A sheet of mass that changes nothing but the answer

A quasar behind a galaxy is seen several times, and the images do not arrive together: light along the longer path is later, and the delay is a distance divided by the speed of light. Measure the delay by watching the quasar vary and seeing the variation repeat in the other image, model the lens, and a distance falls out — a distance measured with a stopwatch, with no ladder under it.

The measurement is beautiful and it has one structural difficulty that no amount of data removes.

A galaxy is weighed by the light that bends past it, and the weighing is exact for the mass inside the Einstein radius. What is not determined is how the mass is arranged outside it — and the time delay cares about that arrangement while the images do not.

An exact symmetry, not an approximation #

The lens equation relates a source position to an image position through the deflection produced by the mass. Write the mass as a convergence $\kappa$ — the surface density in units of the critical value — and the equation is invariant under

$$\kappa \to \lambda\kappa + (1-\lambda), \qquad \beta \to \lambda\beta,$$

where $\beta$ is the source’s position, which is unobservable. Every image position scales by $\lambda$ along with the source, so the observed configuration — image positions relative to the lens, and therefore all the angles and ratios — is unchanged exactly.

This is not a near-degeneracy that better data narrows. It is an exact symmetry of the equations for any $\lambda$, and no measurement of image positions, image shapes or flux ratios can constrain it at all.

What breaks the symmetry is the time delay, which scales as $\lambda$. So the delay measures $\lambda$ times the quantity wanted, and the quantity wanted is the Hubble constant.

It is worth seeing why the delay is not invariant, because the asymmetry is the whole of the measurement. The arrival time at an image has two contributions: a geometric one, from the extra path length of a bent ray, and a gravitational one, from the Shapiro delay through the lens’s potential. Both scale with the overall depth of the potential, which is exactly what the sheet changes — a sheet adds a term to the potential that is quadratic in position, which contributes a constant to the deflection’s derivative and therefore a uniform stretch. The images, which depend on where the deflection is stationary, do not notice; the times, which depend on the potential’s value, do.

Where a sheet comes from #

A uniform sheet of matter is not a hypothetical. Two things supply one.

The line of sight. The light passes through everything between the source and the observer, not only the lens galaxy. Over a small field the external structure is nearly uniform, so it acts as a sheet — of convergence typically a few per cent, and of either sign relative to the mean.

The lens’s own outskirts. A parameterised lens model — a power-law profile, say — has a particular relationship between the mass inside the Einstein radius and the mass outside it. Choosing a different profile changes the outer mass without changing the images much, which is a mass-sheet transformation in disguise, and it is the version that has caused the most trouble.

The second of those two is the more insidious because it does not look like a sheet. An analyst who fits a lens with a power-law density profile and finds an excellent fit has not tested the mass sheet at all: the power-law family is too rigid to contain the transformation, so the fit is good and the uncertainty is small and the answer is whatever the assumed exponent implies. Adding a single extra parameter that allows the profile to steepen or flatten in the outskirts reopens the degeneracy and widens the error bar by a factor of several. Whether a published uncertainty includes it is the first thing to check.

There is a third source that is worth naming because it is the one nobody can bound from the data at all: mass in the source plane’s own neighbourhood, and structure between the lens and the source. Both act as sheets and both are invisible in the image configuration. Their contribution is estimated from simulations of the matter distribution along a random line of sight, which is a statistical statement about a population rather than a measurement of this system, and it is the reason the external convergence is quoted with an uncertainty of a few per cent rather than with a value.

Why an extended image does not break it #

The obvious hope is that a resolved arc carries more information than four point images, and that fitting the whole surface brightness distribution pins the profile down. It carries a great deal more information and it does not break this degeneracy, for a reason worth following.

Lensing conserves surface brightness. So the observed arc is the source’s surface brightness distribution, sampled at the positions the lens maps to it. Rescaling the source plane by $\lambda$ maps a different source — one scaled in size and repositioned — onto exactly the same arc. The fit is perfect for every value of $\lambda$, with a source that is correspondingly larger or smaller, and nothing about the arc says which source was correct because the source is not observed except through the lens.

