1. Introduction
The phenomenon of neutrino oscillation indicates that neutrinos have nonzero masses and there are mass splittings between different neutrino species [1, 2]. The neutrino oscillation experiments can provide the information about the squared mass differences between the neutrino mass eigenstates. Specifically, the solar and reactor experiments give the result of ${\rm{\Delta }}{m}_{21}^{2}\simeq 7.5\times {10}^{-5}\,{\mathrm{eV}}^{2}$, and the atmospheric and accelerator beam experiments give the result of $| {\rm{\Delta }}{m}_{31}^{2}| \simeq 2.5\times {10}^{-3}\,{\mathrm{eV}}^{2}$ [2, 3]. Therefore, we can get two possible mass hierarchies of the neutrino mass spectrum, i.e. the normal hierarchy (NH) with m1 < m2 ≪ m3 and the inverted hierarchy (IH) with m3 ≪ m1 < m2, where m1, m2, and m3 denote the masses of neutrinos for the three mass eigenstates. However, the absolute masses of neutrinos are still unknown.
In principle, laboratory experiments of particle physics can directly measure the absolute masses of neutrinos, but these experiments have always been facing great challenges [4-12]. Compared with these particle physics experiments, cosmological observations are more prone to be capable of measuring the absolute masses of neutrinos [13-15], since massive neutrinos can leave rich signatures on the cosmic microwave background (CMB) anisotropies and the large-scale structure (LSS) formation at different epochs of the cosmic evolution [16]. Thus, we can extract useful information on neutrinos from these available cosmological observations.
Recently, the issue of cosmological constraints on the total neutrino mass with the consideration of mass hierarchy using the latest observational data has been discussed in [17]. In [17], the authors discussed the constraints on neutrino mass in several typical dark energy (DE) models, e.g. the Λ cold dark matter (ΛCDM), wCDM, Chevallier-Polarski-Linder (CPL), and holographic dark energy (HDE) models. It was found that, compared to the ΛCDM+$\sum {m}_{\nu }$ model, larger neutrino masses are favored in the wCDM+$\sum {m}_{\nu }$ and CPL+$\sum {m}_{\nu }$ models, and the most stringent upper limits are obtained in the HDE+$\sum {m}_{\nu }$ model. Moreover, in [17], it was also confirmed that the NH case is more favored by current cosmological observations than the IH case. For more relevant studies on constraining the total neutrino mass by using cosmological observations, see e.g. [18-69].
Furthermore, the impacts of interaction between DE and cold dark matter (CDM) on constraining neutrino mass have also been considered. For example, in the scenario of vacuum energy interacting with cold dark matter, which is abbreviated as the IΛCDM scenario in this work, the constraint on $\sum {m}_{\nu }$ becomes $\sum {m}_{\nu }\lt 0.10\,\mathrm{eV}$ (2σ) for $Q=\beta H{\rho }_{\mathrm{de}}$, $\sum {m}_{\nu }\lt 0.20\,\mathrm{eV}$ (2σ) for $Q=\beta H{\rho }_{{\rm{c}}}$ [70], and $\sum {m}_{\nu }\lt 0.214\,\mathrm{eV}$ (2σ) for $Q=\beta {H}_{0}{\rho }_{{\rm{c}}}$ [71]. When the mass hierarchies of neutrinos are considered in the IΛCDM model [72, 73], the results showed that the degenerate hierarchy (DH) case gives the smallest upper limit of the neutrino mass and the NH case is more favored over the IH case. In the present work, we will revisit the constraints on the total neutrino mass in the IΛCDM scenario after the Planck 2018 data release. We will consider more forms of interaction term Q, and also adopt the mass hierarchies of neutrinos in this work.
In the so-called ‘interacting dark energy' (IDE) scenario, some direct, nongravitational coupling between DE and dark matter is assumed and its cosmological consequences have been widely studied [74-114]. Theoretically speaking, the consideration of such an interaction is helpful in solving the cosmic coincidence problem [76-78, 87, 89], but actually what is more important is to detect such an interaction using the cosmological observations. The impacts of interactions between DE and dark matter on the CMB [89, 106] and LSS [75, 83, 87, 90, 101, 106] have been studied in-depth.
In this paper, we only consider the simplest class of models in the IDE scenario, i.e. the IΛCDM models, in which the vacuum energy with w = −1 serves as DE. In this scenario, the energy conservation equations of the vacuum energy and the cold dark matter satisfy1 ) and (2 ), it is known that β > 0 means cold dark matter decaying into DE, β < 0 means DE decaying into cold dark matter, and β = 0 indicates no interaction between vacuum energy and cold dark matter.
