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We study the expected redshift evolution of galaxy cluster abundance between 0 < z < 3 in different cosmologies, including the effects of the cosmic equation of state parameter w=p/rho. Using the halo mass function obtained in recent large scale numerical simulations, we model the expected cluster yields in a 12 deg² Sunyaev-Zeldovich Effect (SZE) survey and a deep 10⁴ deg² X-ray survey over a wide range of cosmological parameters. We quantify the statistical differences among cosmologies using both the total number and redshift distribution of clusters. Provided that the local cluster abundance is known to a few percent accuracy, we find only mild degeneracies between w and either Omegaₘ or h. As a result, both surveys will provide improved constraints on Omegaₘ and w. The Omegaₘ-w degeneracy from both surveys is complementary to those found either in studies of CMB anisotropies or of high-redshift Supernovae (SNe). As a result, combining these surveys together with either CMB or SNe studies can reduce the statistical uncertainty on both w and Omegaₘ to levels below what could be obtained by combining only the latter two data sets. Our results indicate a formal statistical uncertainty of about 3% (68% confidence) on both Omegaₘ and w when the SZE survey is combined with either the CMB or SN data; the large number of clusters in the X-ray survey further suppresses the degeneracy between w and both Omegaₘ and h. Systematics and internal evolution of cluster structure at the present pose uncertainties above these levels. We briefly discuss and quantify the relevant systematic errors. By focusing on clusters with measured temperatures in the X-ray survey, we reduce our sensitivity to systematics such as non-standard evolution of internal cluster structure.
Haiman et al. (Fri,) studied this question.