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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.

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 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 ${\theta }_E/(2|\theta |)$.