The evolution of the phase-space density profile in dark matter ( DM ) halos is investigated by means of constrained simulations , designed to control the merging history of a given DM halo .
Halos evolve through a series of quiescent phases of a slow accretion intermitted by violent events of major mergers .
In the quiescent phases the density of the halo closely follows the NFW profile and the phase-space density profile , Q ( r ) , is given by the Taylor & Navarro power law , r ^ { - \beta } , where \beta \approx 1.9 and stays remarkably stable over the Hubble time .
Expressing the phase-space density by the NFW parameters , Q ( r ) = Q _ { s } ( r / R _ { s } ) ^ { - \beta } , the evolution of Q is determined by Q _ { s } .
We have found that the effective mass surface density within R _ { s } , \Sigma _ { s } \equiv \rho _ { s } R _ { s } , remains constant throughout the evolution of a given DM halo along the main branch of its merging tree .
This invariance entails that Q _ { s } \propto R { { } _ { s } ^ { -5 / 2 } } and Q ( r ) \propto \Sigma { { } _ { s } ^ { -1 / 2 } } R { { } _ { s } ^ { -5 / 2 } } \Big ( r / R _ { s } \Big ) ^ { - \beta } .
It follows that the phase-space density remains constant , in the sense of Q _ { s } = const . , in the quiescent phases and it decreases as R { { } _ { s } ^ { -5 / 2 } } in the violent ones .
The physical origin of the NFW density profile and the phase-space density power law is still unknown .
Yet , the numerical experiments show that halos recover these relations after the violent phases .
The major mergers drive R _ { s } to increase and Q _ { s } to decrease discontinuously while keeping Q _ { s } \times R { { } _ { s } ^ { 5 / 2 } } = const .
The virial equilibrium in the quiescent phases implies that a DM halos evolves along a sequence of NFW profiles with constant energy per unit volume ( i.e. , pressure ) within R _ { s } .