Running vacuum model in non-flat universe

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Chao-Qiang Geng, Yan-Ting Hsu, Lu Yin and Kaituo Zhang. Running vacuum model in non-flat universe[J]. Chinese Physics C.
Chao-Qiang Geng, Yan-Ting Hsu, Lu Yin and Kaituo Zhang. Running vacuum model in non-flat universe[J]. Chinese Physics C. shu
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Received: 2020-03-19
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    沈阳化工大学材料科学与工程学院 沈阳 110142

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Running vacuum model in non-flat universe

    Corresponding author: Kaituo Zhang, ktzhang@ahnu.edu.cn
  • 1. School of Fundamental Physics and Mathematical Sciences Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024
  • 2. International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou
  • 3. Department of Physics, National Tsing Hua University, Hsinchu, Taiwan 300
  • 4. National Center for Theoretical Sciences, Hsinchu, Taiwan 300
  • 5. Department of Physics, Anhui Normal University, Wuhu, Anhui 241000
  • 6. Collaborative Innovation Center of Light Manipulations and Applications, Shandong Normal University, Jinan 250358

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, $a$ represents the scalar factor, and $c_{0}$ is a constant defined by the boundary conditions. We study the CMB power spectra with several sets of $\nu$ and $K$ in RVM. By fitting the cosmological data, we find that the best fitted $\chi^2$ value of RVM is slightly smaller than that of $\Lambda$CDM in the non-flat universe, along with the constraints of $\nu\leqslant O(10^{-4})$ (68% C.L.) and $|\Omega_K=-K/(aH)^2|\leqslant O(10^{-2})$ (95% C.L.). In particular, our results favor the open universe in both $\Lambda$CDM and RVM. In addition, we show that the cosmological constraints of $\Sigma m_{\nu}=0.256^{+0.224}_{-0.234}$ (RVM) and $\Sigma m_{\nu}=0.257^{+0.219}_{-0.234}$ ($\Lambda$CDM) at 95% C.L. for the neutrino mass sum are relaxed in both models in the spatially curved universe.

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    1.   Introduction
    • Since the discovery that our universe has been expanding at an accelerated rate at recent time from the type Ia supernova data [1-3], many dark energy models have been proposed to explain these phenomena [4-9]. The simplest one is the $ \Lambda{\rm{CDM}} $ model, in which $ \Lambda $ represents the cosmological constant term. However, $ \Lambda{\rm{CDM}} $ encounters some difficulties, mainly the "fine tuning" [10,11] and "coincidence" [12-14] problems.

      The running vacuum model (RVM) [15,16] has been introduced in order to solve the "coincidence problem", where the cosmological constant term is assumed to be varying with the Hubble parameter $ H $. This model links the existence of dark energy to the theoretical mechanism of the quantum field, which may trigger the primordial inflation scenario [17], and fit with the observational data better than $ \Lambda{\rm{CDM}} $ [18]. In the literature, the spatially flat RVM has been extensively investigated [18-35].

      Recently, the Planck Legacy 2018 analysis by Valentino, Melchiorri and Silk in Ref. [36] has suggested that the universe is closed at 99% C.L. [37]. Similar conclusions have been obtained by Park and Ratra in the context of non spatially-flat DE models obtained [38-41]. Nevertheless, when the BAO data set is included together with the CMB, the evidence in favor of a non spatially-flat universe disappears completely [36,42,43]. These interesting results encourage us to study RVM in a non-flat universe [44,45] besides the flat one [18]. With the involvement of the non-zero spatial curvature, it is inevitable to encounter the degeneracies between curvature and other parameters. One of them is the famous "geometrical degeneracy" [46,47] on CMB power spectra, caused by different sets of parameters that lead to same value of the angular diameter distance of the last scattering. On the other hand, when fitting with the observational data, the non-zero spatial curvature also broadens the constraints of the cosmological parameters [48].

      In this work, we concentrate on the running cosmological constant in the non-flat universe, $ \Lambda = 3\nu (H^{2}+ K/a^2)+c_0 $, where $ \nu $ and $ c_{0} $ are the model parameters, $ a $ represents the scalar factor and $ K $ corresponds to the spatial curvature constant. We first study the CMB power spectra in this non-flat RVM and discuss the degeneracy between $ \nu $ and the density parameter of curvature $ \Omega_K = -K/(aH)^2 $. We then constrain the cosmological parameters of both non-flat RVM and $ \Lambda{\rm{CDM}} $ with the observational data by using the Markov chain Monte Carlo (MCMC) method, and compare the results with those in the flat universe. The effectiveness of RVM versus $ \Lambda{\rm{CDM}} $ in the non-flat universe is also tested based on the minimal $ \chi^2 $ values.

