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Analytical covariances for catalogue-based pseudo- $C_\ell$ s

This paper presents a method implemented in the NaMaster code to accurately estimate the Gaussian covariance of catalogue-based pseudo- $C_\ell$ s by combining the Narrow-Kernel Approximation for distinct source pairs with exact treatment of self-pairs, effectively addressing overlap and discrete sampling effects across both dense and sparse cosmological datasets.

Original authors: Kevin Wolz, Elyas Farah, Robert Reischke, David Alonso, Andrina Nicola

Published 2026-07-17

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CC BY 4.0

Original authors: Kevin Wolz, Elyas Farah, Robert Reischke, David Alonso, Andrina Nicola

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible tapestry woven from the threads of galaxies, dark matter, and the mysterious forces that shape them. To understand how this tapestry is put together, astronomers don't just look at individual stars; they try to measure the "texture" of the whole sky. They look for patterns in how matter is clumped together or how light bends around massive objects. One of the most powerful tools for this is called the "angular power spectrum." Think of it like a musical score for the cosmos: just as a piece of music can be broken down into low bass notes and high treble notes, the sky can be broken down into large-scale patterns and tiny, fine-grained details. By reading this cosmic score, scientists can learn about the history of the universe, how fast it is expanding, and what it is made of.

However, reading this score is tricky because we can't see the whole sky at once. Our view is blocked by the Milky Way's dust, by the limitations of our telescopes, and by the fact that we can only see a finite number of galaxies. This creates a "mask" over our view, distorting the music. Furthermore, many of the most exciting new data sources aren't smooth pictures of the sky at all; they are lists of specific points, like a guest list for a party where we only know where the guests are standing, not the space between them. These are called "catalogues." When scientists try to measure the cosmic music from these lists, they face a new problem: how do we know how much our measurement might be wrong? In science, knowing the "uncertainty" is just as important as the measurement itself. If you guess the temperature is 70 degrees, you need to know if it's actually 69 or 75. Without a reliable way to calculate this uncertainty for these point-based lists, we can't trust the cosmic conclusions we draw from them.

This is where the paper by Kevin Wolz and his team comes in. They have developed a clever new mathematical recipe to calculate these uncertainties for data that comes in the form of point-like catalogues, such as lists of galaxies or fast radio bursts. Previously, scientists had to turn these lists into blurry, pixelated maps (like turning a high-resolution photo into a low-res pixel art) to do the math, which introduced errors and lost information. The authors show that you can skip the blurry map entirely. Instead, they treat each point in the catalogue as if it were a tiny, fuzzy cloud rather than a sharp dot. By doing this, they can calculate the "noise" or uncertainty of the measurement directly from the list of points, without ever needing to create a pixelated image.

The team tested their method against two other approaches: a "brute-force" method that calculates every single possible pair of points (which is incredibly accurate but so slow it's impossible for large datasets) and computer simulations that act as a "ground truth" for the universe. They found that their new method is a perfect middle ground. It is almost as accurate as the super-slow brute-force method but runs fast enough to be useful for real-world data. They validated this across a wide variety of scenarios, from dense crowds of galaxies (like those seen in the Dark Energy Survey) to very sparse lists of fast radio bursts (where the sources are few and far between). The results showed that their method correctly predicts the uncertainty in almost every case, even for complex situations where the data is noisy or the sky coverage is patchy.

The paper explicitly rules out the idea that you must pixelate the sky to get accurate results for these types of data. They demonstrate that the old way of binning points into squares introduces unnecessary numerical errors. They also show that while some approximations in their method might struggle if the data is extremely sparse or if the sky mask is incredibly complex, for the vast majority of realistic cosmological scenarios, their approach is robust. The authors are confident in their findings because they didn't just guess; they ran 32 different types of simulations, including cross-correlations between different types of data, and their new formula matched the simulation results with high precision. They even found that their method works well for "momentum fields" (where the density of galaxies is weighted by their motion) and galaxy clustering, proving it's a versatile tool for the future of astronomy.

In essence, this paper provides a new, faster, and more accurate way to measure the "error bars" on our cosmic measurements when the data comes from lists of points. It allows scientists to trust their measurements of the universe's structure more than ever before, without having to wait for supercomputers to crunch numbers for weeks. By treating the points in the catalogue as fuzzy clouds rather than sharp dots, they've smoothed out the rough edges of the math, making it possible to hear the cosmic music clearly, even when the audience is scattered across the sky.

Technical Summary: Analytical Covariances for Catalogue-Based Pseudo- $C_\ell$ s

Problem Statement
Cosmological analyses increasingly rely on fields sampled at the discrete positions of astrophysical sources (e.g., cosmic shear, galaxy clustering, fast radio bursts) rather than continuous pixelated sky maps. While recent developments have enabled the estimation of angular power spectra for these catalogue-based fields without constructing pixelated maps, a robust method to estimate the covariance of these spectra has been lacking. Standard analytical approaches, such as the Narrow-Kernel Approximation (NKA) and its improved variant (iNKA), rely on the product of sky masks in real space. For catalogue-based fields, where masks are represented by Dirac delta functions at discrete source positions, these products either vanish (for cross-correlations between distinct catalogues) or diverge (for auto-correlations due to coincident delta functions). Furthermore, the discrete nature of catalogues introduces Poisson-like shot noise (self-pair contributions) that significantly impacts statistical uncertainties, particularly in sparse samples or noise-dominated regimes like cosmic shear. Existing methods often require computationally expensive brute-force simulations or fail to accurately capture these discrete effects.

