Astronomy & Celestial Mechanics
the sky, computed
Concept

Lens model — where it appears

A parameterised description of a gravitational lens's mass distribution, fitted to the observed image positions and shapes. Its assumed radial profile decides quantities the images do not constrain, which is why two models fitting the same data give different time-delay distances.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as line of sight structure — the same set of essays touches all of them, so they are one junction rather than several.

A delay of 81 days, and a sheet nobody can see that moves H₀ to 82.4. Above: the arrival-time surface of a lensed source, along the line through the lens. The curve is the Fermat potential in days — the geometric cost of taking a longer path, minus the gravitational cost of climbing out of the potential — and the images sit at its stationary points, at -1.20″ and 2.04″, which for an isothermal sphere is β ± θ_E. Fermat's principle is doing all of the work here: light does not take the shortest path or the quickest one, it takes every stationary one, and the number of images is the number of stationary points. The vertical distance between the two is 81 days, and it is measurable — the source is a quasar, quasars vary, and the same wiggle appears in one image and then the other. That single number carries an absolute distance: the delay is D_Δt/c times a dimensionless function of the lens model, and D_Δt goes as 1/H₀, so a monitoring campaign gives the Hubble constant with no rung of any ladder beneath it. Below: the two light curves, shifted by exactly that delay. The dashed curve is the second image with the delay removed, and the agreement is the measurement. What the picture also shows is the reason the answer keeps moving. The second arrival-time curve is the same lens with a uniform sheet of convergence added and the source moved to compensate: every image sits at the same place, every flux ratio is the same, every image shape is the same, and the delay is λ = 0.85 times as long. A lens model fitted to positions alone cannot see the sheet, and inferring H₀ from the same delay under it gives 82.4 instead of 70 — a 15 per cent shift with no observable attached. Breaking it needs a mass measured some other way: the velocity dispersion of the deflector, or a count of everything else along the line of sight.

A distance measured with a stopwatch

The images of a lensed quasar sit at the stationary points of an arrival-time surface, and the height between two of them is a delay in days. That delay is proportional to a distance, and the distance is proportional to one over the Hubble constant — so a flickering quasar gives H₀ with no ladder under it.

galaxies · Strong lensing A sheet of mass that changes 32 km/s/Mpc and no image. The mass-sheet degeneracy, drawn as what it does and does not change. Adding a uniform sheet of convergence and rescaling the unobservable source position leaves every image position relative to the lens, every image shape and every flux ratio exactly as it was — the transformation is an exact symmetry of the lens equation, not an approximation. What it does change is the time delay between images, in proportion, so a Hubble constant inferred from a measured delay is multiplied by the inverse of the rescaling. Across the range of sheets that a plausible line of sight can supply, the inferred Hubble constant moves by 32 kilometres a second per megaparsec — which is larger than the disagreement between the early and late measurements the technique is meant to arbitrate. Nothing in the lensing data can fix it; the constraint has to come from the lens galaxy's stellar kinematics or from the environment along the line of sight.

A sheet of mass that changes nothing but the answer

A gravitational lens's images are unchanged by adding a uniform sheet of matter and rescaling the source. Every position, every shape and every flux ratio stays exactly as it was; only the time delays change, in proportion. So a Hubble constant measured from a delay is multiplied by a number the lensing itself cannot determine.

galaxies · Strong lensing

Named alongside it

The objects these essays reach for when they reach for this one.

ConvergenceHubble constantLine of sight structureThe mass-sheet degeneracyTime-delayEinstein radiusExact degeneracyFermat-potentialQuasar variabilitySource planeStationary pointStellar kinematics

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