Gravitational Lensing

1Light falls, too

In general relativity, light travels along the straightest possible paths (null geodesics) through curved spacetime. Near a mass those paths bend. A light ray passing a compact mass M at impact parameter ξ, far outside the mass, is deflected by

where R_S = 2GM/c² is the Schwarzschild radius. This is a weak-field result, valid when ξ ≫ R_S, which is true for every lens on this page. For light grazing the Sun it gives 1.75″. If you treat light as Newtonian particles moving at c, you get exactly half of that, 0.87″. Johann von Soldner did that calculation around 1801, and Einstein got the same half value in 1911, before he finished general relativity (0.84″ and 0.83″ with the solar data of their day). The extra factor of 2 comes from the curvature of space itself. In 1919, Dyson, Eddington and Davidson measured stars near the eclipsed Sun and found the full value. That result made Einstein famous overnight.

For every lens on this page the deflection is tiny: $\widehat{\alpha }$ is at most a few tens of arcseconds. That allows two very good approximations. First, the thin-lens approximation: all the bending happens in a single plane, because the lens is far thinner than the distances involved. Second, all angles are small, so sin θ ≈ tan θ ≈ θ.

2The lens equation

Put the observer, the lens and the source on a line, at angular-diameter distances D_d (to the lens, the "deflector"), D_s (to the source) and D_ds (from lens to source). Note that in an expanding universe D_ds ≠ D_s − D_d. The source sits at true angular position β on the sky. We see it at angle θ instead. Simple geometry gives the lens equation:

It maps each image position θ to exactly one source position β. Going the other way, from β back to θ, is what makes lensing interesting: one β can have several solutions θ. Each solution is an image. The simulators on this page do exactly this: for every pixel θ they compute β = θ − α(θ) and look up the colour of the source there. This is called inverse ray-tracing.

3The Einstein radius

If the source sits exactly behind a round lens (β = 0), symmetry turns its image into a ring, the Einstein ring. Its angular radius θ_E sets the scale of everything in strong lensing. For a point mass:

A point mass always makes two images, one on each side. Galaxies are better described by the singular isothermal sphere (SIS). Its density falls as ρ ∝ r⁻², which is what gives galaxies their flat rotation curves. Its strength is set by the stellar velocity dispersion σ_v. It deflects every ray by the same angle, θ_E:

For any round lens, the mean surface density inside the Einstein radius equals the critical surface density. So the Einstein radius directly measures the projected mass inside it, independent of how that mass is distributed:

This is why lensing is so good at weighing dark matter: it doesn't care whether the mass shines.

Einstein-radius calculator

4Magnification, critical curves and caustics

It is convenient to write the deflection as the gradient of a lensing potential ψ, α = ∇ψ, which itself obeys a 2D Poisson equation, ∇²ψ = 2κ. Here κ = Σ/Σ_cr is the convergence: the projected surface density in units of the critical one. How a small patch of the source gets distorted is described by the Jacobian of the lens equation:

Convergence κ magnifies images the same way in every direction. Shear γ stretches them along one direction. Lensing never creates or destroys photons, and it conserves surface brightness. So a magnified image is not brighter per square arcsecond. It is bigger, and the total flux grows by |μ|. Images with μ < 0 are mirror images: they have negative parity.

Where det A = 0, the magnification formally becomes infinite. In the image plane these points form the critical curves (red in the simulators). Mapped back to the source plane, they become the caustics (yellow). Caustics divide the source plane into regions with different numbers of images. Each time a source crosses a caustic, the image count changes by two: a pair of images is born or merges on the critical curve. A tangential critical curve (1 − κ − γ = 0) stretches images tangentially into arcs. A radial one (1 − κ + γ = 0) stretches them radially. Smooth parts of a caustic are folds (two images merge), and the sharp points are cusps (three images merge).

5Time delays and the Hubble constant

Each image is a different path from the source to us, and different paths take different times. The arrival time, relative to an undeflected ray, is the Fermat potential:

The first term is the geometric delay (bent paths are longer). The second is the Shapiro delay (light passing through a potential well is slowed down). Images form where t is stationary: at minima, saddle points and maxima of the arrival-time surface. That is Fermat's principle, applied to gravity. If the source varies, as a quasar or a supernova does, the variation shows up in each image at a different time. A measured delay gives D_Δt, and with it the Hubble constant H₀, in a single step and without any distance ladder. Sjur Refsdal proposed exactly this in 1964. You can try it in the time-delay box. The catch is that you need the lens potential ψ, and the images alone do not quite pin it down. That is the subject of the next section.

