The unequal-time matter power spectrum: impact on weak lensing
  observables
  We investigate the impact of a common approximation on weak lensing power
spectra: the use of single-epoch matter power spectra in integrals over
redshift. We disentangle this from the closely connected Limber's
approximation. We derive the unequal-time matter power spectrum at one-loop in
standard perturbation theory and effective field theory to deal with non-linear
physics. We compare these formalisms and conclude that the unequal-time power
spectrum using effective field theory breaks for larger scales. As an
alternative, we introduce the midpoint approximation. We also provide, for the
first time, a fitting function for the time evolution of the effective field
theory counterterms based on the Quijote simulations. Then we compute the
angular power spectrum using a range of approaches: the Limber's approximation,
and the geometric and midpoint approximations. We compare our results with the
exact calculation at all angular scales using the unequal-time power spectrum.
We use DES Y1 and LSST-like redshift distributions for our analysis. We find
that the use of the Limber's approximation in weak lensing diverges from the
exact calculation of the angular power spectrum on large-angle separations,
$\ell < 10$. Even though this deviation is of order $2\%$ maximum for cosmic
lensing, we find the biggest effect for galaxy clustering and galaxy-galaxy
lensing. We show that not only is this true for upcoming galaxy surveys, but
also for current data such as DES Y1. Finally, we make our pipeline and
analysis publicly available as a Python package called unequalpy.