$\begin{eqnarray}\dot{{\rho }_{\mathrm{de}}}=Q,\end{eqnarray}$
$\begin{eqnarray}\dot{{\rho }_{{\rm{c}}}}=-3H{\rho }_{{\rm{c}}}-Q,\end{eqnarray}$
where ρde and ρc represent the densities of DE (namely, vacuum energy) and cold dark matter, respectively, H is the Hubble parameter, the dot represents the derivative with respect to the cosmic time t, and Q is the energy transfer rate. Usually, the form of Q is assumed to be proportional to the density of DE or dark matter, i.e. $Q=\beta H{\rho }_{\mathrm{de}}$ or $Q=\beta H{\rho }_{{\rm{c}}}$, where the appearance of H is only for mathematical convenience. In the research area of IDE, another perspective is to consider $Q=\beta {H}_{0}{\rho }_{\mathrm{de}}$ or $Q=\beta {H}_{0}{\rho }_{{\rm{c}}}$ [86], where the appearance of H0 is only for a dimensional consideration. From equations (Different phenomenological models of IΛCDM can be built by assuming different forms of Q. In this work, we will collect the popular forms of Q in the current literature and then focus on the impacts of different forms of Q on constraining the total neutrino mass after the Planck 2018 data release. We will consider the four typical forms of Q: $Q=\beta H{\rho }_{\mathrm{de}}$, $Q=\beta H{\rho }_{{\rm{c}}}$, $Q=\beta {H}_{0}{\rho }_{\mathrm{de}}$ and $Q=\beta {H}_{0}{\rho }_{{\rm{c}}}$. The mass hierarchies of neutrinos are also considered in this work. In addition, we also wish to see whether some hint of the existence of nonzero interaction can be found in these IΛCDM models by using the latest observational data.
This paper is organized as follows. In section 2 , we introduce the cosmological observations used in this work and briefly describe the analysis method. In section 3 , we report the constraint results and then make some relevant discussions. The issue of H0 tension will also be discussed in this section. Conclusion is given in section 4 .
2. Method and data
In the IΛCDM model, there are seven basic cosmological parameters $\{{\omega }_{b},\,{\omega }_{c},\,100{\theta }_{\mathrm{MC}},\,\tau ,\,{n}_{s},\,\mathrm{ln}({10}^{10}{A}_{s}),\beta \}$, where ωb is the present density of baryons, ωc is the present density of cold dark matter, θMC is the ratio between the sound horizon to the angular diameter distance at the decoupling epoch, τ is the Thomson scattering optical depth to reionization, ns is the scalar spectral index, As is the amplitude of primordial scalar perturbation power spectrum, and β is the dimensionless coupling constant describing the coupling strength between vacuum energy and dark matter.
For the IΛCDM model there is a problem of early-time perturbation instability, because in the IDE models, the cosmological perturbations of DE will be divergent in a part of the parameter space, which ruins the IDE cosmology in the perturbation level. The origin of the difficulty is that we know little about the nature of DE, so we do not know how to treat the spread of sounds in DE fluid which has a negative equation of state. To overcome the problem of perturbation instability, in 2014, Yun-He Li, Jing-Fei Zhang, and Xin Zhang established an effective theoretical framework for IDE cosmology based on the extended version of the parameterized post-Friedmann (PPF) approach, which can safely calculate the cosmological perturbations in the whole parameter space of an IDE model. About the extended PPF method, see [115-119], and the original PPF method is introduced in [120, 121]. In this work, we will employ the extended PPF method [115-119] to calculate the cosmological perturbations in the IΛCDM model.
We use the modified version of the publicly available Markov-chain Monte Carlo package CosmoMC [122] to constrain the neutrino mass and other cosmological parameters. We monitor the convergence of the generated MCMC chains by using the Gelman-Rubin parameter R [123], requiring $R-1\lt 0.01$ for our MCMC chains to be considered as converged. When considering the neutrino mass splitting, we should note the following rules. For the NH case, the neutrino mass spectrum is
$\begin{eqnarray}({m}_{1},{m}_{2},{m}_{3})=({m}_{1},\sqrt{{m}_{1}^{2}+{\rm{\Delta }}{m}_{21}^{2}},\sqrt{{m}_{1}^{2}+| {\rm{\Delta }}{m}_{31}^{2}| }),\end{eqnarray}$
where m1 is a free parameter; for the IH case, the neutrino mass spectrum is $\begin{eqnarray}\begin{array}{l}({m}_{1},{m}_{2},{m}_{3})\\ \quad =\,(\sqrt{{m}_{3}^{2}+| {\rm{\Delta }}{m}_{31}^{2}| },\sqrt{{m}_{3}^{2}+| {\rm{\Delta }}{m}_{31}^{2}| +{\rm{\Delta }}{m}_{21}^{2}},{m}_{3}),\end{array}\end{eqnarray}$
where m3 is a free parameter; for comparison, the DH case is also considered, in which the neutrino mass spectrum is $\begin{eqnarray}{m}_{1}={m}_{2}={m}_{3}=m,\end{eqnarray}$
where m is a free parameter. Since we have two values of squared mass differences, ${\rm{\Delta }}{m}_{21}^{2}=7.5\times {10}^{-5}\,{\mathrm{eV}}^{2}$ and $| {\rm{\Delta }}{m}_{31}^{2}| =2.5\times {10}^{-3}\,{\mathrm{eV}}^{2}$ [2, 3], the lower limits of NH and IH can be derived by setting the smallest mass be zero. Thus, the input lower bounds of $\sum {m}_{\nu }$ are 0.06 eV for the NH case, 0.10 eV for the IH case, and 0 eV for the DH case, respectively.The current observational data sets we used in this paper include CMB, BAO, SN and H0. For the CMB data, we use the Planck TT, TE, EE spectra at ℓ ≥ 30, the low-ℓ temperature Commander likelihood, and the low-ℓ SimAll EE likelihood, from the Planck 2018 data release [124]. For the BAO data, we consider the measurements from 6dFGS (zeff = 0.106) [125], SDSS-MGS (zeff = 0.15) [126], and BOSS DR12 (zeff = 0.38, 0.51, and 0.61) [127]. For the SN data, we employ the latest Pantheon sample, which is comprised of 1048 data points from the Pantheon compilation [128]. For the H0 data, we use the 2019 local distance ladder measurement of the Hubble constant ${H}_{0}=74.03\pm 1.42\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ from the Hubble Space Telescope [129]. In our analysis, we will use two data combinations, i.e. CMB+BAO+SN and CMB+BAO+SN+H0, to constrain the cosmological parameters.
