The evolution of the two-point correlation function , \xi ( r,z ) , and the pairwise velocity dispersion , \sigma ( r,z ) , for both the matter , \xi _ { \rho \rho } , and halo population , \xi _ { hh } , in three different cosmological models : ( \Omega _ { 0 } , \lambda _ { 0 } ) = ( 1,0 ) , ( 0.2,0 ) and ( 0.2,0.8 ) are described .
If the evolution of \xi is parameterized by \xi ( r,z ) = ( 1 + z ) ^ { - ( 3 + \mbox { $ \epsilon$ } ) } \xi ( r, 0 ) , where \xi ( r, 0 ) = ( r / r _ { 0 } ) ^ { - \gamma } , then \epsilon _ { \rho \rho } ranges from 1.04 \pm 0.09 for ( 1,0 ) and 0.18 \pm 0.12 for ( 0.2,0 ) , as measured by the the evolution of \xi _ { \rho \rho } at 1 Mpc ( from z \sim 5 to the present epoch ) .
For halos , \epsilon depends indeed on their mean overdensity .
Halos with a mean overdensity of about 2000 were used to compute the halo two-point correlation function , \xi _ { hh } , tested with two different group finding algorithms : the friends of friends and the spherical overdensity algorithm .
It is certainly believed that the rate of growth of this \xi _ { hh } will give a good estimate of the evolution of the galaxy two-point correlation function , at least from z \sim 1 to the present epoch .
The values we get for \epsilon _ { hh } range from 1.54 for ( 1,0 ) to -0.36 for ( 0.2,0 ) , as measured by the evolution of \xi _ { hh } from z \sim 1.0 to the present epoch .
These values could be used to constrain the cosmological scenario .

The evolution of the pairwise velocity dispersion for the mass and halo distribution is measured and compared with the evolution predicted by the Cosmic Virial Theorem ( CVT ) .
According to the CVT , \sigma ( r,z ) ^ { 2 } \sim GQ \rho ( z ) r ^ { 2 } \xi ( r,z ) or \sigma \propto ( 1 + z ) ^ { - \mbox { $ \epsilon$ } / 2 } .
The values of \epsilon measured from our simulated velocities differ from those given by the evolution of \xi and the CVT , keeping \gamma and Q constant : \mbox { $ \epsilon$ } = 1.78 \pm 0.13 for ( 1,0 ) or \mbox { $ \epsilon$ } = 1.40 \pm 0.28 for ( 0.2,0 ) .