# Cosmological search for sterile neutrinos after Planck 2018

Lu Feng  College of Physical Science and Technology, Shenyang Normal
University, Shenyang 110034, China  Department of Physics, College of
Sciences, Northeastern University, Shenyang 110819, China  Rui-Yun Guo  School
of Sciences, Xi’an Technological University, Xi’an 710021, China  Jing-Fei
Zhang  Department of Physics, College of Sciences, Northeastern University,
Shenyang 110819, China  Xin Zhang111Corresponding author
<EMAIL_ADDRESS>Department of Physics, College of Sciences,
Northeastern University, Shenyang 110819, China  Frontiers Science Center for
Industrial Intelligence and Systems Optimization, Northeastern University,
Shenyang 110819, China  Key Laboratory of Data Analytics and Optimization for
Smart Industry (Northeastern University), Ministry of Education, China

###### Abstract

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 important 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 $N_{\rm eff}$ 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. [25, 26, 27, 28, 29,
30, 31, 32, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 34, 48, 49,
35, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67,
68, 70, 71, 69, 72, 73, 74, 75] for related studies.

Recently, the latest CMB data, i.e., the Planck 2018 angular power spectra
data, have been released by the Planck collaboration, and some other important
updated observational data, including the baryon acoustic oscillation (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 $\Lambda$ cold dark matter ($\Lambda$CDM)
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 $w$CDM model
and the holographic dark energy (HDE) model in our analysis. These two models
have only one extra parameter compared to $\Lambda$CDM. For the $w$CDM model,
the EoS of dark energy $w$ is a constant. For the HDE model, the energy
density of dark energy is given by
$\rho_{\rm{de}}=3c^{2}M^{2}_{\rm{pl}}R^{-2}_{\rm{eh}}$, where $c$ is a
dimensionless parameter which plays an important role in determining the
evolution of dark energy in the HDE model and $M_{\rm{pl}}$ is the reduced
Planck mass. $R_{\rm{eh}}$ is the future event horizon, defined as
$R_{\rm{eh}}=a(t)\int_{t}^{\infty}\frac{dt^{\prime}}{a(t^{\prime})}=a\int_{a}^{\infty}\frac{da^{\prime}}{Ha^{\prime
2}}$, where $a(t)$ is the scale factor of our universe and $H=\dot{a}/a$ 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
$w(a)=-1/3-(2/3c)\sqrt{\Omega_{\rm de}(a)}$. In this case, the only extra
parameter relative to $\Lambda$CDM is the parameter $c$. For more details of
the HDE model, see e.g., Refs. [77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87,
88, 89, 23, 90, 91, 92]. Although the two dynamical dark energy models are
simple, they are rather representative. In the $w$CDM 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 $\Lambda$CDM, $w$CDM, 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.

##  II Data and method

###  II.1 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
$l\geq 30$, the low-$l$ temperature commander likelihood, and the low-$l$
SimAll EE likelihood from the Planck 2018 data release [93].

The BAO data: We use the measurements from the six-degree-field galaxy survey
sample ($z_{\rm eff}=0.106$) [94], the Sloan Digital Sky Survey Main Galaxy
Sample ($z_{\rm eff}=0.15$) [95], and the Baryon Oscillation Spectroscopic
Survey Data Release 12 ($z_{\rm eff}=$ 0.38, 0.51, and 0.61) [96].

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
$H_{0}=74.03{\pm 1.42}~{}{\rm km}~{}{\rm s}^{-1}~{}{\rm Mpc}^{-1}$ from the
cepheid-supernova distance ladder, reported in Ref. [98].

In what follows, we will use these observational data to place constraints on
the $\Lambda$CDM, $w$CDM, 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. [71, 70].

