We have measured the bias of QSOs as a function of QSO luminosity at fixed redshift (z<1) by cross-correlating them with LRGs in the same spatial volume, hence breaking the degeneracy between QSO luminosity and redshift. We use three QSO samples from 2SLAQ, 2QZ and SDSS covering a QSO absolute magnitude range, -24.5<M_{b_J}<-21.5, and cross-correlate them with 2SLAQ (z~0.5) and AAOmega (z~0.7) photometric and spectroscopic LRGs in the same redshift ranges. The 2-D and 3-D cross-clustering measurements are generally in good agreement. Our (2SLAQ) QSO-LRG clustering amplitude (r_0=6.8_{-0.3}^{+0.1}h^{-1}Mpc) as measured from the semi-projected cross-correlation function appears similar to the (2SLAQ) LRG-LRG auto-correlation amplitude (r_0=7.45pm0.35h^{-1}Mpc) and both are higher than the (2QZ+2SLAQ) QSO-QSO amplitude (r_0simeq5.0h^{-1}Mpc). Our measurements show remarkably little QSO-LRG cross-clustering dependence on QSO luminosity. If anything, the results imply that brighter QSOs may be less highly biased than faint QSOs, the opposite direction expected from simple high peaks biasing models. Assuming a standard LCDM model and values for b_{LRG} measured from LRG autocorrelation analyses, we find b_Q=1.45pm0.11 at M_{b_J}approx-24 and b_Q=1.90pm0.16 at M_{b_J}~-22. We also find consistent results for the QSO bias from a z-space distortion analysis of the QSO-LRG cross-clustering at z~0.55. The dynamical infall results give beta _Q=0.55pm0.10, implying b_Q=1.4pm0.2. Thus both the z-space distortion and the amplitude analyses yield b_Q~1.5 at M_{b_J}~-23. The implied DM halo mass inhabited by QSOs at z~0.55 is sim10^{13}h^{-1}M_{sun}, again approximately independent of QSO luminosity.
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