Cosmological search for sterile neutrinos after Planck 2018

Sterile neutrinos can affect the evolution of the universe, and thus using the cosmological observations can search for sterile neutrinos. In this work, we use the cosmic microwave background (CMB) anisotropy data from the Planck 2018 release, combined with the latest baryon acoustic oscillation (BAO), type Ia supernova (SN), and Hubble constant ($H_0$) data, to constrain the cosmological models with considering sterile neutrinos. In order to test the influences of the properties of dark energy on the {results} of searching for sterile neutrinos, in addition to the $\Lambda$ cold dark matter ($\Lambda$CDM) model, we also consider the $w$CDM model and the holographic dark energy (HDE) model. We find that the existence of sterile neutrinos {is not preferred} when the $H_0$ local measurement is not included in the data combination. When the $H_0$ measurement is included in the joint constraints, it is found that $\Delta N_{\rm eff}>0$ is {favored} at about 2.7$\sigma$ level for the $\Lambda$CDM model and at about 1-1.7$\sigma$ level for the $w$CDM model. However, $m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ still cannot be well constrained and only upper limits can be given. In addition, we find that the HDE model is definitely ruled out by the current data. We also discuss the issue of the Hubble tension, and we conclude that involving sterile neutrinos in the cosmological models cannot truly resolve the Hubble tension.


I. INTRODUCTION
At present, the possible existence of sterile neutrinos is one of the most debated topics in neutrino physics. The recent experiments and the historic anomalies [1][2][3][4][5][6][7][8][9][10][11][12][13][14] seem to point towards the existence of light massive sterile neutrinos with the mass around the eV scale, but some other experiments, such as the result of neutrino oscillation experiment by the Daya Bay and MINOS collaborations [15] and the result of cosmic ray experiment by the IceCube collaboration [16], did not detect such a signal, which casts doubt on this hypothesis. Since the sterile neutrinos have some effects on the evolution of the universe, cosmological observations can provide independent way to search for sterile neutrinos.
However, using the cosmological observations to search for sterile neutrinos depends on cosmological models. In order to fit the cosmological data, one needs to assume a specific cosmological model with some cosmological parameters, and these cosmological parameters could be simultaneously determined in the sense of statistics in the cosmological fit. Therefore, the cosmological searches of sterile neutrinos not only depend on cosmological observations, but also depend on cosmological models. In particular, some recent studies [17][18][19][20][21] have revealed that the properties of dark energy can significantly impact on the cosmological fit results of sterile neutrinos.
On the other hand, currently, one of the most impor- * Corresponding author † Electronic address: zhangxin@mail.neu.edu.cn tant puzzles in cosmology is the Hubble tension [22]. It was found that a significant tension exists between the observations of the early and late universe. One way of relieving the Hubble tension is to consider a dynamical dark energy in a cosmological model [23]. It has been found that the equation of state (EoS) of dark energy is in anti-correlation with the Hubble constant in the cosmological fits using the observation of cosmic microwave background (CMB) anisotropies. However, due to the observations becoming increasingly precise, the EoS of dark energy has been constrained tightly and thus cannot provide enough room for accommodating a rather high value of the Hubble constant [24]. Nevertheless, it is also well known that the existence of sterile neutrinos can also help relieve the Hubble tension, as the parameter N eff of sterile neutrinos is in positive correlation with the Hubble constant. In this circumstance, an obvious further way is to simultaneously consider a dynamical dark energy and sterile neutrinos in a cosmological model [18]. Although such a consideration indeed can effectively relieve the Hubble tension, a deep analysis shows that the current cosmological observations might not favor such models when using the Akaike (or Bayesian) information criterion to assess the fits [24]. Anyway, on one hand, the Hubble constant measurement is useful in searching for sterile neutrinos in cosmology, and on the other hand, the consideration of sterile neutrinos in a cosmological model is also helpful in solving the problem of the Hubble tension. See e.g. Refs.  for related studies. tion (BAO) data, the type Ia supernovae (SN) data, and the Hubble constant H 0 data, have also been released. Thus, it is necessary to make a new analysis for the issue of searching for sterile neutrinos in cosmology using the cosmological observations.
In this work, we will use the latest CMB, BAO, SN, and H 0 data to search for sterile neutrinos. Since the influence of dark energy is important in this issue, we will not only assume a Λ cold dark matter (ΛCDM) model, but also consider dynamical dark energy models in the cosmological fits. To be simple as far as possible, we only consider the simplest dynamical dark energy models in this work. Therefore, we only consider the wCDM model and the holographic dark energy (HDE) model in our analysis. These two models have only one extra parameter compared to ΛCDM. For the wCDM model, the EoS of dark energy w is a constant. For the HDE model, the energy density of dark energy is given by ρ de = 3c 2 M 2 pl R −2 eh , where c is a dimensionless parameter which plays an important role in determining the evolution of dark energy in the HDE model and M pl is the reduced Planck mass. R eh is the future event horizon, defined as Ha 2 , where a(t) is the scale factor of our universe and H =ȧ/a is the Hubble parameter, with the dot denoting the derivative with respect to the cosmic time t. In the HDE model [76], the evolution of EoS is given by w(a) = −1/3−(2/3c) Ω de (a). In this case, the only extra parameter relative to ΛCDM is the parameter c. For more details of the HDE model, see e.g., Refs. [23,[77][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92]. Although the two dynamical dark energy models are simple, they are rather representative. In the wCDM model, the dark energy is either quintessence type (w > −1) or phantom type (w < −1). While in the HDE model, the EoS of dark energy is dynamically evolutionary, and the dark energy can be quintessence type (c > 1) with w always larger than −1 or quintom type (c < 1) with w evolving from w > −1 to w < −1.
In this work, we constrain the ΛCDM, wCDM, and HDE models where sterile neutrinos are considered using the latest cosmological observations, and we discuss the issues of cosmological searches of sterile neutrinos, impacts of properties of dark energy, model comparison, and Hubble tension, based on the cosmological fit results.