###  II.2 Method

In a cosmological model without considering sterile neutrinos, the free
parameter vector is ${\bf
P}=\\{\Omega_{b}h^{2},~{}\Omega_{c}h^{2},~{}100\theta_{\rm
MC},~{}\tau,~{}\ln(10^{10}A_{s}),~{}n_{s},~{}w~{}({\rm or}~{}c)\\},$ where
$\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$ represent the physical baryon density
and the physical cold dark matter density, respectively, $\theta_{\rm MC}$ is
the ratio (multiplied by 100) between the sound horizon $r_{s}$ and angular
diameter distance $D_{\rm A}$ at decoupling, $\tau$ 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 $w$CDM 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 $\Lambda$CDM model and seven independent parameters in total for the
$w$CDM model and the HDE model. The total mass of active neutrinos $\sum
m_{\nu}$ 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 $N_{\rm eff}$ should be involved in the
calculation. When massless sterile neutrinos are considered in the
$\Lambda$CDM model, the $w$CDM model, and the HDE model, these cases are
called the $\Lambda$CDM+$N_{\rm eff}$ model, the $w$CDM+$N_{\rm eff}$ model,
and the HDE+$N_{\rm eff}$ model, respectively. Thus, the $\Lambda$CDM+$N_{\rm
eff}$ model has seven independent parameters, and the $w$CDM+$N_{\rm eff}$
model and the HDE+$N_{\rm eff}$ model have eight independent parameters.

When the sterile neutrinos are considered to be massive, two additional
parameters, $N_{\rm eff}$ and the effective sterile neutrino mass $m_{\nu,{\rm
sterile}}^{\rm eff}$, need to be added in the cosmological models.
Correspondingly, the models considered in this paper are called the
$\Lambda$CDM+$N_{\rm eff}$+$m_{\nu,{\rm sterile}}^{\rm eff}$ model, the
$w$CDM+$N_{\rm eff}$+$m_{\nu,{\rm sterile}}^{\rm eff}$ model, and the
HDE+$N_{\rm eff}$+$m_{\nu,{\rm sterile}}^{\rm eff}$ model, respectively. The
$\Lambda$CDM+$N_{\rm eff}$+$m_{\nu,{\rm sterile}}^{\rm eff}$ model has eight
independent parameters, and the $w$CDM+$N_{\rm eff}$+$m_{\nu,{\rm
sterile}}^{\rm eff}$ and HDE+$N_{\rm eff}$+$m_{\nu,{\rm sterile}}^{\rm eff}$
models have nine independent parameters.

Note here that in the cases of considering sterile neutrinos the prior of
$N_{\rm eff}>3.044$ 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 $\chi^{2}$ of the two data combinations can be written as
$\chi^{2}=\chi^{2}_{\rm CMB}+\chi^{2}_{\rm BAO}+\chi^{2}_{\rm SN}$ and
$\chi^{2}=\chi^{2}_{\rm CMB}+\chi^{2}_{\rm BAO}+\chi^{2}_{\rm
SN}$+$\chi^{2}_{H_{0}}$. In general, the $\chi^{2}$ 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 ($\chi^{2}_{\rm min}$ tends to be smaller), and thus the simple
$\chi^{2}$ 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 $\rm AIC=\chi^{2}_{\rm min}$+2$k$,
where $k$ is the number of parameters. Actually, we only care about the
relative values of AIC between different models, and thus we use $\Delta\rm
AIC=\Delta\chi^{2}_{\rm min}$+$2\Delta k$ to compare models. Here we take the
$\Lambda$CDM model as a reference model for calculating the $\Delta\rm AIC$
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 1 and 2 as well as Figs. 1 and 2.

