Astronomy & Celestial Mechanics
the sky, computed
Concept

Diffusion length — where it appears

The distance a particle random-walks in a given time, √(λct/3) for a mean free path λ. For neutrons before nucleosynthesis it decides whether a density fluctuation survives to the moment the network fires or is smoothed away first.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The damping tail measures a thickness: ℓ_D = 1400 for a shell 80 deep in redshift. The suppression of small-scale structure in the microwave background, against multipole, on logarithmic axes. Every other feature of the power spectrum measures the last-scattering surface as a surface — its distance, the sound horizon written on it, the ruler it provides. This one measures how thick it is. Recombination takes time: the ionised fraction falls over a range of redshift rather than all at once, and while it is falling the photons are still scattering, so each one random-walks. A photon taking N steps of a mean free path λ diffuses √N λ, which is much further than a single step and much less than the whole interval, and any temperature fluctuation smaller than that distance is mixed away before it can be frozen in. What is left is a Gaussian cut-off, drawn here for three shell thicknesses. A thicker shell means more steps and a longer walk, so it damps at a smaller multipole: Δz = 40 gives ℓ_D = 1980, Δz = 80 gives ℓ_D = 1400, Δz = 160 gives ℓ_D = 990. The observed cut-off is near ℓ = 1400, corresponding to a diffusion length of about 0.03 proper megaparsecs at the time — a scale the reader should compare with the sound horizon, some hundred and fifty comoving megaparsecs, which is what the peaks measure. The tail is therefore a genuine probe of the inside of the transition rather than of its position, and because the damping depends on the free-electron density it is also one of the cleanest constraints on anything that changes it: extra relativistic species, a varying fine-structure constant, or an early energy injection all move ℓ_D while leaving the peak positions nearly alone. The normalisation here is set once by the observed value, so what the figure claims is the scaling with thickness and the shape of the cut-off, not the absolute number.

A blur that measures a depth

Every other feature of the microwave background measures the last-scattering surface as a surface — its distance, the ruler written on it, the geometry between. The damping of its small-scale structure measures how thick it is, because a photon random-walking through a finite transition smears away anything smaller than its walk.

cosmology · Microwave background 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 How far an electron can have come, by its energy. The distance an electron or positron can diffuse in its radiative lifetime, 2√(∫D dt), against its energy, for the diffusion coefficient fitted to the nuclei with slopes ⅓ and 0.6, and 1 eV/cm³ of field and radiation. At 100 GeV the horizon is 2.7 or 4.7 kpc; at 1 TeV, 1.24 or 2.95; at 10 TeV, 0.57 or 1.86. The marks are the distances to the nearest middle-aged pulsars and to the Galactic centre. Above a teraelectronvolt the electrons and positrons arriving at the Earth were made within about a kiloparsec — inside the Sun's own neighbourhood of the disc — and a feature in their spectrum is a feature of a handful of nearby sources, not of the Galaxy.

A positron that cannot have come far

A proton at a teraelectronvolt can wander the Galaxy for millions of years. An electron or a positron at the same energy radiates its energy away in a few hundred thousand, and in that time it can diffuse about a kiloparsec. So the rising fraction of positrons among high-energy cosmic rays is not a fact about the Galaxy. It is a fact about a few objects within a thousand parsecs of the Sun.

galaxies · Cosmic rays

Named alongside it

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

Dark matterMean free pathBaryon densityBaryon-to-photon ratioBig bang nucleosynthesisConvexityCosmic raysCritical densityDamping tailDeuterium abundanceEnergy loss timeInverse Compton scattering

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