Cosmology and Astrophysics from Relaxed Galaxy Clusters II: Cosmological
  Constraints
  We present cosmological constraints from measurements of the gas mass
fraction, $f_{gas}$, for massive, dynamically relaxed galaxy clusters. Our data
set consists of Chandra observations of 40 such clusters, identified in a
comprehensive search of the Chandra archive, as well as high-quality weak
gravitational lensing data for a subset of these clusters. Incorporating a
robust gravitational lensing calibration of the X-ray mass estimates, and
restricting our measurements to the most self-similar and accurately measured
regions of clusters, significantly reduces systematic uncertainties compared to
previous work. Our data for the first time constrain the intrinsic scatter in
$f_{gas}$, $(7.4\pm2.3)$% in a spherical shell at radii 0.8-1.2 $r_{2500}$,
consistent with the expected variation in gas depletion and non-thermal
pressure for relaxed clusters. From the lowest-redshift data in our sample we
obtain a constraint on a combination of the Hubble parameter and cosmic baryon
fraction, $h^{3/2}\Omega_b/\Omega_m=0.089\pm0.012$, that is insensitive to the
nature of dark energy. Combined with standard priors on $h$ and $\Omega_b h^2$,
this provides a tight constraint on the cosmic matter density,
$\Omega_m=0.27\pm0.04$, which is similarly insensitive to dark energy. Using
the entire cluster sample, extending to $z>1$, we obtain consistent results for
$\Omega_m$ and interesting constraints on dark energy:
$\Omega_\Lambda=0.65^{+0.17}_{-0.22}$ for non-flat $\Lambda$CDM models, and
$w=-0.98\pm0.26$ for flat constant-$w$ models. Our results are both competitive
and consistent with those from recent CMB, SNIa and BAO data. We present
constraints on models of evolving dark energy from the combination of $f_{gas}$
data with these external data sets, and comment on the possibilities for
improved $f_{gas}$ constraints using current and next-generation X-ray
observatories and lensing data. (Abridged)
