QML-FAST - A Fast Code for low-$\ell$ Tomographic Maximum Likelihood Power Spectrum Estimation

TL;DR

QML-FAST tackles the computational challenge of optimal low-$\ell$ power-spectrum estimation for multiple correlated fields on the sphere by extending the quadratic maximum likelihood formalism and implementing extensive optimizations. The code delivers unbiased, minimum-variance estimates with cross-bin capabilities, cross-field deprojection, and an explicit treatment of unwanted multipoles, while leveraging a real spherical-harmonics basis, sparsity, and contraction-path optimizations to reach practical runtimes. Validation against simulations shows improved performance over pseudo-$C_\\ell$ methods at large scales and robust cross-field inference, including iterative parameter updates to refine fiducial models. The public Python code enables scalable, exact QML analyses for upcoming photometric surveys and CMB experiments, with demonstrated applicability to kSZ velocity reconstruction in a companion work.

Abstract

We present a novel implementation for the quadratic maximum likelihood (QML) power spectrum estimator for multiple correlated scalar fields on the sphere. Our estimator supports arbitrary binning in redshift and multipoles $\ell$ and includes cross-correlations of redshift bins. It implements a fully optimal analysis with a pixel-wise covariance model. We implement a number of optimizations which make the estimator and associated covariance matrix computationally tractable for a low-$\ell$ analysis, suitable for example for kSZ velocity reconstruction or primordial non-Gaussianity from scale-dependent bias analyses. We validate our estimator extensively on simulations and compare its features and precision with the common pseudo-$C_\ell$ method, showing significant gains at large scales. We make our code publicly available. In a companion paper, we apply the estimator to kSZ velocity reconstruction using data from ACT and DESI Legacy Survey and construct full set of QML estimators on 40 correlated fields up to $N_{\text{side}}= 32$ in timescale of an hour on a single 24-core CPU requiring $<256\ \mathrm{Gb}$ RAM, demonstrating the performance of the code.

Figures (10)

• Figure 1: An $N_{\text{side}}=16$ binary mask downgraded from $N_{\text{side}}=32$ and applied to the kSZ analysis in Lai2025kszpaper, the unmasked pixels have a value of 1.
• Figure 2: Comparison of mode-deprojection methods. Left: The full-sky QML estimates and the associated error bars for the $C^{a_2, a_2}_{\ell}$ power spectrum. The red curve shows the fiducial power spectrum given in Eq. \ref{['eq:fiducial auto power w/ noise']} and the purple curve represents the power spectrum estimation from the fullsky Gaussian simulation using hp.anafast. Data points with solid, dashed and dotted error bars give the three different mode deprojection methods, with 'no treatment' stands for simple column/row removal (method 1), 'pinv' stands for pesudo-inverse covariance construction (method 2) and 'scale' for scaling unwanted modes (method 3). Right: Same as left panel, but with the mask shown in Fig. \ref{['fig:mask']} applied to the same set of simulated maps. Green and orange data points are shifted by $\pm0.3$ along the x-axis to avoid overlapping.
• Figure 3: Comparison of (un-)biasedness of mode-deprojection methods. The QML estimates of the $C^{a_2, a_2}_{\ell}$ power spectrum averaged from 5000 masked Gaussian realizations. Each realization is processed with the multi-field QML pipeline and modes below $\ell = 5$ are removed according to the three prescriptions in main text. The fiducial power spectrum is also plotted for reference.
• Figure 4: Influence of the fiducial power on the estimator optimality. The average of 5000 QML estimates, each realization is simulated from the true fiducial power spectrum given by green curve while evaluated with a modified covariance matrix that is computed from the orange curve. The green and orange shaded region highlight the 1-sigma uncertainty obtained from the 5000 simulations. The ratio of the 1-sigma uncertainty between Wrong and True fiducial power is plotted in the bottom, and it is consistently above 1.
• Figure 5: Test of the pseudo-$C_{\ell}$ pipeline. Pseudo-$C_{\ell}$ estimates on the autopower $C^{a,a}_{\ell}$ from averaging $10^5$ pseudo-$C_{\ell}$ outputs. The pseudo-$C_{\ell}$ power spectrum is calculated using PyMaster without apodization to the mask. The red curve shows the fiducial power spectrum convolved with the bandpower window function. There is residual bias beyond $\ell = 46$. The spikiness of the fiducial power is due to the window function.