A hundred lenses and one assumption

One lens gives one Hubble constant, with an uncertainty that is honest only when it includes the mass sheet: with a single dispersion through a single aperture, the sheet alone leaves about nine per cent; with the dispersion resolved into rings, about four. The two values the Hubble constant is contested between, 67.4 and 73.0 km/s/Mpc, are 8.3 per cent apart. No single lens can choose between them.

The natural response is to use many. It is the same response weak lensing makes to a signal thirty times smaller than the noise on any one galaxy: average over enough independent objects and the noise goes, provided it is noise. Errors that are independent from one lens to the next fall as one over the square root of the number of lenses, so a hundred lenses at ten per cent each give one per cent — enough to decide the tension at many standard deviations. The surveys now beginning will find thousands of lensed quasars, and monitoring a few hundred of them for delays is a programme of a decade.

The arithmetic is right for independent errors and wrong for the others, and the question that decides whether lensing can arbitrate the tension is which of its errors are which. The figures here sort them: the line of sight, the internal sheet, the analysis choices, and the calibration that a second kind of lens can supply. The model is simple and the numbers are stated with it; what it establishes is the shape of the answer — a slope and a floor — and roughly where the floor sits.

The line of sight is different for every lens #

Light from the quasar passes through everything between it and the observer, not just the lens galaxy. Groups, filaments and voids along the way add their own convergence, and over the few arcseconds the images span it is nearly uniform: an external sheet, which changes the delays and nothing else. Unlike the internal sheet, it is not part of the galaxy and the stars’ dispersion does not see it. It is estimated from the galaxies counted around each lens, compared with simulated sightlines that have the same counts, and the residual uncertainty after that comparison is two or three per cent per lens.

The distribution of external convergence is skewed. Most sightlines pass through slightly less matter than average, because most of the volume of the universe is in underdense regions; a minority pass through a group or a cluster and pick up a large positive convergence. One unlucky lens can be wrong by several per cent — the sample in the figure that starts near minus six per cent began with such a sightline — but every lens’s sightline is drawn independently, and the average converges as $0.025/\sqrt N$. After a hundred and twenty lenses, the five samples agree with zero to within two tenths of a per cent.

That convergence depends on one thing every lens shares: the distribution the sightlines are drawn from. If the simulations used to translate galaxy counts into convergence have the wrong amount of structure, or if lenses are not a random draw of sightlines, every lens’s correction is wrong in the same direction. The second is not hypothetical. A lens is more likely to be found where extra matter along the line of sight has boosted the lensing, so a sample selected by being lenses is biased towards overdense sightlines — the same family of selection that makes a flux-limited sample brighter than the population it came from. The bias is estimated at about a per cent, and the calibration of the line of sight as a whole is assigned an uncertainty of about one per cent shared by every lens.

The delay itself, and a smaller lens inside the lens #

The delays are measured by watching each image of the quasar flicker and finding the shift that lines the flickers up, and their errors are the most obviously independent of any in the problem: different quasars, different monitoring seasons, different weather. Two or three per cent per lens is typical after several years of monitoring, and it averages away as the figures assume.

There is one effect in the delays that is subtle enough to be worth naming, because at first sight it looks like it should be shared. The light of each image passes through the lens galaxy’s stars, and the stars themselves act as lenses — microlenses, with Einstein radii of micro-arcseconds — magnifying different parts of the quasar’s accretion disc by different amounts. The disc’s brightness varies from its centre outwards with a light-travel delay of a day or so across it, so a microlens that favours the outer disc shifts the apparent timing of that image’s variability. The shift is a few tenths of a day to a day, it differs from image to image and from lens to lens, and it changes slowly as the stars move. For a lens with a delay of a hundred days it is a per cent; for one with a delay of ten days it is ten. It is random from lens to lens, so it averages, but it is a reason to prefer lenses with long delays, and a reason the shortest-delay systems carry the largest per-lens errors.

An assumption about orbits that every lens shares #

The internal sheet is different. It is constrained lens by lens, by the stars’ dispersion, and each lens’s constraint has a random part — the measurement error — and a part that depends on how the kinematics were analysed. Every analysis has to say something about the anisotropy of the stars’ orbits: a prior on $\beta$, a functional form for how it varies with radius. If every lens in a sample is analysed with the same prior and the prior is wrong, every lens’s mass-sheet parameter is pushed the same way.