A. Data
In this paper, for the observational data, we consider the following data sets.
The CMB data: We use the CMB likelihood including the 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 [93].
The SN data: We use the latest Pantheon sample, which is comprised of 1048 data points [97].
The H 0 measurement: We use the local measurement result of H 0 = 74.03±1.42 km s −1 Mpc −1 from the cepheid-supernova distance ladder, reported in Ref. [98].
In what follows, we will use these observational data to place constraints on the ΛCDM, wCDM, and HDE models with and without sterile neutrinos. We will use two data combinations, i.e., CMB+BAO+SN (abbreviated as CBS) and CMB+BAO+SN+H 0 (abbreviated as CBSH), to constrain the cosmological parameters. These usages enable us to conveniently compare with the cosmological fit results of the neutrinos mass obtained in previous works, e.g., Refs. [70,71].

B. Method
In a cosmological model without considering sterile neutrinos, the free parameter vector is P = where Ω b h 2 and Ω c h 2 represent the physical baryon density and the physical cold dark matter density, respectively, θ MC is the ratio (multiplied by 100) between the sound horizon r s and angular diameter distance D A at decoupling, τ is the optical depth to the reionization, A s is the amplitude of the power spectrum of primordial curvature perturbations, n s is the power-law spectral index, w is the EoS parameter of dark energy for the wCDM model, and c is the dimensionless phenomenological parameter for determining the evolution of dark energy in the HDE model. Thus, there are six independent parameters in total for the ΛCDM model and seven independent parameters in total for the wCDM model and the HDE model. The total mass of active neutrinos m ν is fixed at 0.06eV.
In this work, we consider the both cases of massless and massive sterile neutrinos. When the case of massless neutrinos (as the dark radiation) is considered in the cosmological models, one extra free parameter, the effective number of relativistic species N eff should be involved in the calculation. When massless sterile neutrinos are considered in the ΛCDM model, the wCDM model, and the HDE model, these cases are called the ΛCDM+N eff model, the wCDM+N eff model, and the HDE+N eff model, respectively. Thus, the ΛCDM+N eff model has seven independent parameters, and the wCDM+N eff model and the HDE+N eff model have eight independent parameters.
When the sterile neutrinos are considered to be massive, two additional parameters, N eff and the effective sterile neutrino mass m eff ν,sterile , need to be added in the cosmological models.
Correspondingly, the models considered in this paper are called the ΛCDM+N eff +m eff ν,sterile model, the wCDM+N eff +m eff ν,sterile model, and the HDE+N eff +m eff ν,sterile model, respectively. The ΛCDM+N eff +m eff ν,sterile model has eight independent parameters, and the wCDM+N eff +m eff ν,sterile and HDE+N eff +m eff ν,sterile models have nine independent parameters.
Note here that in the cases of considering sterile neutrinos the prior of N eff > 3.044 should be set in the calculations.
We use the CosmoMC package [99] to infer the posterior probability distributions of the sterile neutrino parameters and other cosmological parameters.
The total χ 2 of the two data combinations can be written as In general, the χ 2 comparison is simplest analysis method for comparing different models with the same parameter number. When the comparison is made for models with different numbers of free parameters, a model with more parameters tends to give a better fit to the same data (χ 2 min tends to be smaller), and thus the simple χ 2 comparison is obviously unfair. Therefore, in this work we use the Akaike information criterion (AIC) as an evaluation tool to compare different cosmological models. A model with a smaller value of AIC is believed to be more favored by data. The AIC is defined as AIC = χ 2 min +2k, where k is the number of parameters. Actually, we only care about the relative values of AIC between different models, and thus we use ∆AIC = ∆χ 2 min +2∆k to compare models. Here we take the ΛCDM model as a reference model for calculating the ∆AIC values for other models.

