EFT-Based Field-Level Inference

• EFT-Based Field-Level Inference is a framework that uses symmetry-protected bias expansions and forward models to extract maximal information from high-dimensional cosmological and field data.
• It employs explicit UV regularization, controlled bias expansions, and stochastic modeling to outperform traditional summary-statistic methods with sub-percent parameter accuracy.
• Its Bayesian likelihood formulation combined with advanced sampling techniques ensures robust and systematic inference across cosmology, collider physics, and lattice gauge theory.

Effective Field Theory (EFT)-Based Field-Level Inference refers to the methodology of inferring cosmological or quantum field theory parameters directly from high-dimensional field data (e.g., galaxy density fields, lattice field configurations) using forward models and likelihoods built from Effective Field Theory principles. This paradigm integrates symmetry-protected information, robust error control, and explicit bias modeling into Bayesian field-level inference architectures across cosmology, collider physics, and lattice gauge theory. Distinctive features include explicit UV regularization, controlled bias expansions, stochasticity modeling, and fully field-level, non-summarizing likelihoods, rendering it a systematically improvable, maximally informative alternative to summary-statistic-based constraints.

1. Foundations of EFT-Based Field-Level Inference

The core motivation of EFT-based field-level inference is to extract maximal physical information from the complete high-dimensional tracer or field data—such as the three-dimensional galaxy number density or detector-level particle four-vectors—by constructing a controlled path from first-principle initial conditions to observations, with all relevant physical effects encoded in the forward model. The field-level likelihood is constructed on the basis of renormalized bias expansions, explicit stochastic modeling, and rigorous handling of non-linearities, with the following key steps:

• Forward Modeling: Initial (often Gaussian) field realizations are evolved through physical dynamics (e.g., Lagrangian or Eulerian perturbation theory, or MC event generation).
• EFT Expansion: The observable field (e.g., $\delta_g$ for galaxies) is expanded in all operators allowed by symmetries and locality, with bias coefficients determined by the large-scale environment.
• Stochastic Modeling: Residuals from deterministic modeling are treated with additive or multiplicative stochastic fields, whose covariance is analytically tractable at low $k$ and can be extended perturbatively.
• Regularization: All operators and likelihoods are constructed with explicit high-$k$ (ultraviolet) cutoffs. Regularization absorbs divergent loop corrections and guarantees the perturbative control of the expansion.
• Likelihood: The field-level likelihood is a (modified) Gaussian in the residuals, with full incorporation of cutoff-scale effects and analytic marginalization (where feasible) over nuisance parameters.
• Bayesian Inference: Inference is performed on the joint posterior for initial fields, bias and noise parameters, and cosmological or EFT coefficients, often requiring efficient high-dimensional sampling techniques (e.g., HMC, block-Gibbs, microcanonical samplers).

This approach leverages the symmetry protection and flexibility of EFT to ensure robustness to unknown small-scale dynamics and observational systematics (Elsner et al., 2019, Schmidt et al., 2018, Schmidt et al., 2020, Nguyen et al., 5 Mar 2024, Ivanov et al., 2 Dec 2024, Ivanov, 10 Mar 2025, Mohrman, 2023, Appelquist et al., 2017).

2. Mathematical Structure and Forward Model Construction

The mathematical formalism involves constructing a forward model that maps initial conditions (often high-dimensional latent fields) and physical parameters to predicted density or observable fields. Its essential ingredients are:

• EFT Bias Expansion: For galaxy clustering and related cosmological applications, the observable tracer density is written (in real or Fourier space) as:

$$\delta_g(x) = b_1\,\delta(x) + \frac{b_2}{2}[\delta^2](x) + b_{\mathcal G_2}[\mathcal G_2](x) + b_{\Gamma_3}[\Gamma_3](x) - b_{\nabla^2\delta}\nabla^2\delta(x) + \epsilon(x)$$

where the operators include quadratic and cubic terms, tidal/shear components, higher-derivative contributions, and stochastic noise (Sullivan et al., 6 May 2025, Ivanov et al., 2 Dec 2024).

• Redshift-Space Distortions (RSD) and Fingers of God (FoG): Mapping into redshift-space is performed via non-linear shifts and velocity expansions, leading to additional counterterms (e.g., $c_{\mu^2}\mu^2k^2\delta_1$, $b_4\mu^4k^4\delta_1$) capturing non-perturbative velocities.
• Stochastic Covariance: The noise model is consistently constructed, e.g.,

$$P_{\rm err}(k,\mu) = \frac{1}{\bar n}\left[1 + \alpha_0 + \alpha_1 \frac{k^2}{k_S^2} + \alpha_2\,\mu^2\,\frac{k^2}{k_S^2}\right]$$

with parameters derived from simulations or physically motivated analytic predictions (Ivanov, 10 Mar 2025, Sullivan et al., 6 May 2025, Ivanov et al., 2 Dec 2024).