      This paper is organized as follows. In Sec. 2, we introduce the non-flat running vacuum model and derive the background evolution equations. We compare the CMB power spectra of RVM in the non-flat universe along with the Planck 2018 data, and show the constraints of the cosmological parameters in Sec. 3. Our conclusions is presented in Sec. 4.

    2.   Evolution of RVM in curved universe
    • We start with the Einstein field equation of RVM, given by

      $ R_{\alpha\beta}-\frac{1}{2} g_{\alpha\beta} R+g_{\alpha\beta} \Lambda = \kappa^2 T_{\alpha\beta}, $

      (1)

      where $ \kappa^2 = 8\pi G $ is set to be 1 for simplicity, $ R = g^{\alpha\beta}R_{\alpha\beta} $ represents the Ricci scalar, $ T_{\alpha\beta} $ stands for the energy-momentum tensor for matter and radiation, and $ \Lambda $ corresponds to the dynamical cosmological constant.

      The spatially isotropic and homogeneous universe can be described by the Robertson-Walker metric:

      $ {\rm d}s^{2} = -{\rm d}t^{2}+a^2(t)\left\{\frac{{\rm d}r^2}{1-Kr^2} +r^2 {\rm d}\theta^2+r^2\sin^2\theta {\rm d}\phi^2 \right\}\,, $

      (2)

      where $ a $ is the scale factor, while $ K $ is a constant describe the spatial curvature with $ K = 1,0,-1 $ corresponding to the closed, flat, and open universe, respectively. Then, the Friedmann equations can be expressed as

      $ H^{2} = \frac{1}{3}(\rho_m + \rho_r + \rho_{\Lambda})-\frac{K}{a^2} \,, $

      (3)

      $ \dot{H} = - \frac{1}{2} (\rho_m + \rho_r + \rho_{\Lambda} + P_m+ P_r + P_{\Lambda})+\frac{K}{a^2}, \, $

      (4)

      where $ \rho_{m, r, \Lambda} $ ($ P_{m, r, \Lambda} $) are the energy densities (pressures) of matter, radiation and dark energy, respectively, and $H = {\rm d}a/(a{\rm d}t)$ represents the Hubble parameter. We note that $ \rho_{\Lambda} = \kappa^{-2} \Lambda $, and the density parameters are parameters are given by

      $ \Omega_{m, r} = \frac{\rho_{m, r}}{3H^2} , $

      (5)

      $ \Omega_{\Lambda} = \frac{\Lambda}{3H^2} , $

      (6)

      $ \Omega_K = -\frac{K}{a^2 H^2}. \, $

      (7)

      In the non-flat universe, the running cosmological constant term is set to be

      $ \Lambda = 3\nu H^{2}+3\nu \frac{K}{a^2}+c_0\,, $

      (8)

      where $ \nu $ is a non-negative model parameter to ensure that the energy density of dark energy is positive in the early universe, $ c_0 $ is given by $ c_0 = -3\nu ( H_0^2+K)+\Lambda_0 $ with $ H_0 $ and $ \Lambda_0 $ the present values of the Hubble parameter and cosmological constant, respectively. The model becomes to be $ \Lambda{\rm{CDM}} $ when $ \nu = 0 $. The corresponding equations of state in this model can be defined as

      $ w_{m, r, \Lambda} = \frac{P_{m, r, \Lambda}}{\rho_{m, r, \Lambda}} = 0, \frac{1}{3}, -1 \,. $

      (9)

      For the energy transformations from dark energy to matter and radiation, the modified continuity equations are given by

      $ \dot{\rho}_{m, r}+3 H(1+w_{m, r})\rho_{m, r} = Q_{m, r} \,, $

      (10)

      $ \dot{\rho}_\Lambda+3 H(1+w_\Lambda)\rho_\Lambda = - Q \,, $

      (11)

      with $ Q_{m}+Q_{r} = Q $, $ Q_{m, r} $ can be written as

      $ Q_{m, r} = -\frac{\dot{\rho}_{\Lambda}(\rho_{m, r}+P_{m, r})}{\rho_{m}+\rho_{r}+P_m+P_r} = 3\nu H(1+w_{m, r})\rho_{m, r}\,. $

