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Matter density perturbation and power spectrum in running vacuum model

We investigate the matter density perturbation $δ_m$ and power spectrum $P(k)$ in the running vacuum model (RVM) with the cosmological constant being a function of the Hubble parameter, given by $Λ= Λ_0 + 6 σH H_0+ 3νH^2$, 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 $δ_m$ and find a specific scale $d_{cr}=2 π/k_{cr}$, 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 $δ_m$ has the same behavior as that in the $Λ$CDM model, while for the latter, the growth of $δ_m$ is frozen (greatly enhanced) when $ν+ σ>(<)0$ due to the couplings between radiation, matter and dark energy. It is clear that the observational data rule out the cases with $ν<0$ and $ν+ σ<0$, while the allowed window for the model parameters is extremely narrow with $ν, |σ| \lesssim \mathcal{O}(10^{-7})$.