HubbleParameter

The Hubble parameter, often denoted H(t), is a measure of the rate at which the universe expands at a given time. It is defined as H(t) = (da/dt)/a, where a(t) is the scale factor in the Friedmann–Lemaître–Robertson–Walker metric. The parameter thus characterizes the proportional growth of distances in the cosmos and forms the core of the expansion history in general relativity.

The current value, H0, is called the Hubble constant. It is usually expressed in units of velocity per distance, most commonly kilometers per second per megaparsec (km/s/Mpc). The precise value of H0 is a topic of active research, with different measurement methods yielding slightly different results. Typical modern estimates place H0 around 67–74 km/s/Mpc, with a notable tension between early-universe measurements (e.g., from the cosmic microwave background) and local distance indicators.

In cosmology, the time dependence of H(t) is encoded in the expansion history through H(z) = H0 E(z), where E(z) is determined by the energy content of the universe. In the standard ΛCDM model, E(z) = sqrt(Ωr(1+z)^4 + Ωm(1+z)^3 + Ωk(1+z)^2 + ΩΛ). For a flat, matter-dominated early universe with negligible radiation and curvature, E(z) ≈ sqrt(Ωm(1+z)^3 + ΩΛ). The Hubble parameter thus connects redshift observations to the evolution of the scale factor and the inferred age and fate of the universe.

Historically named after Edwin Hubble, the parameter remains fundamental for constraining cosmological parameters, including matter density, dark energy, and spatial curvature, by linking observations to the cosmic expansion rate.