Astronomy & Celestial Mechanics
the sky, computed
Concept

Spatial curvature — where it appears

The departure of the geometry of space from flat on the largest scales, measured by the parameter Ωk. Positive curvature makes the angles of a large triangle sum to more than 180 degrees, and the measured value is within 0.2 per cent of zero.

Named by 2 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 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

Named alongside it

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

Acoustic peaksAcoustic scaleAge of the universeAngular power spectrumBaryon acoustic oscillationsBaryon loadingCosmic varianceCritical densityDegeneracyHubble constantMultipolePhoton baryon fluid

All concepts