Perturbative and numerical study of nonlinear relativistic effects in weak lensing

Abstract

The standard weak lensing formalism assumes that the lensing map relating the observed image of a source to its intrinsic shape depends only on the deflection angle. We show that this description is incomplete beyond linear perturbation theory, even when only scalar perturbations are present at first order. Using the Jacobi map formalism, we derive expressions for the rotation field, shear B-modes, and their angular power spectra at second order in relativistic perturbation theory. In the standard formalism, rotation and shear B-modes share the same spectrum, however, this degeneracy is broken once the parallel transport of the Sachs basis is consistently taken into account. We quantify this correction numerically, finding a difference of about $5\%$ on large angular scales $\ell \sim 5$ for sources at redshift $z_\mathrm{s} = 0.5$. We also investigate frame-dragging effects, which are usually neglected in weak lensing. We present the first analytical derivation of the corresponding impact on the angular power spectrum of shear B-modes and show that it becomes the dominant contribution on scales $\ell \lesssim 10$. While both Sachs-basis rotation and frame dragging significantly affect shear B-modes on large scales, their effect on the observed galaxy ellipticity is of order $1\%$, making these nonlinear relativistic corrections challenging to detect in practice. Our results are supported by relativistic simulations of weak lensing observables, including the first numerical study of frame dragging in the power spectra of the lensing convergence and cosmic shear.

Figures (8)

• Figure 1: Dimensionless power spectra for the metric perturbations $\Psi$, $B_\alpha$, and $h_{\alpha\beta}$ at redshift $z=0.25$ from perturbation theory up to 1-loop order. The 1-loop calculation for $\Psi$, which we do with CLASS-PTChudaykin:2020aoj here, also contains a contribution from the correlation between first and third order perturbations, sometimes denoted as $P_{13}$ in the literature. This contribution allows the 1-loop result to become negative, indicated with a dashed line style in the plot. The full power spectrum of $\Psi$ (not shown in the plot) is given by the sum of linear and 1-loop result. $B_\alpha$ and $h_{\alpha\beta}$ vanish at first order, hence their power spectra are given by their $P_{22}$ contribution only, i.e. the autocorrelation of the second-order perturbations. Their expressions can be found in App. \ref{['ap:numer-Four']}.
• Figure 2: Angular power spectra of scalar-induced convergence and shear E-modes (top panels) and their relative difference (bottom panels). Left: Simulated spectra are computed from the fully relativistic Jacobi map. The theoretical prediction for the convergence includes the relativistic corrections from Eq. \ref{['eq:true-kappa']}. Right: Simulated spectra are extracted from maps of the deflection angle. Theoretical predictions are estimated using the standard weak lensing formalism and do not include relativistic corrections to the convergence. The source redshift is $z_{\rm s} = 0.5$.
• Figure 3: Angular power spectra of scalar rotation and shear B-modes (top panels) and their relative difference (bottom panels). Left: Simulated spectra are computed from the fully relativistic Jacobi map. Right: Simulated spectra are extracted from maps of the deflection angle. The source redshift is $z_{\rm s} = 0.5$.
• Figure 4: Contributions to the angular power spectrum of the scalar shear B-modes. Dotted lines indicate negative values of the relativistic corrections. Left: Source redshift is $z_{\rm s} = 0.5$. Right: Source redshift is $z_{\rm s} = 1$.
• Figure 5: Frame-dragging contributions to the weak lensing observables at source redshift $z_{\rm s} = 0.5$ and $z_{\rm s} = 1$. Left: Lensing convergence (solid lines) and shear E-modes (dotted lines). Right: Shear B-modes. Frame dragging does not contribute at leading order to the rotation, and its contribution is approximately ten orders of magnitude smaller than the frame-dragging contribution to the shear B-modes. For this reason, it is not shown in this plot.