The only way an extended image could break the symmetry is if the source’s true size were known independently. It is not, and it is not knowable: a quasar host galaxy at redshift two has no standard size, and assuming one would be importing exactly the sort of prior the technique exists to avoid.

There is a larger family of transformations with the same property, found by asking what else leaves an observed configuration invariant. Allowing the source-plane rescaling to depend on position — a general source-position transformation rather than a uniform one — produces a family that leaves the images almost unchanged and changes the delays, of which the mass sheet is the one exactly-invariant member. In practice such transformations are constrained by requiring the lens’s density to stay physically sensible, which is a weaker constraint than a measurement and a real one.

So the situation is that the degeneracy is not a single parameter to be measured but a direction in a large model space, and its exactness is what forces the analysis outside the lensing data entirely. Every route out — the dynamics, the environment, the weak shear — measures mass by some means other than the deflection of light, and each imports its own systematic in exchange. An error budget added in quadrature is the right description of what results: a measurement whose precision comes from combining several mediocre constraints on the same quantity, none of which would be trusted alone.

The kinematic route deserves one caution of its own in that light. A velocity dispersion of a lens galaxy is measured from the width of its integrated absorption lines, and that width contains rotation, anisotropic orbits and any unresolved substructure along with the mass. The problem has the same shape as a dispersion inflated by orbits nobody resolved: a width is being read as a mass, and everything else that contributes to a width is a bias.

What breaks it #

Three observations constrain $\lambda$, and each brings a different systematic.

Stellar kinematics of the lens. The velocity dispersion of the lens galaxy’s stars measures its mass within a radius, by dynamics rather than by lensing. Since the sheet changes the mass and the lensing does not see it, a dynamical mass is exactly the missing constraint. It comes with the difficulties of any dispersion measurement — the orbital anisotropy is unknown, and it is degenerate with the mass profile in its own right.

The environment. Counting galaxies along the line of sight, and comparing the count with a simulated distribution, estimates the external convergence statistically. It is a weak constraint per system and it improves with a larger sample.

Weak lensing of the field. Measuring the shear of background galaxies around the lens constrains the mass in its outskirts directly, which is what the sheet is.

None of them is precise for a single system. Their combination reaches a few per cent, which is the current uncertainty on a Hubble constant from a single well-studied lens.

What was actually measured #

Three results stand out, and the third is what changed the field’s confidence.

The first analyses, and their error bars. Early time-delay measurements of the Hubble constant gave values with quoted uncertainties of a few per cent, obtained with lens models flexible enough to fit the images and not flexible enough to explore the mass-sheet direction. Those uncertainties were later shown to be optimistic.

The reanalyses with flexible profiles. Allowing the lens’s radial profile to vary freely — which is the mass-sheet direction in a different parameterisation — widened the uncertainty on a single system from a few per cent to eight or more. The central value moved less than the error bar grew, which is the expected behaviour when a degeneracy is being explored rather than a bias corrected.

And the kinematic constraint, applied properly. Combining spatially resolved stellar kinematics with the lensing brings the uncertainty back down, at the cost of importing the anisotropy assumption. Current results from samples of several lenses give a Hubble constant on the high side, consistent with the distance ladder and in mild tension with the microwave background — but with an error bar that depends on how the kinematics were modelled.

There is also a fourth result worth recording because it is the sharpest available demonstration that the degeneracy is not academic. Two independent teams analysed the same set of lenses with the same delay measurements and different modelling choices, and reported Hubble constants differing by about five per cent — with individually quoted uncertainties smaller than that difference. Neither analysis contained an error. The difference was entirely in how much freedom each allowed the lens’s radial profile, which is to say in how much of the mass-sheet direction each explored. That comparison is the field’s own measurement of the systematic, and it is larger than any single team’s error bar.

Where the picture stops #

The picture stops in three places, and the second is the one that generalises.

The transformation is exact only for a perfectly uniform sheet. A real external structure is not uniform across the image separation, so its higher-order terms do affect the images slightly. Exploiting that is possible in principle and the effect is small — it is not a route to breaking the degeneracy in practice.