Table 1: Fitting results for the cosmological models by using the CBS data. We quote $\pm 1\sigma$ errors, but for the parameters that cannot be well constrained, we quote the 95.4% CL upper limits. Here, $H_{0}$ is in units of ${\rm km}/{\rm s}/{\rm Mpc}$ and $m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ is in units of eV. Note also that the minimal value of $\chi^{2}$ for the $\Lambda$CDM model is $\chi^{2}_{\rm min}=3806.994$. |  | No sterile neutrinos |  | Massless sterile neutrinos |  | Massive sterile neutrinos |
---|---|---|---|---|---|---|---
Parameter |  |  $\Lambda$CDM |  $w$CDM | HDE |  |  $\Lambda$CDM |  $w$CDM | HDE |  |  $\Lambda$CDM |  $w$CDM | HDE |
$N_{\rm eff}$ |  | … | … | … |  | $<3.46$ | $<3.47$ | $<3.58$ |  | $<3.45$ | $<3.40$ | $<3.56$ |
$m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ |  | … | … | … |  | … | … | … |  | $<0.579$ | $<0.651$ | $<0.479$ |
$H_{0}$ |  | $67.69\pm 0.43$ | $68.31^{+0.83}_{-0.82}$ | $67.86^{+0.79}_{-0.80}$ |  | $68.67^{+0.64}_{-1.02}$ | $69.05^{+0.96}_{-1.13}$ | $68.90^{+1.00}_{-1.20}$ |  | $68.15^{+0.50}_{-1.00}$ | $68.78^{+0.91}_{-1.06}$ | $68.64^{+0.96}_{-1.30}$ |
$w$ |  | … | $-1.029\pm 0.033$ | … |  | … | $-1.020^{+0.034}_{-0.033}$ | … |  | … | $-1.037\pm 0.036$ | … |
$c$ |  | … | … | $0.636^{+0.027}_{-0.029}$ |  | … | … | $0.649^{+0.028}_{-0.032}$ |  | … | … | $0.639^{+0.029}_{-0.033}$ |
$\sigma_{8}$ |  | $0.810\pm 0.007$ | $0.819\pm 0.013$ | $0.794\pm 0.012$ |  | $0.818^{+0.008}_{-0.010}$ | $0.824\pm 0.014$ | $0.800\pm 0.013$ |  | $0.796^{+0.024}_{-0.014}$ | $0.801^{+0.026}_{-0.018}$ | $0.789^{+0.021}_{-0.016}$ |
$H_{0}$ tension |  | $4.27\sigma$ | $3.48\sigma$ | $3.80\sigma$ |  | $3.44\sigma$ | $2.91\sigma$ | $2.95\sigma$ |  | $3.91\sigma$ | $3.11\sigma$ | $3.14\sigma$ |
$\Delta\chi^{2}$ |  | 0 | $-0.534$ | 19.162 |  | $-0.240$ | $-0.585$ | 18.909 |  | $-0.272$ | $-0.617$ | 19.249 |
$\Delta\rm AIC$ |  | 0 | 1.466 | 21.162 |  | 1.760 | 3.415 | 22.909 |  | 3.728 | 5.383 | 25.249 |
Table 2: Fitting results for the cosmological models by using the CBSH data. We quote $\pm 1\sigma$ errors, but for the parameters that cannot be well constrained, we quote the 95.4% CL upper limits. Here, $H_{0}$ is in units of ${\rm km}/{\rm s}/{\rm Mpc}$ and $m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ is in units of eV. Note also that the minimal value of $\chi^{2}$ for the $\Lambda$CDM model is $\chi^{2}_{\rm min}=3824.875$. |  | No sterile neutrinos |  | Massless sterile neutrinos |  | Massive sterile neutrinos |
---|---|---|---|---|---|---|---
Parameter |  |  $\Lambda$CDM |  $w$CDM | HDE |  |  $\Lambda$CDM |  $w$CDM | HDE |  |  $\Lambda$CDM |  $w$CDM | HDE |
$N_{\rm eff}$ |  | … | … | … |  | $3.51\pm 0.17$ | $3.38^{+0.16}_{-0.20}$ | $3.53\pm 0.19$ |  | $3.54\pm 0.18$ | $3.35^{+0.12}_{-0.28}$ | $3.55\pm 0.20$ |
$m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ |  | … | … | … |  | … | … | … |  | $<0.121$ | $<0.413$ | $<0.089$ |
$H_{0}$ |  | $68.25\pm 0.42$ | $69.77^{+0.75}_{-0.74}$ | $69.37\pm 0.71$ |  | $70.64^{+0.99}_{-0.98}$ | $70.92\pm 0.93$ | $71.09^{+0.98}_{-0.99}$ |  | $70.59^{+1.01}_{-1.00}$ | $70.80^{+0.95}_{-1.07}$ | $71.10\pm 1.00$ |
$w$ |  | … | $-1.078\pm 0.031$ | … |  | … | $-1.047\pm 0.033$ | … |  | … | $-1.068\pm 0.038$ | … |
$c$ |  | … | … | $0.600\pm 0.024$ |  | … | … | $0.636^{+0.027}_{-0.032}$ |  | … | … | $0.631^{+0.029}_{-0.032}$ |
$\sigma_{8}$ |  | $0.807\pm 0.007$ | $0.833^{+0.012}_{-0.013}$ | $0.807\pm 0.012$ |  | $0.829^{+0.010}_{-0.011}$ | $0.839\pm 0.013$ | $0.816\pm 0.013$ |  | $0.821^{+0.015}_{-0.012}$ | $0.822^{+0.025}_{-0.016}$ | $0.812\pm 0.014$ |
$\Delta N_{\rm eff}>0$ |  | … | … | … |  | $2.73\sigma$ | $1.67\sigma$ | $2.55\sigma$ |  | $2.74\sigma$ | $1.09\sigma$ | $2.52\sigma$ |
$H_{0}$ tension |  | $3.90\sigma$ | $2.65\sigma$ | $2.94\sigma$ |  | $1.96\sigma$ | $1.83\sigma$ | $1.70\sigma$ |  | $1.97\sigma$ | $1.89\sigma$ | $1.69\sigma$ |
$\Delta\chi^{2}$ |  | 0 | $-6.659$ | 15.148 |  | $-6.471$ | $-8.904$ | 11.190 |  | $-6.497$ | $-9.089$ | 10.736 |
$\Delta\rm AIC$ |  | 0 | $-4.659$ | 17.148 |  | $-4.471$ | $-4.904$ | 15.190 |  | $-2.497$ | $-3.089$ | 16.736 |

