  (Abridged) 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 clusters drawn from the ROSAT All-Sky
Survey, and incorporates extensive follow-up observations using the Chandra
X-ray Observatory. 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, sigma_8=0.82 +- 0.05, and w=-1.01 +-
0.20, marginalizing over conservative allowances for systematic uncertainties.
These constraints agree well and are competitive with independent data in the
form of cosmic microwave background (CMB) anisotropies, type Ia supernovae
(SNIa), cluster gas mass fractions (fgas), baryon acoustic oscillations (BAO),
galaxy redshift surveys, and cosmic shear. The combination of our data with
current CMB, SNIa, fgas, and BAO data yields Omega_m=0.27 +- 0.02, sigma_8=0.79
+- 0.03, and w=-0.96 +- 0.06 for flat, constant w models. For 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
w_0=-0.88 +- 0.21 and w_et=-1.05 +0.20 -0.36. The combined data provide
constraints equivalent to a DETF FoM of 15.5. Our results highlight the power
of X-ray studies to constrain cosmology. However, the new statistical framework
we apply to this task is equally applicable to cluster studies at other
wavelengths.