3. Results and discussion
In this section, we report the constraint results of cosmological parameters for these IΛCDM+$\sum {m}_{\nu }$ models. The fitting results are listed in table 1 for the ΛCDM+$\sum {m}_{\nu }$ model and tables 2-5 and figures 1-4 for the four IΛCDM+$\sum {m}_{\nu }$ models. For convenience, the IΛCDM models with the interaction terms $Q=\beta H{\rho }_{\mathrm{de}}$, $Q=\beta H{\rho }_{{\rm{c}}}$, $Q=\beta {H}_{0}{\rho }_{\mathrm{de}}$ and $Q=\beta {H}_{0}{\rho }_{{\rm{c}}}$ are denoted as ‘IΛCDM1', ‘IΛCDM2', ‘IΛCDM3', and ‘IΛCDM4', respectively. In these tables, we show the best fit values with ±1σ errors of the cosmological parameters, but for the total neutrino mass $\sum {m}_{\nu }$, which cannot be well constrained, the 2σ upper limits are given.
Figure 1. Observational constraints (68.3% and 95.4% confidence level) on the IΛCDM1+$\sum {m}_{\nu }$ ($Q=\beta H{\rho }_{\mathrm{de}}$) model by using the CMB+BAO+SN (left) and CMB+BAO+SN+H0 (right) data combinations, respectively. |
Figure 2. Observational constraints (68.3% and 95.4% confidence level) on the IΛCDM2+$\sum {m}_{\nu }$ ($Q=\beta H{\rho }_{{\rm{c}}}$) model by using the CMB+BAO+SN (left) and CMB+BAO+SN+H0 (right) data combinations, respectively. |
Figure 3. Observational constraints (68.3% and 95.4% confidence level) on the IΛCDM3+$\sum {m}_{\nu }$ ($Q=\beta {H}_{0}{\rho }_{\mathrm{de}}$) model by using the CMB+BAO+SN (left) and CMB+BAO+SN+H0 (right) data combinations, respectively. |
Figure 4. Observational constraints (68.3% and 95.4% confidence level) on the IΛCDM4+$\sum {m}_{\nu }$ ($Q=\beta {H}_{0}{\rho }_{{\rm{c}}}$) model by using the CMB+BAO+SN (left) and CMB+BAO+SN+H0 (right) data combinations, respectively. |
Table 1. Fitting results for the ΛCDM+$\sum {m}_{\nu }$ model by using the CMB+BAO+SN and CMB+BAO+SN+H0 data combinations, respectively. Here, H0 and $\sum {m}_{\nu }$ are in units of km s−1 Mpc−1 and eV, respectively. |
| Data | CMB+BAO+SN | CMB+BAO+SN+H0 | ||||
|---|---|---|---|---|---|---|
| Model | ΛCDM+$\sum {m}_{\nu }^{\mathrm{NH}}$ | ΛCDM+$\sum {m}_{\nu }^{\mathrm{IH}}$ | ΛCDM+$\sum {m}_{\nu }^{\mathrm{DH}}$ | ΛCDM+$\sum {m}_{\nu }^{\mathrm{NH}}$ | ΛCDM+$\sum {m}_{\nu }^{\mathrm{IH}}$ | ΛCDM+$\sum {m}_{\nu }^{\mathrm{DH}}$ |
| Ωm | 0.3126 ± 0.0063 | 0.3150 ± 0.0060 | 0.3097 ± 0.0063 | 0.3044 ± 0.0056 | 0.3069 ± 0.0056 | 0.3015 ± 0.0056 |
| ${H}_{0}$ | 67.48 ± 0.47 | 67.26 ± 0.45 | 67.75 ± 0.49 | 68.11 ± 0.43 | 67.88 ± 0.43 | 68.40 ± 0.44 |
| σ8 | ${0.801}_{-0.008}^{+0.011}$ | ${0.793}_{-0.008}^{+0.010}$ | ${0.812}_{-0.008}^{+0.013}$ | ${0.801}_{-0.008}^{+0.009}$ | ${0.792}_{-0.008}^{+0.009}$ | ${0.813}_{-0.008}^{+0.010}$ |
| $\sum {m}_{\nu }$ | $\lt 0.156$ | $\lt 0.185$ | <0.123 | <0.125 | <0.160 | <0.082 |
Table 2. Fitting results for the IΛCDM1+$\sum {m}_{\nu }$ ($Q=\beta H{\rho }_{\mathrm{de}}$) model by using the CMB+BAO+SN and CMB+BAO+SN+H0 data combinations, respectively. Here, H0 and $\sum {m}_{\nu }$ are in units of km s−1 Mpc−1 and eV, respectively. |
