4.1. Current Status of SZE/X-ray Distances

To date, there are 41 distance determinations to 26 different galaxy clusters from analysis of SZE and X-ray observations. All of these SZE/X-ray distances use ROSAT X-ray data and model the cluster gas as a spherical isothermal model (Cavaliere & Fusco-Femiano, 1976; Cavaliere & Fusco-Femiano, 1978). The ROSAT data do not warrant a more sophisticated treatment. In Figure 6 we show all SZE-determined distances from high signal-to-noise ratio SZE experiments. The uncertainties shown are statistical at 68% confidence. The theoretical angular-diameter distance relation is shown for three cosmologies assuming H₀ = 60 km s^-1 Mpc^-1.

There are currently three homogeneously analyzed samples of clusters with SZE distances: (1) a sample of seven nearby (z < 0.1) galaxy clusters observed with the OVRO 5 m telescope (green solid triangles) that finds H₀ = 66⁺¹⁴⁺¹⁵_-11-15 (Mason et al., 2001; Myers et al., 1997), (2) a sample of five intermediate-redshift (0.14 < z < 0.3) clusters from the RT interferometer that finds H₀ = 65⁺⁸⁺¹⁵_-7-15 (cyan solid squares) (Jones et al., 2003), and (3) a sample of 18 clusters with 0.14 < z < 0.83 from interferometric observations by the OVRO and BIMA SZE imaging project, which infers H₀ = 60⁺⁴⁺¹⁸_-4-13 (blue solid stars) (Reese et al., 2002). The above Hubble constants assume a _m = 0.3, = 0.7 cosmology, and the uncertainties are statistical followed by systematic at 68% confidence. The treatment of uncertainties varies among the three cluster samples.

A fit to the ensemble of 41 SZE-determined distances yields H₀ 61 ± 3 ± 18 km s^-1 Mpc^-1 for an _m = 0.3, = 0.7 cosmology, where the uncertainties are statistical followed by systematic at 68% confidence. Since many of the clusters are at high redshift, the best-fit Hubble constant will depend on the cosmology adopted; the best-fit Hubble constant shifts to 57 km s^-1 Mpc^-1 for an open _m = 0.3 universe, and to 54 km s^-1 Mpc^-1 for a flat _m = 1 geometry. The systematic uncertainty, discussed below, clearly dominates. The systematic uncertainty is approximate because it is complicated by shared systematics between some distance determinations. For example, including multiple distance determinations to a single cluster overstresses any effects of asphericity and orientation from that galaxy cluster.

Statistical uncertainty includes contributions from the ICM shape parameters, the electron temperature, point sources in the field, and the cooling functions, which depends on T_e, metallicity, and the column density. The largest sources of statistical uncertainty are the ICM shape parameters (from model fitting) and the X-ray determined electron temperature (D_A T_e^-2). Uncertainty from model fitting is roughly 20% in the distance and is dominated by the uncertainty in the central decrement. The contribution from T_e on the distances varies greatly from ~ 5% to ~ 30%, with 10%-20% being typical values. As expected, nearby cluster temperatures are more precisely determined than those of distant galaxy clusters.