Astronomy & Celestial Mechanics
the sky, computed
Concept

Critical density — where it appears

The density at which the expansion is exactly marginal, about five hydrogen atoms per cubic metre, used as the unit every other density is quoted in. It is computed from the Hubble constant alone, and every component of the universe is quoted as a fraction of it rather than in kilograms per cubic metre.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

A quadratic and a linear, crossing at 149 objects. The two rates that decide whether a shell at 900 km is stable, against how many objects are in it. Production goes as N² — every collision needs two objects, so the number of collisions is proportional to the square of the population, and each one is taken here to make 1600 trackable fragments. Removal goes as N, because drag acts on each object independently and takes 1,195 years to do it at this altitude. A quadratic and a linear cross exactly once, at 149 objects in this shell, and above that crossing the population grows with nothing launched. The shell presently holds about 2,280, which is 15 times the crossing. Every number on the production side is uncertain by a factor of a few — the fragment yield most of all, and the cross-section is calibrated against an observed collision rate rather than measured — so the position of the crossing carries that uncertainty with it. The shape does not, and the shape is the argument: a quadratic overtakes a linear once and never comes back, the crossing falls as the altitude rises because the lifetime is in the denominator, and what results is a threshold rather than a trend.

A collision rate that needs no collision

The flux through an orbital shell is a gas-kinetic calculation with no orbits in it. Production goes as the square of the population and removal goes as the first power, so a quadratic overtakes a linear once and never comes back — and which side of that a shell is on is decided by its altitude.

spaceflight · Orbital debris 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.

cosmology · Hubble constant 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.

cosmology · Density parameters A lumpy universe makes more of both, and one of them was already too much. The deuterium–lithium plane, with the curve a homogeneous universe traces as its baryon density varies and the points a two-zone universe reaches at a fixed mean density of η₁₀ = 6.13. The dense zone occupies 15 per cent of the volume, and the contrast between the zones runs from 1 — which is the homogeneous case — to 100. Every mixture lies up and to the right of its own homogeneous point, and that is not a modelling choice: an abundance is measured per baryon, so what a telescope averages is η times the abundance — and for both deuterium and lithium that product is a convex function of the density, whose average therefore exceeds its value at the average. The excess is large. At a contrast of 100 the mixture gives D/H = 1.13e-4 against 2.51e-5 smooth, and ⁷Li/H = 1.77e-8 against 4.70e-10, a factor of 37.7. The extra deuterium is exactly what the proposal was for: it lets the mean baryon density be raised while the observed D/H is still matched, which in the 1980s was the one way to make the baryons account for all the matter that dynamics required. The extra lithium is what it costs. The measured abundance was already a factor of 2.9 below the homogeneous prediction, and every step toward the lumpy universe that fixes the density makes that discrepancy worse — which is why the answer to "the baryons are lumpy" turned out to be that the extra matter is not baryons.

The universe that was lumpy at one second

If the baryons were unevenly spread when the network fired, each region ran its own nucleosynthesis and what is observed is an average. For a decade that was the one way to make ordinary matter account for all the matter — and the reason it fails is a theorem about convex curves.

cosmology · Nucleosynthesis 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.

cosmology · Density parameters 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.

cosmology · Density parameters Every curvature has a Hubble constant that leaves the peaks where they are. The Hubble constant a universe needs in order to put the acoustic peaks of the microwave background at the measured angle, against its spatial curvature Ωₖ, with the physical densities of baryons, matter and radiation held at their fitted values so that the sound horizon is unchanged. Along this line the angle θ* — the sound horizon over the distance to last scattering — is exactly constant and the peak positions do not move. The flat member has H₀ = 67.4 km/s/Mpc; a closed universe with Ωₖ = −0.044 needs 53.5, with a matter density of 0.50; an open one with Ωₖ = +0.010 reaches the locally measured 73. The peaks alone therefore measure a combination of curvature and expansion rate, not either one, and the statement that the universe is flat to a few parts in a thousand needs a second distance measured somewhere else.

A flatness that needs a second distance

The universe is said to be flat to two parts in a thousand, and the figure is usually credited to the microwave background. On its own the background measures one angle, and one angle fixes one distance — so every curvature has a Hubble constant that reproduces the peaks exactly. Flatness is where that line crosses a second measurement, and it is only as good as the crossing.

cosmology · Density parameters Whether an excited ion radiates depends only on what reaches it first. The rate per second at which an oxygen ion in the state that gives the 5007 Å line loses its energy, by radiating and by collision, against electron density, both logarithmic. Radiating takes about 51 seconds whatever the surroundings. Collisions speed up with density and cross at 6.8e+5 per cubic centimetre. The air at the Earth's surface, 2.7×10¹⁹ molecules per cubic centimetre, has a collision rate about 4×10¹³ times the radiative one, so the ion loses its energy without radiating. A nebula at 10⁴ per cubic centimetre is sixty times below the crossing. The line has a tiny transition probability; the atom was never forbidden to make it, and it requires only the emptiness in which there is time.

An element that was only empty space

In 1864 William Huggins turned a spectroscope on a planetary nebula and saw a bright green line that matched nothing on Earth. It was given a name, nebulium, and it was treated as a new element for sixty years, until the periodic table had no room left for it. In 1927 Ira Bowen showed it was ordinary oxygen, emitting by a route that collisions close long before any laboratory vacuum could reach it. The line was never forbidden. It simply needs about a minute of being left alone, and nothing on Earth gets that.

starlight · Spectra

Named alongside it

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

Baryon densityAge of the universeCoincidence problemCosmic inventoryDark matterDensity parameterEquation of stateHubble constantMean free pathAcoustic scaleAnthropic boundBallistic coefficient

All concepts