Astronomy & Celestial Mechanics
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A dispersion that weighs the sheet only if the orbits are known

A lens's images cannot tell how much of its mass is in the galaxy and how much is spread through a sheet around it, and the time delay — and so the Hubble constant — scales with the answer. The stars' velocity dispersion can tell, because orbits feel mass in three dimensions where light only integrates along a line. One dispersion through one aperture weighs the sheet only to sixteen per cent, because the shape of the orbits moves it as much; the same dispersion measured in rings weighs it to four.

Assumes Strong lensing and Velocity dispersion.

16 min read 5 figures Measuring the unreachableLight is the only messenger

A quasar seen four times around a foreground galaxy is a distance measured with a stopwatch: the images’ light arrives at different times, the delays are differences of path length, and a path length is a distance multiplied by geometry the lens model supplies. The Hubble constant falls out with no chain of calibrated candles under it, which is why time-delay lenses were recruited into the argument over whether the local and early-universe values of the constant disagree.

The measurement has one structural weakness, and it is exact. Adding a uniform sheet of mass to a lens, while rescaling the unseen source, leaves every image position, every image shape and every flux ratio exactly as it was. Writing the transformation with a parameter λ\lambda, the convergence becomes λκ+(1−λ)\lambda\kappa + (1 - \lambda) and the time delays are multiplied by λ\lambda; so for a measured delay, the inferred Hubble constant is proportional to λ\lambda, and nothing in the images says what λ\lambda is.

That essay named the observation that breaks the symmetry — the stars’ velocity dispersion — and noted that it comes with difficulties of its own. What follows solves for them. A lens and its sheet are put through the spherical Jeans equation, which turns a mass distribution and a shape of stellar orbits into the dispersion a spectrograph would measure, and the question is how well that dispersion weighs the sheet: with one number, and with a profile.

Light integrates along a line; orbits feel a sphere

The sheet is invisible to the images because lensing responds to projected mass: the light bends according to the mass inside a cylinder along the line of sight, and the images fix the mass inside the cylinder whose radius is the Einstein radius. A mass-sheet transformation, in the internal form that matters here, moves mass along the line of sight. Some of the galaxy’s own concentrated mass is replaced with a broad, cored distribution that projects as an almost uniform sheet across the images, and inside the Einstein radius the two exchange exactly.

The same mass inside the Einstein ring, and a different mass inside a sphere. For a lens under the internal mass-sheet transformation with λ = 0.85, 1, 1.1: the projected mass inside a cylinder of radius R (dashed), which is what lensing weighs, and the mass inside a sphere of radius r (solid), which is what the stars' orbits respond to, both divided by the λ = 1 lens's mass inside a sphere of one Einstein radius. At the Einstein radius every projected mass is the same, to 0.1 per cent, because that is what the images fix. Inside a sphere of half the stars' effective radius the three-dimensional masses are 0.86, 1.00, 1.10 times the λ = 1 value. The transformation moves mass along the line of sight — out of the galaxy and into a broad core that projects as an almost uniform sheet — and the stars, which orbit in three dimensions, feel the difference that the light, which integrates along the line of sight, does not.
Fig. 1 Enclosed mass for a lens transformed with λ=0.85\lambda = 0.85, 1 and 1.1. Dashed: the mass inside a cylinder, which lensing weighs; it is the same at the Einstein radius to 0.1 per cent for all three. Solid: the mass inside a sphere, which the stars’ orbits respond to. At half the stars’ effective radius it is 0.86, 1.00 and 1.10 times the untransformed value.

The lens in the figure is the simplest model that does the job: a singular isothermal sphere — a galaxy whose mass grows in proportion to radius, which is close to what lens galaxies are measured to be — transformed by replacing a fraction 1−λ1 - \lambda of it with a cored component of six Einstein radii, wide enough to be uniform to a few per cent across the images. Inside a cylinder of one Einstein radius the three versions agree to a tenth of a per cent. Inside a sphere of half the stars’ effective radius they disagree by the full factor: 0.86, 1.00 and 1.10.

That is the opening dynamics has. A star orbiting in the galaxy feels the mass inside the sphere through which it moves, not the mass in a column along a line of sight that runs out to the source. The transformation that is exactly invisible to light is plainly visible to orbits, in proportion to λ\lambda, and a measurement of how fast the stars are moving is in principle a measurement of λ\lambda — and so of the Hubble constant from that lens.

