Running vacuum model in non-flat universe

Abstract: We investigate observational constraints on the running vacuum model (RVM) of $\Lambda=3\nu (H^{2}+K/a^2)+c_0$ in the spatially curved universe, where $\nu$ is the model parameter, $K$ corresponds to the spatial curvature constant, and $c_{0}$ is a constant defined by the boundary conditions. As $\dot{\Lambda} \ne 0$, there are energy exchanges between vacuum, matter and radiation in RVM. We study the "geometrical degeneracy" of RVM on the CMB power spectra. By fitting the cosmological data, we find that the values of $\chi^2$ in RVM and $\Lambda$CDM are similar to each other for the non-flat universe. Explicitly, we obtain the constraints of $\nu\leq O(10^{-4})$ (68 $\%$ C.L.) and $|\Omega_K|\leq O(10^{-2})$ (95 $\%$ C.L.) in our study. In addition, we show that the cosmological constraints of $\Sigma m_{\nu}=0.416^{+0.311}_{-0.407}$ (RVM) and $\Sigma m_{\nu}=0.497^{+0.335}_{-0.387}$ ($\Lambda$CDM) at 95$\%$ C.L. for the neutrino mass sum are relaxed in both models in the spatially curved universe.