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.

中文翻译：

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在非平坦宇宙中运行真空模型

我们研究了空间弯曲宇宙中 $\Lambda=3\nu (H^{2}+K/a^2)+c_0$ 运行真空模型 (RVM) 的观测约束，其中 $\nu$ 是模型参数，$K$对应空间曲率常数，$c_{0}$是边界条件定义的常数。如 $\dot{\Lambda} \ne 0$，RVM 中存在真空、物质和辐射之间的能量交换。我们研究了 CMB 功率谱上 RVM 的“几何简并性”。通过拟合宇宙学数据，我们发现对于非平坦宇宙，RVM 和 $\Lambda$CDM 中的 $\chi^2$ 值彼此相似。明确地，我们得到 $\nu\leq O(10^{-4})$ (68 $\%$ CL) 和 $|\Omega_K|\leq O(10^{-2})$ (95 $\%$ CL) 在我们的研究中。此外，我们证明了 $\Sigma m_{\nu}=0.416^{+0.311}_{-0 的宇宙学约束。