A search for sterile neutrinos with the latest cosmological observations

We report the result of a search for sterile neutrinos with the latest cosmological observations. Both cases of massless and massive sterile neutrinos are considered in the $\Lambda$CDM cosmology. The cosmological observations used in this work include the Planck 2015 temperature and polarization data, the baryon acoustic oscillation data, the Hubble constant direct measurement data, the Planck Sunyaev-Zeldovich cluster counts data, the Planck lensing data, and the cosmic shear data. We find that the current observational data give a hint of the existence of massless sterile neutrino (as dark radiation) at the 1.44$\sigma$ level, and the consideration of an extra massless sterile neutrino can indeed relieve the tension between observations and improve the cosmological fit. For the case of massive sterile neutrino, the observations give a rather tight upper limit on the mass, which implies that actually a massless sterile neutrino is more favored. Our result is consistent with the recent result of neutrino oscillation experiment done by the Daya Bay and MINOS collaborations, as well as the recent result of cosmic ray experiment done by the IceCube collaboration.

The possibility of the existence of light sterile neutrinos has been motivated to explain the anomalies of short-baseline neutrino experiments [31][32][33][34][35][36][37][38][39][40]. It seems that the fully thermalized (∆N eff ≈ 1) sterile neutrinos with eV-scale mass are needed to explain these results [41][42][43]. The cosmological observations play an important role in constraining the total mass of the active neutrinos (see, e.g., Refs. ). In fact, since the sterile neutrinos have some effects on the evolution of the universe, the cosmological observations can also provide independent evidence in searching for sterile neutrinos [15].
Recently, some important updated observational data were released. For example, Riess et al. [67] reported the new result of local measurement of the Hubble constant, H 0 = 73.00±1.75 km s −1 Mpc −1 , which is 3.3σ higher than the result of 66.93±0.62 km s −1 Mpc −1 derived by the Planck mission based on the ΛCDM model with m ν = 0.06 eV using the latest Planck CMB data. Moreover, the BAO measurements were also updated for the CMASS and LOWZ galaxy samples, done by the Data Release 12 (DR12) [68] of SDSS-III BOSS (Baryon Oscillation Spectroscopic Survey).
The aim of this work is to search for the sterile neutrinos by using the latest cosmological observations. We shall investigate how the sterile neutrinos can relieve the tensions in the current observations and whether the current cosmological data can provide evidence for the existence of sterile neutrino. This paper is organized as follows. In Sec. II, we introduce the analysis method and the observational data we use in this paper. The results are given and discussed in Sec. III. Conclusion is given in Sec. IV.

II. ANALYSIS METHOD AND OBSERVATIONAL DATA
In this section, we introduce the analysis method and the observational data that will be used in this paper.

A. Analysis method
We will consider the both cases of massless and massive sterile neutrinos in the framework of ΛCDM cosmology. The massless sterile neutrino serves as the dark radiation, and thus when this case is considered, an additional parameter N eff needs to be added in the model; this case is called ΛCDM+N eff model in this paper. When the sterile neutrino is considered to be massive, then one needs to add another extra parameter, m eff ν,sterile , in the model; this case is thus called ΛCDM+N eff +m eff ν,sterile model in this paper. In the case of massive sterile neutrino, the true mass of a thermally distributed sterile neutrino reads m thermal sterile = (N eff − 3.046) −3/4 m eff ν,sterile . In order to avoid a negative m thermal sterile , N eff must be larger than 3.046 in a universe with sterile neutrinos. In this work, the active neutrino mass is kept at 0.06 eV (i.e., the minimal-mass normal hierarchy is assumed).
We place constraints on the ΛCDM cosmology with sterile neutrinos by using the current observational data. The conventions used in this paper are consistent with those adopted by the Planck collaboration [69], i.e., those used in the camb Boltzmann code [70]. There are six independent cosmological parameters in the base ΛCDM model, where ω b ≡ Ω b h 2 and ω c ≡ Ω c h 2 are the present-day baryon and cold dark matter densities, respectively, θ MC is the ratio between the sound horizon and the angular diameter distance at the decoupling epoch, τ is the Thomson scattering optical depth due to reionization, A s is the amplitude of initial curvature perturbation power spectrum at k = 0.05 Mpc −1 , and n s is its spectral index. In addition, there are two additional free parameters, N eff and m eff ν,sterile , for describing the sterile neutrino. Thus, there are seven independent parameters in total for the ΛCDM+N eff model, and there are eight independent parameters in total for the ΛCDM+N eff +m eff ν,sterile model. Other parameters, such as Ω m , σ 8 , H 0 , and so on, are the derived parameters. We use the CosmoMC package [72] to infer the posterior probability distributions of parameters. Flat priors for the base parameters are used. The prior ranges for the base parameters are chosen to be much wider than the posterior ranges in order not to affect the results of parameter estimation.

