  Starting in late 2013, the eROSITA telescope will survey the X-ray sky with
unprecedented sensitivity. Assuming a detection limit of 50 photons in the
(0.5-2.0) keV energy band with a typical exposure time of 1.6 ks, we predict
that eROSITA will detect ~ 9.3 X 10^4 clusters of galaxies more massive than 5
X 10^13 Msun/h, with the currently planned all-sky survey. Their median
redshift will be z ~ 0.35. We perform a Fisher-matrix analysis to forecast the
constraining power of eROSITA on the LambdaCDM cosmology and, simultaneously,
on the X-ray scaling relations for galaxy clusters. Special attention is
devoted to the possibility of detecting primordial non-Gaussianity. We consider
two experimental probes: the number counts and the angular clustering of a
photon-count limited sample of clusters. We discuss how the cluster sample
should be split to optimize the analysis and we show that redshift information
of the individual clusters is vital to break the strong degeneracies among the
model parameters. For example, performing a "tomographic" analysis based on
photometric-redshift estimates and combining 1- and 2-point statistics will
give marginal 1-sigma errors of Delta(sigma_8) ~ 0.036 and Delta(Omega_m) ~
0.012 without priors, and improve the current estimates on the slope of the
luminosity-mass relation by a factor of 3. Regarding primordial
non-Gaussianity, eROSITA clusters alone will give Delta(f_NL) = 9, 36, 144 for
the local, orthogonal and equilateral model, respectively. Measuring redshifts
with spectroscopic accuracy would further tighten the constraints by nearly 40
per cent (barring f_NL which displays smaller improvements). Finally, combining
eROSITA data with the analysis of temperature anisotropies in the cosmic
microwave background by the Planck satellite should give sensational
constraints on both the cosmology and the properties of the intracluster
medium.
