Systematic Errors in Future Weak Lensing Surveys: Requirements and Prospects for Self-Calibration

TL;DR

The paper develops a unified framework to quantify how three generic weak-lensing systematics—redshift errors, multiplicative shear calibration, and additive shear errors—affect cosmological constraints. Using a Fisher-matrix forecast for DES, SNAP, and LSST with tomographic power spectra (and optionally bispectra), the authors derive stringent but survey-appropriate requirements for redshift bias control, shear calibration, and additive-error modeling. A key result is that self-calibration is achievable for many nuisance modes, and that combining power spectrum and bispectrum information substantially mitigates degradation (20–30%) for poorly constrained dark-energy parameters, though fixed, well-measured combinations like $w={\rm const}$ gain less from self-calibration. Additive errors remain the most challenging, requiring precise characterization (mean additive shear around $2\times10^{-5}$) to avoid dominating the error budget, with space-based surveys like SNAP offering advantages due to reduced atmospheric effects. Overall, PS+BS provides a robust route to suppress systematic impacts, enabling future weak-lensing surveys to reach their cosmological potential while highlighting areas—particularly additive systematics and photometric redshift calibration—that demand dedicated effort and simulations.

Abstract

We study the impact of systematic errors on planned weak lensing surveys and compute the requirements on their contributions so that they are not a dominant source of the cosmological parameter error budget. The generic types of error we consider are multiplicative and additive errors in measurements of shear, as well as photometric redshift errors. In general, more powerful surveys have stronger systematic requirements. For example, for a SNAP-type survey the multiplicative error in shear needs to be smaller than 1%(fsky/0.025)^{-1/2} of the mean shear in any given redshift bin, while the centroids of photometric redshift bins need to be known to better than 0.003(fsky/0.025)^{-1/2}. With about a factor of two degradation in cosmological parameter errors, future surveys can enter a self-calibration regime, where the mean systematic biases are self-consistently determined from the survey and only higher-order moments of the systematics contribute. Interestingly, once the power spectrum measurements are combined with the bispectrum, the self-calibration regime in the variation of the equation of state of dark energy w_a is attained with only a 20-30% error degradation.

Figures (9)

• Figure 1: Select three modes (first, second and seventh) of perturbation to the distribution of galaxies $n(z)$ for the DES (left) and SNAP (right), shown together with the original unperturbed distribution (dashed curve in each panel). These three modes correspond to the modes of perturbation in the $z_p-z_s$ relation shown in Fig. \ref{['fig:deltaz_vs_zs']} in the Appendix. Note that SNAP's fiducial $n(z)$ is broader, making the resulting wiggles in $n_p(z_p)$ less pronounced and the weak lensing measurements therefore less susceptible to the redshift biases. Note also that the allowed perturbations to $n(z)$ are not only located near the peak of the distribution, but can also have significant wiggles near the tails of the distribution.
• Figure 2: Degradation in the cosmological parameter accuracies as a function of our prior knowledge of $\delta z \equiv z_p - z_s$. We assume equal Gaussian priors to each redshift bin centroid, shown on the x-axis. For example, to have less than $\sim 50\%$ degradation in $\Omega_M$, $\sigma_8$ or $w$, we need to control the redshift bias to about $0.003\,(f_{\rm sky}/f_{\rm sky, fid})^{-1/2}$ or better for the DES and SNAP, and to about $0.0015\,(f_{\rm sky}/f_{\rm sky, fid})^{-1/2}$ or better for the LSST. For the varying equation of state parameterization, the requirements for the best measured combination of $w_0$ and $w_a$ ($F(w_0, w_a)$) are identical to those for $w={\rm const}$, while the requirements on $w_0$ and $w_a$ individually are somewhat less stringent.
• Figure 3: Degradation in marginalized errors in $\Omega_M$, $\sigma_8$ and $w={\rm const}$, as well as $w_0$ and $w_a$, as a function of our prior knowledge of the redshift bias coefficients $f_i$ for the DES, SNAP and LSST. We use $N_{\rm cheb}=30$ parameters $f_i$ that describe the bias in redshift and give equal prior to each of them, shown on the x-axis. For the DES and SNAP, knowledge of $f_i$ to better than $0.001\,(f_{\rm sky}/f_{\rm sky, fid})^{-1/2}$, corresponding to redshift bias of $|z_p-z_s|\lesssim 0.001\,(f_{\rm sky}/f_{\rm sky, fid})^{-1/2}$ for each Chebyshev mode, is desired as it leads to error degradations of about 50% or less. For LSST the requirement is about a factor of two stronger. These results are corroborated by computing the bias in cosmological parameters as discussed in the text.
• Figure 4: Degradation in marginalized errors in $\Omega_M$, $\sigma_8$ and $w={\rm const}$, as well as $w_0$ and $w_a$, as a function of our prior knowledge of the shear multiplicative factors. We give equal prior to multiplicative factors in all redshift bins, and show results for the DES, SNAP and LSST. For example, existence of the multiplicative error of $0.01\,(f_{\rm sky}/f_{\rm sky, fid})^{-1/2}$ (or 1% in shear for the fiducial sky coverages) in each redshift bin leads to 50% increase in error bars on $\Omega_M$, $\sigma_8$ and $w$ for the DES and SNAP, and about a 100% degradation for LSST.
• Figure 5: Degradation in the cosmological parameter accuracies as a function of the fiducial value of the additive shear errors $b_i$, assuming no prior on the $b_i$. We show results for SNAP and for several values of the correlation coefficients between different bins, $\rho$, and marginalizing over the spatial power law exponent $\alpha$ which has fiducial value $\alpha=0$. The results are insensitive to various details, as discussed in the text, and the only exception is the possibility that the additive effect on the cross-power spectra is negligibly suppressed (i.e. that the cross-power correlation is close to 100%). We conclude that the mean additive shear will need to be known to about $\sim 2\times 10^{-5}$, corresponding to shear variance of $\sim 10^{-4}$ on scales of $\sim 10$ arcmin.