| Data | CMB+BAO+SN | CMB+BAO+SN+H0 | ||||
|---|---|---|---|---|---|---|
| Model | IΛCDM+$\sum {m}_{\nu }^{\mathrm{NH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{IH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{DH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{NH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{IH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{DH}}$ |
| Ωm | 0.285 ± 0.029 | 0.279 ± 0.030 | ${0.292}_{-0.030}^{+0.029}$ | 0.235 ± 0.027 | 0.229 ± 0.027 | ${0.243}_{-0.027}^{+0.028}$ |
| ${H}_{0}$ | 68.08 ± 0.81 | ${68.08}_{-0.82}^{+0.83}$ | 68.14 ± 0.83 | ${69.64}_{-0.72}^{+0.73}$ | ${69.62}_{-0.73}^{+0.75}$ | ${69.67}_{-0.75}^{+0.74}$ |
| β | ${0.10}_{-0.11}^{+0.10}$ | 0.13 ± 0.11 | ${0.07}_{-0.11}^{+0.10}$ | ${0.257}_{-0.097}^{+0.096}$ | 0.286 ± 0.098 | 0.215 ± 0.099 |
| ${\sigma }_{8}$ | ${0.870}_{-0.088}^{+0.058}$ | ${0.884}_{-0.094}^{+0.061}$ | ${0.859}_{-0.085}^{+0.056}$ | ${1.013}_{-0.123}^{+0.076}$ | ${1.031}_{-0.131}^{+0.081}$ | ${0.987}_{-0.117}^{+0.073}$ |
| $\sum {m}_{\nu }$ | <0.189 | <0.218 | <0.151 | <0.177 | <0.209 | <0.138 |
3.1. Neutrino mass
Firstly, we use the CMB+BAO+SN data combination to constrain these models. In the ΛCDM+$\sum {m}_{\nu }$ model, we obtain $\sum {m}_{\nu }\lt 0.156\,\mathrm{eV}$ for the NH case, $\sum {m}_{\nu }\lt 0.185\,\mathrm{eV}$ for the IH case, and $\sum {m}_{\nu }\lt 0.123\,\mathrm{eV}$ for the DH case, as shown table 1. In the IΛCDM1+$\sum {m}_{\nu }$ model, the constraint results are $\sum {m}_{\nu }\lt 0.187\,\mathrm{eV}$ for the NH case, $\sum {m}_{\nu }\,\lt 0.218\,\mathrm{eV}$ for the IH case, and $\sum {m}_{\nu }\lt 0.151\,\mathrm{eV}$ for the DH case (see table 2); in the IΛCDM2+$\sum {m}_{\nu }$ model, the results are $\sum {m}_{\nu }\lt 0.190\,\mathrm{eV}$ for the NH case, $\sum {m}_{\nu }\lt 0.223\,\mathrm{eV}$ for the IH case, and $\sum {m}_{\nu }\lt 0.149\,\mathrm{eV}$ for the DH case (see table 3); in the IΛCDM3+$\sum {m}_{\nu }$ model, we get $\sum {m}_{\nu }\lt 0.179\,\mathrm{eV}$ for the NH case, $\sum {m}_{\nu }\lt 0.208\,\mathrm{eV}$ for the IH case, and $\sum {m}_{\nu }\lt 0.140\,\mathrm{eV}$ for the DH case (see table 4); in the IΛCDM4+$\sum {m}_{\nu }$ model, the constraint results become $\sum {m}_{\nu }\lt 0.202\,\mathrm{eV}$ for the NH case, $\sum {m}_{\nu }\lt 0.235\,\mathrm{eV}$ for the IH case, and $\sum {m}_{\nu }\lt 0.156\,\mathrm{eV}$ for the DH case (see table 5). We find that, the constraint results of $\sum {m}_{\nu }$ are looser in the four IΛCDM+$\sum {m}_{\nu }$ models than those in the ΛCDM+$\sum {m}_{\nu }$ model. When considering the three mass hierarchies, we find that the constraint results of $\sum {m}_{\nu }$ are tightest in the DH case and loosest in the IH case (see the left panels in figures 1-4); actually this is mainly because in the NH and IH cases there are lower limits for the total neutrino mass.
