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×N_b feature covariance matrix C is essential to obtain accurate parameter confidence intervals. When 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≤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 ≤N_s≤200 and 1 ≤N_r≲10⁵. Previous work [S. Dodelson and M. D. Schneider, Phys. Rev. D 88, 063537 (2013)] 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_r²) 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, 240 Mpc /h sized 512³-particle N -body simulation (N_s=1 ) can be repeatedly recycled to produce as many as N_r=few×10⁴ 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.