Testing the cosmic shear spatially-flat universe approximation with generalized lensing and shear spectra

We introduce the Generalised Lensing and Shear Spectra (GLaSS) code which is available for download from https://github.com/astro-informatics/GLaSS It is a fast and flexible public code, written in Python, that computes generalized spherical cosmic shear spectra. The commonly used tomographic and spherical Bessel lensing spectra come as built-in run-mode options. GLaSS is integrated into the Cosmosis modular cosmological pipeline package. We outline several computational choices that accelerate the computation of cosmic shear power spectra. Using GLaSS, we test whether the assumption that using the lensing and projection kernels for a spatially-flat universe—in a universe with a small amount of spatial curvature—negligibly impacts the lensing spectrum. We refer to this assumption as the spatially-flat universe approximation, that has been implicitly assumed in all cosmic shear studies to date. We confirm that the spatially-flat universe approximation has a negligible impact on Stage IV cosmic shear experiments.

Figure 1

Time to compute $50\ell$-modes in GLaSS against grid resolution $N$ (i.e., number of tomographic bins, or $k$-modes in 3D cosmic shear) on a single 2.7 GHz Intel i5 Core. Spectra for resolutions up to $N=400$ can all be computed in less than $1.5s$. This is a sufficient resolution to evaluate the tomographic cosmic shear power spectra and recover all information from the 3D shear field [8]. At high-$\ell$, the computation time, $t$, is dominated by the matrix multiplications and hence scales as $t\propto N^{2.87}$.

Figure 2

Top: The cross-correlated lensing spectrum, $C_{\ell }$, between tomographic bins 9 and 10, using the spatially-flat kernels. Middle: Relative change in $C_{\ell }$ for different kernels with ${\mathrm{\Omega }}_k=-0.08$. The fiducial spectrum, $C_{\ell }$, uses the spatially-flat kernels. $CP$, FP, CL and FL respectively denote when the spatially-curved projection kernel, spatially-flat projection kernel, spatially-curved lensing kernel, and spatially-curved lensing kernel, are used. The sample variance for Euclid-like (ECV) and the cosmic variance for a theoretical all sky survey (ASCV) are also shown. The relative change in $C_{\ell }$ is smaller than the sample variance for a Euclid-like survey for all $\ell <3000$. Bottom: Same as the middle figure, but with ${\mathrm{\Omega }}_k=0.08$.