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We investigate the spatial clustering of dark matter halos, collapsing from$1-4 \sigma$ fluctuations, in the redshift range $0 - 5$ using N-bodysimulations. The halo bias of high redshift halos ($z \geq 2$) is found to bestrongly non-linear and scale-dependent on quasi-linear scales that are largerthan their virial radii ($0.5-10$ Mpc/h). However, at lower redshifts, thescale-dependence of non-linear bias is weaker and and is of the order of a fewpercent on quasi-linear scales at $z \sim 0$. We find that the redshiftevolution of the scale dependent bias of dark matter halos can be expressed asa function of four physical parameters: the peak height of halos, thenon-linear matter correlation function at the scale of interest, an effectivepower law index of the {\it rms} linear density fluctuations and the matterdensity of the universe at the given redshift. This suggests that thescale-dependence of halo bias is not a universal function of the dark matterpower spectrum, which is commonly assumed. We provide a fitting function forthe scale dependent halo bias as a function of these four parameters. Our fitreproduces the simulation results to an accuracy of better than 4 % over theredshift range $0\leq z \leq 5$. We also extend our model by expressing thenon-linear bias as a function of the linear matter correlation function. It isimportant to incorporate our results into the clustering models of dark matterhalos at any redshift, including those hosting early generations of stars andgalaxies before reionization.

The clustering of dark matter halos: scale-dependent bias on quasi-linear scales