Matter density perturbation and power spectrum in running vacuum model
  We investigate the matter density perturbation $\delta_m$ and power spectrum
$P(k)$ in the running vacuum model (RVM) with the cosmological constant being a
function of the Hubble parameter, given by $\Lambda = \Lambda_0 + 6 \sigma H
H_0+ 3\nu H^2$, in which the linear and quadratic terms of $H$ would originate
from the QCD vacuum condensation and cosmological renormalization group,
respectively. Taking the dark energy perturbation into consideration, we derive
the evolution equation for $\delta_m$ and find a specific scale $d_{cr}=2
\pi/k_{cr}$, which divides the evolution of the universe into the sub and
super-interaction regimes, corresponding to $k \ll k_{cr}$ and $k \gg k_{cr}$,
respectively. For the former, the evolution of $\delta_m$ has the same behavior
as that in the $\Lambda$CDM model, while for the latter, the growth of
$\delta_m$ is frozen (greatly enhanced) when $\nu + \sigma >(<)0$ due to the
couplings between radiation, matter and dark energy. It is clear that the
observational data rule out the cases with $\nu<0$ and $\nu + \sigma <0$, while
the allowed window for the model parameters is extremely narrow with $\nu,
|\sigma| \lesssim \mathcal{O}(10^{-7})$.
