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This is the first of a series of papers in which we derive simultaneous constraints on cosmological parameters and X-ray scaling relations using observations of the growth of massive, X-ray flux-selected galaxy clusters. Our data set consists of 238 cluster detections from the ROSAT All-Sky Survey, and incorporates follow-up observations of 94 of those clusters using the Chandra X-ray Observatory or ROSAT. Here we describe and implement a new statistical framework required to self-consistently produce simultaneous constraints on cosmology and scaling relations from such data, and present results on models of dark energy. In spatially flat models with a constant dark energy equation of state, w, the cluster data yield Omega_m = 0.23 +- 0.04, sigma8 = 0.82 +- 0.05, and w = -1.01 +- 0.20, incorporating standard priors on the Hubble parameter and mean baryon density of the Universe, and marginalizing over conservative allowances for systematic uncertainties. These constraints agree well and are competitive with independent data in the form of cosmic microwave background anisotropies, type Ia supernovae, cluster gas mass fractions, baryon acoustic oscillations, galaxy redshift surveys, and cosmic shear. The combination of our data with current microwave background, supernova, gas mass fraction, and baryon acoustic oscillation data yields Omega_m = 0.27 +- 0.02, sigma8 = 0.79 +- 0.03, and w = -0.96 +- 0.06 for flat, constant w models. The combined data also allow us to investigate evolving w models. Marginalizing over transition redshifts in the range 0.05-1, we constrain the equation of state at late and early times to be respectively w0 = -0.88 +- 0.21 and w_et = -1.05^{+0.20}_{-0.36}, again including conservative systematic allowances. The combined data provide constraints equivalent to a Dark Energy Task Force figure of merit of 15.5. Our results highlight the power of X-ray studies, which enable the straightforward production of large, complete, and pure cluster samples and admit tight scaling relations, to constrain cosmology. However, the new statistical framework we apply to this task is equally applicable to cluster studies at other wavelengths.

Confidence contours in the w / Omega_m plane.

Confidence contours in the w / Omega_m plane.

Joint 68.3 and 95.4 per cent confidence regions for parameters of the constant w model. Constraints on Omega_m and w from the XLF (purple, including all systematic allowances in Table 1 of the paper) are compared with those from cluster fgas data (red; Allen et al. 2008, including conservative systematic allowances), 5-year WMAP data (blue; Dunkley et al. 2009, marginalized over the SZ signal), SNIa data (green; Kowalski et al. 2008, including their treatment of systematics), and BAO observations (brown; Percival07, also using our standard priors on h and Omega_b h^2). Results from combining these 5 data sets are shown in gold.

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