One number for the whole universe
The cosmological principle is so restrictive that the entire geometry of a homogeneous, isotropic universe reduces to a single function of time: the scale factor a(t). Think of a fixed grid painted on space, with each galaxy sitting (roughly) at a fixed grid coordinate \chi called its comoving position. Expansion does not move galaxies across the grid; it stretches the grid itself. The real (proper) distance between two galaxies is their fixed comoving separation times a(t).
Proper distance = fixed comoving separation × the scale factor. All the time dependence lives in a(t).
Conventionally we set a=1 today. Differentiate the distance and Hubble's law falls out immediately: \dot d = \dot a\,\chi = (\dot a/a)\,d, so the recession speed is proportional to distance with proportionality constant H = \dot a/a. Hubble's 'constant' is just the present-day value of a quantity H(t) that changes over cosmic time.
The Hubble parameter is the fractional rate of expansion; H_0 = H(today).
Redshift is stretched space
The scale factor gives redshift a beautifully literal meaning. A photon's wavelength is carried along by the grid, so it stretches by exactly the same factor the universe does between emission and observation. With a=1 today:
One plus the redshift is exactly the factor by which the universe has grown since the light was emitted.
The Friedmann equations
What sets the pace of a(t)? Gravity. Feed the homogeneous, isotropic metric into Einstein's field equations and you get the Friedmann equations, the equations of motion for the whole universe. The first one relates the expansion rate to the energy density \rho, the spatial curvature k, and the cosmological constant \Lambda:
The first Friedmann equation: the square of the expansion rate is driven by density, curvature and the cosmological constant.
The second Friedmann equation governs the acceleration of the expansion, and it hides a shock: pressure adds to gravity. A positive pressure (ordinary matter, radiation) makes the expansion decelerate more strongly, while a sufficiently negative pressure — or a positive \Lambda — makes the expansion accelerate.
The acceleration equation. Note pressure p appears alongside density — negative pressure can drive acceleration.
Critical density and the shape of space
Set \Lambda=0 for a moment and ask: how much density makes space exactly flat (k=0)? The Friedmann equation gives a special value, the critical density. Comparing the actual density to it defines the density parameter \Omega=\rho/\rho_c.
Omega compares the density to the critical value; Omega and the spatial curvature k determine the geometry.
The total \Omega (summing matter, radiation and dark energy as effective densities) fixes the geometry: \Omega>1 gives a closed, positively curved space (triangles bulge past 180^\circ); \Omega=1 gives flat space; \Omega<1 gives an open, saddle-shaped space. Observations — chiefly the microwave background — find \Omega_{\text{total}}\approx 1 to within a percent: the universe is spatially flat, an astonishing and telling result.
The three eras of the universe
Different ingredients dilute differently as space expands, and that is what carves cosmic history into eras. Matter thins out with volume, \rho_m\propto a^{-3}. Radiation thins faster, \rho_r\propto a^{-4} — one power from volume plus one from each photon redshifting to lower energy. The cosmological constant (dark energy) does not dilute at all: its density stays fixed as space grows.
How each ingredient dilutes with expansion. Whichever falls slowest eventually dominates.
Follow the logic forward. When a was tiny, radiation (a^{-4}) dominated — the radiation era. As space grew, matter overtook it — the matter era, when galaxies formed. And because dark energy never dilutes, it inevitably wins in the end — the dark-energy era we have just entered, in which the expansion is accelerating. The same three densities, read through these scaling laws, are the whole plot of cosmic history. Guide 4 asks what the dark ingredients actually are.