Hubble's Law: Distance Ladder, Redshift and the Expanding Universe

Hubble's law. Galaxies recede from us with a speed proportional to their distance, v = H0 d, because space itself expands: every pair of galaxies is carried apart, so an observer anywhere sees the same law. Measuring it needs two things per galaxy, a redshift (easy) and a distance (hard). That is the distance ladder: Cepheid variables in nearby galaxies, whose period fixes their luminosity (Leavitt law, M_V = -2.43 (log P - 1) - 4.05), calibrate Type Ia supernovae (M about -19.25), which reach to z of 1 and beyond. The distance follows from the distance modulus mu = m - M = 5 log10(d / 10 pc).

Expansion history. For a flat universe with matter and a cosmological constant, H(z) = H0 sqrt(Om (1+z)^3 + OL). The comoving distance is D_C = (c/H0) integral of dz/E(z), the luminosity distance is D_L = (1+z) D_C, and the scale factor obeys da/dt = H0 a E(a). Supernovae at z around 0.5 appear fainter than a matter-only universe predicts: that was the 1998 discovery of accelerated expansion. The sim fits Omega_m to the simulated supernovae with a free magnitude offset (as real analyses do, since M and H0 are degenerate), and fits H0 from the nearby ones. The Hubble sphere, radius c/H(a), is where the recession speed reaches c; galaxies beyond it recede faster than light, which general relativity allows because it is the space between us that grows.

What is simplified. Galaxies are placed at random in a sphere of comoving radius 4 Gpc (more of them nearby, which a flux-limited survey also gives), all at rest in the Hubble flow: no peculiar velocities (about 300 km/s in reality, which dominates below 30 Mpc), no clustering, no dust or K-corrections, no Malmquist bias, no supernova stretch and colour corrections. The scatter is Gaussian in magnitude. The map is drawn in proper distance at the chosen scale factor, not along the light cone, so the picture shows the geometry of the expansion rather than what a telescope sees; the time-lapse beyond a = 1 is an extrapolation of the same model. The galaxy images are Blender renders of generic spiral, barred, elliptical and irregular galaxies, not real objects, and the telescope is a modelled Hubble Space Telescope drawn far larger than the scale of the map. The Planck-like and distance-ladder presets only show how the two published ranges of H0 (about 67 and 73 km/s/Mpc) change the age and the diagram, they are not a statement about the tension.

Numerical checks (run on load): comoving distance against the exact Einstein-de Sitter formula; the age against the closed form for flat Lambda-CDM; an RK4 integration of the Friedmann equation against its exact solution; the recovery of H0 from 120 simulated skies; and the Leavitt-law round trip.