Beyond Limber: Efficient computation of angular power spectra for galaxy
  clustering and weak lensing
  Angular two-point statistics of large-scale structure observables are
important cosmological probes. To reach the high accuracy required by the
statistical precision of future surveys, some of these statistics may need to
be computed without the commonly employed Limber approximation; the exact
computation however requires integration over Bessel functions, and a
brute-force evaluation is slow to converge. We present a new method based on
our generalized FFTLog algorithm for the efficient computation of angular power
spectra beyond the Limber approximation. The new method significantly
simplifies the calculation and improves the numerical speed and stability. It
is easily extended to handle integrals involving derivatives of Bessel
functions, making it equally applicable to numerically more challenging cases
such as contributions from redshift-space distortions and Doppler effects. We
implement our method for galaxy clustering and galaxy-galaxy lensing power
spectra. We find that using the Limber approximation for galaxy clustering in
future analyses like LSST Year 1 and DES Year 6 may cause significant biases in
cosmological parameters, indicating that going beyond the Limber approximation
is necessary for these analyses.
