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

1. Problem Statement

The Euclid mission aims to map the large-scale structure of the universe using a photometric survey of approximately $1.5 \times 10^9$ galaxies. A primary goal is to measure the angular power spectra ($C_\ell$) of galaxy number counts to infer cosmological parameters via Markov Chain Monte Carlo (MCMC) analysis.

• The Challenge: Calculating the full theoretical angular power spectrum (using codes like CLASS or CAMB) is computationally expensive, making it infeasible for the thousands of iterations required by MCMC.
• Current Standard: The Limber approximation is widely used to speed up calculations (by $\sim$100x) by replacing the integral over the line-of-sight wavenumber with a Dirac delta function. However, it is known to fail for narrow redshift bins and at large scales (low multipoles, $\ell \lesssim 100$), often introducing errors of a factor of 2 or more.
• The Question: Is the Limber approximation sufficiently accurate for Euclid's specific data releases (DR1 and DR3), or does it introduce biases that exceed observational uncertainties? If not, can the flat-sky approximation provide a viable, faster alternative?

2. Methodology

The authors compared three methods for computing the angular power spectra of galaxy number counts:

1. Full Calculation: The exact solution using the Boltzmann solver CLASS, including density fluctuations, redshift-space distortions (RSD), and relativistic effects (lensing magnification).
2. Limber Approximation: The standard approximation used in many analysis pipelines (e.g., CLOE).
3. Flat-Sky Approximation: An improved approximation that treats the sky as a plane and retains the parallel component of the wavenumber ($k_\parallel$), while approximating the perpendicular component. This method uses a "recalibrated" version where $\ell$ is replaced by $\ell + 1/2$ to improve accuracy.

Simulation Setup:

• Data Source: The Flagship simulation, a high-resolution N-body simulation ($3600 h^{-1}$ Mpc box) populated with galaxies using Halo Occupation Distribution (HOD) and sub-halo abundance matching.
• Scenarios:
  • DR1 (Data Release 1): A conservative scenario with 6 equi-populated redshift bins and a magnitude cut of $I_E < 23.5$.
  • DR3 (Data Release 3): The final mission scenario with 13 equi-populated redshift bins and a deeper magnitude cut of $I_E < 24.5$.
• Statistical Assessment: The authors generated 1,000 noisy mock power spectra based on the theoretical covariance matrix (Gaussian approximation). They calculated the $\chi^2$ statistic for the Limber and flat-sky approximations relative to the "true" full-sky result to determine if the approximations could be statistically distinguished from the truth given the survey's error budget.

3. Key Contributions

• Quantitative Validation for Euclid: The paper provides the first rigorous, simulation-based assessment of the Limber approximation's validity specifically for Euclid's photometric galaxy clustering across different data release configurations.
• Performance of Flat-Sky Approximation: It demonstrates that the flat-sky approximation (specifically the recalibrated version) recovers the full angular power spectrum with an accuracy better than 5% across all scales and redshift bins, even for the narrow bins of DR3.
• Computational Trade-off Analysis: The study highlights the computational cost-benefit analysis. While the flat-sky approximation is slower than Limber, it offers a significant speed-up (factor of $\sim$2) compared to the full calculation, whereas Limber is $\sim$200x faster.

4. Results

DR1 Configuration (6 Bins)

• Limber Performance: The Limber approximation is sufficiently accurate. The deviations from the full calculation fall within the $1\sigma$ observational error bars.
• Statistical Test: The $\Delta\chi^2$ distribution for Limber peaks around 12 ($p \approx 0.038$), which is marginally rejected at the 95% confidence level, but the flat-sky approximation performs nearly identically to the full calculation ($\Delta\chi^2 \approx 0.2$, $p \approx 0.39$).
• Conclusion: For the initial, wider bins of DR1, the computational speed of Limber makes it a viable choice, as the bias is not statistically significant relative to the survey noise.

DR3 Configuration (13 Bins)

• Limber Performance: The Limber approximation fails. The narrow redshift bins and deeper survey depth make the approximation invalid.
  • Relative errors exceed 5% and often exceed the $1\sigma$ error bars at low multipoles ($\ell \lesssim 100$).
  • The failure is driven by the inability of Limber to correctly model RSD terms and the $k_\parallel$ dependence in narrow bins.
• Statistical Test: The $\Delta\chi^2$ distribution for Limber peaks around 410 ($p < 0.001$), strongly rejecting the null hypothesis that it is consistent with the full calculation.
• Flat-Sky Performance: The flat-sky approximation remains highly accurate ($\Delta\chi^2 \approx 5$, $p \approx 0.131$), failing to reject the null hypothesis. It recovers the spectrum within the observational uncertainties.
• Conclusion: For the final DR3 release, the Limber approximation introduces a biased result that cannot be used for precision cosmology. The flat-sky approximation is required.

Physical Insights

• The errors in Limber are most pronounced for auto-correlations of high-redshift bins and cross-correlations between disparate bins.
• The failure is largely attributed to the Redshift Space Distortions (RSD) and density terms, which Limber approximates poorly in narrow bins. The lensing magnification (L) terms remain reasonably well-approximated by Limber even in DR3.

5. Significance

• Pipeline Recommendations: The authors recommend that Euclid's data analysis pipelines (such as CLOE) must switch to the flat-sky approximation for the final data release (DR3) to avoid biased cosmological constraints.
• Hybrid Approach: A potential strategy for future analysis is a hybrid approach: using the Limber approximation for wide bins (DR1) or for lensing-dominated terms (L-L, D-L) where it remains accurate, and using the flat-sky approximation for the critical density and RSD terms in narrow bins.
• Computational Optimization: While the flat-sky approximation is currently only $\sim$2x faster than the full calculation (due to the need for triple integrals), the paper notes that further optimizations (e.g., pre-computing the $k$-integral or using BLAST-like techniques) could make it a practical standard for MCMC analysis.
• Spectroscopic Survey Implications: The findings also suggest that for the spectroscopic survey's 6x2pt analysis (combining spectroscopic clustering with weak lensing), the flat-sky approximation is necessary if narrow redshift bins are used, as Limber would be insufficient.

In summary, this paper establishes that while the Limber approximation is adequate for the early, wider-bin Euclid data, it is fundamentally inadequate for the final, high-precision dataset, necessitating the adoption of the flat-sky approximation to ensure unbiased cosmological results.