Euclid preparation: The flat-sky approximation for the clustering of Euclid's photometric galaxies

TL;DR

This paper evaluates the accuracy of flat-sky versus Limber approximations for the angular clustering of Euclid's photometric galaxies. By applying the approximations to six DR1-like bins and thirteen DR3-like bins, the authors show that Limber is sufficient for DR1 but fails to reproduce the full result for DR3, where the flat-sky approximation remains accurate to better than $5\%$ and enables faster computations. Using the Flagship Euclid simulation, full CLASS-based calculations, and a Gaussian covariance-based mock-observations framework, the study provides a rigorous comparison across tomographic binings and multipoles up to $\ell\leq 300$, highlighting the regimes where each approximation is reliable. The results support adopting the flat-sky approach for DR3 analyses and motivate hybrid or precomputed-k approaches to further accelerate parameter inference in Euclid-like surveys.

Abstract

We compare the performance of the flat-sky approximation and Limber approximation for the clustering analysis of the photometric galaxy catalogue of Euclid. We study a 6 bin configuration representing the first data release (DR1) and a 13 bin configuration representative of the third and final data release (DR3). We find that the Limber approximation is sufficiently accurate for the analysis of the wide bins of DR1. Contrarily, the 13 bins of DR3 cannot be modelled accurately with the Limber approximation. Instead, the flat-sky approximation is accurate to below $5\%$ in recovering the angular power spectra of galaxy number counts in both cases and can be used to simplify the computation of the full power spectrum in harmonic space for the data analysis of DR3.

Figures (7)

• Figure 1: The number density bins simulated for the photometric surveys of , in six bins and 13 bins equi-populated configurations. The numbering convention we use for the $i$th bin is shown in the coloured blocks above the maxima of each bin.
• Figure 2: Upper Panel: Equal redshift angular power spectrum for the lowest redshift bin of , using six equi-populated bins. In black solid line we show the full calculation result from CLASS, and in dashed purple the Limber approximation result. The green dashed line corresponds to the recalibrated version of the flat-sky approximation. Lower Panel: Relative error (in %) of each approximation from the full calculation result.
• Figure 3: Upper Panel: Equal redshift angular power spectrum for the lowest redshift bin of , using 13 equi-populated bins. In black solid line we show the full calculation result from CLASS, and in dashed purple the Limber approximation result. The green dashed line corresponds to the recalibrated version of the flat-sky approximation. Lower Panel: Relative error (in %) of each approximation from the full calculation result.
• Figure 4: Equal redshift angular power spectrum for the $5^{\rm th}$ redshift bin of (DR1). In black solid line we show the full calculation result from CLASS, and in dashed purple the Limber approximation result. The green dashed line corresponds to the recalibrated version of the flat-sky approximation. The various grey lines represent the contributions, from observational effects, to the exact solution (calculated using CLASS). The L-L contribution in this correlation is too insignificant to be visible on these axes.
• Figure 5: Relative errors in the equal redshift angular power spectrum for the $1^{\rm st}$, $4^{\rm th}$, $8^{\rm th}$, and $13^{\rm th}$ redshift bins of , using 13 equi-populated bins (DR3). In black solid line we show the full calculation result from CLASS, and in dashed purple the Limber approximation result. The green dashed line corresponds to the recalibrated version of the flat-sky approximation. The relative error associated with the Gaussian covariance calculated at $1\sigma$ for the given survey configuration is shown in the grey contour. Note that the $y$-axis ranges are $[-50,50]$, in contrast to \ref{['EucBin1x1of6', 'EucBin1x1of13']}.