Table 3. Fitting results for the IΛCDM2+$\sum {m}_{\nu }$ (Q = β H ρc) model by using the CMB+BAO+SN and CMB+BAO+SN+H0 data combinations, respectively. Here, H0 and $\sum {m}_{\nu }$ are in units of km s−1 Mpc−1 and eV, respectively. |
| Data | CMB+BAO+SN | CMB+BAO+SN+H0 | ||||
|---|---|---|---|---|---|---|
| Model | IΛCDM+$\sum {m}_{\nu }^{\mathrm{NH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{IH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{DH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{NH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{IH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{DH}}$ |
| ${{\rm{\Omega }}}_{{\rm{m}}}$ | 0.3085 ± 0.0080 | 0.3092 ± 0.0081 | 0.3077 ± 0.0081 | 0.2953 ± 0.0071 | ${0.2960}_{-0.0072}^{+0.0071}$ | ${0.2946}_{-0.0072}^{+0.0071}$ |
| ${H}_{0}$ | 67.83 ± 0.64 | 67.74 ± 0.65 | 67.92 ± 0.65 | 68.92 ± 0.60 | ${68.83}_{-0.60}^{+0.61}$ | ${69.01}_{-0.60}^{+0.61}$ |
| β | ${0.0011}_{-0.0012}^{+0.0013}$ | ${0.0014}_{-0.0013}^{+0.0012}$ | 0.0005 ± 0.0013 | 0.0024 ± 0.0012 | 0.0028 ± 0.0012 | 0.0019 ± 0.0012 |
| ${\sigma }_{8}$ | ${0.806}_{-0.012}^{+0.014}$ | ${0.800}_{-0.012}^{+0.014}$ | ${0.814}_{-0.013}^{+0.014}$ | ${0.815}_{-0.012}^{+0.013}$ | ${0.809}_{-0.012}^{+0.013}$ | ${0.824}_{-0.012}^{+0.014}$ |
| $\sum {m}_{\nu }$ | $\lt 0.190$ | <0.223 | <0.149 | <0.170 | <0.202 | <0.126 |
Table 4. Fitting results for the IΛCDM3+$\sum {m}_{\nu }$ ($Q=\beta {H}_{0}{\rho }_{\mathrm{de}}$) model by using the CMB+BAO+SN and CMB+BAO+SN+H0 data combinations, respectively. Here, H0 and $\sum {m}_{\nu }$ are in units of km s−1 Mpc−1 and eV, respectively. |
| Data | CMB+BAO+SN | CMB+BAO+SN+H0 | ||||
|---|---|---|---|---|---|---|
| Model | IΛCDM+$\sum {m}_{\nu }^{\mathrm{NH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{IH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{DH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{NH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{IH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{DH}}$ |
| Ωm | ${0.282}_{-0.035}^{+0.036}$ | 0.276 ± 0.035 | 0.291 ± 0.036 | ${0.223}_{-0.031}^{+0.032}$ | 0.217 ± 0.033 | 0.233 ± 0.032 |
| ${H}_{0}$ | 68.03 ± 0.81 | ${67.96}_{-0.80}^{+0.79}$ | 68.10 ± 0.83 | 69.57 ± 0.72 | 69.50 ± 0.74 | ${69.63}_{-0.72}^{+0.73}$ |
| β | 0.14 ± 0.16 | 0.18 ± 0.16 | ${0.08}_{-0.17}^{+0.16}$ | ${0.37}_{-0.14}^{+0.13}$ | 0.40 ± 0.14 | 0.31 ± 0.14 |
| ${\sigma }_{8}$ | ${0.886}_{-0.118}^{+0.069}$ | ${0.899}_{-0.122}^{+0.072}$ | ${0.868}_{-0.113}^{+0.068}$ | ${1.072}_{-0.172}^{+0.095}$ | ${1.10}_{-0.19}^{+0.10}$ | ${1.036}_{-0.156}^{+0.091}$ |
| $\sum {m}_{\nu }$ | $\lt 0.179$ | <0.208 | <0.140 | <0.166 | <0.198 | <0.128 |
Table 5. Fitting results for the IΛCDM4+$\sum {m}_{\nu }$ ($Q=\beta {H}_{0}{\rho }_{{\rm{c}}}$) model by using the CMB+BAO+SN and CMB+BAO+SN+H0 data combinations, respectively. Here, H0 and $\sum {m}_{\nu }$ are in units of km s−1 Mpc−1 and eV, respectively. |
| Data | CMB+BAO+SN | CMB+BAO+SN+H0 | ||||
|---|---|---|---|---|---|---|
| Model | IΛCDM+$\sum {m}_{\nu }^{\mathrm{NH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{IH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{DH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{NH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{IH}}$ | IΛCDM+$\sum {m}_{\nu }^{\mathrm{DH}}$ |
| ${{\rm{\Omega }}}_{{\rm{m}}}$ | 0.299 ± 0.016 | 0.297 ± 0.016 | ${0.302}_{-0.017}^{+0.016}$ | 0.272 ± 0.013 | 0.269 ± 0.013 | 0.275 ± 0.013 |
| ${H}_{0}$ | 68.05 ± 0.80 | ${68.07}_{-0.82}^{+0.83}$ | 68.07 ± 0.81 | 69.58 ± 0.72 | 69.58 ± 0.73 | ${69.58}_{-0.72}^{+0.71}$ |
| β | 0.043 ± 0.047 | ${0.058}_{-0.048}^{+0.047}$ | ${0.024}_{-0.048}^{+0.047}$ | 0.111 ± 0.043 | 0.128 ± 0.043 | 0.092 ± 0.044 |
| ${\sigma }_{8}$ | ${0.814}_{-0.021}^{+0.019}$ | 0.812 ± 0.019 | ${0.820}_{-0.020}^{+0.019}$ | 0.840 ± 0.019 | 0.837 ± 0.019 | 0.845 ± 0.019 |
| $\sum {m}_{\nu }$ | $\lt 0.202$ | $\lt 0.235$ | <0.156 | <0.202 | <0.239 | <0.162 |
Then, we consider the data combination involving the latest local measurement of the Hubble constant H0 to constrain these models. By using the CMB+BAO+SN+H0 data combination, we find that the constraint results of $\sum {m}_{\nu }$ are looser in the four IΛCDM+$\sum {m}_{\nu }$ models than those in the ΛCDM+$\sum {m}_{\nu }$ model, and when considering the three mass hierarchies, the constraint results of $\sum {m}_{\nu }$ are tightest in the DH case and loosest in the IH case. These conclusions are consistent with the case using the CMB+BAO+SN data combination. Additionally, we also find that the constraints on $\sum {m}_{\nu }$ become slightly tighter for using CMB+BAO+SN+H0 than CMB+BAO+SN.
