Matter density perturbation and power spectrum in running vacuum model

TL;DR

This paper analyzes the matter density perturbations and power spectrum in a running vacuum model where the cosmological constant varies with the Hubble parameter, revealing scale-dependent behaviors and tight observational constraints on model parameters.

Contribution

It derives the evolution equations for matter perturbations in the running vacuum model considering dark energy perturbations and identifies a critical scale dividing different evolutionary regimes.

Findings

Growth of matter perturbations is scale-dependent with a critical scale $k_{cr}$.

Observational data exclude models with negative $ u$ or $ u + \sigma$.

Allowed parameter range for $ u$ and $\sigma$ is extremely narrow, around $10^{-7}$.

Abstract

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 $\nu, |\sigma| \lesssim \mathcal{O}(10^{-7})$.