Astronomy & Celestial Mechanics
the sky, computed
Concept

Critical surface density — where it appears

The projected surface density at which a gravitational lens becomes strong enough to produce multiple images, c²D_s/4πGD_lD_ls. It depends only on the geometry of the three distances, which is why converting a measured shear into a mass requires the redshifts of both the lens and the source.

Named by 3 essays across 2 fields — 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 One cluster weighed two ways, and the redshift where the two swap places. The fractional error on the mass of a single cluster of 3·10¹⁴ solar masses, on a logarithmic axis, against the cluster's redshift. Weighed by the shapes of the galaxies behind it (rising curves), the error is 50 per cent at redshift 0.3 and 750 per cent at 1.5 for a survey of today's depth, 10 galaxies per arcmin²; 26 per cent at redshift 0.3 and 187 per cent at 1.5 for a survey of the next depth, 30 galaxies per arcmin²: the galaxies far enough behind the cluster to be lensed by it run out. Weighed by the dipole it draws on the microwave background (nearly flat curves), the error is 461 per cent at 0.3 and 337 per cent at 1.5 for a map with 10 μK-arcmin of noise; 224 per cent at 0.3 and 153 per cent at 1.5 for a map with 3 μK-arcmin of noise — never good for one cluster, and almost the same at every distance, because the source is at redshift 1090. The curves cross at z = 1.22 (10 galaxies per arcmin², 10 μK-arcmin), z = 0.89 (10 galaxies per arcmin², 3 μK-arcmin), z = 1.77 (30 galaxies per arcmin², 10 μK-arcmin), z = 1.41 (30 galaxies per arcmin², 3 μK-arcmin). Below the crossing a galaxy survey weighs each cluster better; above it, only the background weighs it at all.

A mass scale checked near and carried far

A cluster can be weighed by the galaxies behind it or by the microwave background behind it, and the two share no systematic error. They do not share a redshift range either: the galaxies run out at about one, the background never does, and a comparison can only be made where both still work.

cosmology · Sunyaev zeldovich

Named alongside it

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

ConvergenceExcess surface densityGravitational lensingThe mass-sheet degeneracyPhotometric redshiftShape noiseCluster mass functionCosmic shearThe cross componentDark matterDeflection angleEinstein radius

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