Methodology
The authors present a generalization of the improved Narrow-Kernel Approximation (iNKA) to compute the disconnected (Gaussian) covariance of pseudo- $C_\ell$ estimators for catalogue-based fields. The core of the method involves two key innovations:

Regularization via Gaussian Smoothing: To handle the divergence or vanishing of mask products, the authors treat each discrete source not as a point but as a finite-sized "cloud" with a characteristic angular size $\theta_c$ . This size is determined by the mean interparticle distance of the catalogue and the maximum multipole $\ell_{\text{max}}$ of interest. Mathematically, this corresponds to applying a Gaussian smoothing kernel $\phi_\ell$ to the harmonic coefficients of the mask. This regularization allows for the stable calculation of real-space mask products required by the iNKA formalism.
Separation of Self-Pairs and Distinct Pairs: The covariance is decomposed into contributions from distinct source pairs and self-pairs (where $i=j$ $i = j$ ).
- Distinct Pairs: The contribution from distinct pairs is approximated using the standard iNKA approach, treating the smoothed masks as continuous fields.
- Self-Pairs: The contribution from self-pairs is calculated exactly. This term acts as an effective source of white noise (shot noise) and is treated as an inhomogeneous noise component in the covariance structure. The authors derive exact expressions for the "signal-signal," "signal-noise," and "noise-noise" terms, ensuring that the shot noise floor is correctly accounted for without approximation.

The method also addresses specific cases such as "momentum fields" (fields weighted by local density) and galaxy clustering, where the mask may be defined by random catalogues rather than the data itself. The formalism is implemented in the public code NaMaster, introducing new field object types (NmtFieldCatalog, NmtFieldCatalogMomentum, NmtFieldCatalogClustering) to handle these discrete fields seamlessly.

Key Contributions

Analytical Covariance Estimator: The derivation of a closed-form analytical estimator for the Gaussian covariance of catalogue-based power spectra, avoiding the need for massive simulation ensembles.
Exact Treatment of Shot Noise: A rigorous separation of self-pair (shot noise) and distinct-pair contributions, allowing the noise component to be calculated exactly while the signal component utilizes the efficient iNKA approximation.
Generalization to Momentum and Clustering: Extension of the formalism to handle momentum fields and galaxy clustering, including the specific treatment of random catalogues used to define completeness masks.
Brute-Force Benchmark: Implementation of an exact "brute-force" covariance calculation based on the direct sum of all source pairs (scaling as $O(N_{\text{sources}}^4 \ell_{\text{max}}^4)$ ), optimized using the jax library, to serve as a validation benchmark for sparse samples.

Results and Validation
The authors validate the method against empirical covariances derived from 1,000 simulations across various regimes:

Sparse Catalogues (FRBs): Tested on Fast Radio Burst (FRB) dispersion measures with low source densities ( $\sim 10^{-3} \text{ arcmin}^{-2}$ ). The iNKA covariance showed excellent agreement with both the empirical simulation covariance and the exact brute-force calculation, even in the presence of significant shot noise.
Dense, Noise-Dominated Catalogues (Cosmic Shear): Validated on a Dark Energy Survey (DES) Year-3 like sample with high source density ( $\sim 1-10 \text{ arcmin}^{-2}$ ) and shape noise. The analytical covariance accurately reproduced the sample variance for both E-modes and B-modes up to $\ell \approx 10,000$ , including the off-diagonal elements.
Galaxy Clustering: Tested on mock LRG samples. The method accurately predicted the variance and correlation structure, with the Gaussian approximation holding well despite the underlying log-normal and Poisson nature of the clustering field.
General Combinations: A comprehensive suite of 32 validation cases covering auto- and cross-correlations of spin-0 and spin-2 fields (tracer catalogues, momentum fields, and standard maps) confirmed that the method performs robustly across different spin modes and field combinations. Kolmogorov-Smirnov tests on $\chi^2$ distributions generally showed good agreement between analytical and empirical covariances.

Significance and Claims
The paper claims that this method provides a highly accurate and computationally efficient alternative to simulation-based covariance estimation for catalogue-based power spectra. By generalizing the iNKA to handle discrete sources and exactly treating self-pair shot noise, the authors demonstrate that the approach remains robust across a wide range of scenarios, from sparse, noise-dominated observables (like FRBs) to dense, shape-noise-dominated fields (like cosmic shear).

The significance lies in enabling rigorous statistical analysis for current and future Large Scale Structure (LSS) surveys, particularly in multi-probe analyses where the number of power spectra and the volume of data make the generation of high-fidelity simulation ensembles for covariance estimation impractical. The authors note that while the method is robust, its accuracy relies on the Narrow-Kernel Approximation for the signal component, which may degrade for very steep power spectra, highly complex masks, or significant E/B asymmetry in spin fields. However, for the regimes tested, the method represents a significant improvement over standard approximations, offering a reliable tool for propagating statistical uncertainties to cosmological parameter constraints.

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