      (12)

      By combining Eqs. (3)-(12), we derive the energy densities as functions of the scale factor:

      $ \rho_{m}(a) = \rho_{m}^{(0)}a^{-3\xi} , $

      (13)

      $ \rho_{r}(a) = \rho_{r}^{(0)}a^{-4\xi} , $

      (14)

      $ \rho_{\Lambda}(a) = \rho_{\Lambda}^{(0)}+\left(\xi^{-1}-1\right)\left [\rho_m^{(0)}\left(a^{-3\xi}-1\right)+\rho_r^{(0)}\left(a^{-4\xi}-1\right)\right] , $

      (15)

      where $ \rho_{m,r.\Lambda}^{(0)} $ are current values and $ \xi = (1-\nu) $. Consequently, the Friedmann equation defined in Eq. (3) can be rewritten as

      $ H^{2} \!=\! H_{0}^{2}\left\{\frac{1}{\xi}\left[\Omega_{m}^{0}\left(a^{-3\xi}\!-\!1\right)+\Omega_{r}^{0}\left(a^{-4\xi}\!-\!1\right)\right]+\Omega_{K}^{0}\left(a^{-2}\!-\!1\right)\!+\!1\right\} , $

      (16)

      where $ \Omega_{m,r,K}^{0} $ are current values of density parameters.

    3.   Numerical calculations
    • To study the degeneracy between the cosmological parameters, we first modify the CAMB [49] program to generate theoretical CMB power spectra for both models of RVM and $ \Lambda{\rm{CDM}} $. The results are presented in Sec. III A. We then use the CosmoMC package [50], which is a Markov Chain Monte Carlo (MCMC) engine exploring the cosmological parameter space, to constrain RVM and $ \Lambda{\rm{CDM}} $ from the observational data. For simplification, we take $ \Omega_K $ afterward to represent the density parameter of curvature at the present time except those specifically indicated.

    • 3.1.   CMB power spectra of the models

    • There is "geometrical degeneracy" between curvature and other parameters for CMB power spectra. To see this effect, we compare CMB power spectra of RVM and $ \Lambda{\rm{CDM}} $ with different $ \nu $ and $ \Omega_K $ along with the observational data from Planck 2018 [43]. From the previous studies in the literature [18,33,34] with $ 0 \leqslant \nu \leqslant O(10^{-3}) $ in RVM for the flat universe and the result of $ -0.007 \geqslant \Omega_K \geqslant -0.095 $ at 99% C.L. in Ref. [36], we choose $ 0 \leqslant \nu < 0.01 $ and $ 0 \geqslant \Omega_K \geqslant -0.01 $ to see the degeneracy between $ \nu $ and $ \Omega_K $ on CMB power spectra. Furthermore, the $ \Lambda{\rm{CDM}} $ model is recovered when $ \nu = 0 $ and $ \Omega_K = 0 $ in Eq. (8).

      In Fig. 1, we present the CMB power spectra for the TT, EE and TE modes from the CAMB package. It can be seen that $ 0\leqslant\nu\leqslant O(10^{-3}) $ (solid lines) and $ 0 \geqslant \Omega_K \geqslant -O(10^{-2}) $ (dashed lines) fit well with the data from Planck 2018. The residues with respect to $ \Lambda{\rm{CDM}} $ are plotted in Fig. 2. We find that the geometrical degeneracy with $ (\nu, \Omega_K) = (0.001,0) $ (green solid line) and $ (0.0, -0.01) $ (purple dashed line) has the similar results on CMB power spectra. However, only $ \nu $ can cause strong suppressions on the TT mode spectra when $ \nu > 0 $. In addition, the effects of $ \nu $ and $ K $ show additive property on CMB power spectra (red dash-dotted line).

      Figure 1.  (color online) Power spectra of CMB TT, EE and TE for RVM and $ \Lambda{\rm{CDM}} $ in the flat and non-flat universe along with the Planck 2018 data.

      Figure 2.  (color online) Residuals of $ \Delta D_{\ell}^{TT} $, $ \Delta D_{\ell}^{EE} $ and $ \Delta D_{\ell}^{TE} $ in RVM with respect to $ \Lambda{\rm{CDM}} $ for CMB power spectra, respectively, along with the observational data from Planck 2018.

