Combining Optical and X-ray Observations of Galaxy Clusters to Constrain
  Cosmological Parameters
  Galaxy clusters have their unique advantages for cosmology. Here we collect a
new sample of 10 lensing galaxy clusters with X-ray observations to constrain
cosmological parameters.The redshifts of lensing clusters lie between 0.1 and
0.6, and the redshift range of their arcs is from 0.4 to 4.9. These clusters
are selected carefully from strong gravitational lensing systems which have
both X-ray satellite observations and optical giant luminous arcs with known
redshift. Giant arcs usually appear in the central region of clusters, where
mass can be traced with luminosity quite well. Based on gravitational lensing
theory and cluster mass distribution model we can derive an Hubble constant
independent ratio between two angular diameter distances. One is the distance
of lensing source and the other is that between the deflector and the source.
Since angular diameter distance relies heavily on cosmological geometry, we can
use these ratios to constrain cosmological models. Meanwhile X-ray gas
fractions of galaxy clusters can also be a cosmological probe. Because there
are a dozen parameters to be fitted, we introduce a new analytic algorithm,
Powell's UOBYQA (Unconstrained Optimization By Quadratic Approximation), to
accelerate our calculation. Our result proves that this algorithm is an
effective fitting method for such continuous multi-parameter constraint. We
find an interesting fact that these two approaches are sensitive to
$\Omega_{\Lambda}$ and $\Omega_{M}$ separately. Combining them we can get quite
good fitting values of basic cosmological parameters:
$\Omega_{M}=0.26_{-0.04}^{+0.04}$, and $\Omega_{\Lambda}=0.82_{-0.16}^{+0.14}$ .
