Simple simulations suggest that the phase space structure of haloes identified in cosmological calculations is invariant under the dynamics induced by sinking substructure satellites — the background expands so as to leave the total distribution unchanged .
We use a Fokker-Planck formulation to show that there are long lived solutions for densities \rho \sim r ^ { \gamma } and -2 \leq \gamma \la - 1 ; indices between -1 and -1.5 corresponding to the inner cusps of cosmological haloes , where the coupling is strongest ; steeper ones to intermediate radii .
We recover the exact solutions found by Evans & Collett ( 1997 ) ; reinterpret them in terms of well defined background-satellite interaction ; and show that these , and all other solutions , are valid for any mass spectrum of substructure , because the governing equation is linear in their mass weighed phase space distribution .
If the spatial distribution of substructure has a milder cusp than the total , the system expands ; when the background has a milder cusp there is compression .
It is not possible for the individual distributions to retain their original form : light particles are driven out of low energy states , being replaced by the sinking massive ones .
If the clumps are considered solid , this takes the form of an exponential instability , with characteristic timescale of the order of the dynamical friction time , leading to a low energy cutoff in the distribution function of the background and a constant density core .
We show that there are long lived solutions with such a cutoff .
They would correspond to a situation whereas the clumps are made of dense baryonic material .
When stripping is important , as in the case of dissipationless substructure , it is likely that this situation is reversed — the cutoff is now in the clump distribution function .
A simple description suggests that this renders equilibria even more long lived .
In all cases it is possible to find solutions that are long lived from the thermodynamical ( energy transfer ) perspective .
In systems without stripping the only truly stable solutions however are isothermal spheres , but there are double power law solutions that may be relevant if stripping is involved .
The results in this paper suggest that halo profiles similar to those found in dissipationless cosmological simulations are approximately invariant under the interaction induced by the presence of substructure satellites — a necessary condition for the observed ‘ universality ’ .
In addition , the total profile , including baryons , should also be invariant ; provided the latter are initially in the form of dense clumps , whose distribution follows that of the dark matter .