Why a galaxy would carry a sheet at all

A uniform sheet sounds like a mathematical device, and the external version of it — mass along the line of sight, from groups and filaments unrelated to the lens — is genuinely uniform across a few arcseconds. The internal version is a statement about the galaxy’s own profile, and there are physical reasons for it to differ from the simple forms a lens model assumes.

A massive elliptical is a mixture: stars concentrated towards the centre, dark matter spread more widely, and the total close to isothermal over the radii lensing samples — the “bulge–halo conspiracy” that makes a single power law fit lens after lens. Close to isothermal is not exactly isothermal. Gas that cooled into the centre before the stars formed pulled the dark matter inwards with it; feedback from the central black hole may have pushed some of it out again; and the dark matter’s own profile turns over from a steep inner slope to a steeper outer one somewhere near the radii where the images are. Any departure of that kind changes the mass outside the Einstein radius relative to the mass inside, which over the small region the images sample is indistinguishable from a sheet. Simulations of massive galaxies with these processes included give internal mass-sheet parameters spread over several per cent about one, and the sign of the deviation is not known in advance.

So the parameter is not a pathology to be ruled out but a real property of each lens, of the size that matters: a few per cent, varying from galaxy to galaxy, on the scale of the tension the measurement was meant to arbitrate.

One number that knows two things

What the spectrograph measures is a line-of-sight velocity dispersion: the width of the stellar absorption lines, averaged over whatever aperture the light was collected through and weighted by the brightness there. The Jeans equation connects it to the mass, and the connection runs through the shape of the orbits.

For a spherical system in equilibrium the radial velocity dispersion σr\sigma_r satisfies

d(νσr2)dr+2βr νσr2=−ν GM(r)r2,\frac{\mathrm{d}(\nu\sigma_r^2)}{\mathrm{d}r} + \frac{2\beta}{r}\,\nu\sigma_r^2 = -\nu\,\frac{GM(r)}{r^2},

with ν\nu the density of stars and β=1−σt2/σr2\beta = 1 - \sigma_t^2/\sigma_r^2 the anisotropy: zero for isotropic orbits, positive for orbits that plunge radially, negative for orbits that circle. The mass appears once and the anisotropy once, and what is observed is a projection of σr\sigma_r and σt\sigma_t onto the line of sight, which mixes them in proportions that depend on radius. That is the mass–anisotropy degeneracy that every dispersion measurement carries, here with the mass-sheet parameter as the mass.

One velocity dispersion, three orbit shapes, and the λ each allows. The luminosity-weighted velocity dispersion inside one Einstein radius, relative to that of the untransformed isotropic lens, against the mass-sheet parameter λ, for constant anisotropies β = −0.3, 0, 0.3 (negative: more tangential orbits; positive: more radial). The dispersion rises with λ — about 4.3 per cent for a change of 0.1 in λ — because a larger λ puts more of the mass inside the galaxy and less in the sheet. The shaded band is a measurement of 5 per cent. With the anisotropy unknown, it allows λ from 0.82 to beyond 1.15, the edge of the range drawn, and the inferred Hubble constant, proportional to λ, inherits that range: at least 16 per cent either way. The anisotropy moves the dispersion by as much as a change of 0.11 in λ, so a single dispersion constrains a combination of the mass and the orbits, not the mass.
Fig. 2 The dispersion inside one Einstein radius, relative to the untransformed isotropic lens, against λ\lambda, for anisotropies β=−0.3\beta = -0.3, 0 and 0.3. It rises by about 4.3 per cent for a change of 0.1 in λ\lambda. A measurement to five per cent (shaded), with β\beta unknown, allows λ\lambda from 0.82 to beyond 1.15: at least sixteen per cent either way in the Hubble constant.

The stars are a Hernquist profile with an effective radius 1.2 times the Einstein radius, typical of the massive ellipticals that make the best time-delay lenses, and the aperture is one Einstein radius. The dispersion rises with λ\lambda, because a larger λ\lambda keeps more of the mass inside the galaxy, but slowly: 4.3 per cent per 0.1 in λ\lambda. Velocity dispersion goes as the square root of mass, which halves the leverage, and the mass the sheet moves is partly outside the region the stars sample.

