LEFTfield: Lagrangian EFT Cosmological Inference

• LEFTfield is a computational framework for cosmology that models full field-level Bayesian inference using Lagrangian perturbation theory and effective field theory.
• It employs systematic bias expansions and counterterms to robustly capture nonlinear gravitational evolution and mitigate theoretical uncertainties.
• The approach significantly improves BAO and σ8 constraints compared to traditional pipeline methods by leveraging non-Gaussian, higher-order information from the full density field.

LEFTfield

LEFTfield (“Lagrangian EFT-based forward model at the field level”) is a computational framework and methodology for forward modeling cosmological large-scale structure, specifically the nonlinear evolution and clustering of galaxies and halos, within the Effective Field Theory of Large Scale Structure (EFTofLSS). Unlike traditional approaches that constrain cosmological parameters using summary statistics such as the power spectrum or the bispectrum, LEFTfield enables full field-level Bayesian inference by explicitly modeling the mapping from Gaussian initial conditions through nonlinear gravitational evolution, biasing, and stochasticity to observable tracer fields. By incorporating a systematic Lagrangian bias expansion and marginalizing over both initial conditions and nuisance parameters, LEFTfield robustly accounts for nonlinear mode coupling, non-Gaussian information, and theoretical uncertainties, providing significant improvements (factors of 1.2–1.4 for BAO, 3–5 for σ₈) in parameter constraints over standard pipelines (Babić et al., 19 May 2025, Babić et al., 1 Jul 2024, Nguyen et al., 5 Mar 2024).

1. Theoretical Foundation: Lagrangian-space EFT and Perturbative Forward Modeling

LEFTfield is grounded in the Lagrangian-space EFTofLSS, wherein the density and velocity fields are separated into “long modes” (modeled perturbatively up to a cutoff Λ) and “short modes” (integrated out, with effects encoded via counterterms, bias operators, and stochastic noise) (Porto et al., 2013). Unlike Eulerian approaches that require expansion in both small-scale nonlinearity and the large IR bulk displacement (which is $\mathcal{O}(1)$ at the BAO scale), LEFTfield leverages the IR-resummed structure of Lagrangian perturbation theory (LPT). Displacements are kept exponentiated, resumming the effect of long-wavelength flows and rendering BAO predictions more robust to IR systematics.

The evolved positions of tracers are computed as:

$$\mathbf{x}(\mathbf{q}) = \mathbf{q} + \Psi(\mathbf{q}; \delta_{\rm in}),$$

where $\Psi$ is constructed via LPT (typically up to 2LPT–3LPT), with additional EFT counterterms to absorb UV sensitivity. For instance, the inclusion of a “sound-speed” counterterm,

$$\Psi_{\rm EFT}(k) = - i\,c_s^2 k\,\delta_{\rm in}(k),$$

ensures consistent renormalization of small-scale effects (Babić et al., 19 May 2025).

Mass and biased tracer fields are then assigned to Eulerian space via displacement of the Lagrangian (initial) grid, with the final deterministic tracer density constructed from a local expansion in scalar bias operators, e.g.,

$$\delta_g(\mathbf{x}) = b_1\,\delta(\mathbf{x}) + \tfrac{b_2}{2} \delta^2(\mathbf{x}) + b_{K^2} K^2(\mathbf{x}) + b_{\nabla^2}\nabla^2\delta(\mathbf{x}) + \cdots$$

and noise modeled as a Gaussian field with a scale-dependent power spectrum.

2. Numerical Implementation and Best-Practice Guidelines

LEFTfield employs a multi-grid pipeline to maximize accuracy-to-cost ratio in the field-level forward model, with a sharp-$k$ cutoff $\Lambda$ to include only perturbatively controlled modes (Stadler et al., 17 Sep 2024). The implementation involves:

• Gridding Gaussian initial conditions at resolution $N_{\rm G,in} \approx \lceil \Lambda L / \pi \rceil$,
• Computing LPT displacements on a Fourier-padded grid ($N_{\rm G,fwd}$),
• Performing coordinate transformation $q\to x$ (Lagrangian to Eulerian) on a finer grid $N_{\rm G,Eul} \geq 3N_{\rm G,in}/2$ (using the 3/2 rule to minimize aliasing),
• Assigning mass and bias fields to the Eulerian grid using a non-uniform FFT (NUFFT) for high-precision density mapping.

Best-practice recommendations ensure that all numerical errors (from truncation, aliasing, and FFT-size mismatch) remain subdominant to model errors from finite truncation of LPT or bias expansion. For forward modeling up to $k_{\max}\leq0.2\,h/$Mpc, typical execution times are $\sim$0.2–3 s per $8(h^{-1}\mathrm{Gpc})^3$ box on current multi-core CPUs (Stadler et al., 17 Sep 2024).

3. Bayesian Field-Level Likelihood and Posterior Inference

In LEFTfield analyses, the full joint posterior over initial conditions, bias and EFT parameters, noise parameters, cosmological parameters, and the key parameter of interest (e.g., BAO dilation $\alpha$ or amplitude $\sigma_8$) is sampled numerically:

$$P(\delta_{\rm in}, \theta_{\rm bias}, \theta_{\rm noise}, \alpha \mid d) \propto P(d \mid \delta_{\rm in}, \theta_{\rm bias}, \theta_{\rm noise}, \alpha)\, P(\delta_{\rm in})\,P(\theta_{\rm bias})\,P(\theta_{\rm noise})\,P(\alpha).$$

The likelihood is Gaussian in the density field residuals:

$$-2\ln P(d | ...) = \sum_{\rm voxels} [d(x) - F(\delta_{\rm in}, \theta_{\rm bias}, \alpha)(x)]^\top N^{-1} [d(x) - F(...)(x)] + \ln|2\pi N|,$$

where $N$ is the pixel/noise covariance matrix, often assumed diagonal for simplicity. The forward operator $F$ predicts the observable tracer field for each sample.

