Astronomy & Celestial Mechanics
the sky, computed
Concept

Silk damping — where it appears

The erasure of small-scale temperature structure in the microwave background by photons random-walking during recombination. Anything smaller than the diffusion length is mixed away, leaving a Gaussian cut-off in the power spectrum whose scale measures how thick the last-scattering shell is.

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

The acoustic peaks, and where the geometry says they should be. The temperature angular power spectrum of the microwave background. The drawn curve is a monotone interpolation through the published positions and heights of the six peaks and five troughs of the Planck 2018 TT measurement — it is a representation of data, and nothing between two extrema is claimed. The marks along the top are not: they are computed from this collection's own cosmology as ℓₐ(m − 0.267), where ℓₐ = π × 13866 / 144.43 = 301.6 — π times the comoving distance to last scattering divided by the sound horizon there — is the angle the sound horizon subtends at last scattering turned into a multipole. The two agree to 2.8 per cent at worst across six peaks, which is the whole of what makes this a measurement of geometry: a wave of known physical wavelength, seen at a known distance, is a protractor. The first peak at ℓ = 220 corresponds to about 0.82 degrees on the sky — roughly twice the width of the full Moon, which is the largest hot and cold patch the sky has.

A standing wave frozen at one instant

The temperature of the microwave background varies across the sky by one part in a hundred thousand, and the sizes of the patches are not random. There is a preferred angular scale near one degree, and it is a sound wave that stopped ringing four hundred thousand years after the beginning.

cosmology · Microwave background One ionising photon per atom, and where that happens. The number of photons energetic enough to ionise hydrogen, per baryon, against temperature — with temperature falling to the right, so the universe ages to the right. The quantity is the fraction of a Planck distribution above 13.6 eV, integrated numerically, divided by the 6.13·10⁻¹⁰ baryons there are per photon. It crosses one at 5,844 K, which is z = 2143, and it falls by 19 orders of magnitude across the plot because it is the tail of an exponential. That crossing is the answer to why recombination waits until three thousand kelvin. At hydrogen's own ionisation temperature of 157,803 K there are two billion ionising photons per atom and the gas has no chance; the temperature has to fall by a factor of forty before the supply runs out, and it runs out suddenly because an exponential tail does. The number that sets the factor of forty is not an energy at all — it is the photon-to-baryon ratio, which is to say the entropy per baryon, which is to say a number fixed long before any of this and measured today from the second acoustic peak and from deuterium alike.

The surface the background actually is

Hydrogen ionises at 157,800 kelvin and the universe became transparent at 3,000. The factor of fifty between them is not an error, and it is not about energy — it is about there being two billion photons for every atom, so the far tail of the distribution can keep the gas ionised long after the typical photon has become useless.

cosmology · Microwave background 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 running the measurement cannot yet tell from zero. The running of the spectral tilt, dnₛ/d ln k, against the tilt itself. The ellipses are the microwave-background measurement, nₛ = 0.9649 ± 0.0042 and a running of −0.0045 ± 0.0067, at one and two standard deviations. The three curves are what slow-roll inflation predicts once the tilt is fixed: a running of −(1 − nₛ)²/β, with β = 2 for the plateaus and every α-attractor (and for φ²), 4/3 for the steepest power law still drawn and 4 for a cubic hilltop. At the measured tilt they give −0.0009 to −0.0003, all within a band thinner than a line at this scale, 11 times smaller than the measurement's error.

A drift in the tilt that is the tilt squared

The primordial spectrum's tilt has been measured to half a per cent. Whether the tilt itself changes with scale has not, and slow-roll inflation says why it will be hard: the running is fixed by the tilt, at about six ten-thousandths, ten times below the present error. The one measurement that reaches far enough along the spectrum to feel it — the μ-distortion of the microwave background's colour — would need a precision no planned instrument has.

cosmology · Inflation

Named alongside it

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

Primordial fluctuationsRecombinationSound horizonThomson scatteringVisibility functionAcoustic peaksAngular power spectrumBaryon loadingDamping tailDiffusion lengthE-foldsFreeze out

All concepts