The anisotropy moves the same number as much. Changing β\beta from −0.3-0.3 to +0.3+0.3 shifts the aperture dispersion by as much as a change of 0.11 in λ\lambda. A five per cent dispersion then allows λ\lambda anywhere from 0.82 to beyond the 1.15 the figure draws — and the Hubble constant from this lens, proportional to λ\lambda, is uncertain by at least sixteen per cent from this source alone. The difference between the two contested values of the Hubble constant is eight per cent.

There is a well-known escape from the mass–anisotropy degeneracy, and it is worth seeing why it does not apply. For a galaxy whose dispersion profile is flat and whose dispersion is averaged over all its light, the mass inside about the three-dimensional half-light radius is nearly independent of the anisotropy: the one radius where the orbits stop mattering. A lens’s dispersion is not that measurement. It comes from an aperture inside the Einstein radius, typically smaller than the galaxy’s half-light radius, so it weights the centre where radial and tangential orbits project very differently; and the quantity wanted is not the mass at one radius but how the mass is distributed between the galaxy and the sheet, which is a statement about two radii at once.

A profile where an aperture gives a number

The degeneracy is between how the mass is distributed and how the orbits are shaped, and they leave different marks in how the dispersion changes with radius.

Three lenses with one aperture dispersion and three dispersion profiles. Line-of-sight velocity dispersion against projected radius, relative to the aperture value, for three combinations of mass-sheet parameter and anisotropy — λ = 1 with β = 0.1; λ = 0.9 with β = −0.25; λ = 1.06 with β = 0.35 — whose dispersions inside one Einstein radius agree to within 7 per cent before being scaled to agree exactly, as a single measured number would make them. Resolved, they are different curves: they differ by up to 18 per cent at the same radius, with the combination of radial orbits and a larger λ falling most steeply and the combination of tangential orbits and a smaller λ falling least. An integral-field spectrograph that measures the dispersion in several rings separates the three where an aperture cannot, because the anisotropy and the mass move the profile in different ways while moving the aperture average the same way.
Fig. 3 Dispersion against projected radius for three lenses whose aperture dispersions agree once scaled to one value: λ=1\lambda = 1 with β=0.1\beta = 0.1; λ=0.9\lambda = 0.9 with β=−0.25\beta = -0.25; λ=1.06\lambda = 1.06 with β=0.35\beta = 0.35. They differ by up to 18 per cent at the same radius — radial orbits with a larger λ\lambda falling most steeply, tangential orbits with a smaller λ\lambda least.

The three lenses in the figure are chosen to lie along the degeneracy: each pairs a mass-sheet parameter with the anisotropy that brings its aperture dispersion close to the others’, and the residual few per cent is then scaled away, as a single measurement would never see it. Drawn as profiles, they are three different curves. Radial orbits concentrate the line-of-sight motion at the centre, where the plunging orbits are seen end-on, and let it fall outside; tangential orbits do the opposite. A smaller λ\lambda puts more of the mass in a broad core that keeps the outer dispersion up. Combined along the degeneracy, the effects that cancel in an aperture average separate in radius, by up to 18 per cent at the same projected distance.

An integral-field spectrograph, which records a spectrum at every point of an image, measures exactly this: the dispersion in rings around the lens galaxy rather than one number through one aperture. For lenses at redshifts of a half or more, the rings are a fraction of an arcsecond apart, and the measurement needs the resolution of adaptive optics on the largest ground-based telescopes or the stability of a space telescope.

What the spectrograph actually measures

A dispersion is not read directly from a spectrum. The absorption lines of a galaxy are the lines of its stars, broadened by their motions, and the dispersion is found by convolving a template — a spectrum of stars at rest — with a distribution of velocities until it matches. The template matters. A galaxy’s stars are a mixture of ages and compositions, and a template library that does not represent the mixture well leaves residuals in the line shapes that the fit absorbs into the width. Different libraries applied to the same high-quality spectrum of a lens galaxy give dispersions that differ by two or three per cent.

Two or three per cent in dispersion is five or six per cent in σ2\sigma^2, which is what the Jeans equation responds to, and through the slope in the aperture figure — 4.3 per cent in dispersion for 0.1 in λ\lambda — it is five to seven per cent in λ\lambda. That is a systematic error comparable to the whole statistical uncertainty of a single aperture measurement, and unlike the anisotropy, it is not reduced by measuring the dispersion in rings, because every ring is fitted with the same templates. Analyses now fit each spectrum with several independent libraries and carry the spread among them as part of the uncertainty, which is the honest treatment and also a reminder that the precision of a kinematic measurement can be limited by knowledge of stars rather than by photons.

