tokens: bias evolution scheme of Basilakos Plionis (2001 , 2003) , which is based on linear perturbation theory and given by ( Basilakos et al. 2011 ) with The constants (the present day bias factor) and depend on the host dark matter halo mass and for the CDM model are given by where ( Basilakos et al. 2012 ). This bias evolution model has been thoroughly tested and found to rate very well in reproducing -body simulation data ( Basilakos et al. 2008 ) as well as fitting observational data ( Papageorgiou et al. 2012 ). Fig. 1 Redshift evolution of the comoving clustering length for the two different parametrizations of clustering evolution. For the case of our model (Eq. ( 13 )), shown as black and blue curves, the different evolution behaviours shown correspond to the indicated parameters ). The specific values of these parameters have been chosen such as to resemble the and models of the traditional parametrization of Eq. ( 7 ), shown in red, which correspond to the constant in comoving coordinates clustering (for ) and to the stable clustering models (for details see the text). We therefore see that through the dependence of the bias factor on the dark matter halo mass, one expects a different clustering evolution for dark matter halos of different mass for the same values of and . In order to appreciate our parametrization of the clustering evolution we present in Fig. 1 the evolution of the comoving clustering length for models that have been purposely selected to resemble some based models. As can be seen the model with is equivalent with the constant in comoving coordinates clustering model () up to very large redshifts (). No other ( combination can provide such an equivalence up to such high redshifts. However, as we can again see in Fig. 1 , one can find () combinations that correspond to the comoving clustering model, but up to , as the example shown with . In fact one can derive the degeneracy between the and parameters such that the clustering evolution is constant in comoving coordinates (), but up to , since beyond this redshift the equivalence is unattainable. In Fig. 2 we present the corresponding 1, 2, and 3 contours in the ) parameter space of the solution that provides the above equivalence with the model. The range is dictated by the uncertainty that we have imposed on , which is taken to be of , typical of current accurate measurements. The degeneracy is clear and it can only be broken if we either impose a value of , for example from the expected slope of the power spectrum on these scales, or from an independent estimate of the mass of the dark matter halos in which the tracers (in our case X-ray selected AGN) reside. Fig. 2 Contour ranges of 1, 2 and 3 in the parameter plane for the case where our clustering evolution scheme corresponds to the constant in comoving coordinates model (imposed up to in this example). The strong degeneracy of the parameters is evident. Similarly, larger values of provide clustering evolution
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