Extended Limber Approximation

TL;DR

This paper introduces a systematic $1/( abla\ell+1/2)$ expansion of the angular cross-power spectrum to extend the traditional Limber approximation. The authors derive the leading and next-to-leading terms, showing how the second-order correction, involving derivatives of the projection kernels and the power spectrum, significantly improves accuracy for narrow or weakly overlapping redshift distributions. They establish the proper flat-sky mapping with the $\ell+1/2$ convention and demonstrate, through several examples, that the extended formula reduces errors from $O(1/\ell^2)$ to $O(1/\ell^4)$ and provides practical convergence criteria. The framework is applicable to galaxy, weak-lensing, and CMB cross- and auto-correlations, and highlights the importance of using $\ell+1/2$ for precision.

Abstract

We develop a systematic derivation for the Limber approximation to the angular cross-power spectrum of two random fields, as a series expansion in 1/(\ell+1/2). This extended Limber approximation can be used to test the accuracy of the Limber approximation and to improve the rate of convergence at large \ell's. We show that the error in ordinary Limber approximation is O(1/\ell^2). We also provide a simple expression for the second order correction to the Limber formula, which improves the accuracy to O(1/\ell^4). This correction can be especially useful for narrow redshift bins, or samples with small redshift overlap, for which the zeroth order Limber formula has a large error. We also point out that using \ell instead of (\ell+1/2), as is often done in the literature, spoils the accuracy of the approximation to O(1/\ell).

Figures (3)

• Figure 1: Left Panel: The galaxy auto-correlation for a narrow redshift bin ($\sigma_{z}=0.01$) at redshifts $z_0=0.3$ (upper black curves) and $z_0=0.6$ (lower cyan/gray curves). The solid lines are the exact $C_{\ell}$ (Equation (\ref{['Clexact']})) the dotted lines are the $0^{th}$ order Limber approximation, the dashed lines are the Limber approximation keeping the first $\mathcal{O}(\nu^{-2})$ term in Equation (\ref{['ClexLimber']}). Right panel: The difference between the Limber approximation at $0^{th}$ and $2^{nd}$ order (in $1/\nu$) and the exact angular power spectrum in redshift bins at $z_0=0.3$ (black) and $z_0=0.6$ (cyan/gray). The radii of convergence for the Limber expansion are roughly at $\ell \sim 15$ and $30$ respectively.
• Figure 2: Left panel: angular cross-power spectra between two samples with Gaussian width $\sigma_z=0.05$. Upper black curves are for the cross power spectrum between bins at $z_0=0.3$ and $z_0=0.4$, lower cyan/gray curves are for more widely separated bins with $z_0=0.3$ and $z_0=0.5$. Solid lines show the exact power spectrum, dotted the $0^{th}$ order Limber formula and dashed the Limber approximation to $2^{nd}$ order in $1/\nu$. Right panel: the difference between the curves shown on left.
• Figure 3: Left panel: angular cross-power spectrum between two redshift bins centered at $z=0.4$ with different widths $\sigma_1=0.10$ and $\sigma_2=0.01$. Solid lines show the exact power spectrum, dotted the $0^{th}$ order Limber calculation and dashed the Limber approximation to $2^{nd}$ order in $1/\nu$. Right panel: the difference between the curves shown at left.