Concept

Closed universe — where it appears

A universe of positive spatial curvature, finite in volume and with no boundary, like the surface of a sphere in one more dimension. Without a cosmological constant it must eventually recollapse. Measurements of the present universe find a curvature consistent with zero.

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

Also named here as friedmann equation — the same set of essays touches all of them, so they are one junction rather than several.

Two galaxies that stopped moving apart, and the mass that did it. The separation of two galaxies on a radial orbit that began together at the big bang, against cosmic time, solved so that after 13.797 Gyr they are 770 kpc apart and approaching at 110 km/s — the present separation and approach speed of the Milky Way and the Andromeda galaxy. The curve is a cycloid, r = A(1 − cos θ) and t = B(θ − sin θ), and only one cycloid passes through that point with that slope. It rose to 1037 kpc, turned round when the universe was 8.5 Gyr old, and on this purely radial orbit the two meet 3.3 Gyr from now. Its period fixes the mass: A³/(GB²) = 4.2·10¹² solar masses. The dashed curve is the same calculation with the cosmological constant's outward push included, integrated rather than solved; to arrive at the same place at the same speed against that push it needs 4.76·10¹² solar masses, 13 per cent more. Far more than the stars of the two galaxies, it is the timing argument's measurement of the Local Group's dark matter.

The age of the universe weighs the Local Group

The Andromeda galaxy is approaching the Milky Way, and in an expanding universe that means the two once moved apart, stopped and turned round. One radial orbit passes through their present separation with their present speed after exactly the age of the universe, and its period fixes the mass that turned them — four trillion suns, twenty-five times what their stars can account for.

cosmology · Expansion

The static universe is a saddle in the plane of size and speed. Every history of a closed universe of dust and an exactly balancing cosmological constant, as curves of constant energy in the plane of size against rate of change of size, in units of the static radius and 1/√(4πGρ). The two thick curves cross at the static point: they are the histories that approach it for an infinite time or leave it after an infinite time. Above them a universe expands from a big bang through forever or contracts from forever; to the left it rises from a big bang and falls back; to the right it contracts, bounces, and expands. No history other than the point itself stays near it.

A balance that is a saddle

In 1917 Einstein added a constant to his equations to hold the universe still, and the constant does exactly that: there is one size at which the attraction of matter and the repulsion of the constant cancel. Eddington showed in 1930 that the balance is a pencil on its point. A thousandth out and it is gone in seven units of its own free-fall time. The instability is not a flaw of that one model. It is why the real universe could cross the same balance without stopping.

cosmology · Dark energy

Named alongside it

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

Cosmological constant Friedmann equation Dark matter Deceleration parameter Einstein static universe Local group Peculiar velocity Radial orbit Saddle point Spherical collapse Timing argument Turnaround

All concepts