Astronomy & Celestial Mechanics
the sky, computed
Concept

Cosmic variance — where it appears

The irreducible uncertainty that comes from having only one universe and a finite volume of it to observe. It sets a floor on any measurement of a large-scale quantity, and no amount of extra observing lowers it because there are no more independent volumes to look at.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

One curve the temperature fixes, and one line the polarisation adds. The plane of the optical depth to reionisation against the amplitude of the primordial fluctuations. The temperature power spectrum of the microwave background measures the product of the amplitude and the exponential of minus twice the optical depth, so it constrains a curve rather than a point: more electrons scattering the photons out is indistinguishable from fewer fluctuations to begin with, and the two trade along the drawn locus across a factor of 1.22 in amplitude. What breaks it is the polarisation at the largest angular scales, where rescattered photons regenerate a signal whose amplitude is proportional to the optical depth itself rather than to its exponential. That constraint is nearly vertical here, it comes from a handful of multipoles at the very largest scales, and it is the single hardest measurement the microwave background has demanded — because at those scales the Galaxy's own polarised emission is larger than the signal.

An amplitude and a depth that arrive multiplied

The microwave background's temperature fluctuations are the primordial ones damped by everything that scattered them since. The damping is uniform, so a smaller starting amplitude and more scattering produce identical maps — and separating them requires a signal from a handful of the largest angular scales, where the Galaxy's own emission is larger than what is being measured.

cosmology · Reionisation Residuals of 112 per cent nearby and 2.2 far out. Deviations from a pure Hubble flow, in per cent, against distance. Each galaxy carries a peculiar velocity of a few hundred kilometres a second — part a coherent bulk flow shared with its neighbours and part a random dispersion — and that velocity is added to its recession. Since the recession grows with distance and the peculiar velocity does not, the fractional error falls as one over the distance: it is 112 per cent at 5 megaparsecs and 2.2 at 250. The practical consequence is a lower cut-off on any Hubble-constant measurement: below about 40 megaparsecs the motions dominate, and the coherent part does not average away over a sample because neighbouring galaxies share it. Choosing that cut-off is one of the analysis decisions a local expansion rate depends on.

A residual that is somebody else's velocity

A redshift is not a distance until the galaxy's own motion has been removed, and galaxies move at a few hundred kilometres a second. Nearby that is comparable to the expansion itself, so the local Hubble diagram's scatter is motions rather than measurement — and the motions are shared between neighbours, so they do not average away.

cosmology · Dark energy One stream after three different populations of dark subhaloes. The density of stars along the middle sixteen kiloparsecs of a thin stream, relative to its mean, after four billion years of encounters with dark subhaloes drawn from a mass function falling as the −1.9 power of the mass between 10⁵ and 10⁸ solar masses. Top: the abundance cold dark matter predicts — in this realisation 26 encounters, 4 of them above a million solar masses — which leave 32 per cent of the stream's length more than 30 per cent below the mean density. Middle: 0.3 of that abundance, 8 encounters and 7 per cent of the length in gaps. Bottom: none at all, where the only variation is the counting noise of the 60,000 model stars. No single dip in the top trace is a clean gap: they overlap, and a large one and several small ones are superposed at every point. What differs between the three is the statistics of the whole trace, not any feature in it.

A spectrum that counts what no single gap can

One gap in a stellar stream measures a subhalo's mass multiplied by an unknown time. A stream crossed by dozens of them measures something better — how many subhaloes of each size there are — but only as a statistic of the whole stream's density, where no feature can be pointed to and one stream is one throw of the dice.

galaxies · Stellar streams A velocity error that grows with distance. The uncertainty in one galaxy's peculiar velocity, v = cz − H₀d, against its distance in Mpc/h, when the distance comes from an indicator with a fractional error of 5 per cent, 20 per cent, 25 per cent (type Ia supernovae, 5%; Tully–Fisher, 20%; fundamental plane, 25%). The redshift is exact to a few kilometres a second, so the whole error is the distance error times H₀, and it grows in proportion to the distance. The horizontal line is the rms velocity of a galaxy along one direction in linear theory for the model drawn, 302 km/s. Each galaxy's own velocity is measured to better than its own size only inside 60 Mpc/h for 5 per cent, 15 Mpc/h for 20 per cent, 12 Mpc/h for 25 per cent; beyond that a single velocity is mostly noise, and a velocity survey is useful only as an average over many galaxies.

Velocities that see past the edge of the map

A redshift survey infers the motions of galaxies from the distortions of its own map. A peculiar-velocity survey measures them directly, one galaxy at a time, by subtracting a distance from a redshift — and every one of those measurements is dominated by its own error beyond a few tens of megaparsecs. What makes the survey worth doing is that a velocity answers for structure far larger than the volume it is measured in.

cosmology · Large-scale structure 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 The spectrum ahead is the spectrum behind, shifted. The difference between the local power spectrum of the microwave background measured in the direction of the observer's motion and in the opposite direction, as a percentage of the average, against multipole, at 369.8 km/s. Ahead, the pattern is compressed and every feature moves to a higher multipole by a factor 1 + β; behind, to a lower one by 1 − β, with β = 1.234e-3. The difference is largest where the spectrum is steepest — 2.30 per cent at most in this range — and passes through zero at every peak and trough, where a small shift changes nothing. The spectrum is an interpolation through the measured peaks and troughs, so its steepness between them is known only to that resolution. A per cent of a quantity estimated from millions of independent modes is a measurable number.

A speed read from the finest spots in the map

The dipole in the microwave sky is taken to be the observer's motion and subtracted. The same motion also squeezes the small-scale pattern towards the direction of travel by a part in a thousand, and that squeeze measures the speed a second time without using the dipole at all — which is the only way to ask how much of the dipole is the universe's own.

cosmology · Microwave background How deep a hole the Hubble tension needs, and how deep holes are. The typical departure of the matter density from the mean, averaged over a sphere — one standard deviation, σ, from the linear power spectrum with σ₈ = 0.81 — against the sphere's radius, both on logarithmic axes. It falls from 0.55 at 20 Mpc to 0.022 at 300 Mpc. A local expansion rate 8.3 per cent above the global one, the size of the Hubble tension, needs the sphere the supernovae sample to be underdense by 0.47 (dashed), with a growth rate of 0.53. Across the radii the supernovae span (shaded, 100–600 Mpc) that is 4 to 67 standard deviations.

A hole too deep for the sky to have made

If the Milky Way sat inside a vast underdense region, everything around it would be expanding faster than the universe as a whole, and the locally measured Hubble constant would come out high — which is what Cepheids and supernovae find. The idea needs no new physics. It needs a void about half empty across the six hundred megaparsecs the supernovae sample, and the same cosmology that predicts the microwave background prices such a void at dozens of standard deviations.

cosmology · Hubble constant

Named alongside it

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

Peculiar velocityPower spectrumBulk flowDegeneracyGrowth rateHubble constantAberrationAcoustic scaleAge of the universeAngular power spectrumBaryon acoustic oscillationsBoost

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