Unequal-Time Correlators for Cosmology
  Measurements of the power spectrum from large-scale structure surveys have to
date assumed an equal-time approximation, where the full cross-correlation
power spectrum of the matter density field evaluated at different times (or
distances) has been approximated either by the power spectrum at a fixed time,
or in an improved fashion, by a geometric mean $P(k; r_1, r_2)=[P(k; r_1) P(k;
r_2)]^{1/2}$. In this paper we investigate the expected impact of the geometric
mean ansatz, and present an application in assessing the impact on weak
gravitational lensing cosmological parameter inference, using a perturbative
unequal-time correlator. As one might expect, we find that the impact of this
assumption is greatest at large separations in redshift $\Delta z > 0.3$ where
the change in the amplitude of the matter power spectrum can be as much as $10$
percent for $k > 5h$Mpc$^{-1}$. However, of more concern is that the
corrections for small separations, where the clustering is not close to zero,
may not be negligibly small. In particular, we find that for a Euclid- or
LSST-like weak lensing experiment the assumption of equal-time correlators may
result in biased predictions of the cosmic shear power spectrum, and that the
impact is strongly dependent on the amplitude of the intrinsic alignment
signal. To compute unequal-time correlations to sufficient accuracy will
require advances in either perturbation theory to high $k$-modes, or extensive
use of simulations.