3.2. Coupling parameter
In this subsection, we discuss the fitting results of the coupling parameter β in these four IΛCDM+$\sum {m}_{\nu }$ models by using the CMB+BAO+SN and CMB+BAO+SN+H0 data combinations, respectively.
First, we constrain the IΛCDM1+$\sum {m}_{\nu }$ model (see table 2) using the CMB+BAO+SN data combination, and we obtain $\beta ={0.10}_{-0.11}^{+0.10}$ for the NH case, β = 0.13 ± 0.11 for the IH case, and $\beta ={0.07}_{-0.11}^{+0.10}$ for the DH case. It is shown that a positive value of β is favored and β > 0 is at the 0.91σ, 1.18σ, and 0.64σ levels for the three mass hierarchy cases, respectively. Furthermore, we constrain this model by using the CMB+BAO+SN+H0 data combination, and we obtain $\beta ={0.257}_{-0.097}^{+0.096}$ for the NH case, β = 0.286 ± 0.098 for the IH case, and β = 0.215 ± 0.099 for the DH case. Now, β > 0 is obtained at the 2.65σ, 2.92σ, and 2.17σ levels, respectively. This indicates that cold dark matter decaying into DE is favored when using the CMB+BAO+SN+H0 data combination.
For the IΛCDM2+$\sum {m}_{\nu }$ model (see table 3), we obtain $\beta ={0.0011}_{-0.0012}^{+0.0013}$ for the NH case, $\beta ={0.0014}_{-0.0013}^{+0.0012}$ for the IH case, and $\beta =0.0005\pm 0.0013$ for the DH case, by using the CMB+BAO+SN data combination. Thus, β > 0 is favored at the 0.92σ, 1.08σ, and 0.38σ levels, respectively. When using the CMB+BAO+SN+H0 data combination, we obtain β = 0.0024 ± 0.0012 for the NH case, β = 0.0028 ± 0.0012 for the IH case, and β = 0.0019 ± 0.0012 for the DH case, which indicates that a positive value of β can be detected at the 2.00σ, 2.33σ, and 1.58σ levels, respectively.
As for the IΛCDM3+$\sum {m}_{\nu }$ model (see table 4), we obtain β = 0.14 ± 0.16 for the NH case, β = 0.18 ± 0.16 for the IH case, and $\beta ={0.08}_{-0.17}^{+0.16}$ for the DH case, by using the CMB+BAO+SN data combination. Therefore, the positive values of β are favored and β > 0 is preferred at the 0.88σ, 1.13σ, and 0.50σ levels, respectively. When using the CMB+BAO+SN+H0 data combination, we obtain $\beta ={0.37}_{-0.14}^{+0.13}$ for the NH case, β = 0.40 ± 0.14 for the IH case, and β = 0.31 ± 0.14 for the DH case, respectively. And β > 0 is detected at the 2.64σ, 2.90σ, and 2.21σ levels, respectively, which indicates cold dark matter decaying into DE.
Finally, we show the constraint results of IΛCDM4+$\sum {m}_{\nu }$ model (see table 5). We obtain β = 0.043 ± 0.047 for the NH case, $\beta ={0.058}_{-0.048}^{+0.047}$ for the IH case, and $\beta ={0.024}_{-0.048}^{+0.047}$ for the DH case, by using the CMB+BAO+SN data combination. So, a positive value of β is favored and β > 0 is at the 0.91σ, 1.21σ, and 0.50σ levels, respectively. When using the CMB+BAO+SN+H0 data combination, we obtain β = 0.111 ± 0.043 for the NH case, β = 0.128 ± 0.043 for the IH case, and β = 0.092 ± 0.044 for the DH case. Now, β > 0 is preferred at the 2.58σ, 2.98σ, and 2.09σ levels, respectively. The conclusion is the same as the above three cases, i.e. cold dark matter decaying into DE is supported by the CMB+BAO+SN+H0 data combination.
In summary, for all the IΛCDM+$\sum {m}_{\nu }$ models considered in this paper, the values of β are greater by using the CMB+BAO+SN+H0 data combination than using CMB+BAO+SN data combination. We can also intuitively obtain this conclusion by comparing the left and right panels of figures 1-4. Additionally, when using CMB+BAO+SN+H0 data combination, β > 0 is favored at more than 1σ level in all the IΛCDM+$\sum {m}_{\nu }$ models, which indicates that cold dark matter decaying into DE is supported in these models.
