Institute of Astronomy

Flat-sky angular power spectrum revisited

SpeakerTalk DateTalk Series
Zucheng Gao3 May 2023Institute of Astronomy Seminars


We revisit the flat-sky approximation for evaluating the angular power spectra of projected random fields by retaining information about the correlations along the line of sight. For the case of projections with broad, overlapping radial window functions, these line-of-sight correlations are suppressed and are ignored in the commonly adopted Limber approximation. However, retaining the correlations is important for narrow window functions or unequal-time spectra but introduces significant computational difficulties due to the highly oscillatory nature of the integrands involved. We deal with the integral over line-of-sight wave-modes in the flat-sky approximation analytically, using the FFTlog expansion of the 3D power spectrum. This results in an efficient computational method with a performance time comparable to the Limber approximation, which is a substantial improvement compared to any full-sky approaches. We apply our results to galaxy clustering (with and without redshift-space distortions) and CMB lensing observables in a flat
ΛCDM universe. In the case of galaxy clustering, we find excellent agreement with the full-sky results on large (percent-level agreement) and intermediate (subpercent agreement) scales, dramatically outperforming the Limber approximation for both wide and narrow window functions, and in equal- and unequal-time cases. In the case of CMB lensing, the flat-sky approach yields subpercent agreement for multipoles ℓ ≳ 20 with discrepancies gradually increasing to tens of percent at smaller ℓ. We further analyse these angular power spectra by isolating the projection effects due to the observable and survey-specific window functions, separating them from the effects due to integration along the line of sight and unequal-time mixing in the 3D power spectrum. All of the angular power spectrum results presented in this paper are obtained using a Python code implementation, which we make publicly available.


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