III. RESULTS AND DISCUSSION
In this section, we report the fitting results of the cosmological models and discuss the implications of these results in the searches for sterile neutrinos using the latest observational data. The fitting results are given in Tables I and II as well as Figs. 1

and 2.
A. The case of massless sterile neutrinos In the universe, the total energy density of radiation is given by where ρ γ is the photon energy density. The effective number of relativistic species in the standard three-generation neutrino cosmology is N eff = 3.044 [100][101][102].
The presence of sterile neutrinos in the universe leads to ∆N eff = N eff − 3.044 > 0. In the case of massless sterile neutrinos, the only parameter of sterile neutrinos is N eff and so we take the fit result of ∆N eff > 0 as the preference of the existence of massless sterile neutrinos.    In the Planck 2018 results [93], the case of taking N eff as a free parameter is considered, and for the ΛCDM+N eff model the result of N eff = 2.92 +0. 36 −0.37 (95% C.L., Planck TT,TE,EE+lowE) is obtained. Obviously, if we set the requirement of N eff > 3.044 for this case, then only a 95% C.L. upper limit for ∆N eff can be obtained. Therefore, the existence of massless sterile neutrinos is not preferred by using only the Planck 2018 temperature and polarization power spectra.
Here, in the aspect of observational data, we consider BAO, SN, and H 0 data in addition to the Planck 2018 CMB data, and in the aspect of cosmological models, we consider wCDM and HDE in addition to ΛCDM. We wish to see what results can be given.
In Table I, we show the fitting results given by using the CBS data. We can clearly see that in this case for all the models only an upper limit on N eff can be obtained, indicating that the existence of massless neutrinos is not favored by using the CBS data, no matter what dark energy model is considered.
Changes happen when the H 0 local measurement is added in the data combination. Table II shows the case of using the CBSH data. We find that in this case ∆N eff > 0 is favored at 2.73σ, 1.67σ, and 2.55σ level for ΛCDM+N eff , wCDM+N eff , and HDE+N eff , respectively.
Thus, we find that the existence of massless sterile neutrinos is preferred when the H 0 local measurement is included in the data combination. This is because N eff is in positive correlation with H 0 when using the CMB data. As a result, a higher H 0 prior leads to a result of higher N eff . The correlation between N eff and H 0 can be clearly seen in Fig. 1. In addition, the results of using AIC as an assess tool to compare dark energy are also shown in the tables. We can clearly see that the HDE model is definitely excluded by the current observations, since its ∆AIC values are too high (around 23 in the case of CBS and around 15 in the case of CBSH). This confirms the previous results in Refs. [24,92]. We find that the wCDM model can well fit the current data. In particular, we notice that, in the case of CBSH, ∆AIC = −4.9 for wCDM+N eff . We also find that w < −1 is favored at around 1σ level (w = −1 is outside the 1σ limit), indicating that the ΛCDM model is not preferred by the CBS and CBSH data.