3.3. The H0 tension
In this subsection, we discuss the issue of H0 tension between the Planck observation of the CMB power spectra and the local measurement based on the method of distance ladder. In the IΛCDM scenario, β > 0 (in the convention defined in this work) leads to the vacuum energy behaving as an effective phantom, and thus a larger cosmic expansion rate compared with ΛCDM can be obtained. We therefore can use this scenario to discuss the issue of relaxing the Hubble tension. The detailed fitting results are given in tables 1-5 and figures 5-7.
Figure 5. The one-dimensional posterior distributions for the parameter H0 for the ΛCDM+$\sum {m}_{\nu }$ (NH, IH, and DH) model and IΛCDM1+$\sum {m}_{\nu }$ (NH, IH, and DH) model with $Q=\beta H{\rho }_{\mathrm{de}}$ by using the CMB+BAO+SN data combination. The result of the local measurement of Hubble constant (${H}_{0}=74.03\pm 1.42\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$) is shown by the hotpink band. |
In table 1, we show the constraint results of the ΛCDM+$\sum {m}_{\nu }$ model by using the CMB+BAO+SN data combination. We obtain ${H}_{0}=67.48\pm 0.47\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the NH case, ${H}_{0}=67.26\pm 0.45\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the IH case, and ${H}_{0}=67.75\pm 0.49\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the DH case, which are 4.38σ 4.54σ and 4.18σ lower than the direct measurement of the Hubble constant (${H}_{0}=74.03\,\pm 1.42\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$). So, we investigate whether the H0 tension can be solved or relieved in the IDE scenario. In tables 2-5, we show the constraint results of the IΛCDM1+$\sum {m}_{\nu }$, IΛCDM2+$\sum {m}_{\nu }$, IΛCDM3+$\sum {m}_{\nu }$, and IΛCDM4+$\sum {m}_{\nu }$ models from the CMB+BAO+SN data combination. In the IΛCDM1+$\sum {m}_{\nu }$ model, we obtain ${H}_{0}\,=68.08\,\pm 0.81\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the NH case, ${H}_{0}\,={68.08}_{-0.82}^{+0.83}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the IH case, and ${H}_{0}=68.14\,\pm 0.83\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the DH case; in the IΛCDM2+$\sum {m}_{\nu }$ model, we obtain ${H}_{0}=67.83\pm 0.64\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the NH case, ${H}_{0}=67.74\pm 0.65\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the IH case, and ${H}_{0}=67.92\pm 0.65\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the DH case; in the IΛCDM3+$\sum {m}_{\nu }$ model, we obtain ${H}_{0}=68.03\,\pm 0.81\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the NH case, ${H}_{0}={67.96}_{-0.80}^{+0.79}\,\mathrm{km}\,{{\rm{s}}}^{-1}{\mathrm{Mpc}}^{-1}$ for the IH case, and ${H}_{0}=68.10\,\pm 0.83\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the DH case; in the IΛCDM4+$\sum {m}_{\nu }$ model, we obtain ${H}_{0}=68.05\pm 0.80\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the NH case, ${H}_{0}={68.07}_{-0.82}^{+0.83}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the IH case, and ${H}_{0}=68.07\,\pm 0.81\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the DH case. For these cases, the tensions with the Hubble constant direct measurement are at the 3.64σ level, 3.62σ level, 3.58σ level, 3.98σ level, 4.03σ level, 3.91σ level, 3.67σ level, 3.74σ level, 3.61σ level, 3.67σ level, 3.62σ level, and 3.65σ level, respectively.
Then, we show the constraint results of these models by using the CMB+BAO+SN+H0 data combination (see tables 1-5). In the ΛCDM+$\sum {m}_{\nu }$ model, we obtain ${H}_{0}=68.11\,\pm 0.43\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the NH case, ${H}_{0}=67.88\,\pm 0.43\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the IH case, and ${H}_{0}=68.40\,\pm 0.44\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the DH case, which indicates that the tensions with the Hubble constant direct measurement are at the 3.99σ level 4.14σ level and 3.79σ level, respectively. In the IΛCDM1+$\sum {m}_{\nu }$ model, we obtain ${H}_{0}={69.64}_{-0.72}^{+0.73}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the NH case, ${H}_{0}\,={69.62}_{-0.73}^{+0.75}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the IH case, and ${H}_{0}\,={69.67}_{-0.75}^{+0.74}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the DH case; in the IΛCDM2+$\sum {m}_{\nu }$ model, we obtain ${H}_{0}=68.92\,\pm 0.60\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the NH case, ${H}_{0}={68.83}_{-0.60}^{+0.61}\,\mathrm{km}\,{{\rm{s}}}^{-1}{\mathrm{Mpc}}^{-1}$ for the IH case, and ${H}_{0}={69.01}_{-0.60}^{+0.61}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the DH case; in the IΛCDM3+$\sum {m}_{\nu }$ model, we obtain ${H}_{0}=69.57\,\pm 0.72\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the NH case, ${H}_{0}\,=69.50\pm 0.74\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the IH case, and ${H}_{0}\,={69.63}_{-0.72}^{+0.73}\,\mathrm{km}{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the DH case; in the IΛCDM4+$\sum {m}_{\nu }$ model, we obtain ${H}_{0}=69.58\pm 0.72\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the NH case, ${H}_{0}=69.58\pm 0.73\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the IH case, and ${H}_{0}={69.58}_{-0.72}^{+0.71}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ for the DH case. The tensions with the Hubble constant direct measurement are at the 2.75σlevel, 2.75σ level, 2.72σ level, 3.31σ level, 3.36σ level, 3.25σ level, 2.80σ level, 2.83σ level, 2.76σ level, 2.80σ level, 2.79σ level, and 2.80σ level, respectively.
