  X-ray luminosities and surface brightness profiles of the hot gas haloes of
simulated disc galaxies at redshifts z=0-2 are presented. The galaxies are
extracted from fully cosmological simulations and correspond in mass to the
Milky Way. We find that the bolometric X-ray luminosities of the haloes
decrease by a factor 4-10 from z~1 to z~0, reflecting the decrease in the rate
at which hot halo gas cools out on to the disc. At all redshifts, most of the
emission is found to originate within 10-15 kpc of the disc. When combined with
models in which the evolution of disc X-ray luminosity is dominated by X-ray
binaries, the predicted halo luminosities at z~1 show good agreement with
constraints from spiral galaxies in Chandra Deep Field data. There is an
indication that haloes with a metal abundance of 0.3 solar overpredict observed
X-ray luminosities at z~1, suggesting that halo metallicities are lower than
this value. Prospects for direct detection of the haloes of Milky Way-sized
galaxies with current and future X-ray instrumentation are discussed. It is
found that XEUS should be able to single out the halo emission of highly
inclined Milky Way-sized disc galaxies out to approximately z=0.3. For such
galaxies in this redshift interval, we estimate a lower limit to the surface
density of detectable haloes on the sky of ~10 deg^(-2). More generally, owing
to their luminosity evolution, the optimum redshifts at which to observe such
haloes could be 0.5<z<1, depending on their assembly history.
