Synthetic simulations of the extragalactic sky seen by eROSITA. I.
  Pre-launch selection functions from Monte-Carlo simulations
  Studies of galaxy clusters provide stringent constraints on models of
structure formation. Provided that selection effects are under control, large
X-ray surveys are well suited to derive cosmological parameters, in particular
those governing the dark energy equation of state. We forecast the capabilities
of the all-sky eROSITA (the extended ROentgen Survey with an Imaging Telescope
Array) survey to be achieved by the early 2020s. We bring special attention to
modeling the entire chain from photon emission to source detection and
cataloguing. The selection function of galaxy clusters for the upcoming eROSITA
mission is investigated by means of extensive and dedicated Monte-Carlo
simulations. Employing a combination of accurate instrument characterization
and of state-of-the-art source detection technique, we determine a cluster
detection efficiency based on the cluster fluxes and sizes. Using this eROSITA
cluster selection function, we find that eROSITA will detect a total of $\sim
10^5$ clusters in the extra-galactic sky. This number of clusters will allow
eROSITA to put stringent constraints on cosmological models. We show that
incomplete assumptions on selection effects, such as neglecting the
distribution of cluster sizes, induce a bias in the derived value of
cosmological parameters. Synthetic simulations of the eROSITA sky capture the
essential characteristics impacting the next-generation galaxy cluster surveys
and they highlight parameters requiring tight monitoring in order to avoid
biases in cosmological analyses.
