Astronomy & Celestial Mechanics
the sky, computed
Series

Hubble constant — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Four expansion histories that agree exactly today. The scale factor against time, with the present at zero and every model normalised to a = 1 and an expansion rate of 67.36 km/s/Mpc there. That normalisation is the figure: four universes that are indistinguishable from a measurement made now, separated entirely by what is behind and ahead of them. Each curve is integrated from da/dt = aH₀E(a) rather than from its own closed form, so the four are compared through one routine. The age each implies is where its curve meets zero: empty (Ω = 0) 14.52 Gyr, matter only (Ω = 1) 9.68 Gyr, ΛCDM (Planck 2018) 13.80 Gyr, closed (Ω = 2) 8.29 Gyr. The empty universe's 14.52 Gyr is the Hubble time 1/H₀ exactly, which is what makes it the natural thing to measure an acceleration against. The closed model turns over and is stopped at its turnaround.

    One number that is an age, a size and a density

    The slope of the velocity–distance relation has units of inverse time, so it can be read as an age, multiplied by c to give a length, or squared to give a density. All three readings are natural, all three are quoted, and not one of them is the quantity it appears to be.

    part 1 · cosmology
  2. H₀: nine determinations in two families. Published determinations of H₀, each with its quoted one-sigma interval, sorted into two families — measured locally, in the late universe, against inferred from z ≈ 1100 through a model. The shaded band behind each family is that family's inverse-variance weighted mean: 72.66 ± 0.75 across 5 of them, against 67.40 ± 0.41 across 4. The difference is 5.26 ± 0.85 km/s/Mpc, which is 6.2 standard deviations, computed here from the quoted errors alone. That number is an upper bound on the significance rather than the significance: the determinations within each family share calibrations, samples and in two cases the same supernovae, so they are not independent, and a correlated pair combines to something wider than the formula used here gives. What the figure does establish is that the split is not one discrepant measurement against a consensus — it is two internally consistent groups, and the grouping is by method rather than by result.

    The same constant, measured twice, five sigma apart

    Cepheids and supernovae give an expansion rate of about 73 kilometres per second per megaparsec. The microwave background gives 67.4. Both quote errors near one per cent, both have been rebuilt from scratch by rival teams, and the gap between them has grown as the measurements have improved.

    part 2 · cosmology
  3. Take 8 per cent off the horizon and the tension is gone. The Hubble constant against the sound horizon at recombination, along the locus the microwave background's measured angular scale fixes. What is measured is an angle — the angular size of the horizon, to a part in three thousand — and an angle is a length divided by a distance, so extracting an expansion rate requires the length. That length is computed from the physics of the first four hundred thousand years: the baryon density, the radiation density, the number of relativistic species and the recombination history. Change any of those and the locus is unchanged while the point on it moves. The horizontal band is the late-universe measurement from the distance ladder, and it meets the locus at 136 megaparsecs — 8 per cent shorter than the standard model gives. That is the arithmetic behind every proposal to resolve the disagreement by changing the early universe rather than the late one.

    A constant that is an angle divided by a length

    The microwave background does not measure an expansion rate. It measures one angle — the apparent size of the sound horizon at recombination — to a part in three thousand, and converting that angle into a rate requires the horizon's physical length, which is computed from a model of the first four hundred thousand years rather than observed.

    part 3 · cosmology
  4. The Hubble constant the microwave background implies, with an early component. The Hubble constant that keeps the microwave background's measured angular sound horizon fixed, against the peak fraction of early dark energy at z = 3500, with the baryon density held at its standard value and the matter fraction at the 0.315 that late-time distances measure, so that the dark-matter density rises with the Hubble constant. With none, it is 67.4 km/s/Mpc and the horizon 144.3 Mpc. At 5 per cent it is 68.7; at 10 per cent 70.2, with the horizon at 138.5 Mpc; the local value of 73 is reached at 19 per cent. The shrinkage of the horizon is what raises the constant: a shorter ruler at the same angle must be nearer, and nearer at a fixed matter fraction means faster expansion. The extra dark matter this requires is the cost the published fits also find: at the value that reaches 73, the physical matter density is 18 per cent higher than in the standard model.

