Weak Gravitational Lensing

TL;DR

The work provides a comprehensive overview of weak gravitational lensing as a cosmological probe, detailing the theoretical foundations (deflection, lensing potential, convergence, and shear) and the practical measurement pipeline (PSF handling, ellipticity estimation, and calibration). It connects lensing observables to the matter distribution through the convergence and power spectra, implements the Limber approximation for projecting 3D $P_{\delta\delta}(k,z)$ into 2D statistics, and discusses convergence reconstruction via Kaiser–Squires. It also covers major systematics (intrinsic alignments, baryonic feedback, redshift and shear biases) and outlines the robust 3×2pt framework (galaxy clustering, galaxy–galaxy lensing, and cosmic shear) used by current Stage III surveys, with forecasts for Stage IV experiments. The discussion emphasizes Bayesian inference, non-Gaussian information, and cross-correlations with other probes as key directions to enhance cosmological constraints, enabling stringent tests of $\Lambda$CDM and potential extensions in the era of precision cosmology.

Abstract

This chapter provides a comprehensive overview of weak gravitational lensing and its current applications in cosmology. We begin by introducing the fundamental concepts of gravitational lensing and derive the key equations for the deflection angle, lensing potential, convergence, and shear. We explore how weak lensing can be used as a cosmological probe, discussing cosmic shear, galaxy-galaxy lensing, and their combination with galaxy clustering in the 3$\times$2pt analysis. The chapter covers the theoretical framework for modeling lensing observables, shear estimation techniques, and major systematic effects such as intrinsic alignments and baryonic feedback. We review the current results of weak lensing cosmology from major surveys and outline prospects for future advancements in the field.

Figures (14)

• Figure 1: Sketch of a gravitational thin lens system. The optical axis runs from the observer through the center of the lens. The angle between the source S and the optical axis is $\beta$, the angle between the image I and the optical axis is $\theta$. The light ray towards the image is bent by $\delta \theta$, measured at the lens. The deflection angle $\alpha$ is measured at the observer.
• Figure 2: The Born approximation assumes that the perturbations to the light path due to gravitational lensing by the structure along the line of sight are negligible and that we can compute the deflection angle integrating along the straight line back to the image position.
• Figure 3: Inverse critical surface density ($\Sigma^{-1}_\mathrm{crit}$) as a function of lens redshift ($z_l$) for various source redshifts ($z_s$). The plot shows how the lensing efficiency changes with lens redshift for different source redshifts, ranging from 0.5 to 1.5. Each line represents a different source redshift, with colors indicating the $z_s$ value according to the colorbar on the right. This figure illustrates the dependence of lensing strength on the relative distances between the observer, lens, and source. A rule of thumb is that the lensing efficiency peaks at around half the redshift between the observer and the source.
• Figure 4: The left side of the image demonstrates the effects of the Jacobian matrix elements -- shear $\gamma = (\gamma_1, \gamma_2)$ and convergence $\kappa$ -- on an initially circular background object. The right side illustrates the key components that define an ellipse: the semi-major axis $a$, semi-minor axis $b$, and the position angle $\varphi$.
• Figure 5: The massive foreground cluster (Abell 1689) causes the images of the background galaxies to be distorted, forming arcs, due to strong gravitational lensing. The arcs are tangentially aligned, and so their ellipticity is oriented tangent to the direction of the foreground mass, in this case the cluster, represented by the lens point $L$. Image taken with the Advanced Camera for Surveys on board the Hubble Space Telescope. Credits: NASA, ESA, the Hubble Heritage Team (STScI/AURA), J. Blakeslee (NRC Herzberg Astrophysics Program, Dominion Astrophysical Observatory), and H. Ford (JHU).