Cosmic shear covariance: the log-normal approximation

Abstract

Context. Accurate estimates of the errors on the cosmological parameters inferred from cosmic shear surveys require accurate estimates of the covariance of the cosmic shear correlation functions. Aims. We seek approximations to the cosmic shear covariance that are as easy to use as the common approximations based on normal (Gaussian) statistics, but yield more accurate covariance matrices and parameter errors. Methods. We derive expressions for the cosmic shear covariance under the assumption that the underlying convergence field follows log-normal statistics. We also derive a simplified version of this log-normal approximation by only retaining the most important terms beyond normal statistics. We use numerical simulations of weak lensing to study how well the normal, log-normal, and simplified log-normal approximations as well as empirical corrections to the normal approximation proposed in the literature reproduce shear covariances for cosmic shear surveys. We also investigate the resulting confidence regions for cosmological parameters inferred from such surveys. Results. We find that the normal approximation substantially underestimates the cosmic shear covariances and the inferred parameter confidence regions, in particular for surveys with small fields of view and large galaxy densities, but also for very wide surveys. In contrast, the log-normal approximation yields more realistic covariances and confidence regions, but also requires evaluating slightly more complicated expressions. However, the simplified log-normal approximation, although as simple as the normal approximation, yields confidence regions that are almost as accurate as those obtained from the log-normal approximation. The empirical corrections to the normal approximation do not yield more accurate covariances and confidence regions than the (simplified) log-normal approximation. Moreover, they fail to produce positive-semidefinite data covariance matrices in certain cases, rendering them unusable for parameter estimation. Conclusions. The log-normal or simplified log-normal approximation should be used in favour of the normal approximation for parameter estimation and parameter error forecasts. More generally, any approximation to the cosmic shear covariance should ensure a positive-(semi)definite data covariance matrix.

The bispectrum covariance beyond Gaussianity

“…Indeed, as Kayo et al (2001) have shown, the distribution of the density fluctuations in the universe can rather well be described by a log-normal distribution. Moreover, Taruya et al (2002) and Hilbert et al (2011) have shown that the same holds true for the cosmic shear convergence field, which serves as justification for using these particular non-Gaussian fields for our study. Furthermore, Hilbert et al (2011) employed the log-normal ansatz to calculate the covariance of the power spectrum of cosmic shear fields; they showed that this model provides fairly accurate results, when compared to ray-tracing simulations.…”

“…Moreover, Taruya et al (2002) and Hilbert et al (2011) have shown that the same holds true for the cosmic shear convergence field, which serves as justification for using these particular non-Gaussian fields for our study. Furthermore, Hilbert et al (2011) employed the log-normal ansatz to calculate the covariance of the power spectrum of cosmic shear fields; they showed that this model provides fairly accurate results, when compared to ray-tracing simulations. However, we caution that the log-normal approach has yet not been shown to give a realistic approximation for higher-order correlation functions and that the log-normal is not completely determined by its moments (Coles & Jones 1991;Carron & Neyrinck 2012).…”

“…As shown by Taruya et al (2002) and Hilbert et al (2011), the cosmic shear convergence distribution is rather well described by a log-normal; from their results we can estimate characteristic values of α.…”

“…Furthermore, the deviation of the diagonal elements from those predicted by the Gaussian approximation are negligible for α < ∼ 0.5, and below 10% for α < ∼ 1. From simulations of convergence fields in weak lensing shown in Taruya et al (2002) and Hilbert et al (2011), we found that realistic values of α are not larger than ∼0.7 on angular scales of about 4 ; hence, the Gaussian approximation is expected to yield an accurate description of the bispectrum covariance -which is fortunate, given the complexity of the full expression of this covariance.…”

Retrieving the three-dimensional matter power spectrum and galaxy biasing parameters from lensing tomography

“…-The assumption of Gaussian statistics for the cosmic shear fields is inaccurate for nonlinear scales (roughly ten arcmin or smaller), which leads to an underestimation of the shear noise covariance (Semboloni et al 2007;Hartlap et al 2009;Hilbert et al 2011). The adopted log-normal statistics (Bouchet et al 1993) of the galaxy clustering, on the other hand, is presumably a good approximation of the galaxygalaxy lensing and galaxy clustering covariance.…”