The rings also have to be resolved. A lens galaxy at a redshift of a half with an Einstein radius of an arcsecond has its rings a fifth of an arcsecond apart, comparable to the blurring of the best natural seeing. On the ground the measurement needs adaptive optics, whose blurring has a sharp core and broad wings that mix light between rings and have to be modelled; from space the blurring is smaller and stable. Either way the profile in the figures is the profile after deconvolution, and the deconvolution is another place a systematic can enter.

Where the mass and the orbits are left

The plane of λ\lambda and β\beta shows what each kind of measurement leaves.

Where one dispersion and a resolved profile leave the mass and the orbits. The plane of the mass-sheet parameter λ and the orbital anisotropy β. Light shading: the region within Δχ² = 2.3 of a mock lens with λ = 1 and β = 0.1, given one aperture dispersion measured to five per cent — a long diagonal valley, because a larger λ and more tangential orbits trade against each other. Dark shading: the same, given the dispersion in six rings between a quarter and two Einstein radii, each to three per cent. Marginalised over β with a flat prior, and with λ allowed between 0.8 and 1.15, the aperture measurement gives λ = 0.990 ± 0.091 and the resolved profile λ = 1.000 ± 0.040 — the Hubble constant from this lens to 9 and 4 per cent.
Fig. 4 Regions within Δχ² = 2.3 of a mock lens with λ=1\lambda = 1 and β=0.1\beta = 0.1. Light: one aperture dispersion to five per cent — a valley across most of the plane, along which more tangential orbits trade against a larger λ\lambda. Dark: the dispersion in six rings to three per cent each — a closed region around the true values. Marginalised over β\beta: λ=0.990±0.091\lambda = 0.990 \pm 0.091 from the aperture, 1.000±0.0401.000 \pm 0.040 from the rings.

With one aperture dispersion the allowed region is a broad valley running across almost the whole plane: any amount of sheet can be balanced by a suitable shape of orbits, and only the extremes — very radial orbits with a large λ\lambda, very tangential ones with a small λ\lambda — are excluded. Marginalising over the anisotropy with a flat prior, the mass-sheet parameter comes out at 0.990±0.0910.990 \pm 0.091. With six rings the region closes into a compact patch around the truth, and λ=1.000±0.040\lambda = 1.000 \pm 0.040: the resolved profile measures the anisotropy itself, rather than leaving it to be averaged over, and with the orbits known, the dispersion weighs the mass.

The comparison is between one aperture at five per cent and six rings at three per cent each, which is not a fixed amount of telescope time. Six rings at three per cent need more photons than one aperture at five, and what they buy is not merely more precision on the same quantity but a second quantity — the anisotropy — that turns the dispersion from a degenerate measurement into a direct one.

What it does to the Hubble constant from one lens

The two posteriors translate directly into Hubble constants, since H0H_0 is proportional to λ\lambda.

The Hubble constant from one lens, with one dispersion and with six. The posterior on the Hubble constant from one mock lens whose true λ gives H₀ = 70 km/s/Mpc, with everything except the mass-sheet parameter and the anisotropy assumed known — the time delay, the lens model's other parameters, the external convergence — and a flat prior on β from −0.4 to 0.5. With a single aperture dispersion to five per cent: 69.3 ± 6.4. With the dispersion in six rings to three per cent each: 70.0 ± 2.8. The difference between the two values the Hubble tension is about — 67 and 73 — is 9 per cent; the single dispersion leaves an uncertainty comparable to it from this source alone, and the resolved profile one 2.3 times smaller.
Fig. 5 The Hubble constant from one mock lens with true value 70 km/s/Mpc, with the delay, the rest of the lens model and the line of sight taken as known. One aperture dispersion (bars): 69.3 ± 6.4. Six rings (curve): 70.0 ± 2.8. Dotted: 67.4 and 73.

The single aperture dispersion leaves the Hubble constant from this lens at 69.3±6.469.3 \pm 6.4 km/s/Mpc, an uncertainty comparable to the whole gap between 67.4 and 73.0. The resolved profile gives 70.0±2.870.0 \pm 2.8, two and a third times tighter, and that is with every other source of error in the measurement — the delay, the rest of the lens model, the mass along the line of sight — set to zero. A real lens adds those in quadrature, and the per-lens error that results is the starting point for any sample of lenses.