    • 3.2.   Global fitting

    • In order to constrain the cosmological parameters of RVM and $ \Lambda{\rm{CDM}} $ in the non-flat universe, we use the CosmoMC package with a MCMC engine to explore the parameter space with the combinations of the observational data sets, which include the CMB temperature fluctuation from Planck 2018 with TT, TE, EE, low-$ l $ polarization from SMICA [43,51-53], BAO data from 6dF Galaxy Survey [54] and BOSS [55], supernova(SN) data from the JLA compilation [56], the weak lensing (WL) data from CFHTLenS [57] and direct large scale structure (LSS) formation data, and the data points of $ f\sigma_8 $ listed in Table 1. The priors of parameters are given in Table 2. Due to the tension between the geometry data (SNIa, BAO etc.) and growth data (WL, $ f\sigma_8 $) [74], we choose the two combinations of CMB+BAO+SN and CMB+BAO+SN+WL+$ f\sigma_8 $ in our fits. To calculate the best fitted values of $ \chi^2 $, we use that

      $ z $ $ f\sigma_8 $ Ref. $ z $ $ f\sigma_8 $ Ref. $ z $ $ f\sigma_8 $ Ref.
      $ 1 $ $ 1.36 $ $ 0.482 \pm 0.116 $ [58] $ 10 $ $ 0.59 $ $ 0.488 \pm 0.06 $ [66] $ 19 $ $ 0.35 $ $ 0.440 \pm 0.05 $ [61,69]
      $ 2 $ $ 0.8 $ $ 0.470 \pm 0.08 $ [59] $ 11 $ $ 0.57 $ $ 0.444 \pm 0.038 $ [67] $ 20 $ $ 0.32 $ $ 0.394 \pm 0.062 $ [67]
      $ 3 $ $ 0.78 $ $ 0.38 \pm 0.04 $ [60] $ 12 $ $ 0.51 $ $ 0.452 \pm 0.057 $ [64] $ 21 $ $ 0.3 $ $ 0.407 \pm 0.055 $ [65]
      $ 4 $ $ 0.77 $ $ 0.490 \pm 0.18 $ [61,62] $ 13 $ $ 0.5 $ $ 0.427 \pm 0.043 $ [65] $ 22 $ $ 0.25 $ $ 0.351 \pm 0.058 $ [68]
      $ 5 $ $ 0.73 $ $ 0.437 \pm 0.072 $ [63] $ 14 $ $ 0.44 $ $ 0.413 \pm 0.080 $ [63] $ 23 $ $ 0.22 $ $ 0.42 \pm 0.07 $ [60]
      $ 6 $ $ 0.61 $ $ 0.457 \pm 0.052 $ [64] $ 15 $ $ 0.41 $ $ 0.45 \pm 0.04 $ [60] $ 24 $ $ 0.17 $ $ 0.51 \pm 0.06 $ [61,70]
      $ 7 $ $ 0.60 $ $ 0.390 \pm 0.063 $ [63] $ 16 $ $ 0.4 $ $ 0.419 \pm 0.041 $ [65] $ 25 $ $ 0.15 $ $ 0.49 \pm 0.15 $ [71]
      $ 8 $ $ 0.6 $ $ 0.433 \pm 0.067 $ [65] $ 17 $ $ 0.38 $ $ 0.430 \pm 0.054 $ [64] $ 26 $ $ 0.067 $ $ 0.423 \pm 0.055 $ [72]
      $ 9 $ $ 0.60 $ $ 0.43 \pm 0.04 $ [60] $ 18 $ $ 0.37 $ $ 0.460 \pm 0.038 $ [68] $ 27 $ $ 0.02 $ $ 0.36 \pm 0.04 $ [73]

      Table 1.  data points of $ f\sigma_8 $.

      Parameter Prior
      RVM parameter $ \nu $ $ 0.0 \leqslant \nu \leqslant 3.0\times10^{-4} $
      Curvature parameter $ \Omega_K $ $ -0.25 \leqslant \Omega_K \leqslant 0.2 $
      Baryon density $ 0.5 \leqslant 100\Omega_bh^2 \leqslant 10 $
      CDM density $ 0.1 \leqslant 100\Omega_ch^2 \leqslant 99 $
      Optical depth $ 0.01 \leqslant \tau \leqslant 0.8 $
      Neutrino mass sum $ 0 \leqslant \Sigma m_{\nu} \leqslant 2 $ eV
      $ \dfrac{\mathrm{Sound \; horizon}}{\mathrm{Angular \; diameter \; distance}} $ $ 0.5 \leqslant 100 \theta_{MC} \leqslant 10 $
      Scalar power spectrum amplitude $ 2 \leqslant \ln \left( 10^{10} A_s \right) \leqslant 4 $
      Spectral index $ 0.8 \leqslant n_s \leqslant 1.2 $

      Table 2.  Priors for cosmological parameters with the non-flat RVM of $ \Lambda = 3\nu (H^{2}+ K/a^2)+c_0 $.