The figure uses the Jeans calculation that set the per-lens errors. With one aperture dispersion per lens, the prior drags each lens’s inferred anisotropy towards isotropy, and the degeneracy between anisotropy and mass sheet converts that into a bias of 1.2 per cent in $\lambda$. Averaged over a thousand lenses, the random part has gone and the bias remains, exactly. With six rings per lens the bias is 0.5 per cent, because the rings measure the anisotropy rather than inheriting it from the prior.

Nothing in the sample reveals the bias. The lenses agree with one another; their average is precise; and it is wrong by an amount set by an assumption that was the same in every analysis. The only defences are to measure the quantity the assumption replaces — which is what resolved kinematics do — or to test the assumption against an independent sample in which it can be checked.

A slope and a floor #

Put the independent errors and the shared ones together and the precision of a sample of lenses has a definite shape.

Each curve starts at a single lens’s error and falls with the square root of the sample, then flattens onto its floor. With one aperture dispersion per lens, the independent error is 10.1 per cent, dominated by the sheet, and the floor — the line-of-sight calibration and the orbital bias together — is 1.6 per cent; the random part falls to the floor at about forty lenses, and beyond that each additional lens buys almost nothing. With resolved rings, the independent error is 5.9 per cent per lens and the floor 1.1 per cent, reached at about thirty.

The two curves differ in more than their starting point. Better kinematics lower the slope, so fewer lenses are needed to reach a given precision, and they lower the floor, because the shared bias came from an assumption the rings replace. The improvement at the floor is the one no amount of sample size could have bought. A survey that finds a thousand lenses and measures each the old way stops improving at 1.6 per cent in this model; one that measures fifty with resolved spectroscopy reaches about 1.4.

The numbers belong to the model and are not a forecast for any particular programme. The per-lens errors are idealised; the shared errors include only the two sources the figures compute, and a real analysis has others — the choice of lens-model family, the population’s spread of internal sheets, the photometry of the lensed host galaxy. Each adds to the floor rather than to the slope. What the figure establishes is that there is a floor, that it is set by what every lens has in common, and that measuring each lens better is the way to lower it.

Lenses with no delays, used to weigh the sheets of lenses with them #

The internal sheet is a property of the lens galaxy, not of the quasar behind it. Lens galaxies without a variable background source — a galaxy lensing another galaxy into an arc or ring — are far more numerous than lensed quasars, they are brighter and nearer on average, and their kinematics can be measured with more photons. They have no delays and say nothing about the Hubble constant directly. What they can say is what the population of lens galaxies is like: how large the internal sheet typically is, and how the anisotropy behaves.

The figure takes seven lenses with delays, measured with one aperture dispersion each, and adds a calibrating sample of lenses without delays but with resolved kinematics. Even ten of them shrink the uncertainty from 4.1 to 2.9 per cent, because they measure the population’s mean sheet better than the seven lenses’ own dispersions can. Beyond twenty, adding more calibrating lenses does nothing, and the reason is the assumption the calibration rests on: that lenses chosen because a quasar behind them varies and lenses chosen because they are bright, nearby and ringed by an arc have the same internal structure. Allowing that assumption two per cent of uncertainty puts a floor under the calibration exactly as the anisotropy prior put one under the delays.

That is close to how the measurement developed. In 2020 a joint analysis of seven time-delay lenses allowed their internal sheets to be free and constrained them with the kinematics alone, and the uncertainty in the Hubble constant grew from the two per cent of power-law models to about eight. Adding thirty-three lenses without delays, with their own kinematics, brought it back to about five per cent and moved the central value from 74.5 to 67.4 km/s/Mpc — towards the early-universe value, by more than one standard deviation, through an assumption that the two samples shared their structure. The model here is more optimistic than that analysis, because it leaves out the population’s spread of sheets and several other sources; the direction of each effect is the same.

How many lenses it takes to tell 67 from 73 #

The question the programme was recruited for can be put as a number of lenses.

To distinguish the two values at three standard deviations, the uncertainty has to fall to 2.8 per cent; at five standard deviations, to 1.7. With one aperture dispersion per lens the first takes twenty lenses, and the second takes 338 — because 1.7 per cent sits just above that design’s floor of 1.6, and approaching a floor from above takes a sample that grows without limit. With resolved kinematics the two targets need six lenses and twenty-four.

The asymmetry is the whole lesson of the ensemble. Every design reaches three standard deviations with a sample of tens; only a design whose floor lies well below the target reaches five, and the floor is set by how each lens is measured, not by how many there are. Twenty-four lenses with resolved kinematics do what three hundred with single dispersions barely do. The independent arbiter of the Hubble tension that lensing was meant to be is available — but at a cost per lens, in spectroscopy, rather than a cost in numbers.

