A deep ROSAT survey - IV. The evolution of X-ray-selected QSOs
  We report on a new estimate of the QSO X-ray luminosity function and its evolution with redshift based on a sample of 107 QSOs detected at faint X-ray fluxes, $S{\rm(0.5-2\,keV)}>4\times10^{-15}\,$\ergcms , with the {\it ROSAT} X-ray satellite. For $q_0=0.5$, the X-ray evolution of QSOs in this sample is consistent with strong luminosity evolution, $L_{\rm X} ^*(z) \propto (1+z)^{3.25\pm0.1}$, at low redshifts ($z<1.60$) and a constant comoving space density at higher redshifts. The derived rate of evolution at low redshifts is thus significantly higher than that obtained previously for the {\it Einstein} Extended Medium Sensitivity Survey (EMSS). Indeed, most luminosity evolution models provide a very poor fit (rejected at the 99 per cent confidence level) when applied to the combined EMSS and {\it ROSAT} samples, although a polynomial evolution model, $L_{\rm X} ^*(z) \propto 10^{(1.14\,z -0.23\,z^2)}$, provides an adequate fit for $q_0=0$. For $q_0=0.5$, a simple power-law luminosity evolution model with a redshift cut-off ($L_{\rm X} ^*(z) \propto (1+z)^{2.51\pm0.1}$, 
B.J.Boyle (IoA, Cambridge), T.Shanks (Durham), I.Georgantopoulos   (Leicester) G.C.Stewart (Leicester) and R.E.Griffiths (JHU)