• Explicit UV Cutoff: All fields and operators are sharp-$k$ filtered or otherwise cutoff, absorbing UV sensitivity into the counterterms.

For lattice and collider EFTs, analogous principles apply: for instance, SMEFT analyses at detector-level employ a quadratic parameterization of Monte Carlo weights in Wilson coefficients, which enter the event-level forward simulation and are mapped to (binned) observable yields for inference (Mohrman, 2023).

3. Likelihood Specification and Bayesian Framework

The generic field-level likelihood in the EFT approach is Gaussian (or corrected Gaussian) in the difference between observed and forward-modeled fields:

$$\ln \mathcal L = -\frac{1}{2}\sum_{k,\mu} \left\{ \ln[2\pi P_{\rm err}(k, \mu)] + \frac{|\delta_g^{\rm obs}(k, \mu) - \delta_g^{\rm EFT}(k, \mu; \theta)|^2}{P_{\rm err}(k, \mu)} \right\}$$

with $P_{\rm err}$ containing the full stochastic covariance structure. Corrections for cross-correlated noise components and non-Gaussianities can be systematically included via functional expansions and higher-point moments (Cabass et al., 2019).

The joint posterior for all parameters and initial conditions,

$$P(\theta, \textrm{IC} \mid \textrm{data}) \propto \mathcal L(\textrm{data} \mid \theta, \textrm{IC}) P(\theta) P(\textrm{IC}),$$

is sampled using a combination of analytic marginalization (available when the likelihood is linear in bias/nuisance parameters), and advanced Monte Carlo or microcanonical methods (e.g., Hamiltonian or Langevin MC) for the high-dimensional initial condition space (Kostić et al., 2022, Nguyen et al., 5 Mar 2024, Chen et al., 18 Aug 2025, Mohrman, 2023).

4. Performance, Convergence, and Validation

EFT-based field-level inference schemes are validated by demonstrating:

• Convergence of Parameter Estimation: Systematic offsets in recovered cosmological parameters (e.g., $\sigma_8$, BAO scale $r_s$, growth rate $f$) decrease to sub-percent levels as the expansion order increases and the cutoff scale is maintained within the perturbative regime (Schmidt et al., 2020, Babić et al., 1 Jul 2024, Schmidt, 2020).
• Superiority over Summary Statistics: Bayesian field-level inference outperforms summary-statistics-based pipelines (e.g., $P+B$ analyses) by factors of $1.2$–$5$ in constraint tightness on key parameters for the same data vectors and $k_{\max}$, by accessing higher-order and configuration-dependent information encoded in the field realization.
• Controlled Theoretical Error: Theoretical uncertainties are dominated by order truncation ($\sim(k/k_{\rm NL})^{n}$), missing higher-order operators, and forward-model limitations, all of which can be systematically reduced or estimated quantitatively.
• Consistency under Model Misspecification: Robustness is established by generating synthetic data at higher resolution/model complexity ($\Lambda_0 > \Lambda$) and validating unbiased recovery of parameters when analyzed at lower $\Lambda$ with the matched field-level likelihood (Kostić et al., 2022).
• Cosmic Variance Cancellation: Paired simulations—sharing initial phases but with distinct halo or galaxy assignments—allow for measurement of transfer functions with $\ll1\%$ errors, directly revealing the deterministic, stochastic, and convolutional structure of the field-level biasing (Sullivan et al., 6 May 2025, Ivanov et al., 2 Dec 2024).

5. Simulation-Based and Analytic Priors for EFT Parameters

A key innovation is the construction of simulation-based (and analytic) priors for the EFT parameter vector, derived from populations of halo catalogs under varying HOD, HMQ, or decorated HOD prescriptions:

• Simulation-Based Priors: High-fidelity mock catalogs yield empirical multivariate densities $P(\vec{b}_{\rm EFT})$ for bias and stochastic parameters, improving inference efficiency by replacing broad, ineffective priors with realistic distributions constrained by halo formation physics (Ivanov et al., 2 Dec 2024).
• Analytic Priors: When the dependence of EFT parameters on halo/galaxy formation can be modeled semi-analytically (e.g., via peak-background split), complete analytic simulation-based priors are constructed as mass- or HOD-parameter integrals, with stochastic components calibrated empirically (Ivanov, 10 Mar 2025).
• Cosmology–Halo Occupation Degeneracy: Variations in cosmological parameters (e.g., $\sigma_8$) can be almost entirely absorbed by corresponding reparametrizations of HOD parameters, justifying the robust cosmology-independence of such priors over a wide range of scenarios.

