CosmoForge I: A unified framework for QML power spectrum estimation and pixel-based likelihood analysis

Abstract: Optimal power spectrum estimation on the largest angular scales of the cosmic microwave background relies on the Quadratic Maximum Likelihood (QML) estimator. Existing public implementations, however, each address only a subset of the problem and none combine power spectrum estimation with a self-consistent pixel-space likelihood within a single framework. We present CosmoForge, a public Python framework that unifies QML power spectrum estimation and pixel-based Gaussian likelihood evaluation for spin-0 and spin-2 fields on the sphere, with general (non-diagonal) noise covariances. The framework is split into three installable packages: CosmoCore (infrastructure), QUBE (Fisher and QML estimation), and PICSLike (pixel-space likelihood). A common interface exposes two interchangeable computation bases $-$ a harmonic basis built on the Sherman-Morrison-Woodbury identity and a direct pixel-space basis $-$ selecting whichever is cheaper for the configuration at hand. Exact algorithmic optimisations reduce the Fisher cost to $\mathcal{O}(\ell_{\rm max}^4)$ for arbitrary noise covariances, with Numba JIT compilation of the hot kernels and MPI parallelisation of the likelihood scan. CosmoForge reproduces the Planck low-$\ell$ Fortran reference implementation across both the QML and pixel-space likelihood pipelines, consistently with double-precision arithmetic. Native multipole binning and three output normalisations (deconvolved, decorrelated, window-convolved) are exposed through a single code path, and the same covariance infrastructure powers both QML estimation and likelihood evaluation. CosmoForge offers a general-purpose, modular, and validated tool for the optimal analysis of large-scale data on the sphere. It is publicly available, pip-installable, and extensible to non-CMB observables.

• The paper introduces CosmoForge I, a unified Python toolkit that combines QML power spectrum estimation with pixel-based Gaussian likelihood analysis for spherical data.
• It leverages both pixel and harmonic implementations, adapting to computational demands while effectively handling non-diagonal noise and mask-induced mode coupling.
• Extensive validation, including Monte Carlo simulations and Planck cross-checks, demonstrates unbiased, minimum-variance estimates and robust cosmological parameter inference.

CosmoForge I: A Unified Framework for QML Power Spectrum Estimation and Pixel-Based Likelihood Analysis

Introduction and Motivation

CosmoForge addresses a central challenge in large-scale cosmological data analysis: unifying optimal Quadratic Maximum Likelihood (QML) power spectrum estimation with pixel-based Gaussian likelihood inference for scalar and tensor fields on the sphere, accounting for realistic, non-diagonal noise covariance structures. Historically, optimal CMB power spectrum analyses at low multipoles ($\ell$) have depended on implementations that handle either estimation or likelihood but not both within one coherent, validated framework. Furthermore, no public software previously combined support for both pixel- and harmonic-basis computation, generalized (possibly correlated) noise, spin-2 fields, and native binning or decorrelation of power spectrum estimates. CosmoForge aims to fill this gap with a modular, extensible, and validated Python-based toolkit.

QML and Pixel-Likelihood Formalism

CosmoForge implements the canonical QML estimator, which provides unbiased, minimum-variance estimates of the angular power spectrum $\{C_\ell\}$ by constructing quadratic estimators normalized by the Fisher information matrix. Both pixel-space and harmonic-basis (Sherman–Morrison–Woodbury, SMW) implementations are supported, with automatic selection of the optimal path depending on computational cost. This flexibility is crucial for scaling analyses from large, low-resolution (pixel-dominated) datasets to smaller, high-resolution cases where harmonic methods are more tractable.

The framework accommodates spin-0 and spin-2 fields, supporting the full set of auto- and cross-spectra, and allows for non-block diagonal, spatially correlated noise covariance — a requirement for realistic CMB likelihoods and polarization analyses.

Key Aspects of QML Implementation

• Signal and Noise Modeling: Generalizes to arbitrary combinations of fields (e.g., T, Q, U for the CMB; galaxy surveys).
• Covariance Assembly: Efficient construction of the total covariance $C = S + N$ in both pixel and harmonic representations.
• Estimator Optimality: Saturating the Cramér–Rao bound for unbiased spectrum estimation, with rigorous handling of mask-induced mode coupling and noise anisotropy.
• Multipole Binning and Normalisation Options: Supports binned bandpower estimation, Fisher-weighted decorrelation, and window-function-convolved outputs.
• Cross-Correlation Modes: Accommodates both joint (block-structured) and independent (cross-QML) approaches for cross-spectra.

The direct pixel-based likelihood computes the true Gaussian log-likelihood $$\ln \mathcal{L} = -\frac{1}{2}[d^\top C^{-1} d + \ln|C|]$$, providing a rigorous end-to-end path from maps to parameter inference.

Architecture and Implementation

CosmoForge is partitioned into three installable packages: CosmoCore (computational infrastructure), QUBE (QML estimation pipeline), and PICSLike (pixel-based likelihood evaluation). All core algorithms are implemented in Python but numerically accelerated via Numba JIT and optimized BLAS/LAPACK routines, ensuring practical feasibility on both cluster and workstation hardware. MPI parallelization is implemented for both Fisher trace evaluation and parameter-grid likelihood sweeps.

(Figure 1)

Figure 1: Architecture and data flow of CosmoForge, showing interchangeable pixel and harmonic computation bases driving both QML estimation (QUBE) and pixel-likelihood (PICSLike) via a unified interface.

