  Measuring the 3D distribution of mass on galaxy cluster scales is a crucial
test of the LCDM model, providing constraints on the nature of dark matter.
Recent work investigating mass distributions of individual galaxy clusters
(e.g. Abell 1689) using weak and strong gravitational lensing has revealed
potential inconsistencies between the predictions of structure formation models
relating halo mass to concentration and those relationships as measured in
massive clusters. However, such analyses employ simple spherical halo models
while a growing body of work indicates that triaxial 3D halo structure is both
common and important in parameter estimates. We recently introduced a Markov
Chain Monte Carlo (MCMC) method to fit fully triaxial models to weak lensing
data that gives parameter and error estimates that fully incorporate the true
shape uncertainty present in nature. In this paper we apply that method to weak
lensing data obtained with the ESO/MPG Wide-Field Imager for galaxy clusters
A1689, A1835, and A2204, under a range of Bayesian priors derived from theory
and from independent X-ray and strong lensing observations. For Abell 1689,
using a simple strong lensing prior we find marginalized mean parameter values
M_200 = (0.83 +- 0.16)x10^15 M_solar/h and C=12.2 +- 6.7, which are marginally
consistent with the mass-concentration relation predicted in LCDM. The large
error contours that accompany our triaxial parameter estimates more accurately
represent the true extent of our limited knowledge of the structure of galaxy
cluster lenses, and make clear the importance of combining many constraints
from other theoretical, lensing (strong, flexion), or other observational
(X-ray, SZ, dynamical) data to confidently measure cluster mass profiles.
(Abridged)
