Optimal time-decay estimates for the compressible Navier-Stokes-Poisson equations without additional smallness assumptions

The present paper is dedicated to the large time asymptotic behavior of global strong solutions near constant equilibrium (away from vacuum) to the compressible Navier-Stokes-Poisson equations. Precisely, we present that under the same regularity assumptions as in \cite{SX2}, a \textit{different} time-decay framework of the $\dot{B}_{p,1}^{s}$ norm of the critical global solutions is established. The proof mainly depends on the pure energy argument \textit{without the spectral analysis}, which allows us to remove \textit{the usual smallness assumption of low frequencies of initial data}.