Astronomy & Celestial Mechanics
the sky, computed
Concept

Halo mass function — where it appears

The number density of dark matter haloes per interval of mass at a given epoch. Its exponential tail at high mass is extremely sensitive to the amplitude of density fluctuations, which is why counts of massive clusters constrain cosmology.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Galaxy formation is inefficient nearly everywhere. Star formation efficiency — the stellar mass a halo has made, divided by the 0.157 of its mass that is baryons — against halo mass, by abundance matching. The n-th most numerous halo is assigned the n-th most numerous galaxy and nothing else is assumed: the halo count is a Sheth–Tormen mass function integrated from a linear power spectrum, the galaxy count is a measured double Schechter, and the matching is a monotone map between two cumulative counts. The curve peaks at 22 per cent, at a halo mass of 10^11.89 M☉, and falls to 0.6 per cent at the bottom of the range and 0.3 per cent at the top. A halo of the Milky Way's mass, 1.3·10¹² M☉, sits near the peak at 21 per cent — and near the peak means near the best any halo has ever managed. The two sides are two different problems and the figure cannot tell them apart: below the peak the shallow potential lets supernovae drive gas out, above it the gas falling in is shock-heated and cannot cool fast enough. What the abundance-matching assumption cannot show is scatter — it assigns one galaxy mass per halo mass by construction, and the real relation has about 0.15 dex of spread that this method is blind to by definition.

Two counts that are not the same shape

Dark matter halos are counted by a calculation that knows nothing about stars. Galaxies are counted by a survey. Laid on the same axes the two curves disagree at both ends and agree nowhere, and the knee is where the two disagreements hand over.

galaxies · Luminosity function 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 Haloes at the centre of a void, and in an average place. The number of dark matter haloes per unit volume and per factor e in mass, against halo mass in solar masses per h, in the universe on average (dashed) and inside a sphere of 5 Mpc/h radius at the centre of a void whose density there is 20 per cent of the mean (solid), from the excursion-set theory at redshift zero. The void's sphere holds 9.2e+12 solar masses per h of matter altogether, and no halo can be heavier than that. At the dwarf end the void holds 38 per cent of the average abundance per unit volume — more than its 20 per cent share of the matter. At the mass of the Milky Way's halo, 10¹² M☉/h, it holds 1.62 per cent. An emptiness of galaxies like the Milky Way is guaranteed by the theory; an emptiness of dwarfs is not.

A void is not short of dwarf haloes

The centre of a large void holds a fifth of the average density of matter and almost no galaxies, and it is natural to read the two facts as one. They are not. The theory of how haloes form says a void turns its matter into dwarf-sized haloes at least as readily as anywhere else — it is short of big haloes, not small ones — so a void empty of dwarf galaxies is a statement about which haloes were allowed to light up.

cosmology · Large-scale structure The number of lit dwarfs at a void's centre, against the date it was reionised. The number of haloes heavier than 3.2·10⁹ M☉/h that had collapsed from the matter of a 5 Mpc/h sphere by the redshift at which that sphere was reionised, on a logarithmic axis, against that redshift — for a sphere at the centre of a void with 20 per cent of the mean density today (solid) and an average sphere of the same size (dashed). If only those haloes could make stars, they are the sphere's dwarf galaxies. Reionised at 5.3: 4.3 in the void sphere, 195 in the average one; reionised at 6: 2.0 in the void sphere, 121 in the average one; reionised at 7.7: 0.22 in the void sphere, 29 in the average one. Each unit of delay in the date multiplies the void's count by 3.2 and the average sphere's by 2.1: the void's dwarfs sit further out in the tail of the distribution, and a tail is steep. A count of a few galaxies at a void centre is a statement about when that void's gas was heated, to within a fraction of a unit of redshift — provided the threshold is known.

A void dated by the dwarfs that lit in time

Reionisation did not heat the universe all at once, and the centres of the large voids were among the last places it reached. A dwarf halo there could make stars only if it had grown heavy enough before its own gas was heated — so the number that did is a clock for one void, and a clock that runs faster the emptier the void.

cosmology · Large-scale structure

Named alongside it

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

Dark matter haloStructure growthCosmic voidDwarf galaxyPress schechterReionisationSpherical collapseAbundance matchingAngular-diameter distanceBaryon fractionCluster mass functionCooling time

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