  The compressible Navier-Stokes-Poisson system takes the form of usual
Navier-Stokes equations coupled with the self-consistent Poisson equation,
which is used to simulate the transport of charged particles under the electric
field of electrostatic potential force. In this paper, we focus on the large
time behavior of global strong solutions in the $L^{p}$ Besov spaces of
critical regularity. By exploring the dissipative effect arising from Poisson
potential, we posed the new regularity assumption of low frequencies and then
establish a \textit{sharp} time-weighted inequality, which leads to the optimal
time-decay estimates of the solution. Indeed, we see that the decay of density
is faster at the half rate than that of velocity, which is a different
ingredient in comparison with the situation of usual Navier-Stokes equations.
Our proof mainly depends on tricky and non classical Besov product estimates
with respect to various Sobolev embeddings.