B. Observational data
In this paper, the data sets we use include the cosmic microwave background (CMB), the baryon acoustic oscillations (BAO), the Hubble constant (H 0 ), the Planck Sunyaev-Zeldovich (SZ), the Planck lensing, and the weak lensing (WL) observations.
The CMB data: We use the Planck 2015 CMB temperature and polarization data [73] in our calculations.
We consider the combination of the likelihood at 30 ≤ ≤ 2500 in the temperature (TT), the crosscorrelation of temperature and polarization (TE), and the polarization (EE) power spectra and the Planck low-likelihood in the range of 2 ≤ ≤ 29, which is denoted as "Planck TT,TE,EE+lowP", following the nomenclature of the Planck collaboration [69].
The BAO data: In order to break the geometric degeneracy, it is necessary to consider the BAO data. We use the LOWZ (z eff = 0.32) and CMASS (z eff = 0.57) samples of BOSS DR12 [68], as well as the 6dFGS (six-degree-field galaxy survey) (z eff = 0.106) sample [74] and the SDSS MGS (main galaxy sample) (z eff = 0.15) sample [75].
The H 0 measurement: We use the recently measured new local value of the Hubble constant, H 0 = 73.00±1.75 km s −1 Mpc −1 , reported in Ref. [67].
The SZ data: The counts of rich clusters of galaxies are from the sample of Planck thermal Sunyaev-Zeldovich (SZ) cluster observation [76].
The Lensing data: We use the Planck lensing data [77], which provide additional information at low redshift.
The WL data: We use the cosmic shear data of weak lensing (WL) from the CFHTLenS survey [78].
In what follows, we will use these observational data to place constraints on the ΛCDM cosmology with sterile neutrinos. We will compare the ΛCDM, ΛCDM+N eff , and ΛCDM+N eff +m eff ν,sterile models under the uniform data sets. The basic data combination adopted in this paper is the Planck TT,TE,EE+lowP+BAO combination. In order to show the impacts from the other astrophysical observations on measuring the properties of the sterile neutrino, we also further combine the H 0 +SZ+Lensing+WL data in the analysis.
Thus, in our analysis, we use the two data combinations: (i) Planck TT,TE,EE+lowP+BAO, and (ii) Planck TT,TE,EE+lowP+BAO+H 0 +SZ+Lensing+WL. For convenience, we occasionally use the abbreviations "CMB+BAO" and "CMB+BAO+other" for them in the paper. In the next section, we will report and discuss the fitting results of the cosmological models in the light of these data sets.