    A ruler shortened by something already gone

    The microwave background gives the Hubble constant only through a computed length, the distance sound travelled before the universe became transparent. Add a component of energy that exists for a few thousand years before recombination and then vanishes faster than radiation, and the universe expands a little faster while the sound is travelling, the ruler comes out a few per cent shorter, and the inferred expansion rate rises toward the local value. The price is paid in dark matter and in the finest details of the oldest light.

    part 4 · cosmology
  5. The end of the red giant branch, found as an edge. The luminosity function of a synthetic galaxy halo — the number of red giants against apparent magnitude in the I band, smoothed — and its derivative, which is what the edge detector computes, searched over the magnitudes brighter than the branch's crowded lower part, as every application of the method does. 1582 stars are drawn: red giants rising in number towards fainter magnitudes as ten to the 0.3 times the magnitude, ending at the tip at magnitude 24, with a sixth as many asymptotic giants thinning out above it, all measured with errors of 0.08 magnitudes. The derivative peaks sharply where the counts jump, here at 24.09 — 0.09 from the true tip. The red giant branch ends where helium ignites in a degenerate core of almost fixed mass, so the tip's absolute magnitude is close to −4.05 for old, metal-poor stars, and the edge's apparent magnitude is a distance.

    An edge that needs no pulsation

    The local measurement of the expansion rate rests on a calibration step, and for decades that step was the Cepheid variables. The tip of the red giant branch replaces it with an edge — the brightness at which old stars ignite helium, nearly the same in every galaxy — found by counting stars rather than modelling them. It gives a slightly lower expansion rate, and whether the difference is a flaw in one calibration or in both is the local half of the Hubble tension.

    part 5 · cosmology
  6. How deep a hole the Hubble tension needs, and how deep holes are. The typical departure of the matter density from the mean, averaged over a sphere — one standard deviation, σ, from the linear power spectrum with σ₈ = 0.81 — against the sphere's radius, both on logarithmic axes. It falls from 0.55 at 20 Mpc to 0.022 at 300 Mpc. A local expansion rate 8.3 per cent above the global one, the size of the Hubble tension, needs the sphere the supernovae sample to be underdense by 0.47 (dashed), with a growth rate of 0.53. Across the radii the supernovae span (shaded, 100–600 Mpc) that is 4 to 67 standard deviations.

    A hole too deep for the sky to have made

    If the Milky Way sat inside a vast underdense region, everything around it would be expanding faster than the universe as a whole, and the locally measured Hubble constant would come out high — which is what Cepheids and supernovae find. The idea needs no new physics. It needs a void about half empty across the six hundred megaparsecs the supernovae sample, and the same cosmology that predicts the microwave background prices such a void at dozens of standard deviations.

    part 6 · cosmology
  7. The flow round the Local Group, shell by shell. Radial velocity against distance from the Local Group's centre today, for spherical shells of matter in a standard expanding background with the cosmological constant, round a group whose mass inside the zero-velocity surface is 1×10¹² M☉, 2×10¹² M☉, 4×10¹² M☉. Inside the zero-velocity radius — 0.79 Mpc, 1.00 Mpc, 1.26 Mpc — the shells are falling in; outside it they recede, slowly at first and then at the Hubble rate (dashed, 67.7 km/s per Mpc). At 1.5 Mpc the three groups give 75, 57, 30 km/s against 102 for an undisturbed flow; by 5 Mpc the curves differ by 10 km/s and have nearly rejoined the line. The group's pull is written into the first two or three megaparsecs of the flow, and only there.

    A flow too quiet for the lumps it runs between

    The galaxies within ten megaparsecs of the Milky Way recede almost exactly as the Hubble law says, scattering about it by a few tens of kilometres a second. The matter they move through is lumpy at the factor-of-several level, and elsewhere galaxies move relative to their neighbours at three hundred. The quiet is a measurement, it limits what the group's own pull can be read from, and it says something about the place rather than about the law.

    part 7 · cosmology

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