  In this study we predict the total distributions of powerful (FR II) active
double-lobed radio galaxies and ghost sources, and their observable
distribution in the X-ray sky. We develop an analytic model for the evolution
of the lobe emission at radio and X-ray energies. During jet activity, a double
radio source emits synchrotron radiation in the radio and X-ray emission due to
inverse-Compton (IC) upscattering by gamma~10^3 electrons of the cosmic
microwave background. After the jets switch off, the radio luminosity (due to
higher gamma electrons) falls faster than the X-ray luminosity and for some
time the source appears as an IC ghost of a radio galaxy before becoming
completely undetectable in the X-ray. With our model, for one set of typical
parameters, we predict radio lobes occupy a volume fraction of the universe of
0.01, 0.03, 0.3 at z=2 (during the quasar era) of the filamentary structures in
which they are situated, for typical jet lifetimes 5*10^7 yr, 10^8 yr, 5*10^8
yr; however since the inferred abundance of sources depends on how quickly they
fall below the radio flux limit the volume filling factor is found to be a
strong function of radio galaxy properties such as energy index and minimum
gamma factor of injected particles, the latter not well constrained by
observations. We test the predicted number density of sources against the
Chandra X-ray Deep Field North survey and also find the contribution to the
unresolved cosmic X-ray background by the lobes of radio galaxies. 10-30 per
cent of observable double-lobed structures in the X-ray are predicted to be IC
ghosts. The derived X-ray luminosity function of our synthetic population shows
that double-lobed sources have higher space densities than X-ray clusters at
redshifts z>2 and X-ray luminosities above 10^44 erg s^-1.
