Astronomy & Celestial Mechanics
the sky, computed
Concept

Spectral distortion — where it appears

A departure of a spectrum from a pure Planck shape, produced by processes that move photons in frequency without creating or destroying them. The thermal Sunyaev–Zel'dovich effect is the best-measured example, with its characteristic null at 217 GHz.

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

A blackbody at 2.7255 kelvin, filling the sky. The Planck function at 2.7255 K in the units the measurement is reported in, with the peak marked where Wien's law in frequency puts it: x = hν/kT = 2.8214, so ν = 160.2 GHz and the intensity there is 384 MJy per steradian. The points are drawn at the twenty-one frequencies across the FIRAS band, displaced from the curve by a Gaussian of 50 parts per million of the peak, which is the root-mean-square deviation the instrument actually reported. At the scale of this plot that displacement is a fifth of a pixel and the points sit on the line — which is the entire finding. Nothing else in astronomy is a blackbody to a part in twenty thousand: a stellar spectrum is a blackbody with absorption lines cut into it and a continuum that is the wrong shape at both ends. A thermal spectrum this exact requires that the radiation was once in equilibrium with matter, which requires that the universe was once opaque, which requires that it was once hot and dense.

The most perfect blackbody ever measured

The whole sky glows at 2.7255 kelvin, and its spectrum matches the Planck curve to fifty parts in a million. Nothing else in astronomy is thermal to a part in twenty thousand, and a spectrum that exact is not a description of the radiation — it is a constraint on the history of everything that could have disturbed it.

cosmology · Microwave background One cluster moved to z = 1.8: one signal unchanged, the other 57 times fainter. The same cluster of galaxies placed at a series of redshifts, with two ways of detecting it compared. The Sunyaev–Zel'dovich decrement is flat, because it is a fraction of the microwave background and the microwave background has the same surface brightness at every redshift a cluster can sit at — moving the cluster further away shrinks it on the sky but does not make its shadow shallower. The X-ray surface brightness of the identical object falls as (1+z)⁻⁴, the dimming every surface brightness suffers in an expanding universe, and by z = 1.8 it is 57 times below where it started. This is why the cluster surveys that reach the early universe are millimetre surveys: an X-ray telescope's cluster catalogue thins out with distance and a Sunyaev–Zel'dovich catalogue is limited only by how large the cluster looks and by how faint a fractional distortion the instrument can measure. What the effect cannot supply on its own is a distance — the signal that does not know how far away the cluster is also cannot say — so every one of these clusters still needs a redshift measured the ordinary way, from a spectrum of a galaxy inside it.

A shadow that does not get fainter with distance

Every other way of finding a cluster of galaxies gets harder the further away the cluster is. One does not. A cluster's hot gas scatters about one microwave background photon in a hundred to a higher frequency, and because the result is a fraction of a background rather than a flux from a source, the same cluster is exactly as detectable at ten billion light years as at one.

cosmology · Sunyaev zeldovich At 15 keV the decrement is 9 per cent shallower and the null has moved to 224.4 GHz. The thermal Sunyaev–Zel'dovich distortion at one fixed Compton parameter, 10⁻⁴, computed with the relativistic kinetic equation expanded to second order in kTₑ/mₑc² for gas at 5, 10, 15 keV, against the non-relativistic shape that is the same for every temperature. All are scaled to the non-relativistic curve's largest excursion. Heating the gas at fixed y does two things to the spectrum. The decrement becomes shallower — by 9.3 per cent at its deepest point for 15 keV — and the increment becomes lower and broader, by 17.3 per cent at its peak, because fast electrons scatter photons over a wider spread of frequencies than slow ones and some of the boost is carried to frequencies above the drawn range. The crossing moves up, from 217.5 GHz to 224.4 GHz. A cluster's temperature is therefore written into the shape of its distortion and not only its amplitude, which is what a thermometer needs; and a Compton parameter read off one frequency with the non-relativistic shape is biased low by the drawn amount, which is what a mass estimate does not need. The expansion is good to well under a per cent below 15 keV; above 20 it has to be replaced by the exact integral.

A null that moves with the temperature

The frequency at which a cluster's hot gas vanishes from the microwave sky was derived for slow electrons. The electrons in a massive cluster move at a quarter of the speed of light, the null drifts half a gigahertz per keV, and what is left at the old frequency reads as a velocity as large as the ones being sought.

cosmology · Sunyaev zeldovich 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.

Compton y parameterCosmic microwave backgroundIntracluster mediumInverse Compton scatteringThe Sunyaev–Zel'dovich effectAngular-diameter distanceBlackbody spectrumCluster mass functionE-foldsEntropy per baryonHydrostatic biasHydrostatic mass

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