6The mass-sheet degeneracy

Take any lens model that reproduces the observed images: a convergence κ(θ) and a source position β. Now build a second model. Scale the lens down by a factor λ and add a uniform sheet of convergence 1 − λ on top. A uniform sheet pulls every ray towards the centre by an angle proportional to its distance, here (1 − λ)θ, just like a uniform disk would. So the new lens equation is simply λ times the old one:

The images land exactly where they were. Only the source has moved, from β to λβ, and nobody ever gets to see the source without the lens in front of it. The Jacobian of the lens equation is multiplied by λ as well (A → λA), so every image gets the same extra magnification 1/λ². That leaves the flux ratios untouched, and flux ratios are all we can measure, since we don't know how bright the source really is. Even the shapes of extended arcs come out identical, as long as the source is also λ times smaller. No image you will ever take can tell the two lenses apart. Their arrival-time surfaces do differ, though: t(θ) is scaled by λ, and so is every delay.

A measured delay therefore only fixes H₀/λ. If the true lens has λ = 0.9 and you model it with λ = 1, your H₀ comes out 10% too high, and nothing in the data warns you. Move λ below: the image positions and the flux ratio stay put, while the source, the delay and the inferred H₀ all move.

Where would such a sheet come from? There are two places. Outside the lens, galaxies and dark matter along the line of sight add a small external convergence κ_ext, typically a few percent. The lens model absorbs it, which amounts to λ = 1 − κ_ext, so the true H₀ is (1 − κ_ext) times the one the model gives. It is estimated by counting galaxies around the lens and comparing with simulations, or from weak lensing. Inside the lens, a sheet that goes on forever is not physical. For λ > 1 it would even turn negative far out (the shaded region in the plot, which the curve reaches for λ > 1.2). But any mass component that is nearly uniform across the images and fades further out does the same job where it matters. In practice it is a change in the radial density profile near θ_E. Assuming a profile shape, for example a power law, quietly fixes λ.

Breaking the degeneracy takes information that does not come from lensing a single source. Stellar kinematics of the lens galaxy help, because σ_v² scales roughly with λ: the sheet changes the mass the stars feel. So do sources at several redshifts behind the same lens, and sources of known brightness, such as a lensed type Ia supernova, whose absolute magnification measures λ directly. Falco, Gorenstein & Shapiro pointed out the degeneracy in 1985, for the Twin Quasar. It matters in practice: H0LiCOW measured H₀ = ${73.3}_{-1.8}^{+1.7}$ km/s/Mpc from six lenses, assuming standard mass profiles. When TDCOSMO (Birrer et al. 2020) dropped that assumption and let the kinematics alone constrain λ, seven lenses, six of them the same, gave ${74.5}_{-6.1}^{+5.6}$. The central value barely moved, but the uncertainty more than tripled. You can walk through the whole argument in the time-delay box, where λ is the "Mass sheet" slider and the orange band is the kinematics.

7Strong, weak and micro

Strong lensing means multiple images, arcs and rings. It happens only where κ ≳ 1, in the cores of massive galaxies and clusters. It is what the first two simulators show.

Weak lensing happens everywhere else. Each background galaxy is sheared by a percent or so, far below the natural scatter in galaxy shapes. Averaging over thousands of galaxies reveals a coherent pattern, the reduced shear g = γ/(1 − κ). From it we can map the mass of clusters (including the dark matter in the Bullet Cluster) and of the cosmic web itself ("cosmic shear", a key science goal of Euclid and the Rubin Observatory).

Microlensing is strong lensing by stars, with images far too close together to resolve. Because the lens and source move relative to each other, the total magnification changes with time. For a point lens it is:

Here u = β/θ_E, and t_E = θ_E/μ_rel is the Einstein-crossing time, typically a few weeks. This symmetric curve is the "Paczyński curve" (Paczyński 1986). A planet orbiting the lens star adds small caustics of its own. A source that crosses or passes near one of them produces a short anomaly. Microlensing is most sensitive to planets near the Einstein ring (a few AU), including cold, low-mass planets that other methods struggle to find.

8What these simulators leave out

• The deep field puts every background galaxy at the same redshift, the "source z" slider. Real galaxies sit at many redshifts and each one would see a different θ_E. The few stars with diffraction spikes are in our own Milky Way. In reality they sit in front of any lens and would not be lensed at all.
• The deep-field image is a finite, pixelated patch of sky. Rays that land outside it come back black.
• All lenses are thin and use simple analytic profiles: isothermal ellipsoids (optionally cored), point masses and external shear. Real lens models (with GLEE, lenstronomy, Lenstool and friends) use more flexible mass profiles. They also model the PSF, noise and the light of the source.
• The time-delay box assumes the intrinsic light curve of the quasar is known. Real analyses have to infer it together with the delays. Microlensing by stars in the lens galaxy is mimicked by a small, slow, independent wobble in each image.
• Light is treated with geometric optics. Wave effects (relevant for gravitational waves, not for starlight) are ignored.