B. The case of massive sterile neutrinos
In the case of massive sterile neutrinos, two extra parameters, N eff and m eff ν,sterile , need to be considered in the cosmological models. Here, the requirement of N eff > 3.044 still holds.
From Table I, we can see that when using the CBS data neither N eff nor m eff ν,sterile can be determined, no matter what dark energy model is considered. Only upper limits on N eff and m eff ν,sterile can be obtained. From Table II, we can see that, when the H 0 measurement is included in the data combination, for all the dark energy models, N eff can be effectively constrained. But even in this case, m eff ν,sterile still cannot be well constrained, and only upper limits can be given.
The preference of ∆N eff > 0 is at 2.74σ, 1.09σ, and 2.52σ level for the ΛCDM+N eff +m eff ν,sterile model, the wCDM+N eff +m eff ν,sterile model, and the HDE+N eff +m eff ν,sterile model, respectively, in the case of using the CBSH data.
For the constraints on m eff ν,sterile , we find that the impact of dark energy is somewhat evident. Compared with ΛCDM, the constraint in wCDM is evidently looser, and that in HDE is slightly tighter. This also confirms the previous results in e.g. Refs. [19,34,71]. We also show the main results in Fig. 2.
As the same in the above subsection, we also find that the HDE model is excluded by the current data because its AIC values are very high in the cosmological fits. The wCDM model is favored by the data, and w = −1 is also outside the 1σ limit, indicating that the ΛCDM model is not preferred by the CBS and CBSH data in the case of massive sterile neutrinos.

C. The Hubble tension
We wish to check if the Hubble tension can be effectively relieved when sterile neutrinos are considered.
From Tables I and II, we can see that in the ΛCDM model (without considering sterile neutrinos) the H 0 tension is at 4.3σ level for the case of using the CBS data, and at 3.9σ level for the case of using the CBSH data. Thus, considering the H 0 measurement as a prior can slightly relieve the Hubble tension, but the tension still exists for ΛCDM at around 4σ level.
In the case of using the CBSH data, we find that the HDE model with considering sterile neutrinos is the best one in relieving the Hubble tension (the tension is relieved to about 1.7σ). However, since the HDE model is excluded by the observational data (as assessed by the AIC tool), we do not consider this model in this issue. Actually, for the ΛCDM and wCDM models with sterile neutrinos, we find that they can also relieve the Hubble tension to less than 2σ. We also find that such models are favored by the current data. However, although the H 0 tension can be relieved to some extent, the σ 8 tension is slightly worsened (from 0.81 to about 0.83-0.84 in the case of massless sterile neutrinos; massive sterile neutrinos can slightly relieve this to about 0.82).

IV. CONCLUSION
We use the Planck 2018 CMB anisotropy data combined with the latest BAO, SN, and H 0 data to search for sterile neutrinos in cosmology. In addition to the ΛCDM model, we also consider the wCDM model and the HDE model to show how properties of dark energy affect the fit results of sterile neutrinos.
We use the data combinations of CBS and CBSH to constrain the cosmological models. Based on the constraints, we can get the following conclusions.
(i) In the case of using the CBS data, no preference of existence of sterile neutrinos can be given, because only the upper limits on N eff can be obtained. In this case, the wCDM model is most favored by the data; in the wCDM model, w = −1 is outside the 1σ limit, indicating that the ΛCDM model is not preferred by the data; the HDE model is definitely excluded by the current data because its AIC values are very high.
(ii) In the case of using the CBSH data, ∆N eff > 0 is favored at about 2.7σ level for the ΛCDM model and at about 1-1.7σ level for the wCDM model. But even in this case, m eff ν,sterile still cannot be well constrained and only upper limits can be given.
(iii) When the H 0 local measurement is included in the data combination, the cosmological models of considering sterile neutrinos can effectively relieve the Hubble tension, with the tension relieved to less than 2σ. However, this does not mean that the Hubble tension can truly be resolved by this consideration, because when the local measurement of H 0 is removed in the data combination, the H 0 tension is then restored to about 3-4σ.