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Systematics mitigation for catalogue-based angular power spectra

This paper extends catalogue-based angular power spectrum formalism to mitigate systematic biases through template deprojection and transfer function corrections, demonstrating its accuracy on simulations and real data via an implementation in the NaMaster code.

Original authors: Thomas Cornish, David Alonso, Boris Leistedt, Kevin Wolz

Published 2026-03-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Thomas Cornish, David Alonso, Boris Leistedt, Kevin Wolz

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 you are an astronomer trying to understand the "music" of the universe. In cosmology, this music is called the Angular Power Spectrum. It's a graph that tells us how much "clumping" or structure exists in the universe at different sizes, from giant cosmic webs to tiny galaxy clusters.

For decades, astronomers have had a problem: the data they collect doesn't come in neat, continuous pictures (like a photo). Instead, it comes as a list of dots (a catalogue), where each dot is a specific galaxy or quasar at a specific location.

The Old Way: The "Pixelated Photo" Problem

Traditionally, to analyze this list of dots, scientists would force them into a grid, like turning a high-resolution photo into a low-resolution pixelated image. They would count how many dots fell into each square (pixel) and draw a map.

The Analogy: Imagine trying to measure the texture of a fine silk scarf by taking a photo of it with a camera that has very large, blocky pixels. You lose all the fine details. The "pixels" blur the edges, and you can't hear the high notes of the universe's music. This is called aliasing and finite-resolution effects.

The New Way: The "Direct List" Approach

A few years ago, a new method was invented (by Wolz et al.) that allowed scientists to skip the pixelated photo entirely. They could analyze the list of dots directly, preserving every tiny detail. It was like listening to the raw audio file instead of a compressed, low-quality MP3.

The Problem: While this new method was great for hearing the music, it had a blind spot. It couldn't easily remove "static" or "noise" caused by systematic errors (like dust in the telescope lens, atmospheric interference, or uneven survey coverage). In the old pixelated method, there was a tool called Template Deprojection that could subtract this noise. But it didn't work on the new "list-only" method.

The Solution: Cleaning the List Without Losing the Song

This paper by Thomas Cornish and colleagues is like a master mechanic who figured out how to install that noise-cleaning tool into the new "list-only" engine.

Here is how they did it, using some creative metaphors:

1. The "Noise-Canceling Headphones" (Template Deprojection)

Imagine you are listening to a song, but there is a loud hum (systematic noise) in the background. You have a recording of just that hum (a "template").

The Old Method: You would try to subtract the hum from the pixelated photo.
The New Method: The authors figured out how to subtract the hum directly from the list of notes. They mathematically "project" the noise out of the data, leaving only the pure cosmic song.

2. The "Side Effect" (Deprojection Bias)

Here is the tricky part: When you subtract the noise, you accidentally subtract a tiny bit of the real song too. It's like using a noise-canceling headphone that is too good; it silences the bass notes of the music along with the hum.

The Analogy: If you try to remove a stain from a shirt by scrubbing too hard, you might wear a hole in the fabric. The "hole" in the data is called mode loss.

3. The "Repair Kit" (Transfer Functions)

The authors developed two ways to fix the holes they accidentally made:

Method A (The Analytical Fix): They derived a precise mathematical formula to calculate exactly how much of the song was lost and added it back in. This works perfectly for the "static" (shot noise) part of the data.
Method B (The Simulation Fix): For the rest of the song, they used a clever trick. They ran thousands of computer simulations of the universe, applied the same noise-removal process, and measured exactly how much signal was lost. They created a "Transfer Function"—essentially a correction map that says, "If you hear a note at this volume after cleaning, the original note was actually this loud."

The Results: A Clearer Picture of the Universe

The team tested their new method on:

Simulated Data: They created fake universes with known "songs" and added heavy noise. Their method successfully cleaned the noise and recovered the original song perfectly.
Real Data: They applied it to the Quaia catalogue (a real list of quasars). The results matched the old, pixelated methods perfectly, proving that the new method is just as reliable but much more precise because it doesn't blur the data.

Why This Matters

This paper is a game-changer for future space missions like the Vera C. Rubin Observatory. These telescopes will collect billions of data points.

The Benefit: By using this new method, astronomers won't have to blur their data into pixels to clean it. They can keep the data in its raw, high-definition form, remove the systematic errors, and hear the "music" of the universe with unprecedented clarity.

In a nutshell: The authors took a high-tech "list-based" music player that couldn't filter out static, figured out how to add a noise-canceling filter, and invented a way to fix the filter so it doesn't accidentally mute the music. The result? A crystal-clear view of the cosmos.

1. Problem Statement

Modern cosmology relies heavily on measuring the angular power spectrum ( $C_\ell$ ) of projected fields (e.g., galaxy clustering, weak lensing, CMB). While optimal estimators exist, they are computationally expensive. The pseudo- $C_\ell$ (PCL) method offers a faster alternative by assuming a diagonal covariance matrix, but traditionally requires data to be binned into pixelized maps.

The Limitation of Pixelization: Binning discrete sources (like galaxies or quasars) into pixels introduces finite-resolution effects and aliasing, particularly at small scales (high $\ell$ ).
The Catalogue-Based Solution: Recent work (Wolz et al. 2025) extended PCL to work directly on discrete catalogues (sums of Dirac delta functions), avoiding pixelization biases. However, this formalism lacked a mechanism to mitigate systematic biases (e.g., observational artifacts, dust, stellar density variations) which are critical for high-precision cosmology.
The Challenge: Standard systematic mitigation techniques, such as template deprojection (removing modes correlated with contaminant templates), introduce a "deprojection bias" (loss of signal modes) and alter the shot noise component. These effects were previously only analytically tractable for continuous maps, not discrete catalogues.

