Running Vacuum Cosmological Models: Linear Scalar Perturbations
  In cosmology, phenomenologically motivated expressions for running vacuum are commonly parametrized as linear functions $\Lambda(H^2)$ or $\Lambda(R)$. Such kind of models assume an equation of state for vacuum given by $\,\overline P_\Lambda=-\,\overline\rho_\Lambda$, relating their background pressure $\,\overline P_\Lambda$ and mean energy density $\,\overline\rho_\Lambda\equiv\Lambda/8\pi G$. This equation of state requires that the dynamic for vacuum is due to the energy exchange with the material species. Most of the approaches to background level consider only the energy exchange between vacuum and the transient dominant material component of the universe. We extend such models assuming the running vacuum as the sum of independent contributions $\,\overline\rho_{\Lambda} =\sum_i\,\overline\rho_{\Lambda i}$, associated with (and interacting with) each of the $i$ material species. We derive the linear scalar perturbations for two running scenarios, modeling its cosmic evolution and identifying their different imprints on the cosmic microwave background anisotropies and the matter power spectrum. In the $\Lambda(H^2)$ scenario the running vacuum are coupled with all the material species in the universe, whereas the $\Lambda(R)$ description only leads to coupling between vacuum and the non-relativistic matter components; which produces different imprints of the two models on the matter power spectrum. A comparison with the Planck 2015 data was made in order to constrain the free parameters of the models. In the case of the $\Lambda(H^2)$ model, it was found that $\Omega_\Lambda=0.705\pm0.027$ and $H_0=69.6\pm2.9\, km\, Mpc^{-1}\, s^{-1}$, which diminish the tension with the low redshift expectations. 
E. L. D. Perico and D. A. Tamayo
astro-ph.CO gr-qc