, Patrone et al., 2023). Operators are organized into a basis that is "closed under renormalization," meaning all divergences can be absorbed into the parameters of operators already present in the expansion. For example, the renormalized quadratic operator is

$$[\delta^2] = \delta^2 - \sigma^2(\Lambda) \left(1 + \frac{68}{21} \delta + ... \right)$$

where $\sigma^2(\Lambda)$ is the variance of the density field up to cutoff $\Lambda$ (Assassi et al., 2015). In the presence of primordial non-Gaussianity or modified gravity, new operators—such as modulating fields $\psi$ (for PNG) or scale-dependent terms—are systematically included.

The Effective Field Theory (EFT) of Large-Scale Structure formalism frames the bias expansion within a larger context, incorporating all operators allowed by symmetries, organizing corrections by powers of $k/k_*$ (where $k_*$ is a breakdown scale set by nonlinearities), and systematically treating higher-order and stochastic terms (Desjacques et al., 2016, Desjacques et al., 2018, Rubira et al., 2023). Renormalization group (RG) methods have been developed to describe the running of bias parameters with respect to the cutoff scale (Rubira et al., 2023), demonstrating full analogy with QFT $\beta$-functions and the resummation of large logarithms.

4. Extensions: Radiative Transfer, Primordial Non-Gaussianity, and Relativistic Effects

Standard bias expansions characterize the nonlocality scale by the Lagrangian radius of halos, $R(M_h)$. However, additional astrophysical effects can introduce new, much larger nonlocality scales. Notably, radiative-transfer (RT) effects from cosmological reionization inject a long-range scale—the mean free path of ionizing radiation $\lambda_\text{eff} \sim 50$–$100~h^{-1}$ Mpc—which gives rise to higher-derivative contributions scaling as $(k\lambda_\text{eff})^n$ (Cabass et al., 2018). The resummation of these RT-induced corrections is critical: attempting to truncate the series perturbatively would lead to a loss of predictivity even at BAO scales ($k \sim 0.1$–$0.2~h/$Mpc). Nonperturbative modeling (e.g., via resummed kernels $f_\text{ion}(k^2)$) is thus required to maintain the robustness of large-scale clustering as a cosmological probe.

Primordial non-Gaussianity (PNG) is incorporated into the bias expansion via additional modulation fields (e.g., $\psi$) that induce a scale-dependent bias $\Delta b(k) \sim k^{-\alpha}$ for various PNG shapes, and new operators such as $\psi\delta$, $\psi^{ij} s_{ij}$ for anisotropic PNG (Assassi et al., 2015). The equivalence principle imposes strong consistency relations, directly connecting the scale-dependent bias in the power spectrum to distinctive multipole moments (e.g., the bispectrum dipole and quadrupole), enabling robust null tests for the primordial origin of observed clustering signatures.

Relativistic corrections enter the bias expansion at second order, modifying the Newtonian energy constraint that relates metric perturbations and the density field (Umeh et al., 2019). However, after performing a local coordinate transformation (removing spurious gauge or mode-coupling), all general relativistic effects are encapsulated by a distortion of the local volume element, and no new bias parameters are required. Thus, GR corrections alter the mapping but do not increase model complexity at second order.

5. Extraction of Bias Parameters: Simulation, Observational Statistics, and Emulation

Bias parameters are measured through several techniques:

• Correlation Function and Power Spectrum Fitting: Bias parameters (e.g., $b_1$, $b_2$, $b_{s^2}$, etc.) are inferred by fitting the observed $P_{gg}(k)$ and bispectrum, using formulae such as

$$P_{gg}(k) = b_1^2 P_{mm}(k) + ... \quad B_{ggg}(k_1,k_2,k_3) = b_1^3 B_{mmm} + b_1^2 b_2 [...]$$

and incorporating shot noise, tidal, and higher-derivative terms (Desjacques et al., 2016, Eggemeier et al., 2018).

