Astronomy & Celestial Mechanics
the sky, computed
Concept

The mass-sheet degeneracy — where it appears

The transformation that adds a uniform sheet of surface density to a lens and rescales the sources, changing no observed image position or shape. It means lensing fixes a mass profile only up to a constant, and breaking it needs a magnification measurement or something dynamical.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

Why an Einstein radius is a mass. The geometry, drawn at an angle some ten thousand times larger than the real one so that anything is visible at all. Light from a source directly behind a lens reaches the observer along every path that passes the lens at the same distance, so the image is a ring rather than a point. The ring's angular radius is θ_E = √(4GM/c² · D_ls/D_l D_s), which for a lens of 1.0×10¹² M☉ at these distances is 2.52 arcseconds and encloses 10 kpc at the lens. Rearranged, it is a mass in terms of an angle and three distances — and the mass so obtained is inside a cylinder rather than a sphere, and assumes nothing whatever about the lens being in equilibrium, which is the assumption every other weighing in this collection makes.

Weighed by the light that bends past it

Every other mass in this collection is measured from something orbiting, which requires the system to have settled down. A gravitational lens weighs whatever is in the way with no such assumption — the light does not care whether the mass is in equilibrium.

galaxies · Strong lensing A coherent one-per-cent distortion, invisible on every galaxy in the picture. 150 background galaxies behind a lens of Einstein radius 14″, each drawn at its own ellipticity: an intrinsic shape with a dispersion of 0.3 per component, plus the reduced shear the lens adds. The strongest shear on any galaxy here is 0.035, one part in 8 of the intrinsic scatter, so no object in this field is measurably distorted and the tangential alignment cannot be seen by eye at all. Averaged over these 150, the mean tangential ellipticity is 0.0088 ± 0.0245 against the 0.0130 the lens model predicts — consistent with the lens and equally consistent with nothing, because 150 galaxies buy a precision of 0.024 and the signal is 0.013. Detecting it at five sigma takes about 13,275 of them, which is not a picture anybody can draw. The cross component — every shape rotated by 45°, which gravitational lensing cannot produce — averages −0.0016 ± 0.0245, consistent with nothing, and that null is what separates a mass from a badly figured optic. The signal is not in any galaxy. It is in the sum, and the whole design of a lensing survey follows from that.

A one-per-cent distortion, and a million galaxies to see it

A galaxy's own shape is unknown and scatters with a dispersion of about 0.3, so a coherent one-per-cent shear is thirty times smaller than the noise on any single measurement. Nothing is ever measured about one object; the estimator is an average, and the whole design of a survey follows from 0.3 over the square root of N.

galaxies · Weak lensing 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.

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 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.

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.

galaxies · Strong lensing How far a sample of lenses can take the Hubble constant. The fractional uncertainty in H₀ from a sample of N lenses, both axes logarithmic: the independent errors of each lens — 2 per cent from its delay, 3 from the rest of its lens model, 2.5 from its line of sight and its own mass sheet as well as its kinematics fix it — fall as 1/√N; the shared errors — 1 per cent from the calibration of the line of sight, and the orbital bias of the previous figure — do not. With one aperture dispersion per lens the independent error is 10.1 per cent per lens and the floor 1.6 per cent, reached at about 42 lenses; with six resolved rings per lens, 5.9 and 1.1 per cent, reached at about 29. Dotted: the two floors. A sample that adds lenses without changing how each is measured stops improving at its floor; the resolved kinematics lower both the slope's starting point and the floor.

A hundred lenses and one assumption

A time-delay lens gives the Hubble constant to five or ten per cent, and the hope is that a hundred give it to one. The errors that differ from lens to lens — its delay, its own line of sight, its own internal sheet — do average away. The errors every lens shares because every lens was analysed the same way do not, and they set a floor that a sample reaches after a few dozen lenses and never goes beneath.

galaxies · Strong lensing Where the quasar counts turn over, the signal changes sign. The local slope s of quasar number counts against limiting magnitude (solid), for counts that steepen to 0.90 at the bright end and flatten to 0.29 at the faint end with a turnover near magnitude 19, and the factor 2.5s − 1 that multiplies twice the convergence in the change of counts (dashed). The factor crosses zero at magnitude 20.1: brighter quasars are found in excess round foreground galaxies, fainter ones in deficit, and quasars near the crossing show no correlation at all. The predicted reversal, observed, is what identifies the correlation as lensing rather than as dust or as quasars and galaxies sitting in the same structures.

A lens counted instead of measured

A foreground mass makes the sources behind it brighter and spreads them over a larger patch of sky. Whether the patch then holds more of them or fewer depends only on how steeply their numbers rise towards faint fluxes, and at one slope the two effects cancel exactly. So a lens can be weighed without measuring a single shape — by counting — and the signal changes sign from bright quasars to faint ones in a way that nothing but lensing can imitate.

galaxies · Weak lensing

Named alongside it

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

ConvergenceTime-delayHubble constantEinstein radiusLine of sight structureStellar kinematicsCritical surface densityGravitational lensingLens modelMagnificationOrbital anisotropySelection effect

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