  This paper describes a serendipitous galaxy cluster survey that we plan to
conduct with the XMM X-ray satellite. We have modeled the expected properties
of such a survey for three different cosmological models, using an extended
Press-Schechter (Press & Schechter 1974) formalism, combined with a detailed
characterization of the expected capabilities of the EPIC camera on board XMM.
We estimate that, over the ten year design lifetime of XMM, the EPIC camera
will image a total of ~800 square degrees in fields suitable for the
serendipitous detection of clusters of galaxies. For the presently-favored
low-density model with a cosmological constant, our simulations predict that
this survey area would yield a catalogue of more than 8000 clusters, ranging
from poor to very rich systems, with around 750 detections above z=1. A
low-density open Universe yields similar numbers, though with a different
redshift distribution, while a critical-density Universe gives considerably
fewer clusters. This dependence of catalogue properties on cosmology means that
the proposed survey will place strong constraints on the values of Omega-Matter
and Omega-Lambda. The survey would also facilitate a variety of follow-up
projects, including the quantification of evolution in the cluster X-ray
luminosity-temperature relation, the study of high-redshift galaxies via
gravitational lensing, follow-up observations of the Sunyaev-Zel'dovich effect
and foreground analyses of cosmic microwave background maps.