      $ \chi^2_c = \sum\limits_{i = 1}^n \frac{(T_c(z_i) - O_c(z_i))^2}{E^i_c} \,, $

      (17)

      where $ c $ denotes the type of the data, $ n $ is the number of the data in each data set, $ T_c $ represents the theoretical value derived form CAMB at the redshift $ z_i $, and $ O_c $ ($ E_c $) corresponds to the observational value (covariance).

      The global fitting results of RVM and $ \Lambda{\rm{CDM}} $ in the non-flat universe are plotted in Figs. 3 and 4, while those listed in Table 3 correspond to the cosmological parameters and $ \nu $, given at 95% and 68% C.L., respectively. Our results show that $ \nu\lesssim 1.39 \times10^{-4} $ at 68%C.L. in the non-flat universe of RVM for the data set of CMB+BAO+SN+WL+$ f\sigma_8 $, which is similar to the previous result of $ 1.54\times10^{-4} $ at 68% C.L. in RVM for the flat universe [33]. Explicitly, we obtain that $ \chi^2_{RVM} = 3472.32\; (3523.74) $ and $ \chi^2_{\Lambda CDM} = 3474.92\; (3524.51) $ when fitting with the data set of CMB+BAO+SN (CMB+BAO+SN+WL+$ f\sigma_8 $), indicating that our results in RVM are consistent with those in $ \Lambda{\rm{CDM}} $ for the non-flat universe. For the density parameter $ \Omega_K $ of the spatial curvature at the present time, our results show that an open universe is preferred instead of the closed one in Ref. [36]. In particular, when the data of WL and $ f\sigma_8 $ are included, both RVM and $ \Lambda{\rm{CDM}} $ favor the open universe with $ |\Omega_K|\leqslant O(10^{-2}) $. Our result of the open universe is consistent with that in Ref. [75].

      Figure 3.  (color online) One and two-dimensional distributions of $ \Omega_b h^2 $, $ \Omega_c h^2 $, $ \tau $, $ \Omega_K $, $ \sum m_\nu $, $ 10^4 \nu $, $ H_0 $, $ \sigma_8 $ for RVM and $ \Lambda{\rm{CDM}} $ in the non-flat universe with the combined data of CMB+BAO+SN, where the contour lines represent 68% and 95% C.L., respectively.

      Parameter CMB+BAO+SN CMB+BAO+SN+WL+$f\sigma_8$
      Model RVM $\Lambda{\rm{CDM}}$ RVM $\Lambda{\rm{CDM}}$
      ${\bf{100\Omega_b h^2}}$ $2.23 \pm 0.03 $ $2.24 \pm 0.03 $ $2.23^{+0.04}_{-0.03}$ $2.23 \pm 0.04$
      ${\bf{100\Omega_c h^2}}$ $12.0 \pm 0.3 $ $12.0 \pm 0.3 $ $11.9 \pm 0.3 $ $11.9 \pm 0.3 $
      ${\bf{100\tau}}$ $5.43^{+1.52}_{-1.45}$ $5.53^{+1.62}_{-1.52}$ $5.22^{+1.56}_{-1.50}$ $5.23^{+1.54}_{-1.58}$
      ${\bf{10^3\Omega_K}}$ $1.55^{+4.91}_{-4.54}$ $0.80^{+4.72}_{-4.53}$ $5.10^{+5.97}_{-5.94}$ $4.53^{+5.92}_{-5.99}$
      ${\bf{\Sigma m_\nu}}$ [eV] $<0.199$ $<0.188$ $0.256^{+0.224}_{-0.234}$ $0.257^{+0.219}_{-0.234}$
      ${\bf{10^4 \nu}}$ $<1.36$ $-$ $<1.39$ $-$
      $H_0$ [km/s/Mpc] $67.8^{+1.4}_{-1.3}$ $67.9^{+1.3}_{-1.2}$ $67.9^\pm 1.4$ $68.1^{+1.4}_{-1.3}$
      $\sigma_8$ $0.808^{+0.027}_{-0.034}$ $0.810^{+0.026}_{-0.031}$ $0.764^{+0.040}_{-0.042}$ $0.765\pm 0.040$
      $\chi^2_{best-fit}$ $3472.32$ $3474.92$ $3523.74$ $3524.51$

