Astronomy & Celestial Mechanics
the sky, computed
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Weak lensing — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · galaxies
  2. Two bands 2.1 standard deviations apart, and neither is a point. The plane of the matter density against the amplitude of matter fluctuations, with the constraints from a weak-lensing survey and from the microwave background drawn as bands. Lensing measures the shear produced by structure along the line of sight, and that shear depends on how much matter there is and on how clumpy it is in a fixed combination — more matter arranged less clumpily gives the same signal. The locus is a power law of exponent one half, and the combination it fixes is written S₈. The two bands are separated by 2.1 standard deviations, and whatever that separation is, it is a statement about the growth of structure between recombination and now rather than about either measurement's precision. Neither band alone determines either quantity, which is why the disagreement is quoted in the combination rather than in the parameters.

    Two parameters that lensing measures as one

    A weak-lensing survey measures how much the shapes of distant galaxies are distorted by the matter in front of them, and that distortion depends on how much matter there is and on how clumpily it is arranged. The two enter as a product. What the survey determines is one number, and the disagreement between surveys and the microwave background is stated in that number because neither measures either quantity alone.

    part 2 · galaxies

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