The cosmological analysis of X-ray cluster surveys; III. 4D X-ray
  observable diagrams
  Despite compelling theoretical arguments, the use of clusters as cosmological
probes is, in practice, frequently questioned because of the many uncertainties
impinging on cluster mass estimates. Our aim is to develop a fully
self-consistent cosmological approach of X-ray cluster surveys, exclusively
based on observable quantities, rather than masses. This procedure is justified
given the possibility to directly derive the cluster properties via ab initio
modelling, either analytically or by using hydrodynamical simulations. In this
third paper, we evaluate the method on cluster toy-catalogues. We model the
population of detected clusters in the count-rate -- hardness-ratio -- angular
size -- redshift space and compare the corresponding 4-dimensional diagram with
theoretical predictions. The best cosmology+physics parameter configuration is
determined using a simple minimisation procedure; errors on the parameters are
derived by scanning the likelihood hyper-surfaces with a wide range of starting
values. The method allows a simultaneous fit of the cosmological parameters, of
the cluster evolutionary physics and of the selection effects. When using
information from the X-ray survey alone plus redshifts, this approach is shown
to be as accurate as the mass function for the cosmological parameters and to
perform better for the cluster physics, as modelled in the scaling relations.
It enables the identification of degenerate combinations of parameter values.
Given the considerably shorter computer times involved for running the
minimisation procedure in the observed parameter space, this method appears to
clearly outperform traditional mass-based approaches when X-ray survey data
alone are available.