From the above constraint results, we find that compared with the ΛCDM+$\sum {m}_{\nu }$ model, the H0 tension can indeed be relieved in the IΛCDM+$\sum {m}_{\nu }$ model. From figures 5, 6 we can clearly see that for whichever neutrino mass hierarchy case, the fitting values of H0 in the IΛCDM+$\sum {m}_{\nu }$ models (here, we take IΛCDM1+$\sum {m}_{\nu }$ with $Q=\beta H{\rho }_{\mathrm{de}}$ as an example) are always much larger than those in the ΛCDM+$\sum {m}_{\nu }$ model. We also find that, the CMB+BAO+SN+H0 data combination (about $2.7-3.4\sigma $ level) is slightly more effective in relieving the H0 tension than the CMB+BAO+SN data combination (about $3.6-4.0\sigma $ level), due to the employment of the H0 prior in the data combination. To visually display the result, we also take the IΛCDM1+$\sum {m}_{\nu }$ model as an example to give this result in figure 7. From these figures, we can clearly see that for whichever hierarchy of the neutrino mass spectrum, the values of H0 are always much larger when adding the H0 data in a cosmological fit. Certainly, the H0 tension problem only can be alleviated to some extent in these cases, but cannot be truly solved. For the issue of H0 tension, further exploration is needed.
Figure 6. The one-dimensional posterior distributions for the parameter H0 for the ΛCDM+$\sum {m}_{\nu }$ (NH, IH, and DH) model and IΛCDM1+$\sum {m}_{\nu }$ (NH, IH, and DH) model with $Q=\beta H{\rho }_{\mathrm{de}}$ by using the CMB+BAO+SN+H0 data combination. The result of the local measurement of Hubble constant (${H}_{0}=74.03\pm 1.42\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$) is shown by the hotpink band. |
Figure 7. The one-dimensional posterior distributions for the parameter H0 for the IΛCDM1+$\sum {m}_{\nu }$ (NH, IH, and DH) model with $Q=\beta H{\rho }_{\mathrm{de}}$ by using the CMB+BAO+SN and CMB+BAO+SN+H0 data combinations, respectively. The result of the local measurement of Hubble constant (${H}_{0}=74.03\pm 1.42\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$) is shown by the hotpink band. |
4. Conclusion
In this work, we have investigated the constraints on the total neutrino mass in the scenario of vacuum energy interacting with cold dark matter by using the latest cosmological observations. We consider four typical models, i.e. IΛCDM1+$\sum {m}_{\nu }$ ($Q=\beta H{\rho }_{\mathrm{de}}$) model, IΛCDM2+$\sum {m}_{\nu }$ ($Q=\beta H{\rho }_{{\rm{c}}}$) model, IΛCDM3+$\sum {m}_{\nu }$ ($Q=\beta {H}_{0}{\rho }_{\mathrm{de}}$) model, and IΛCDM4+$\sum {m}_{\nu }$ ($Q=\beta {H}_{0}{\rho }_{{\rm{c}}}$) model. For the three-generation neutrinos, we consider the NH, IH, and DH cases. We employ the extended version of the PPF approach to calculate the perturbation of DE in the IDE cosmology. We use the Planck 2018 CMB data, the BAO measurements, the SN data of Pantheon compilation, and the local measurement of the Hubble constant H0 from the Hubble Space Telescope to constrain these models.
We find that, compared with the ΛCDM+$\sum {m}_{\nu }$ model, these four IΛCDM+$\sum {m}_{\nu }$ models can provide a much looser constraint on the total neutrino mass $\sum {m}_{\nu }$. When considering the three mass hierarchies, the upper limits on $\sum {m}_{\nu }$ are smallest in the DH case and largest in the IH case. We also find that, the constraints on $\sum {m}_{\nu }$ are slightly tighter by using CMB+BAO+SN+H0 than using CMB+BAO+SN. In addition, in all the IΛCDM+$\sum {m}_{\nu }$ models considered in this paper, the fit values of β are greater using the CMB+BAO+SN+H0 data combination than using the CMB+BAO+SN data combination, and β > 0 is favored at more than 1σ level in all the IΛCDM+$\sum {m}_{\nu }$ models when using the CMB+BAO+SN+H0 data combination, implying the preference of cold dark matter decaying into DE. In addition, we also find that, compared with the ΛCDM+$\sum {m}_{\nu }$ model, the H0 tension can be alleviated to some extent in the IΛCDM+$\sum {m}_{\nu }$ models.