Sampling is performed via Hamiltonian Monte Carlo for the high-dimensional initial conditions and slice or univariate samplers for low-dimensional parameter sets. Analytical marginalization is applied for bias parameters where possible. Convergence is monitored via autocorrelation metrics and standard diagnostics (effective sample size, Gelman–Rubin statistic).

4. Parameter Constraints and Gains over Summary-Statistic-Based Pipelines

LEFTfield field-level inference yields substantial gains in constraining power over traditional pipelines that rely solely on two-point (or low-order $n$-point) statistics. For BAO inference, LEFTfield tightens constraints on the sound horizon scale $r_s$ by a factor of $1.2$–$1.4$ relative to standard Eisenstein et al. (2007) reconstruction pipelines, reducing the uncertainty from the canonical $1\%$ to below $0.7$–$0.8 \%$ on current scales of interest, and up to $50\%$ improvement for heavily nonlinear or nonlinearly biased tracers (Babić et al., 19 May 2025, Babić et al., 1 Jul 2024).

For amplitude inference (e.g., $\sigma_8$), field-level LEFTfield analysis achieves improvement factors of $3$–$5$ over simulation-based inference using the power spectrum and bispectrum, when run on identical data and scales (e.g., reducing uncertainty on $\sigma_8$ from $20\%$ to $5.7\%$ at $k_{\max}=0.10\,h/$Mpc; see Table).

Statistic	$\sigma_\alpha$ (FBI)	$\sigma_\alpha$ (SBI $P,B$)	Improvement Factor
$k_{\max}=0.10$	$5.7\%$	$20.0\%$	$3.5$
$k_{\max}=0.12$	$3.3\%$	$17.0\%$	$5.2$

The improvement derives from the left-over non-Gaussian, higher-order, and off-diagonal information encoded in the full density field, which is discarded by summary statistic approaches, and from the optimal marginalization over EFT and bias parameters (Nguyen et al., 5 Mar 2024, Babić et al., 1 Jul 2024).

5. Extensions: Redshift-space, Velocity Bias, and Survey Complexity

Recent developments have extended LEFTfield to forward modeling in observed redshift space, essential for application to spectroscopic surveys (Stadler et al., 7 Nov 2024, Stadler et al., 2023). The redshift-space distortion (RSD) mapping,

$$x_s(\mathbf{x}) = x + \frac{v_g(x)}{{\cal H}}\hat{n},$$

is implemented alongside a systematic expansion of velocity bias:

$u_g(q) = u(q) + \sum_{\{U\}} \beta_U U(q),$

with $U(q)$ ranging over all independent EFT operators up to specified order. The density and bias expansions are displaced nonlinearly to redshift space, and the anisotropic noise model is incorporated in $k$ and $\mu$.

LEFTfield in redshift space achieves percent-level accuracy in predictions for the redshift-space power spectrum monopole and wedges at $k\leq0.2\,h/$Mpc with 3LPT and higher, and robust detection of velocity bias in halos and galaxies. The computational overhead of RSD is modest ($\sim 1.5\times$ rest-frame) due to one-step displacement optimizations.

Ongoing and future efforts aim to integrate more realistic survey masks, lightcone effects, halo/galaxy selection, and extensions to fully joint inference on $f$, $\sigma_8$, and $r_s$, as well as anisotropic BAO and application to real survey data (Stadler et al., 7 Nov 2024, Stadler et al., 2023, Babić et al., 1 Jul 2024).

6. Limitations, Assumptions, and Outlook

LEFTfield’s accuracy is limited by the validity of the perturbative expansion (LPT order, cutoff $k\leq0.2$–$0.3\,h/$Mpc), the completeness of the bias and velocity-bias expansions, and assumptions on noise properties (usually Gaussian, potentially suboptimal for realistic galaxy samples). Masking, lightcone, and selection effects must be carefully incorporated for application to real data. Model misspecification (e.g., assembly bias, density-dependent noise) may require additional EFT operators or noise terms.

Resource demands remain high (millions of MCMC samples, $10^5$–$10^6$ CPU h), but FFT and GPU optimizations are making field-level inference increasingly tractable for next-generation volumes (Babić et al., 19 May 2025). As survey volumes and precision grow, the advantages of fully Bayesian, forward-modeled, field-level inference—automatic inclusion of all accessible higher-point and non-Gaussian correlations, and robust marginalization of theoretical uncertainties—become essential for extracting percent-level cosmological information from the nonlinear regime.

7. Connections to Related Approaches and Comparisons

LEFTfield distinguishes itself from conventional summary-statistic-based cosmological inference (e.g., BAO reconstruction plus power spectrum template fitting) and from simulation-based inference that relies on approximate summaries. The field-level approach builds directly on advances in EFTofLSS and capitalizes on the full information content of the observed tracer field, matching or outperforming optimal summary-statistic approaches even under ideal conditions, and exceeding them on realistic, nonlinearly biased tracers (Babić et al., 1 Jul 2024, Nguyen et al., 5 Mar 2024).

LEFTfield thus represents a rigorous, fully perturbative, and computationally feasible paradigm for extracting fundamental cosmological parameters—including the sound horizon $r_s$, growth rate $f$, and amplitude $\sigma_8$—from nonlinear structure, and is a cornerstone of future high-precision, large-volume cosmic survey analysis.