The cosmological analysis of X-ray cluster surveys III. 4D X-ray observable diagrams

Context. Despite compelling theoretical arguments, the use of clusters as cosmological probes is, in practice, frequently questioned because of the many uncertainties surrounding cluster-mass estimates. Aims. 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. Methods. We model the population of detected clusters in the count-rate - hardness-ratio - angular size - redshift space and compare the corresponding four-dimensional diagram with theoretical predictions. The best cosmology + physics parameter configuration is determined using a simple minimisation procedure; errors on the parameters are estimated by averaging the results from ten independent survey realisations. The method allows a simultaneous fit of the cosmological parameters of the cluster evolutionary physics and of the selection effects. Results. When using information from the X-ray survey alone plus redshifts, this approach is shown to be as accurate as the modelling of the mass function for the cosmological parameters and to perform better for the cluster physics, for a similar level of assumptions on the scaling relations. It enables the identification of degenerate combinations of parameter values. Conclusions. 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.