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Cosmic shear is the statistically correlated lensing shear of distant galaxies produced by the intervening large-scale matter distribution. Correlations across angle and source redshift constrain clustering, the growth of structure and distance geometry, thereby testing dark matter and dark energy. Intrinsic galaxy alignments and measurement errors must be separated from the lensing signal.

Choose a signed angular coordinate along the source-lens line. The singular isothermal sphere lens equation isFor $\beta >0$, the positive image is ${\theta }_{+}=\beta +{\theta }_E$. A negative image would require ${\theta }_{-}=\beta -{\theta }_E<0$, which is possible only for $\beta <{\theta }_E$. Here $\beta =4$ arcmin and ${\theta }_E=1$ arcmin, so there is only one image, at $\theta =5$ arcmin. Formally using the negative-image formula would produce a positive value, inconsistent with the branch from which it was obtained.

The paper defines its vector deflection as source minus image angle, ${\mathbf{\alpha }}_{\mathrm{paper}}=(\beta -\theta )\hat{\mathbf{\theta }}$. This is the negative of the usual reduced deflection in $\mathbf{\beta }=\mathbf{\theta }-{\mathbf{\alpha }}_{\mathrm{standard}}$. Therefore its relation to lensing convergence isFor the SIS, ${\mathbf{\alpha }}_{\mathrm{paper}}=-{\theta }_E\hat{\mathbf{\theta }}$. The two-dimensional polar-coordinate divergence givesUsing a plus sign with the printed inward deflection would incorrectly give negative convergence.

For a small intrinsically circular source, differentiate the thin gravitational lens equation. The radial and tangential source-image Jacobian matrix eigenvalues areThus the image is stretched tangentially around the lens centre, with no radial stretch in this idealized model. At $\theta =5$ arcmin, ${\lambda }_t=4/5$; the image is an ellipse with tangential-to-radial axis ratio $5/4$. It is not a complete ring. This local description assumes a source small enough that the Jacobian is nearly constant across it.

The signed lensing magnification isSince the lensing shear magnitude equals $\kappa$, it reduces to $\mu =(1-2\kappa {)}^{-1}$. At the image, $\kappa ={\theta }_E/(2\theta )=1/10$, henceThe image has positive parity and is $25\%$ brighter for the same intrinsic source. Gravitational lensing preserves surface brightness, so this flux amplification follows from its larger apparent solid angle. In evaluating the convergence, use the image angle $5$ arcmin, not the source angle $4$ arcmin.

A galaxy cluster deflects background light through its projected gravitational field. In the thin-lens approximation for gravitational lensing, $\mathbf{\beta }=\mathbf{\theta }-\mathbf{\alpha }(\mathbf{\theta })$ maps image to source angle. Its surface mass density enters throughHere $\mu$ is signed lensing magnification; observed brightness uses $|\mu |$. The $D$'s are angular diameter distances to lens, source and between them; ${\gamma }_s$ denotes lensing shear, not a Lorentz factor. strong gravitational lensing creates arcs and multiple images near critical curves. For an axisymmetric lens, a tangential Einstein ring satisfies mean enclosed convergence one, givingThis is projected enclosed mass, not a general three-dimensional mass profile inferred from a single arc.

weak gravitational lensing uses coherent distortions of many background galaxies. Their intrinsic orientations average approximately to zero, allowing an estimate of reduced shear $g={\gamma }_s/(1-\kappa )$ and a projected mass map. The method traces all gravitating material without assuming hydrostatic equilibrium, which is valuable in merging clusters. Accurate work requires correction of the point spread function, calibrated shape measurement, source redshifts, foreground-member rejection and selection effects. Intrinsic shape noise, unrelated line-of-sight structures and lens-model degeneracies remain. The mass-sheet degeneracy leaves reduced shear unchanged under $\kappa \mapsto \lambda \kappa +(1-\lambda )$ and ${\gamma }_s\mapsto \lambda {\gamma }_s$, so absolute mass normalization needs additional information.

In the Bullet Cluster, weak gravitational lensing places the main projected mass peaks near the collisionless galaxies, displaced from the X-ray-emitting gas that contains most observed baryon mass. Comparing the lensing, optical and X-ray maps supports a dominant weakly collisional dark component under the usual gravity model; lensing alone does not identify its particle species. Cluster lensing measures projected gravitational mass independently of gas temperature or dynamical equilibrium.^[ref]

Deflection changes one light ray's direction; focusing concerns relative directions of neighboring light rays and their bundle's cross section. To see the difference for a point-mass gravitational lens, send parallel incident light rays past the lens. A ray at impact parameter $b$ meets the optical axis approximately $f(b)=b/\hat{\alpha }=c^2{b}^2/(4GM)$ beyond the lens. Doubling $b$ quadruples this crossing distance. Thus the lens does not have a common focal length for all rays, although it can produce lensing magnification and an Einstein ring for a specified source-observer geometry.

The point-mass gravitational lens also illustrates why focusing must not be identified just with positive lensing convergence. Away from the central mass, differentiate the reduced deflection $\mathbf{\alpha }={\theta }_E^2\mathbf{\theta }/r^2$: its two-dimensional divergence is zero, so $\kappa =0$ there. Nevertheless its radial and tangential derivatives differ, giving lensing shear of magnitude ${\theta }_E^2/r^2$ and $\det A=1-{\theta }_E^4/r^4$. A vacuum ray bundle can be distorted and magnified. In the same ideal transparent geometrical optics model, surface brightness is preserved; the larger apparent solid angle supplies the increased received light.

The combination of lensing shear and lensing convergence governing the local shape distortion of background sources. Galaxy ellipticities estimate it statistically under a specified ellipticity convention and intrinsic-orientation model. It is invariant under the mass-sheet degeneracy, so shape information alone does not generally fix absolute projected mass normalization.

The projected density $\Sigma (R)={\sigma }^2/(2GR)$ gives cylindrical mass $\pi {\sigma }^{2}R/G$, so the reduced deflection magnitude is the constant displayed angular Einstein radius. The signed thin gravitational lens equation is $\beta =\theta -{\theta }_E\mathrm{sgn}\theta$. A source with $0<\beta <{\theta }_E$ has two images; one with $\beta >{\theta }_E$ has one. The lensing convergence and lensing shear magnitude are both ${\theta }_E/(2|\theta |)$.