Figure 1:  Two-dimensional marginalized posterior contours (68.3% and 95.4%
CL) in the $H_{0}$–$N_{\rm eff}$ plane for the $\Lambda$CDM+$N_{\rm eff}$
model, the $w$CDM+$N_{\rm eff}$ model, and the HDE+$N_{\rm eff}$ model by
using the CBS (a) and CBSH (b) data combinations.

Figure 2:  Two-dimensional marginalized posterior contours (68.3% and 95.4%
CL) in the $m_{\nu,{\rm{sterile}}}^{\rm{eff}}$–$N_{\rm eff}$ and
$H_{0}$–$N_{\rm eff}$ planes for the $\Lambda$CDM+$N_{\rm
eff}$+$m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ model, the $w$CDM+$N_{\rm
eff}$+$m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ model, and the HDE+$N_{\rm
eff}$+$m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ model by using the CBS (a) and CBSH
(b) data combinations.

###  III.1 The case of massless sterile neutrinos

In the universe, the total energy density of radiation is given by

$\rho_{r}=[1+N_{\rm
eff}\frac{7}{8}(\frac{4}{11})^{\frac{4}{3}}]\rho_{\gamma},$

where $\rho_{\gamma}$ is the photon energy density. The effective number of
relativistic species in the standard three-generation neutrino cosmology is
$N_{\rm eff}=3.044$ [100, 101, 102].

The presence of sterile neutrinos in the universe leads to $\Delta N_{\rm
eff}=N_{\rm eff}-3.044>0$. In the case of massless sterile neutrinos, the only
parameter of sterile neutrinos is $N_{\rm eff}$ and so we take the fit result
of $\Delta N_{\rm eff}>0$ as the preference of the existence of massless
sterile neutrinos.

In the Planck 2018 results [93], the case of taking $N_{\rm eff}$ as a free
parameter is considered, and for the $\Lambda$CDM+$N_{\rm eff}$ model the
result of $N_{\rm eff}=2.92^{+0.36}_{-0.37}$ (95% C.L., Planck TT,TE,EE+lowE)
is obtained. Obviously, if we set the requirement of $N_{\rm eff}>3.044$ for
this case, then only a 95% C.L. upper limit for $\Delta N_{\rm eff}$ 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 $w$CDM and HDE in addition to $\Lambda$CDM.
We wish to see what results can be given.

In Table 1, 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
$N_{\rm eff}$ 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 2 shows the case of using the CBSH data. We find that in
this case $\Delta N_{\rm eff}>0$ is favored at $2.73\sigma$, $1.67\sigma$, and
$2.55\sigma$ level for $\Lambda$CDM+$N_{\rm eff}$, $w$CDM+$N_{\rm eff}$, and
HDE+$N_{\rm eff}$, 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 $N_{\rm eff}$ 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
$N_{\rm eff}$. The correlation between $N_{\rm eff}$ 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 $\Delta$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 $w$CDM
model can well fit the current data. In particular, we notice that, in the
case of CBSH, $\Delta{\rm AIC}=-4.9$ for $w$CDM+$N_{\rm eff}$. We also find
that $w<-1$ is favored at around 1$\sigma$ level ($w=-1$ is outside the
1$\sigma$ limit), indicating that the $\Lambda$CDM model is not preferred by
the CBS and CBSH data.

