Precision calculations of the cosmic shear power spectrum projection

TL;DR

This work systematically benchmarks the precision of weak-lensing projection calculations by comparing full spherical projections to a hierarchy of flat-sky and Limber-based approximations. The authors derive and test second-order Limber expansions in both spherical and flat-sky formalisms, showing that the spherical second-order (ExtL2Sph) approach yields sub-percent accuracy for multipoles $\\ell>3$ while remaining computationally fast. Across CFHTLenS data and in forecasted survey regimes, the approximations prove robust against cosmological inferences, with only negligible shifts in parameter constraints when using state-of-the-art approximations. The paper also advocates alternative statistics like COSEBIs and mass aperture to mitigate low-\\ell sensitivity and provides publicly available software (nicaea) for reproducible, high-precision weak-lensing analyses.

Abstract

We compute the spherical-sky weak-lensing power spectrum of the shear and convergence. We discuss various approximations, such as flat-sky, and first- and second- order Limber equations for the projection. We find that the impact of adopting these approximations is negligible when constraining cosmological parameters from current weak lensing surveys. This is demonstrated using data from the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS). We find that the reported tension with Planck Cosmic Microwave Background (CMB) temperature anisotropy results cannot be alleviated. For future large-scale surveys with unprecedented precision, we show that the spherical second-order Limber approximation will provide sufficient accuracy. In this case, the cosmic-shear power spectrum is shown to be in agreement with the full projection at the sub-percent level for l > 3, with the corresponding errors an order of magnitude below cosmic variance for all l. When computing the two-point shear correlation function, we show that the flat-sky fast Hankel transformation results in errors below two percent compared to the full spherical transformation. In the spirit of reproducible research, our numerical implementation of all approximations and the full projection are publicly available within the package nicaea at http://www.cosmostat.org/software/nicaea.

Figures (5)

• Figure 1: The shear power spectrum for different approximations listed in Table \ref{['tab:cases']}. Limber to first order: standard with flat-sky (L1Fl), extended for flat sky (ExtL1Fl), extended hybrid for flat sky (ExtL1FlHyb), and extended in the spherical expansion (ExtL1Sph); second-order Limber approximations: extended flat sky (ExtL2Fl), extended hybrid flat sky (ExtL2FlHyb), and extended spherical expansion (ExtL2Sph); full (exact) spherical projection (FullSph). The left panel shows the total shear power spectrum. The right panel shows the fractional difference resulting from each approximation, relative to the full spherical projection of the shear power spectrum. The two light grey curves on the top show the cosmic variance for KiDS- and Euclid-like surveys with areas of $1,500$ and $15,000$ square degrees, respectively.
• Figure 2: The difference of the two-point shear correlation functions $\xi_+$ (left) and $\xi_-$ (right) of the ExtL2Sph projection relative to the full spherical case (FullSph). Two cases of the shear correlation function transformation for ExtL2Spha are shown, the full spherical case (eq. \ref{['eq:xi_wigner']}, green solid lines), and the flat-sky Hankel transform (eq. \ref{['eqn:xiGG']}, red dashed curves).
• Figure 3: The two-point shear correlation functions $\xi_+$ (left) and $\xi_-$ (right). In the spherical cases (ExtL1Sph, ExtL2Sph, FullSph), $\xi_+$ and $\xi_-$ have been computed using the spherical transform (equation \ref{['eq:xi_wigner']}). For the flat cases the Hankel transform in equation (\ref{['eqn:xiGG']}) was used. The upper panels show the total shear correlation functions for the range of cases listed in Table \ref{['tab:cases']}. The lower panels shows the relative differences to the spherical sky second-order extended Limber approximation, (ExtL2Sph). The theoretical models correspond to the CFHTLenS best-fit cosmological parameters with $\Omega_{\rm m} = 0.279$, $h=0.701$, and $\sigma_8 = 0.79$CFHTLenS-2pt-notomo. For comparison we also show, in the upper panels, the spherical sky second-order extended Limber approximation model for the Planck-best fit cosmology with $\Omega_{\rm m} = 0.3$, $h=0.67$ and $\sigma_8 = 0.83$2015arXiv150201589P.
• Figure 4: Integrand of $\xi_+$ (upper), $\xi_-$ (upper middle), $E_1$ (lower middle, E-COSEBIs) and $\langle M_{\rm ap} \rangle^2$ (lower panel). All integrands are of the form $\ell F(\ell) P(\ell)$, where $F(\ell)$ is the corresponding weight-function for each statistic and $P(\ell)$ is the E-mode convergence power spectrum, with the exception of $\xi_\pm$, for which $P(\ell)$ is equal to the sum of the E and B-mode power spectra. Two cases are shown for each statistic as listed in each caption. For the aperture mass statistic $\theta_{\rm max}=2\theta$ is shown. Note that higher order COSEBIs modes generally probe larger $\ell$-modes, hence here we only show the lowest mode $E_1$. All values are normalized with respect to their maximum value. This figure illustrates how different two-point cosmic shear statistics have different dependences between the angular scales sampled and the $\ell$-range probed.
• Figure 5: The relative differences in percentage of the spherical first- and second-order Limber shear power spectra with respect to the full projection as function of wave mode $\ell$, (see Table \ref{['tab:cases']}). In this figure, the redshift distribution is chosen to be at a single source plane at $z_{\rm S}=1$, with the cosmological parameters (see text) are chosen to match vande2012, see their Figure 7.3 for comparison.