For collider and lattice applications, the analogous step is to use templates, reweighting, or analytic moment formulas to rapidly generate theory predictions as a function of the EFT Wilson coefficients in the parameter domain of interest (Mohrman, 2023, Peng et al., 16 Jan 2025, Appelquist et al., 2017).

6. Applications across Physical Domains

Large-Scale Structure and Cosmology

• Cosmological Parameter Inference: Constraints on $\sigma_8$, $r_s$, the growth rate $f$, and bias coefficients are consistently derived from full 3D field data using Lagrangian forward models (e.g., LEFTfield) with systematic UV regularization and analytic bias marginalization (Nguyen et al., 5 Mar 2024, Stadler et al., 2023, Babić et al., 1 Jul 2024, Sullivan et al., 6 May 2025).
• BAO Scale Determination: EFT-based field-level inference constrains the BAO dilation parameter with $30$–$50\%$ smaller uncertainty than standard post-reconstruction pipelines, by utilizing voxel-level, non-Gaussian information inaccessible to power-spectrum-based approaches, especially at higher $k$ (Babić et al., 1 Jul 2024, Babić et al., 19 May 2025).
• 21-cm Cosmology: The same formalism, augmented with differentiable forward models and gradient-based samplers, enables recovery and inference of modes lost to foregrounds in contaminated 21-cm observations, outperforming summary-statistics frameworks for bias and initial-condition recovery (Chen et al., 18 Aug 2025).

Field Theory and Collider Physics

• SMEFT Detector-Level Inference: Event-level forward modeling in the SMEFT permits simultaneous marginalization over nuisance and signal Wilson coefficients, directly from binned event yields and detector-level observables, with rigorous validation via reweighting and closure tests (Mohrman, 2023).
• Positivity/Unitarity Bounds on EFTs: Loop-level null constraints, implemented as affine and semidefinite constraints on spectral moments, yield parameter islands for Wilson coefficients consistent with unitarity and causality, using a combination of analytic moment techniques and modern semidefinite programming (Peng et al., 16 Jan 2025).

Lattice Gauge Theory

• Dilaton EFT and Lattice Data: Field-level inference employs dilaton–pion EFT Lagrangians, connecting lattice observables to low-energy constants via analytic scaling and loop-corrected formulas, enabling chiral and conformal regime extrapolations with systematic error characterization (Appelquist et al., 2017).

7. Limitations, Open Questions, and Future Prospects

While EFT-based field-level inference achieves major improvements in extracting physical constraints from high-dimensional data, several frontiers and limitations remain:

• Redshift-Space and Observational Effects: Realistic, survey-masked, and redshift-space forward modeling raises algorithmic and modeling complexity, demanding new efficient samplers and window-function treatments (Schmidt, 2020, Stadler et al., 2023).
• High-$k$ Expansion Domain: The strict validity of expansion and operator truncation is limited by UV non-perturbativity; pushing $k_{\max}$ higher requires inclusion of additional operators and more accurate forward models.
• Joint Cosmological and Astrophysical Inference: Marginalization over baryonic effects, assembly bias, and non-standard cosmologies (e.g., beyond-$\Lambda$CDM) is tractable in the current framework, but requires extended calibration and refinement of simulation-based priors (Ivanov, 10 Mar 2025, Ivanov et al., 2 Dec 2024).
• Sampling Scalability: High-dimensional initial condition and parameter space exploration, especially at high resolution, imposes severe computational demands; ongoing advances include mini-batch HMC, variational flows, and other likelihood-free inference schemes (Nguyen et al., 5 Mar 2024, Ivanov et al., 2 Dec 2024).
• Cross-Domain Generalization: There is direct evidence for the applicability of the field-level EFT perspective to lattice field theory, collider SMEFT, and beyond, but systematic translation of error control and inference methods across these domains is an active area of research (Appelquist et al., 2017, Mohrman, 2023, Peng et al., 16 Jan 2025).

Overall, EFT-based field-level inference now forms the backbone of next-generation cosmological, collider, and lattice analyses, enabling model-agnostic, maximally informative, and systematically controlled extraction of physical parameters from complex, high-dimensional field data.