Catastrophic photometric redshift errors: weak lensing survey
  requirements
  We study the sensitivity of weak lensing surveys to the effects of
catastrophic redshift errors - cases where the true redshift is misestimated by
a significant amount. To compute the biases in cosmological parameters, we
adopt an efficient linearized analysis where the redshift errors are directly
related to shifts in the weak lensing convergence power spectra. We estimate
the number Nspec of unbiased spectroscopic redshifts needed to determine the
catastrophic error rate well enough that biases in cosmological parameters are
below statistical errors of weak lensing tomography. While the straightforward
estimate of Nspec is ~10^6 we find that using only the photometric redshifts
with z<=2.5 leads to a drastic reduction in Nspec to ~30,000 while negligibly
increasing statistical errors in dark energy parameters. Therefore, the size of
spectroscopic survey needed to control catastrophic errors is similar to that
previously deemed necessary to constrain the core of the z_s-z_p distribution.
We also study the efficacy of the recent proposal to measure redshift errors by
cross-correlation between the photo-z and spectroscopic samples. We find that
this method requires ~10% a priori knowledge of the bias and stochasticity of
the outlier population, and is also easily confounded by lensing magnification
bias. The cross-correlation method is therefore unlikely to supplant the need
for a complete spectroscopic redshift survey of the source population.