III. RESULTS AND DISCUSSION
In this section, we report the fitting results of the cosmological models (the ΛCDM model, the ΛCDM+N eff model, and the ΛCDM+N eff +m eff ν,sterile model) and discuss the implications of these results in the search for sterile neutrinos. We will discuss the cases of massless and massive sterile neutrinos, respectively, in the two subsections.
Detailed fit values for the three models for cosmological parameters are given in Table I. In the table, we quote the ±1σ errors, but for the parameters that cannot be well constrained, we quote the 95.4% CL upper limits.
When we make comparison for the three models from the statistical point of view, we must be aware of the fact that they have different numbers of parameters. In general, a model with more parameters tends to  give a better fit to the same data, i.e., it tends to have a smaller χ 2 min . Thus, when the comparison is made for models with different parameter numbers, the simple comparison of χ 2 min is not appropriate because it is unfair. A punishment mechanism must be considered for those models with more parameters. The simplest way, for our purpose in this work, is to consider the Akaike information criterion (AIC) [  the definition AIC = χ 2 min + 2k, where k is the number of parameters of a model. A model with a lower AIC is more favored by data. So, when we make model selection for two models, we will calculate the difference of AIC for them, i.e., ∆AIC = ∆χ 2 min + 2∆k. In the case of this paper, we take the ΛCDM model as a reference model, and thus the ΛCDM+N eff model has ∆k = 1 and the ΛCDM+N eff +m eff ν,sterile model has ∆k = 2. They are considered to be more favored over the ΛCDM model provided that the ΛCDM+N eff model has ∆χ 2 < −2 and the ΛCDM+N eff +m eff ν,sterile model has ∆χ 2 < −4. CL) for the ΛCDM+N eff +m eff ν,sterile model, from the constraints of the CMB+BAO and CMB+BAO+other data combinations.

A. The case of massless sterile neutrino
The massless sterile neutrinos serve as the dark radiation, and thus in this case the effective number of relativistic species N eff is treated as a free parameter. The total energy density of radiation in the universe is given by

]ρ γ ,
where ρ γ is the energy density of photons. The standard case of three-generation neutrinos leads to N eff = 3.046 [79,80]. The detection of ∆N eff = N eff − 3.046 > 0 indicates the presence of extra relativistic particle species in the universe, and in this paper we take the fitting result of ∆N eff > 0 as the evidence of the existence of massless sterile neutrinos.
In this subsection, we constrain the ΛCDM+N eff model by using two combinations of data sets, namely, the CMB+BAO and CMB+BAO+other combinations. As mentioned above, hereafter, we shall use "CMB" to denote Planck TT,TE,EE+lowP and use "other" to denote H 0 +SZ+lensing+WL. The free parameters include six base parameters and N eff , and we fix the active neutrino mass m ν = 0.06 eV (two massless and one massive active neutrinos). The results of constraints on N eff can be found in Table I In the right panel of Fig.1, we show the one-dimensional marginalized posterior distributions of N eff for the ΛCDM+N eff model using the two data combinations. We find that the CMB+BAO data can only give an upper limit, N eff < 3.44 (95.4% CL), but the CMB+BAO+other data can well constrain N eff , giving the result of N eff = 3.29 +0.11 −0.17 , which indicates a detection of ∆N eff > 0 at the 1.44σ level. Fitting to the CMB+BAO data, the ΛCDM and ΛCDM+N eff models yield a similar χ 2 min , implying that with this data combination adding the parameter N eff cannot improve the fit. But, when fitting to the CMB+BAO+other data, the ΛCDM+N eff model leads to an increase of ∆χ 2 = −2.172, compared to the ΛCDM model, indicating that in this case the addition of the parameter N eff can evidently improve the fit.
Actually, when we use the information criterion to make a model selection, we have ∆AIC = −0.172 for the ΛCDM+N eff model in this case, which shows that the ΛCDM+N eff model is only slightly better than the ΛCDM model from the statistical point of view.
Therefore, we find that the current CMB+BAO+other data can give a hint of the existence of massless sterile neutrino (as dark radiation) at the 1.44σ level, and the consideration of an extra massless sterile neutrino can indeed relieve the tension between observations and improve the cosmological fit.

