Systematics mitigation for catalogue-based angular power spectra

TL;DR

This work extends the catalogue-based pseudo-$C_lat$ framework to include template deprojection, enabling unbiased angular power spectra measurements directly from discrete sky catalogues. It introduces a transfer-function debiasing approach to correct for deprojection-induced mode loss, derives exact shot-noise corrections for deprojection, and validates the method on simulated sampled fields and galaxy clustering data, including the Quaia catalogue. The authors demonstrate that, after accounting for noise and mode loss, the catalogue-based estimator recovers unbiased $C_lat$ and is consistent with standard map-based pseudo-$C_lat$ results on real data. The method is implemented in NaMaster, and the results support the utility of catalogue-based $C_lat$ estimation for upcoming surveys like LSST, enabling high-resolution analyses without pixelisation biases.

Abstract

Recent work has developed a formalism for computing angular power spectra directly from catalogues containing field values at discrete positions on the sky, thereby circumventing the need to create pixelised maps of the fields, as well as avoiding aliasing and finite-resolution effects. We adapt this formalism to incorporate template deprojection for mitigating systematic biases in the measured angular power spectra. We also introduce an alternative method of mitigating the `deprojection bias' - the loss of modes induced by deprojection - employing simple simulations to compute a transfer function. We find that this approach performs at least as well as existing methods, and is relatively insensitive to how well one can guess the true power spectrum of the observed field, except at the largest scales ($\ell \lesssim 3$). Additionally, we develop exact expressions for the bias introduced by deprojection in the shot-noise component, which further improves the accuracy of this approach. We test our formalism on simulated datasets, demonstrating its applicability both to discretely sampled fields, and to the special case of galaxy clustering, with the survey selection function defined in terms of a random catalogue or as a continuous sky map. After removing the bias in the shot noise and correcting for the remaining mode loss using a transfer function, our formalism produces unbiased measurements of the angular power spectrum in all scenarios tested here. Finally, we apply our formalism to real data and show it produces results consistent with the standard map-based pseudo-$C_\ell$ formalism. We implement our method in the public code NaMaster.

Figures (8)

• Figure 1: Input power spectrum used to validate the approach to contaminant deprojection in the case of sampled fields (black line). The blue line shows the power spectrum of the contaminant maps used, while the horizontal dashed line shows the simulated white noise component. The orange points show the mean power spectrum measured from all contaminated simulations.
• Figure 2: Power spectrum of noise-only simulations with and without contaminant deprojection (black and red points, respectively). The loss of power due to deprojection leads to a negative power spectrum (note that the catalogue-based estimator automatically removes the white noise component), which is accurately predicted by Eq. \ref{['eq:noise_deproj_sampled']} (blue line).
• Figure 3: Mean and standard deviation of the power spectra from contaminated simulations after contaminant deprojection (purple points). The orange points show the result of accounting for the impact of deprojection in the white noise component (via Eq. \ref{['eq:noise_deproj_sampled']}), while the red points show the power spectra after accounting for deprojection bias in both the noise and signal components (the latter via the transfer function approach). The result is an unbiased estimate of the input power spectrum (black line). The bottom panel shows the residuals with respect to the input spectrum normalised by the error in the mean over all simulations.
• Figure 4: Linear deprojection coefficients measured as a function of $\ell_{\rm max}$ from simulations using the randoms-based (teal circles) and mask-based (red squares) approaches. The procedure is outlined in Section \ref{['ssec:res.gcval']}, and details of the templates (indicated at the top of each panel) can be found in Appendix \ref{['sec:appx.templates']}. The dashed black line in each panel shows the input deprojection coefficient used to contaminate the simulated data. Small horizontal offsets have been applied to the data for clarity. Both methods show significant variation in the deprojection coefficients for several contaminants, particularly as one moves to the lowest and highest multipoles. The dotted orange line indicates $\ell_{\rm max} = 100$, which we deem a suitable value for minimising the bias in our measured deprojection coefficients for the majority of templates.
• Figure 5: Means and standard deviations in angular power spectra from contaminated simulations measured at various stages of correction using the randoms-based (top panel) and mask-based (middle panel) approaches. Different marker sizes and linestyles represent the different stages: after contaminant deprojection (small markers, dashed lines); after removing the noise deprojection bias (medium-sized markers, solid lines); after also correcting for deprojection bias in the signal component using the transfer function approach (large markers, dotted lines). The black solid line shows the angular power spectrum used to generate the (non-contaminated) data, and as such is indicative of the 'true' power spectrum. The shaded region begins at $\ell = 2N_{\rm side}$, beyond which we expect aliasing effects to arise from the fact that we have sampled the catalogue from a map with finite resolution. The bottom panel shows the residuals of the fully corrected power spectra with respect to the input power spectrum, normalised by the error in the mean across the simulations. Both the randoms-based (teal circles) and mask-based (red squares) approaches are able to accurately reproduce the input power spectrum, with the randoms-based approach maintaining better accuracy than the mask-based at the higher multipoles. Exacerbated deviations from the input power spectrum after applying the transfer function ($T_\ell$) at the largest scales are due to the data being noise-dominated in this regime (see text).