2. Methodology

The authors extend the catalogue-based PCL formalism (implemented in the code NaMaster) to incorporate template deprojection and correct for the resulting biases.

A. Template Deprojection for Catalogues

The authors adapt the linear deprojection formalism (Rybicki & Press 1992; Elsner et al. 2017) to discrete data:

Sampled Fields: For fields measured at discrete source positions, the deprojection is performed by modifying the field values at those positions: $a^c_i = a_i - \hat{A}_p f^p_i$ . The coefficients $\hat{A}_p$ are calculated using a weighted sum over sources rather than a continuous integral.
Galaxy Clustering: This case is more complex because the "mask" is often defined by a "random catalogue" (a set of points tracing the survey selection function) rather than a continuous map.
- The authors identify that applying standard deprojection directly to the random catalogue leads to biased coefficients because the randoms do not correlate with the data in the same way.
- Solution: They propose truncating the harmonic-space dot product at a maximum multipole $\ell_{\text{max}}$ . This captures large-scale correlations between the galaxy overdensity and contaminants while avoiding issues with pixel window functions and accidental coincidences in the random catalogue.

B. Correcting for Bias

Deprojection removes signal modes, leading to a biased power spectrum. The authors address two types of bias:

Shot Noise Bias: Deprojection alters the uncorrelated noise component (shot noise). The authors derive exact analytical expressions for the change in the noise power spectrum ( $\Delta N_\ell$ ) for both sampled fields and galaxy clustering (including cases with random catalogues).
Signal Mode Loss: The removal of contaminant modes reduces the power of the signal itself.
- Standard Approach: Requires knowledge of the true underlying power spectrum ( $C_\ell^{\text{true}}$ ) to calculate an analytical correction.
- Proposed Alternative: A Transfer Function ( $T_\ell$ ) approach. The authors generate Gaussian simulations with a guessed input power spectrum, apply the same mask and deprojection, and compute the ratio $T_\ell = \langle C_\ell^{\text{after}} \rangle / \langle C_\ell^{\text{before}} \rangle$ . The measured data spectrum is then divided by $T_\ell$ to correct for mode loss.

3. Key Contributions

Formalism Extension: Successfully integrated template deprojection into the catalogue-based PCL framework, allowing for systematic mitigation without pixelization.
Analytical Noise Correction: Derived exact expressions for the shot noise bias introduced by deprojection in discrete catalogues, a previously missing component.
Robust Transfer Function: Validated a simulation-based transfer function method to correct for signal mode loss. They demonstrated that this method is robust and relatively insensitive to the specific guess of the input power spectrum, except at very large scales ( $\ell \lesssim 3$ ).
Handling Random Catalogues: Developed a specific procedure for galaxy clustering where the mask is defined by a random catalogue, including the necessary truncation of multipoles ( $\ell_{\text{max}}$ ) to ensure unbiased coefficient estimation.

4. Results

The authors validated their method using simulated data and real observations from the Quaia quasar catalogue (Gaia-unWISE).

Simulated Sampled Fields: Using 1,000 simulations with 100 contaminant templates, they showed that:
- Without correction, deprojection causes significant power loss (even negative power at low $\ell$ ).
- Applying the analytical shot noise correction recovers most of the power.
- Applying the transfer function correction recovers the remaining signal bias, resulting in an unbiased estimate of the input power spectrum across all scales.
Galaxy Clustering Simulations:
- They determined that $\ell_{\text{max}} \approx 100$ is an optimal cutoff for the deprojection coefficients to balance capturing systematics and avoiding pixelization artifacts.
- Both "randoms-based" (discrete mask) and "mask-based" (continuous map) approaches yielded unbiased results after noise and signal corrections.
- The randoms-based approach maintained better accuracy at high multipoles ( $\ell > 2N_{\text{side}}$ ) compared to the mask-based approach, which suffered from aliasing.
Transfer Function Sensitivity: The transfer function method performed comparably to the standard analytical deprojection bias correction. It remained unbiased even when using a rough guess for the input power spectrum (e.g., measured PCL divided by sky fraction), except at the very largest scales ( $\ell \lesssim 3$ ).
Real Data Application: Applied to the Quaia low-redshift quasar sample. The catalogue-based deprojection results (both randoms-based and mask-based) were consistent with the standard map-based PCL approach, validating the implementation for real-world use.

5. Significance

Precision Cosmology: This work enables the use of high-resolution, pixel-free power spectrum estimators on large-scale structure data while rigorously controlling for systematics. This is crucial for Stage IV surveys (e.g., LSST, Euclid) where systematic errors will dominate statistical errors.
Small-Scale Physics: By avoiding pixelization, the method allows for the exploitation of information at smaller angular scales (higher $\ell$ ), which is often lost or aliased in traditional map-based approaches.
Open Source Implementation: The method is implemented in the public code NaMaster, making it immediately available for the cosmological community to apply to current and future datasets.
Flexibility: The formalism handles both continuous masks and discrete random catalogues, providing a unified framework for various survey analysis strategies.