Sample variance in weak lensing: how many simulations are required?
  Constraining cosmology using weak gravitational lensing consists of comparing
a measured feature vector of dimension $N_b$ with its simulated counterpart. An
accurate estimate of the $N_b\times N_b$ feature covariance matrix $\mathbf{C}$
is essential to obtain accurate parameter confidence intervals. When
$\mathbf{C}$ is measured from a set of simulations, an important question is
how large this set should be. To answer this question, we construct different
ensembles of $N_r$ realizations of the shear field, using a common
randomization procedure that recycles the outputs from a smaller number
$N_s\leq N_r$ of independent ray-tracing $N$--body simulations. We study
parameter confidence intervals as a function of ($N_s,N_r$) in the range $1\leq
N_s\leq 200$ and $1\leq N_r\lesssim 10^5$. Previous work has shown that
Gaussian noise in the feature vectors (from which the covariance is estimated)
lead, at quadratic order, to an $O(1/N_r)$ degradation of the parameter
confidence intervals. Using a variety of lensing features measured in our
simulations, including shear-shear power spectra and peak counts, we show that
cubic and quartic covariance fluctuations lead to additional $O(1/N_r^2)$ error
degradation that is not negligible when $N_r$ is only a factor of few larger
than $N_b$. We study the large $N_r$ limit, and find that a single, 240Mpc$/h$
sized $512^3$-particle $N$--body simulation ($N_s=1$) can be repeatedly
recycled to produce as many as $N_r={\rm few}\times10^4$ shear maps whose power
spectra and high-significance peak counts can be treated as statistically
independent. As a result, a small number of simulations ($N_s=1$ or $2$) is
sufficient to forecast parameter confidence intervals at percent accuracy.
