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 𝑁𝑏 with its simulated counterpart. An accurate estimate of the 𝑁𝑏×𝑁𝑏 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 𝑁𝑟 realizations of the shear field, using a common randomization procedure that recycles the outputs from a smaller number 𝑁𝑠≤𝑁𝑟 of independent ray-tracing 𝑁-body simulations. We study parameter confidence intervals as a function of (𝑁𝑠, 𝑁𝑟) in the range 1≤𝑁𝑠≤200 and 1≤𝑁𝑟≲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/𝑁𝑟) 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/𝑁2𝑟) error degradation that is not negligible when 𝑁𝑟 is only a factor of few larger than 𝑁𝑏. We study the large 𝑁𝑟 limit, and find that a single, 240  Mpc/ℎ sized 5123-particle 𝑁-body simulation (𝑁𝑠=1) can be repeatedly recycled to produce as many as 𝑁𝑟=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 (𝑁𝑠=1 or 2) is sufficient to forecast parameter confidence intervals at percent accuracy.