What a large sample is still good for #

A sample that has reached its floor has not stopped being useful; it has stopped being useful for one number. Beyond the floor, more lenses buy something the average cannot: the ability to split the sample.

A shared error that is the same for every lens is invisible, but most shared errors are not quite the same for every lens. A bias from the line of sight should be larger for lenses in rich environments than for isolated ones; a bias from the anisotropy prior should depend on the galaxy’s mass and shape; a bias from microlensing of the delays should be larger for short delays than long ones. Divided by environment, mass, delay or redshift, a large sample gives several averages, and if they disagree by more than their errors, the disagreement has found a systematic that no single average could. The one-number analysis treats the sample as a population to be averaged; the split analysis treats it as an experiment with controls.

That is the use the coming thousands of lensed quasars will have even if only a few dozen are ever measured well enough to set the Hubble constant. They will measure how the internal sheet varies with the properties of lens galaxies, which is a question about how dark matter and stars arrange themselves in the most massive galaxies, and they will test every shared assumption against subsets in which it ought to fail differently. A floor cannot be averaged through, but it can sometimes be found from inside the sample, and finding it is the first step to measuring it.

What sits under the floor #

The floor in these figures has two components. Real analyses have more, and each is a shared assumption of the same kind.

The lens-model family. Every lens is fitted with some profile for the galaxy’s mass — a power law, a composite of stars and a dark halo — and the family chosen controls how much of the sheet the model can represent. A sample analysed with one family inherits whatever that family cannot represent.

The population of sheets. If lens galaxies’ internal sheets scatter from lens to lens, a sample measures their mean, and the mean must be assumed the same for the calibrating lenses and the lenses with delays. The scatter itself is a parameter, and a population model that gets it wrong biases the mean.

The cosmology the distances assume. A time delay measures a ratio of angular-diameter distances, which depends weakly on the matter density and the dark energy. Combined with any other distance measurement, the dependence is a correlation rather than a floor, but a sample interpreted in one fixed cosmology carries it as a shared error.

The photometry of the host. The lensed quasar’s host galaxy, stretched into an arc, constrains the lens model’s shape; how its light is separated from the quasar’s and the lens’s is an analysis choice made the same way for every system.

None of these averages away, and each is reduced the same way: by measuring, for each lens, the quantity the assumption replaces. The ensemble arithmetic is not a substitute for that. It is the reason it is necessary.

The analysts are a shared error too #

One shared error is not in any equation. A team modelling a lens makes hundreds of choices — which mass profile, how to mask the lens galaxy’s light, how many components to give the host’s arc, when a fit is good enough — and every one of them could be steered, without anyone intending it, towards a Hubble constant the team expected. The same people analyse every lens in a sample, and a preference that nudges each by half a per cent nudges the average by half a per cent.

The defence adopted by the lensing collaborations, as by the supernova and acoustic-scale ones before them, is blinding. The models are built with the Hubble constant, and any quantity from which it could be inferred, hidden from the analysts; every choice is made and frozen; and only then is the value revealed, with no further changes allowed. A blind analysis does not remove the systematic errors of the method. It removes the one that would otherwise be correlated with the answer, and a sample of a hundred lenses analysed blind is the only kind whose average can be trusted not to have converged on what was expected.

That habit is the ensemble lesson applied to people. Independent errors can be left to average. Shared ones have to be found and measured, or prevented, because nothing in the data will ever show them.

Still open: whether the floor sits above the gap #

The time-delay values of the Hubble constant published since 2020 have moved with the assumptions about the mass sheet, from 73 with rigid profiles to between 67 and 74 with the sheet free, depending on what calibrated it. A sample of a few dozen lenses with resolved kinematics, measured from space and with the largest ground-based telescopes, is being assembled now, and its floor in the model here sits near one per cent. If the real floor, with every shared assumption included, is near one per cent, lensing will decide between 67 and 73 at more than five standard deviations, independently of every step of the local chain of calibrated distances and of the physics of the early universe. If it is near three, lensing will be one more measurement in the middle of the disagreement, with a systematic floor of its own, and the arbiter will have to be something else — a distance with no calibration chain under it at all, from gravitational waves, whose shared assumptions are different again — or the early-universe side will have to give, through a ruler shortened by something already gone.