Any flexible model finds the degeneracy. A lens model with enough freedom will explore the mass-sheet direction whether or not it was parameterised as one, and a model with too little freedom will not — and will report a small uncertainty. So the quoted uncertainty on a time-delay Hubble constant is a statement about the model’s flexibility as much as about the data, which is exactly the situation a contrast curve is in with respect to its algorithm.

And the kinematics have a degeneracy of their own. A velocity dispersion constrains the product of a mass and a function of the orbital anisotropy, and the anisotropy is not measured. Spatially resolved kinematics help substantially and do not eliminate it, so breaking one exact degeneracy imports a partial one.

One more belongs on that list, and it is about the delay measurement itself, which this essay has treated as given. Measuring a delay means recognising the same variation in two light curves separated by weeks to months, and quasars vary irregularly. The measurement is a cross-correlation with its own systematics: microlensing by stars in the lens galaxy adds uncorrelated variability to each image, seasonal gaps in the monitoring alias the correlation, and the delay’s uncertainty is a few per cent for a well-monitored system after a decade of observation. That is comparable to the mass-sheet uncertainty, so the two contribute roughly equally to the error budget — which is a healthier situation than one term dominating and is not how it is usually described.

Why an exact degeneracy is a different kind of problem #

The wider point is worth stating because exact degeneracies are rare and behave unlike the approximate ones this collection is otherwise full of.

An approximate degeneracy narrows with better data; an exact one does not. The temperature and gravity that trade in a spectrum are constrained better by better spectra, because the two diagnostics respond slightly differently. The mass sheet does not respond at all: an infinitely precise measurement of every image position, shape and flux constrains $\lambda$ exactly as well as a crude one, which is to say not at all.

The consequence is that the only way forward is a different observable, and the choice of which one determines what the answer depends on. That is why the current best measurements are quoted with their kinematic modelling assumptions attached, and why two groups analysing the same lenses with different anisotropy treatments report different Hubble constants with overlapping errors.

It is also why the honest presentation of such a result is a value and a statement of what was assumed, rather than a value and an error bar. The error bar describes the data’s precision; the assumption describes where the answer came from.

One more observation about what the technique is for. Its value is not that it is the most precise measurement of the Hubble constant — it is not. Its value is that it is independent: no distance ladder, no standard candle, no calibration, and a completely different set of systematics from either the microwave background or the local ladder. Two measurements five sigma apart are not resolved by making either more precise; they are resolved by a third method whose errors are unrelated to both. That is exactly what a time-delay measurement is, and the mass sheet is the price of admission — a systematic that has to be controlled to a few per cent before the method can arbitrate anything.

It is also worth noting how the degeneracy behaves across an ensemble, because that decides whether more systems help. The line-of-sight contribution differs from lens to lens and averages down as the sample grows — it is a random error in that sense. The contribution from the assumed radial profile does not: if every lens is fitted with the same family of profiles, and that family is systematically wrong, every system is wrong in the same direction and no amount of averaging helps. So the two components of the same degeneracy have completely different statistical behaviour, and separating them is what a modern analysis spends most of its effort on. The same distinction between a random and a shared systematic decides how every ensemble measurement in this collection scales with sample size.

There is one further thing worth saying about the technique’s future, because it is unusually well defined. The degeneracy’s two components behave differently with sample size — the line-of-sight part averages down and the profile part does not — so the achievable precision from an ensemble is set by how well the profile assumption can be tested rather than by how many lenses are found. Testing it requires resolved kinematics of the lens galaxies, which requires large telescopes and integral-field spectrographs, and the number of lenses with such data is currently in the tens. The path to a per cent measurement is therefore a spectroscopic programme rather than a monitoring one, which is not what the technique’s name suggests.

Where the ladder goes next #

The immediate next rung is the kinematic measurement: what a resolved velocity field of a lens galaxy adds, how the anisotropy is constrained, and why spatially resolved data are worth so much more than a single dispersion. Above it again sits the ensemble — combining many lenses, each with its own sheet, and whether the sheets average away or share a systematic.