      Table 3.  Fitting results for RVM and $ \Lambda{\rm{CDM}} $ in the non-flat universe, where the cosmological parameters and $ \nu $ are given at 95% and 68% C.L., respectively.

      Figure 4.  (color online) One and two-dimensional distributions of $ \Omega_b h^2 $, $ \Omega_c h^2 $, $ \tau $, $ \Omega_K $, $ \sum m_\nu $, $ 10^4 \nu $, $ H_0 $, $ \sigma_8 $ for RVM and $ \Lambda $CDM in the non-flat universe with the combined data of CMB+BAO+SN+WL+$ f\sigma_8 $, where the contour lines represent 68% and 95% C.L., respectively.

      On the other hand, the best-fit values of the neutrino mass sum, $ \Sigma m_{\nu} $, in the non-flat universe are similar with those in the flat universe when fitting with CMB+BAO+SN [18,76-78]. However, it is interesting to see that the constraints on $ \Sigma m_{\nu} $ are relaxed for the data sets of CMB+BAO+SN+WL+$ f\sigma_8 $, in which the $ f\sigma_8 $ data points play the main role. Note that similar results are also obtained in the flat universe as shown in Ref. [18]. This is because the structure-growth rate of $ f\sigma_8 $ is a unique indicator of massive neutrinos [79,80]. Specifically, we have $ \Sigma m_{\nu} = 0.256^{+0.224}_{-0.234} $ ($ 0.257^{+0.219}_{-0.234} $) eV at 95%C.L., resulting in the non-zero lower bounds of $ \Sigma m_{\nu} \geqslant 0.022\; (0.023) $ eV at 95% C.L. for RVM ($ \Lambda{\rm{CDM}} $) in the spatially curved universe. On the other hand, due to the possible degeneracy among $ \tau $, $ \Omega_K $ and $ \Sigma m_{\nu} $, the constraints on $ \tau $ and $ \Omega_K $ are not improved with the data points of WL+$ f\sigma_8 $ as seen from Table 3. To obtain better constraints, one would use some additional data, such as $ 21\; cm $ emission measurements [81,82], which could fix the parameter $ \tau $ to break the degeneracy of $ \tau $, $ \Omega_K $ and $ \Sigma m_{\nu} $ [79,80]. We note that in our data fitting we do not specify the neutrino mass hierarchy in $ \Sigma m_\nu $. For the cosmological effects of the neutrino mass hierarchy, one can refer to the discussions in the literature [83-85].

    4.   Conclusions
    • We have studied the model with the running cosmological constant of $ \Lambda = 3\nu (H^{2}+ K/a^2)+c_0 $ in the spatially curved universe. We have compared our results for several sets of $ \nu $ and $ \Omega_K $ with the Planck 2018 data in the CMB power spectra. We have found that $ \nu $ and $ \Omega_K $ have similar effects on the CMB power spectra, but only non-zero values of $ \nu $ would lead to large suppressions in the CMB TT mode spectra. In the two combinations of the observational data, we have constrained that $ \nu \leqslant O(10^{-4}) $ together with $ |\Omega_K|\leqslant O(10^{-2}) $. Notably, the constraints on $ \nu $ in the non-flat universe are similar to those in the flat universe. From the best fitted values of $ \chi^2 $, we have shown that RVM is in consistent with $ \Lambda{\rm{CDM}} $. When fitting with the date set of CMB+BAO+SN+WL+$ f\sigma_8 $, we have obtained the non-zero lower bounds of $ \Sigma m_{\nu} \geqslant 0.022 $ and 0.023 eV at 95% C.L. in the non-flat RVM and $ \Lambda{\rm{CDM}} $, respectively, indicating that the involvement of a non-zero $ \Omega_K $ would provide viable constraints on the absolute neutrino masses in cosmological models.

      K. Z. would like to thank Dr. Yangsheng Yuan and Shandong Normal University for part of numerical calculations run by the workstation of their group.

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