The gauge invariant cosmological Jacobi map from weak lensing at leading order

TL;DR

This paper calculates the weak lensing Jacobi map at first order, demonstrating its gauge invariance and symmetry, and clarifies the distinct origins of shear B-modes and E-modes, especially from tensor perturbations.

Contribution

It provides the first-order gauge-invariant and symmetric weak lensing Jacobi map, clarifying the separate contributions of vector and tensor perturbations to shear modes.

Findings

Shear B-modes are induced by vector and tensor perturbations.

The shear B-mode power spectrum is not solely determined by rotation.

The E-mode shear power spectrum is not fully determined by convergence.

Abstract

We compute the weak lensing Jacobi map at first order in perturbation theory and show that it is both, gauge invariant and symmetric. Linear perturbations therefore do not induce any rotation. However, vector and tensor perturbations do induce $B$-modes in the shear. We show that contrary to what is often claimed in the literature, the shear $B$-mode power spectrum is not fully determined by the rotation power spectrum. Also the $E$-mode shear power spectrum is not determined by the convergence power spectrum. While this difference is small for scalar perturbations, it becomes very relevant for tensor perturbations, i.e. gravitational waves.