• Computation Basis Abstraction: A shared interface exposes both pixel and harmonic implementations for covariance, inversion, and derivatives. The SMW basis leverages the low-rank structure of the signal plus noise, reducing cost for high-$\ell$ analyses.
• Field Block-Diagonalization: Automatic detection and exploitation of field group independence at the covariance level minimizes computational overhead in multi-field, multi-spectrum scenarios.

Validation and Statistical Performance

CosmoForge has undergone exhaustive validation:

• Cross-Checks with Planck Low-$\ell$ Reference: All primary outputs (signal and noise covariance, Fisher matrix, QML bandpowers, pixel-likelihood) are consistent to double-precision limits with the Planck collaboration’s Fortran reference implementations.
• Monte Carlo Validation: $10^4$ simulations of CMB polarization fields confirm QML unbiasedness at the $10^{-8}$ level, absence of E-to-B leakage, and variance saturation with Fisher predictions.

  Figure 2: Monte Carlo validation of the QML estimator for the $BB$ power spectrum — mean deviation from input is negligible compared to estimator uncertainty.

• Variance/Optimality vs. Pseudo-$C_\ell$: QML consistently yields tighter error bars, with up to 60% lower variance at low sky fraction compared to state-of-the-art pseudo-$\{C_\ell\}$0 approaches such as NaMaster, and maintains statistical optimality across a wide range of masks and resolutions.

  Figure 3: Standard deviation of the estimated $\{C_\ell\}$1 spectrum versus Fisher-predicted errors as a function of sky fraction; empirical and predicted variances coincide at all tested configurations.

• End-to-End Pixel-Likelihood Inference: Likelihood surfaces for cosmological parameters (e.g., tensor-to-scalar ratio $\{C_\ell\}$2) are unbiased, and upper limits are consistent with simulated truths.

  Figure 4: Mean-$\{C_\ell\}$3 PICSLike likelihood curve for tensor-to-scalar ratio $\{C_\ell\}$4 from $\{C_\ell\}$5 simulations; recovered limits are consistent with input values with high precision.

• Consistency with Alternate Estimators: Empirical bandpower correlation matrices confirm QML outputs are closer to diagonal (uncorrelated), unlike pseudo-$\{C_\ell\}$6 outputs which display strong mask-induced mode coupling.

  Figure 5: Comparison of CosmoForge QML estimator and NaMaster: QML recovers mean spectra with lower variance and uncorrelated bandpowers.

  

  Figure 6: Bandpower correlation matrices — QML (upper triangle) vs. NaMaster pseudo-$\{C_\ell\}$7 (lower triangle) — illustrating the superior decorrelation properties of the QML approach.

Algorithmic and Computational Considerations

Key algorithmic advances of CosmoForge include:

• SMW Reformulation: Reduces the cubic scaling with pixel count to cubic or quartic in mode count, depending on the stage.
• Sparse and Blockwise Fisher Trace Evaluation: Achieves $\{C_\ell\}$8 scaling for the Fisher matrix computation, crucial for tractable analyses at $\{C_\ell\}$9.
• Field Block-Diagonalisation and Multipole Windowing: Automatically exploits block-diagonal structure for both speed and memory savings; multipole switching reduces kernel dimension for targeted parameter inference.
• Numerical Stability: Careful evaluation of projected inverses and traces to prevent catastrophic cancellation in the high signal-to-noise regime.

Empirical performance benchmarks demonstrate practical wall-times — e.g., Fisher matrix plus 10 QML realizations for $C = S + N$0, $C = S + N$1, $C = S + N$2 completed in under 25 minutes on a single multi-core node, with acceptable per-rank memory usage.

Feature Landscape and Code Comparison

CosmoForge unifies capabilities previously distributed among multiple single-purpose codes. Unlike prior tools such as ECLIPSE, xQML, QML-FAST, or legacy Planck Fortran codes, CosmoForge provides in one package:

• Simultaneous support for spin-0 and spin-2 fields, all cross/auto-spectra, and both joint and cross-spectra estimation.
• Native handling of non-diagonal, correlated noise.
• Both harmonic (SMW) and direct pixel-basis computation, automatically selected for cost.
• Decorrelated and window-convolved bandpowers as well as conventional deconvolved estimates.
• Direct pixel-likelihood routines with MPI parallelization.
• Full validation against Planck reference calculations.
• Comprehensive, pip-installable Python packaging, modern YAML-based configuration, and modular design for extensibility to non-CMB science.

Implications and Future Directions

Practically, CosmoForge lowers the barrier for optimal, interpretable, and reproducible estimation in CMB, large-scale structure, and related spherical data applications. The combination of QML and full pixel-likelihood pathways in one framework is particularly pertinent for future missions (e.g., LiteBIRD, Simons Observatory, CMB-S4) with demanding requirements on systematics control, correlated noise, and multi-field cross-analysis.

Theoretically, this unification enables more robust validation of analytic approximations, exploration of systematic cross-correlations, and facilitates methodological extensions such as iterative QML and hybrid optimal-binned analyses. CosmoForge’s design also opens the door to further advances, including integration with GPU computation and cosmological sampling frameworks.

Conclusion

CosmoForge represents a significant step toward the consolidation and modernization of optimal power spectrum and parameter inference pipelines for spherical cosmological fields. By exposing validated, computationally optimized, and feature-complete QML and pixel-likelihood tools with a clean Python interface, CosmoForge directly supports the scientific goals of current and future CMB and large-scale structure experiments, and sets a foundation for new algorithmic developments in spherical data analysis.

Reference:

"CosmoForge I: A unified framework for QML power spectrum estimation and pixel-based likelihood analysis" (2605.21149)