Astronomy & Celestial Mechanics
the sky, computed
Concept

Hubble parameter — where it appears

The expansion rate of the universe at a given time, H = ȧ/a, the fractional rate at which distances between freely moving objects grow. Its present value is the Hubble constant, and its history is what the Friedmann equation predicts from the universe's contents.

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

A sphere reconstructed as a spheroid, and two distortions 0.15 apart. Left: the acoustic scale in the plane of separation across the line of sight against separation along it, one quadrant of it. In the cosmology that actually holds, the sound horizon is a sphere of 99.0 h⁻¹ Mpc and its locus here is a quarter circle. That is the whole content of the Alcock–Paczyński test: nothing about the early universe distinguishes the radial direction from the transverse one, so any departure from a circle is a statement about the observer's arithmetic rather than about the ruler. Converting angles into transverse separations needs the transverse comoving distance and converting redshift intervals into radial ones needs H(z), so assuming distances 1.1 times too large and rates 0.94 times too small returns an ellipse of axis ratio 0.855 — and the ellipticity measures that distance times the expansion rate over c, in which the sound horizon has cancelled. A ruler of unknown length still measures a shape. The third curve is the difficulty: peculiar velocities also distort the same correlation function along the same axis, squashing it by 1/(1+β) = 0.704 for β = 0.42, and a squashing is a squashing. Separating a geometric distortion from a dynamical one is the entire art of the measurement, and it is done by using the fact that they have different dependences on scale — the velocities act on the broad-band shape and the ruler is a feature. Right and below: the two numbers the same feature gives at each redshift. Across the line of sight, the transverse distance divided by the sound horizon; along it, c divided by the expansion rate times the sound horizon. Two functions of the expansion history, from one bump in one correlation function, and their agreement with a single model is one of the sharper consistency tests in the subject.

A ruler measured along and across

The sound horizon is a sphere, and a sphere in a redshift survey is measured twice over — across the line of sight it gives an angle, along it a redshift interval. Two different functions of the cosmology out of one feature, and their ratio is a measurement with no ruler in it at all.

cosmology · Baryon acoustic oscillations How much a galaxy's redshift changes in 10 years, and which way. The change in a galaxy's apparent recession velocity over 10 years of the observer's time, c ż/(1 + z), against the galaxy's redshift, for three universes with the same present expansion rate of 67.36 km/s/Mpc. The drift is (1 + z) H₀ − H(z): positive if the expansion rate at the galaxy's epoch was less than (1 + z) times today's, which is to say if the expansion has been accelerating since. In the empty universe the expansion rate is exactly (1 + z) H₀ at every epoch and nothing drifts at all. In the matter-only universe the expansion has only ever slowed, and every redshift falls: −8.6 cm/s over 10 years at z = 1 and −25.5 at z = 4. In ΛCDM the drift is positive nearby, largest at z = 0.63 where it reaches 2.51 cm/s, changes sign at z = 1.91, and is −5.5 cm/s at z = 4. The whole signal is a few centimetres per second in a decade, against the thirty kilometres per second of the Earth's own orbital motion that has to be removed from every spectrum first.

A redshift that changes while it is watched

A galaxy's redshift is a ratio of two sizes of the universe, and the second one is still growing while the light is being collected. So every redshift drifts, by a few centimetres per second in a decade, and the direction of the drift says whether the expansion has been speeding up since the light left — the one test of that question that needs no distance and no model of any source.

cosmology · Expansion The expansion rate and the acceleration, and which of them reaches inside an orbit. Two quantities per unit distance through cosmic time, both in units of today's H₀². The square of the expansion rate, H², falls steeply from the big bang and is 1.00 today by definition. The acceleration of the expansion, ä/a, is the sum of a matter-and-radiation part, −Ωₘ/(2a³) − Ωᵣ/a⁴, drawn dashed, and the cosmological constant's part, ΩΛ = 0.685, which is the same at every time. The sum was negative — the expansion decelerating — until the universe was 7.7 Gyr old, at a = 0.614 or redshift 0.63, and is 0.527 today. The equation of motion of anything orbiting inside a bound system carries ä/a and never H: the rate at which distant galaxies recede does not appear in it at all. Of ä/a, the matter part is the mean density of the universe, which inside a galaxy or a planetary system is already counted in the mass that is doing the holding, many million times over. What is left is the constant: a fixed outward acceleration per unit distance, ΩΛ H₀², that does not grow with time and does not care how fast the universe is expanding.

An orbit feels the acceleration and never the rate

If space expands, it is natural to ask why the Earth's orbit does not. The answer is not that gravity resists the stretching. It is that the expansion rate never appears in the equation of motion of a bound orbit at all — only the acceleration does, and of that only the cosmological constant's part survives, as a fixed outward push that moves the Earth's orbit once, by twelve picometres, and never again.

cosmology · Expansion Starobinsky: e-folds before the end, against how reheating went. N, the number of e-folds between the pivot scale leaving the Hubble radius and the end of inflation, for the Starobinsky potential, against the temperature at which reheating finished, for 3 equations of state during it. All the lines meet on the right at instant reheating, 2.6 × 10¹⁵ GeV, where N = 55.6. The left edge is 5 MeV, below which nucleosynthesis would not have happened. w = 0, oscillating field: 42.0 at 5 MeV; w = ⅓, like radiation: 55.6 at 5 MeV; w = 1, kination: 69.0 at 5 MeV. The reason is how far the universe stretches while the energy density falls: an oscillating field dilutes like matter, as a⁻³, so for the same fall in density it expands further than radiation would, more of the growth of today's scales happens after inflation, and fewer e-folds of inflation are needed to put them where they are. A stiff epoch with w = 1 dilutes as a⁻⁶, stretches less, and needs more. A radiation-like epoch changes nothing. None of this epoch has been observed; the lines are the arithmetic of energy and entropy, and the spread between them is how much an unobserved history moves a quantity the spectral index depends on.

The epoch nobody saw moves the tilt

Between the end of inflation and the hot universe that made the light elements lies an interval nothing has observed, in which the energy of the inflaton became radiation. How long that took changes how many e-folds before the end the observed scales left — by as many as fourteen — and that moves every model's predicted spectral index by more than the measurement's uncertainty. A potential is never tested by the tilt alone; a potential and a reheating history are tested together.

cosmology · Inflation

Named alongside it

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

Cosmic accelerationDark energyScale factorAlcock paczynskiAngular-diameter distanceAnisotropic clusteringBaryon acoustic oscillationsBig bang nucleosynthesisBound systemComoving distanceConformal timeCorrelation function

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