Astronomy & Celestial Mechanics
the sky, computed
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Density parameters — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The same universe, four times, as a fraction of itself. Each bar is the fractional contribution of the four components to the total density at one epoch, computed from the Planck 2018 parameters by scaling each component from today: radiation as (1+z)⁴, both kinds of matter as (1+z)³, and Λ as a constant. The familiar figure — five per cent baryons, twenty-six dark matter, sixty-nine dark energy — is the top bar and only the top bar. At recombination the same universe is three-quarters dark matter and Λ is one part in ten million; before matter–radiation equality it is mostly radiation. A pie chart of the contents of the universe is therefore a statement about a moment, and the moment is the one it happens to be drawn in.

    A budget whose familiar part is five per cent

    Five per cent ordinary matter, twenty-six per cent dark matter, sixty-nine per cent dark energy. The figures are quoted everywhere and each one comes from a different measurement, the denominator they are fractions of is itself built out of the expansion rate, and the whole chart is a statement about one instant that was a different chart at every earlier time.

    part 1 · cosmology
  2. A delay that counts the gas nothing can see. The mean dispersion measure of a radio pulse against the redshift it comes from, for several shares of the baryons residing in diffuse ionised gas between galaxies. A pulse is delayed by free electrons in proportion to the column it crosses, and that column is an integral over the expansion history of a density the baryon budget fixes — so the only unknown in the whole expression is the share itself. At z = 1 the relation gives 1085 pc cm⁻³ if every baryon is out there and 543 at 50 per cent. The measurement runs the other way: a burst with a known host redshift and a measured delay returns the fraction, and the answer came out consistent with nucleosynthesis. A pulse a millisecond long weighs the half of the ordinary matter that no survey could find, and it does so because the thing that delays it is the thing that does not shine.

    Half the ordinary matter was missing, and a millisecond found it

    Nucleosynthesis fixes how many baryons there are to better than a per cent. Every survey of where they are came up about half short for two decades — and what closed the gap was the delay a radio pulse picks up crossing gas too thin and too hot for any telescope to have seen.

    part 2 · cosmology
  3. A mass measured by the structure it prevents. The fractional suppression of the small-scale matter power spectrum against the sum of the neutrino masses. The relic density follows from the thermal history with no astrophysics in it — the neutrinos’ share of the critical density, times h², is the mass sum divided by 93.14 eV exactly — and the suppression is about eight times the neutrinos' share of the matter, because a particle moving at a large fraction of the speed of light streams out of an overdensity while it is still growing and takes its own gravity with it. The two mass orderings put a floor under the sum at 58 meV and 100 meV; the cosmological upper bound is near 0.12 eV, which is 7.2 per cent of suppression. The floor and the ceiling are within a factor of two of each other, so this is a measurement about to happen or a model about to break, and the quantity it returns is a sum rather than any individual mass.

    A mass measured by what it stopped from forming

    Neutrinos were relativistic in the early universe and are not now, so they are the one entry in the cosmic budget that changes category. What cosmology measures is not their density but the hole they leave — they stream out of a growing clump and take their gravity with them, and the missing structure bounds a particle mass more tightly than any laboratory has.

    part 3 · cosmology
  4. Structure stops forming above about 14 times the observed Λ. The fraction of matter that ever collapses into a bound object, against the cosmological constant in units of the observed one, holding the primordial fluctuation amplitude fixed. Growth of structure stops once Λ dominates the expansion, so the linear growth factor approaches a finite limit rather than rising for ever, and a larger Λ freezes it earlier and smaller. The asymptotic growth factor at the observed Λ is 1.110; at a hundredth of it, 5.152; at a hundred times, 0.239. The collapsed fraction falls from 0.86 to 0.40 to 0.000 across the same range, and passes a tenth of its present value at 14 times the observed constant. The observed value sits a factor of 14 below the largest one that leaves anything at all — which is the whole of the anthropic argument, and that factor is what has to be compared against the 10¹²⁰ by which the naive theoretical estimate misses.

    A coincidence that is a factor of fourteen

    The cosmological constant is famously wrong by a hundred and twenty orders of magnitude, and famously coincidental in sitting near the matter density now. Computing how large it could be and still leave any structure at all turns the second complaint into a number — and the number is fourteen, not a hundred and twenty.

    part 4 · cosmology

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