Impact of Simulation Box Size for Weak Lensing: Replication and Super-Sample Effects

TL;DR

This study quantifies how finite simulation volumes bias weak-lensing statistics and their covariances by comparing a large-volume BIGBOX run to a tiled, replication-prone small-box ensemble (TILED). It demonstrates that replication induces ~10% biases in PDFs and Minkowski functionals and up to ~25% biases in covariances at high source redshift, with extreme cases reaching ∼100% in particular gnomonic-projected patches. After excluding replication-affected lines of sight to isolate SSC, mean statistics agree to ~1% but variances remain ~10–30% higher than in the large box, a discrepancy that persists under realistic noise and smoothing for current and future surveys. The results underscore the need for large simulation volumes to robustly model WL statistics and their covariances, and they provide practical guidance on when tiling may be acceptable for mean estimates versus covariance calculations, as well as how biases scale with box size and redshift.

Abstract

We quantify the bias caused by small simulation box size on weak lensing observables and covariances, considering both replication and super-sample effects for a range of higher-order statistics. Using two simulation suites -- one comprising large boxes ($3750\,h^{-1}{\rm Mpc}$) and another constructed by tiling small boxes ($625\,h^{-1}{\rm Mpc}$) -- we generate full-sky convergence maps and extract $10^\circ \times 10^\circ$ patches via a Fibonacci grid. We consider biases in the mean and covariance of the angular power spectrum, bispectrum (up to $\ell=3000$), PDF, peak/minima counts, and Minkowski functionals. By first identifying lines of sight that are impacted by replications, we find that replication causes a O$(10\%)$ bias in the PDF and Minkowski functionals, and a O$(1\%)$ bias in other summary statistics. Replication also causes a O$(10\%)$ bias in the covariances, increasing with source redshift and $\ell$, reaching $\sim25\%$ for $z_s=2.5$. We additionally show that replication leads to heavy biases (up to O$(100\%)$ at high redshift) when performing gnomonic projection on a patch that is centered along a direction of replication. We then identify the lines of sight that are minimally affected by replication, and use the corresponding patches to isolate and study super-sample effects, finding that, while the mean values agree to within $1\%$, the variances differ by O$(10\%)$ for $z_s\leq2.5$. We show that these effects remain in the presence of noise and smoothing scales typical of the DES, KiDS, HSC, LSST, Euclid, and Roman surveys. We also discuss how these effects scale as a function of box size. Our results highlight the importance of large simulation volumes for accurate lensing statistics and covariance estimation.

Figures (17)

• Figure 1: Geometry of the light-cone analysed in this work. The horizontal axis gives the comoving radial distance from the observer (located at the origin), while the corresponding redshift values are shown on the top axis. Blue dashed square grid: the $L_{\mathrm{box}} = 625\,h^{-1}\mathrm{Mpc}$ cubes used in the tiled ensemble. The shaded blue square in the upper left marks one such individual small box. Red lines: the $10^{\circ}$ half-opening angle of the light-cone, which fully fits up to $z\simeq2$ inside a single $L_{\mathrm{box}} = 3750\,h^{-1}\mathrm{Mpc}$ simulation. The green arc illustrates a shell of constant comoving distance, intersecting the $10^{\circ}$ aperture at $z\simeq3$.
• Figure 2: Visualization of the Fibonacci grid with $N_{\text{patches}} = 273$, where each patch covers approximately $10\times 10\, {\rm deg}^2$. After optimization and masking, the number of patches is reduced to $N_{\text{patches}} = 194$, effectively covering $47\%$ of the sky. Each panel shows the patch distribution from a top view (left), front view (middle), and the front view after masking replication-influenced patches (RIPs) (right). The black patches in the left and middle panels correspond to tiles that are excluded from all analyses, as the HEALPix grids near poles are highly skewed and cause unstable results after gnomonic projection to the 2D plane; the blue tiles on the right panel correspond to the additional tiles that are excluded from parts of our analysis due to being replication-impacted. The patch in the center of the right panel (at $(\theta=\pi/2, \phi=0)$) is especially affected by replication effects.
• Figure 3: Illustration of convergence maps at different redshifts (columns) for the BIGBOX (top) and TILED box (bottom). A clear kaleidoscope pattern can be observed in the TILED box, which becomes larger at higher source redshift, due to replication effects. Such correlated structures do not appear in the BIGBOX.
• Figure 4: Mean (left panels) and variance (right panels) in the BIGBOX simulation of the angular power spectrum ($C_{\ell}$) and the bispectrum components across three configurations: equilateral ($B_{\ell}^{\text{(eq)}}$), isosceles ($B_{\ell}^{\text{(iso)}}$), and squeezed ($B_{\ell}^{\text{(sq)}}$) as function of multipole $\ell$. Results are presented for source redshifts $z_s = 0.5$ (blue), $1.0$ (orange), $1.5$ (green), $2.0$ (red), and $2.5$ (purple). A comparison with $C_\ell$ from Halofit is given by the dotted line. Note, the power spectrum and bispectrum results are normalized by factors of $\ell(\ell+1)/2\pi$ and $\ell^4$ respectively.
• Figure 5: Same as Figure \ref{['fig:mean_std_ell']} but for $\nu$-binned statistics, including the the probability density function (PDF), peak and minima counts, and Minkowski Functionals ($V_0$, $V_1$, $V_2$).