The relation of local measures of Hubble's constant to its global value

Observed large-scale structure and peculiar velocity fields suggest that the local expansion rate, i.e., local value of Hubble's constant, must vary from place to place. To quantify the issue, we examine the distributions of fractional deviations of local values from the global H₀ that observers with perfect distance data would find if they surveyed specified volumes of the universe. We use new, very large-scale (N = 10^7.2 particles) calculations of two popular scenarios for the origin of structure, "CDM" and "PIB," the first having Ω₀ = 1 and the second Ω₀ = 0.2 to simulate such observations. We find that the expected deviations, due to large-scale motions, are larger than quoted observational errors unless very large (H₀R > 3000 km/s) volumes are surveyed. Even perfect sampling and distances of all galaxies within a sphere extending out to the distances of the Virgo and Coma clusters would leave 45% and 3% rms uncertainties, respectively, in the global value of H₀ in the CDM model. There is also a systematic bias between local measures of H₀ and its global value due to observers (galaxies) tending to be in overdense, decelerated regions. Even if the local expansion rate is known to be 80 ± 8 km/s/Mpc out to 30h^-1 Mpc in the North Galactic Cap, the 95% confidence interval on the true global value of H₀ is 50-128 km/s/Mpc in a CDM model. If the universe actually contains coherent structures with sizes H₀R > 10 000 km/s, as suggested by some observations, the actual effects may well be more extreme than those simulated. We show that the local versus global error in an H₀ determination can be roughly estimated by, and is typically slightly smaller than, the angular variance seen over the sky (in octants) in the expansion rate. Moreover, a very rough correction from the local to the global H₀ value can be derived from the over or under density of galaxies within the local volume. We find empirically the approximate relation (ΔH₀/H₀) = -0.6×Δn_galΩ^0.4. Unfortunately, this correction shows both a substantial statistical scatter and is systematically uncertain (depending on both bias and Ω₀). The open PIB model shows typically less than half the local H₀ variations seen in the closed CDM model. Thus, observations of expansion rate variations are potentially useful as probes of cosmic structure formation models.