• Counts-in-Cells Moments: The distribution of galaxy counts in cells of varying sizes yields direct estimates of $b_1$, $b_2$, $b_3$ by comparing the measured moments (variance, skewness, kurtosis) of the galaxy density field to that of the underlying dark matter (Salvador et al., 2018).
• Separate-Universe Simulations: Running large $N$-body simulations with a perturbed background density provides direct "response" measures of the bias parameters, consistent with the peak-background split formalism (Desjacques et al., 2016).
• Hybrid Lagrangian Emulators: Recent approaches combine the perturbative bias expansion (usually in Lagrangian space) with simulation-extracted nonlinear displacement fields and machine-learning emulators to provide rapid and precise predictions of clustering observables (e.g., up to $k \sim 0.7~h/$Mpc with 1–2% error) across cosmological parameter space (Zhou et al., 5 Jun 2025). These models factor in baryonic, neutrino, and dynamical dark energy effects, and are validated against massive simulation suites.
• Field-Level and Forward Modeling: State-of-the-art analyses forward-model the entire galaxy field with sophisticated bias expansions and stochastic corrections, matching the observed field up to small scales necessary for Stage-IV survey constraining power (Collaboration et al., 2023).

6. Practical and Systematic Considerations

Systematic effects lateral to the bias expansion include:

• Redshift-Space Distortions and Selection Effects: The mapping from real to observed (redshift) space requires explicit inclusion of operators dependent on the line-of-sight velocity field and selection effects, leading to the introduction of up to $22$ bias and stochastic parameters at 1-loop order in the effective field theory description (Desjacques et al., 2018).
• Non-Poissonian Stochasticity: Deviations from ideal Poisson shot noise arise due to halo exclusion, satellite occupation, and other effects, introducing scale-dependent noise that must be accounted for with extra stochastic bias parameters and cross-terms (Nicola et al., 2023).
• Impact of Galaxy Formation Physics and Assembly Bias: Differences in galaxy formation modeling (e.g., comparing empirical HOD-based methods, semi-analytic modeling, and hydrodynamics) shift bias parameters and non-Poisson noise, particularly for quenched versus star-forming galaxies. Assembly bias induces systematic shifts in higher-order bias parameters, but these are quantifiable and lead to robust parameter priors across modeling approaches (Shiferaw et al., 9 Dec 2024).
• Radiative-transfer and Very Large Scales: New long-range nonlocality (e.g., from the mean free path of ionizing photons) can introduce scale-dependent deviations in galaxy power spectra that may mimic BAO or neutrino mass suppression (Cabass et al., 2018).

7. Impact, Limitations, and Future Directions

The galaxy bias expansion is a cornerstone of extracting cosmological information from galaxy surveys, allowing systematic inclusion of nonlinearities, nonlocality, stochasticity, primordial non-Gaussianity, and model-specific systematics. Recent advancements extend the expansion to high perturbative order (including two-loop renormalization for 29 deterministic operators), allow robust emulation for wide cosmological parameter spaces (including dark energy and neutrinos), and refine the connection between theory and observations using the principles of effective field theory and the renormalization group (Bakx et al., 18 Jul 2025, Zhou et al., 5 Jun 2025, Rubira et al., 2023).

Ongoing developments address:

• Improved modeling of selection effects, redshift-space distortions, relativistic corrections, and feedback-induced nonlocality.
• Extension of the bias expansion to accurately connect field-level statistics with galaxy formation models through emulators and machine learning.
• The creation of prior distributions for bias parameters informed by an ensemble of physical galaxy formation models, supporting high-precision cosmological parameter inference (Shiferaw et al., 9 Dec 2024).
• Investigation of the convergence properties of various expansion bases (cumulant, polynomial, or Gaussian models), with recent evidence suggesting a nearly Gaussian bias function for dark matter halos (Stücker et al., 3 May 2024, Stücker et al., 3 May 2024).

The robustness and generality of the galaxy bias expansion enable Stage-IV surveys such as Euclid, LSST, DESI, and CSST to exploit the full statistical power of high-resolution, large-volume clustering measurements for fundamental physics, contingent on the careful quantification and marginalization of bias parameter uncertainties.