We study the evolution of the cluster correlation function and its richness-dependence from z = 0 to z = 3 using large-scale cosmological simulations .
A standard flat LCDM model with \Omega _ { m } = 0.3 and , for comparison , a tilted \Omega _ { m } = 1 model , TSCDM , are used .
The evolutionary predictions are presented in a format suitable for direct comparisons with observations .
We find that the cluster correlation strength increases with redshift : high redshift clusters are clustered more strongly ( in comoving scale ) than low redshift clusters of the same mass .
The increased correlations with redshift , in spite of the decreasing mass correlation strength , is caused by the strong increase in cluster bias with redshift : clusters represent higher density peaks of the mass distribution as the redshift increases .
The richness-dependent cluster correlation function , presented as the correlation-scale versus cluster mean separation relation , R _ { 0 } - d , is found to be , remarkably , independent of redshift to z \lesssim 2 for LCDM and z \lesssim 1 for TCDM ( for a fixed correlation function slope and cluster mass within a fixed comoving radius ) .
The non-evolving R _ { 0 } - d relation implies that both the comoving clustering scale and the cluster mean separation increase with redshift for the same mass clusters so that the R _ { 0 } - d relation remains essentially unchanged .
For LCDM , this relation is R _ { 0 } ( z ) \simeq 2.6 \sqrt { d ( z ) } for z \lesssim 2 
( in comoving h ^ { -1 } Mpc scales ) .
TSCDM has smaller correlation scales , as expected .
Evolution in the relation is seen at z \gtrsim 2 
for LCDM and z \gtrsim 1 for TSCDM , where the amplitude of the relations declines .
The evolution of the R _ { 0 } - d relation from z \sim 0 to z \sim 3 provides an important new tool in cosmology ; it can be used to break degeneracies that exist at z \sim 0 and provide precise determination of cosmological parameters .