We discuss how the space of possible cosmological parameters is constrained by the angular diameter distance function DA(z) as measured using the SZ/X-ray method, which combines the Sunyaev-Zeldovich (SZ) effect and X-ray brightness data for clusters of galaxies. New X-ray satellites and ground-based interferometers dedicated to SZ observations should soon lead to DA(z) measurements limited by systematic rather than random error. We analyze the systematic and random error budgets to make a realistic estimate of the accuracy achievable in the determination of (Ωm, ΩΛ, h), the density parameters of matter and cosmological constant, and the dimensionless Hubble constant, using DA(z) derived from the SZ/X-ray method and the position of the first "Doppler" peak in the cosmic microwave background fluctuations. We briefly study the effect of systematic errors. We find that Ωm, ΩΛ, and w are affected, but h is not, by systematic errors that grow with redshift. With as few as 70 clusters, each providing a measurement of DA(z) with a 7% random and 5% systematic error, Ωm can be constrained to ±0.2, ΩΛ to ±0.2, and h to ±0.11 (all at 3 σ). We also estimate constraints for the alternative three-parameter set (Ωm, w, h), where w is the equation-of-state parameter. The measurement of DA(z) provides constraints complementary to those from the number density of clusters in redshift space. A sample of 70 clusters (DA measured with the same accuracy as before) combined with cluster evolution results (or a known matter density) can constrain w within ±0.45 (at 3 σ). Studies of X-ray and SZ properties of clusters of galaxies promise an independent and powerful test for cosmological parameters.
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