This is close to how the subject actually unfolded. Analyses that fitted the lens galaxies with a rigid power-law profile reported Hubble constants to two per cent from a handful of lenses, because a power law cannot represent the transformation and so had silently fixed λ\lambda. When the profile was allowed to absorb an internal sheet and only the kinematics were left to constrain it, the uncertainty from the same seven lenses grew to about eight per cent — almost exactly the size of the effect drawn here, for the same reason. The precision had been in the assumption, and the kinematics, measured as one number per lens, could not replace it. Resolved spectroscopy of lens galaxies, with integral-field instruments on the largest telescopes and in space, is the measurement that has since been made to close that gap one lens at a time.

Two more degeneracies with the same shape

The structure here — a quantity that the primary observable cannot see, constrained by a secondary observable that is itself degenerate with something else — recurs throughout dynamics. A rotation curve fixes a galaxy’s total mass and not its division into disc and halo, because the stellar mass-to-light ratio trades against the halo; the same rotation curve is fitted by three different mass models for the same reason. In each, the remedy is the same as here: a measurement that responds to the hidden quantity in a different way from the quantity it trades against, which for a lens means the radial shape of the dispersion.

And there is a degeneracy the lens does not have. A dispersion measured from single spectra can be inflated by binary stars, each carrying its own orbital velocity — which matters in a dwarf galaxy whose dispersion is a few kilometres a second and not at all in a massive elliptical whose dispersion is two or three hundred. The lens galaxies chosen for time delays are the systems in which the stars’ motions are least contaminated by anything but the mass.

What the model leaves out

The lens is spherical and isothermal and the sheet is a single cored component; real lens galaxies are flattened, have profiles that depart from isothermal, may rotate, and have their own dark-matter halos whose distribution the transformation represents only approximately. The anisotropy is constant with radius, where real galaxies are more radial outside than in. The dispersions are computed without seeing, which blurs the rings into each other and reduces the leverage of a resolved profile from the ground. And the true lens here has exactly the anisotropy assumed in one of the curves; a real analysis has to choose a family of anisotropy profiles, and the choice is itself a systematic shared by every lens it is applied to.

None of these changes the central comparison: one dispersion through one aperture constrains a combination of mass and orbits, and a profile constrains them separately.

Still open: what a sample of lenses can average away

A single lens with resolved kinematics gives the Hubble constant to about four per cent from the mass sheet alone and five or six with everything else included. The tension is eight per cent, so a single lens cannot settle it and the answer has to come from many. That raises the question this calculation cannot answer from one system: which of the errors here are different for every lens and average away as a sample grows, and which are shared — the choice of anisotropy profile, the assumption that lens galaxies’ internal sheets are drawn from one population, the calibration of the mass along the line of sight — and set a floor that no number of lenses goes beneath. That is the arithmetic of the ensemble, and it decides whether lensing can be the independent arbiter it was recruited to be or only one more measurement with its own systematic floor, as every step of the local chain of calibrated distances has turned out to have.

Where this goes next

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. Mass profiles that pinch together near the half-light radius. The enclosed mass of each of the 13 haloes that fit the same flat dispersion of 10 km/s, against radius in units of the stars' three-dimensional half-light radius, 391 parsecs. At a tenth of that radius they span a factor of 17, and at four times it a factor of 2.5; they pinch together near the half-light radius, closest at 1.47 of it, where they agree to 14 per cent. The dot is the estimate 3σ²r/G at the radius where the stars' density falls as r⁻³, 0.94 of the half-light radius; there the fitted masses lie between 0.65 and 1.03 times it. Each model trades a wrong guess about the orbits against a wrong guess about the mass, in opposite senses inside and outside. The one radius where the orbits stop mattering A velocity dispersion weighs a galaxy only if the shape of its stars' orbits is known, and it never is. Yet every model that fits a dwarf galaxy's flat dispersion profile — cored or cusped, radial orbits or tangential — agrees on the mass inside one particular radius, close to the radius that contains half the light. The degeneracy has a node, and the node is where the measurement should be quoted. 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 calibrated distance 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 calibrated distance under it.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Einstein radiusHubble constantIntegral-field spectroscopyJeans equationMass anisotropy degeneracyThe mass-sheet degeneracyOrbital anisotropyStellar kinematicsTime-delayVelocity dispersion