  This work is concerned with the large time behavior of solutions to the
barotropic compressible Navier-Stokes equations in $\mathbb{R}^{d}(d\geq2)$.
Precisely, it is shown that if the initial density and velocity additionally
belong to some Besov space $\dot{B}^{-\sigma_1}_{2,\infty}$ with $\sigma_1\in
(1-d/2, 2d/p-d/2]$, then the $L^p$ norm (the slightly stronger
$\dot{B}^{0}_{p,1}$ norm in fact) of global solutions admits the optimal decay
$t^{-\frac{d}{2}(\frac 12-\frac 1p)-\frac{\sigma_1}{2}}$ for
$t\rightarrow+\infty$. In contrast to refined time-weighted approaches
([11,43]), a pure energy argument (independent of the spectral analysis) has
been developed in more general $L^p$ critical framework, which allows to remove
the smallness of low frequencies of initial data. Indeed, bounding the
evolution of $\dot{B}^{-\sigma_1}_{2,\infty}$-norm restricted in low
frequencies is the key ingredient, whose proof mainly depends on non standard
$L^p$ product estimates with respect to different Sobolev embeddings. The
result can hold true in case of large highly oscillating initial velocities.
