Astronomy & Celestial Mechanics
the sky, computed
Concept

Weak lensing — where it appears

The small, coherent distortion of background galaxy shapes by mass along the line of sight. Averaged over many galaxies it measures total mass whatever state that mass is in, so it calibrates cluster masses and maps the amplitude of structure.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

galaxies · Weak lensing 1020 clusters or 372, and the survey cannot say which parameter moved. The number of clusters per unit redshift a survey finds above an integrated Compton signal of 8·10⁻⁵ arcmin² over 6 per cent of the sky, computed from the Sheth–Tormen halo count grown by the linear growth factor, the comoving volume in each redshift slice, and the calibrated relation between signal and mass. Each curve is one pair of assumptions: the amplitude of structure, σ₈, and the hydrostatic mass bias, 1 − b, which says how far the X-ray masses the relation was calibrated on fall below the true masses. With σ₈ = 0.811 and 1 − b = 0.8 the survey finds 1020; σ₈ = 0.811 with 1 − b = 0.6 gives 372; σ₈ = 0.75 with 1 − b = 0.8 gives 548. The counts fall at low redshift because there is little volume, and at high redshift because massive halos have not yet formed, and the peak sits near z = 0.24. Lowering 1 − b pushes the threshold onto more massive and rarer halos; lowering σ₈ makes every halo rarer. The two lower curves differ in total by 47 per cent and in normalised shape by at most 2 per cent of the peak. A survey that assumed 1 − b = 0.8 would read the 372 clusters of the curve with 1 − b = 0.6 as σ₈ = 0.716, with a redshift distribution that differs from it by at most 12 per cent of the peak — which is the only handle the survey has on the difference, and it is smaller than the counting noise in any redshift bin holding fewer than about 67 clusters. A total count cannot distinguish a universe with less structure from a survey that has misjudged its masses, and it is that degeneracy, not the counting, that has been argued about since the first large catalogue.

Too few clusters, or a scale that reads light

A catalogue selected on the microwave shadow is, past redshift one half, very nearly a catalogue of everything above a fixed mass — so its count by redshift is the growth of structure read almost directly. Almost, because the mass behind the threshold comes from a calibration, and a scale that reads twenty per cent light is indistinguishable from a universe with less in it.

cosmology · Sunyaev zeldovich The matter power spectrum with galaxies' feedback taken out. The ratio of the matter power spectrum with baryonic feedback to the spectrum without it, against wavenumber in h per megaparsec on a logarithmic scale, for three strengths of feedback that span the range the large hydrodynamic simulations disagree over. Gas blown out of haloes by supernovae and active nuclei removes mass from scales below a few megaparsecs, so the ratio falls below one from about 0.5 h per megaparsec and is deepest near 4 h per megaparsec, at 9, 17 and 26 per cent; at still smaller scales gas that has cooled into galaxy centres adds mass back. The marks show the wavenumber that dominates a lensing measurement at multipoles 100, 1000, 5000, taken where the lensing kernel for sources at redshift 1 peaks: 0.09, 0.87, 4.35 h per megaparsec. The shape is a parametric stand-in placed where simulations put the effect, not a simulation.

A measurement of dark matter that needs a model of galaxies

Weak lensing weighs all the matter, dark and ordinary, by how it bends light. But the ordinary matter does not sit where gravity alone would put it — galaxies' supernovae and black holes blow a large part of it out of their haloes — and the scales where that happens are the scales that carry most of a lensing survey's statistical power. Ignored, it reads as a less clumpy universe, in the direction of the disagreement lensing has with the microwave background.

galaxies · Weak lensing A ratio relativity predicts from one density. The ratio EG — written E with a subscript G in the literature — of the lensing around a set of galaxies to their clustering and infall, against redshift. In relativity it is Ωₘ₀/f(z) and depends on nothing else: with Ωₘ₀ = 0.3153, 0.598 today, 0.455 at z = 0.32 and 0.360 at z = 1, falling because f rises towards one as matter comes to dominate. The dashed curves keep light's bending as in relativity and change how hard matter pulls, by −10% and +10% today, growing with the dark energy's share: a weaker pull slows the infall and raises EG. The point is the first measurement, 0.39 ± 0.06 at z = 0.32 from galaxy–galaxy lensing in the Sloan survey.

A ratio in which the galaxies cancel

Matter falls in one gravitational potential and light is bent by the sum of two, and relativity says the two are equal. A survey that measures both the infall of galaxies and the lensing around them can divide one by the other so that the galaxies' bias and the fluctuations' amplitude both cancel — leaving a ratio that relativity predicts from the matter density alone, and that a gravity which bends light differently from the way it pulls matter cannot fake.

cosmology · Dark energy How small an outlier fraction the mean redshift can bear. The shift in a tomographic bin's mean redshift, f·Δz, against the outlier fraction f, for outliers displaced by Δz = 0.5, 1.5, 2.5, both logarithmic. The dashed line is the requirement for a bin at z = 0.8, 0.0036. Outliers half a unit of redshift away must be fewer than 0.72% of the bin; outliers two and a half units away, fewer than 0.14% — or known to that precision, since a known fraction can be corrected. The best photometric redshifts reach outlier fractions of a few per cent, ten to a hundred times above these lines.

A mean that a few per cent of strays can drag

A lensing survey sorts its galaxies into slices of redshift by their colours, and what its cosmology needs from each slice is a single number, the mean redshift, to two parts in a thousand. A few per cent of the galaxies in a slice are catastrophic outliers — young galaxies at redshift three mistaken for old ones at one — and each moves the mean by its full error. Three per cent of strays two redshift units away move it twenty times more than the survey can tolerate, and finding them needs spectra of exactly the galaxies whose spectra fail.

starlight · Magnitudes 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 The clusters scale with the halo and the stars do not. The total mass in a galaxy's globular clusters (lower, with its ±0.3 dex scatter shaded) and its stellar mass (upper), against the mass of its dark-matter halo, all logarithmic. The clusters follow a straight line of slope one, 3.9×10⁻⁵ of the halo, across five and a half decades. The stars follow a curve that bends at both ends round a peak near 5.8×10¹¹ M☉, because star formation is suppressed in small halos and in large ones. Stars outweigh clusters by 12 times in a halo of 10¹⁰ M☉, 879 times at 10¹², 31 times at 3.2×10¹⁴.

A halo weighed by counting its oldest clusters

The globular clusters round a galaxy add up to about four parts in a hundred thousand of the mass of its dark-matter halo — in a dwarf, in the Milky Way, in a giant elliptical and in a cluster of galaxies, across five decades of halo mass. The galaxy's own stars follow no such rule. So a count of a few dozen old star clusters weighs a halo that its stars cannot, and the strangest thing about the rule may be that it is less strange than it looks.

galaxies · Dark matter

Named alongside it

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

Sigma 8Cosmic shearSystematic errorDegeneracyGravitational lensingPhotometric redshiftStructure growthAGN feedbackAngular-diameter distanceBaryonic feedbackCatastrophic outliersCluster mass function

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