###  III.2 The case of massive sterile neutrinos

In the case of massive sterile neutrinos, two extra parameters, $N_{\rm eff}$
and $m_{\nu,{\rm{sterile}}}^{\rm{eff}}$, need to be considered in the
cosmological models. Here, the requirement of $N_{\rm eff}>3.044$ still holds.

From Table 1, we can see that when using the CBS data neither $N_{\rm eff}$
nor $m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ can be determined, no matter what dark
energy model is considered. Only upper limits on $N_{\rm eff}$ and
$m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ can be obtained.

From Table 2, we can see that, when the $H_{0}$ measurement is included in the
data combination, for all the dark energy models, $N_{\rm eff}$ can be
effectively constrained. But even in this case,
$m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ still cannot be well constrained, and only
upper limits can be given.

The preference of $\Delta N_{\rm eff}>0$ is at $2.74\sigma$, $1.09\sigma$, and
$2.52\sigma$ level for the $\Lambda$CDM+$N_{\rm
eff}$+$m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ model, the $w$CDM+$N_{\rm
eff}$+$m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ model, and the HDE+$N_{\rm
eff}$+$m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ model, respectively, in the case of
using the CBSH data.

For the constraints on $m_{\nu,{\rm{sterile}}}^{\rm{eff}}$, we find that the
impact of dark energy is somewhat evident. Compared with $\Lambda$CDM, the
constraint in $w$CDM is evidently looser, and that in HDE is slightly tighter.
This also confirms the previous results in e.g. Refs. [34, 19, 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 $w$CDM model is favored by the data, and $w=-1$ is also
outside the $1\sigma$ limit, indicating that the $\Lambda$CDM model is not
preferred by the CBS and CBSH data in the case of massive sterile neutrinos.

###  III.3 The Hubble tension

We wish to check if the Hubble tension can be effectively relieved when
sterile neutrinos are considered.

From Tables 1 and 2, we can see that in the $\Lambda$CDM model (without
considering sterile neutrinos) the $H_{0}$ tension is at 4.3$\sigma$ level for
the case of using the CBS data, and at 3.9$\sigma$ 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
$\Lambda$CDM at around 4$\sigma$ 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 1.7$\sigma$). 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 $\Lambda$CDM and $w$CDM
models with sterile neutrinos, we find that they can also relieve the Hubble
tension to less than 2$\sigma$. 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 $\sigma_{8}$ 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).

The Hubble tension is by no means easy to be resolved (see e.g. Refs. [21,
103, 104, 105, 106, 24, 107, 108, 109, 110, 73, 74, 111] for various
endeavors). When we remove the $H_{0}$ local measurement from the data
combination, i.e., using the CBS data, we find that the Hubble tension is
still at 3–4$\sigma$ level even sterile neutrinos are considered in the
cosmological models.

##  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 $\Lambda$CDM model, we also consider the $w$CDM 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 $N_{\rm eff}$ can be
obtained. In this case, the $w$CDM model is most favored by the data; in the
$w$CDM model, $w=-1$ is outside the $1\sigma$ limit, indicating that the
$\Lambda$CDM 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, $\Delta N_{\rm eff}>0$ 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. But even in this case,
$m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ 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 2$\sigma$.
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
3–4$\sigma$.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grant Nos. 11947022, 12103038, 11975072, 11835009, 11875102, and 11690021),
the Liaoning Revitalization Talents Program (Grant No. XLYC1905011), the
Fundamental Research Funds for the Central Universities (Grant No. N2005030),
the National Program for Support of Top-Notch Young Professionals, (Grant No.
W02070050), the Natural Science Foundation of Liaoning Province (Grant Nos.
2021-BS-154 and 2021-BS-156), the 2019 Annual Scientific Research Funding
Project of the Education Department of Liaoning Province (Grant No.
LJC201915), the Doctoral Research Project of Shenyang Normal University (Grant
No. BS201844), and the Natural Science Foundation of Shaanxi Provincial
Department of Education (Grant No. 20JK0683).

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