B. The case of massive sterile neutrino
In this subsection, we further consider the case of massive sterile neutrino, i.e., we consider two extra parameters, N eff and m eff ν,sterile , compared to the ΛCDM model. We make a comparison for the ΛCDM model and ΛCDM+N eff +m eff ν,sterile model using the two data combinations. In Fig. 2, we show the posterior distribution contours in the Ω m -H 0 plane, for the two models. In the left panel, we show the case of fitting to the CMB+BAO data, and in the right panel, we show the case of fitting to the CMB+BAO+other data. We find that H 0 is anti-correlated with Ω m in the two models, and the consideration of massive sterile neutrino amplifies the parameter space, in particular, in the direction of H 0 . Actually, the involvement of massive sterile neutrino can indeed give a much higher value of H 0 .
We show the one-dimensional posterior distributions of H 0 for the two models in Fig. 3  14 km s −1 Mpc −1 for the ΛCDM+N eff +m eff ν,sterile model, which indicates that the involvement of massive sterile neutrino improves the tension from 2.58σ to 1.83σ. Thus, in this case, the tension can be largely relieved. But we must admit that the improvement for the tension (from 2.58σ to 1.83σ) is mainly owing to the fact that the introduction of two extra parameters leads to the posterior distribution of H 0 becoming much broader (see the right panel of Fig. 3).
Then we show the constraint results for the parameters N eff and m eff ν,sterile in the ΛCDM+N eff +m eff ν,sterile model (see Table I): We find that N eff cannot be well constrained using only the CMB+BAO data, but the addition of H 0 , SZ, Lensing, and WL data can significantly improve the constraint on N eff , favoring ∆N eff > 0 at the 1.27σ statistical significance. For the mass of sterile neutrino, the CMB+BAO data give m eff ν,sterile < 0.7279 eV (95.4% CL), and further including the H 0 +SZ+Lensing+WL data leads to the result of m eff ν,sterile < 0.2417 eV (95.4% CL). Evidently, adding low-redshift data tightens the constraint on m eff ν,sterile significantly. This indicates that the SZ cluster data (as well as the H 0 , Lensing, and WL data) play an important role in constraining the mass of sterile neutrino. In Fig. 4, we show the one-and two-dimensional marginalized posterior distributions of the parameters N eff , m eff ν,sterile , σ 8 , and H 0 , for the ΛCDM+N eff +m eff ν,sterile model. We find that, in the ΛCDM+N eff +m eff ν,sterile model, σ 8 is anti-correlated with m eff ν,sterile , and H 0 is positively correlated with N eff , which leads to the fact that considering massive sterile neutrinos in cosmology can effectively relieve the tensions among the current observations.
Compared to the ΛCDM model, the ΛCDM+N eff +m eff ν,sterile model does not provide an improved fit to the current observational data. Considering the massive sterile neutrino in cosmology leads to an increase of ∆χ 2 = 0.632 under the CMB+BAO constraint and an increase of ∆χ 2 = 0.828 under the CMB+BAO+other constraint. That is to say, the ΛCDM+N eff +m eff ν,sterile model has ∆AIC = 4.632 and 4.828 for the two datacombination cases, which shows that the massive sterile neutrino is not favored by the current cosmological observations. Therefore, we find that the current CMB+BAO+other data give a rather tight upper limit on m eff ν,sterile and favor ∆N eff > 0 at the 1.27σ level. Together with the constraint results for the massless sterile neutrino, we can conclude that the current observations do not seem to favor a massive sterile neutrino, but favor a massless sterile neutrino in some sense (only at the more than 1σ statistical significance). Our result is consistent with the recent result of neutrino oscillation experiment by the Daya Bay and MINOS collaborations [81], as well as the recent result of cosmic ray experiment by the IceCube collaboration [82].

IV. CONCLUSION
The aim of this work is to search for sterile neutrinos using the latest cosmological observations. We consider the two cases of massless and massive sterile neutrinos, corresponding to the ΛCDM+N eff model and the ΛCDM+N eff +m eff ν,sterile model, respectively. The observational data used in this paper include the Planck TT,TE,EE+lowP data, the BAO data, the H 0 direct measurement, the Planck SZ cluster counts data, the Planck CMB lensing data, and the cosmic shear data.
For the ΛCDM+N eff model, the CMB+BAO+other data give N eff = 3.29 +0.11 −0.17 (68.3% CL), favoring ∆N eff = N eff − 3.046 > 0 at the 1.44σ level. Therefore, there is a hint of the existence of massless sterile neutrinos, from the current cosmological observations. We also find that the addition of the parameter N eff can indeed relieve the tension between the Planck observation and the recent H 0 direct measurement (the tension is reduced to be at the 1.69σ level).
For the ΛCDM+N eff +m eff ν,sterile model, using the CMB+BAO data, we obtain m eff ν,sterile < 0.7279 eV (95.4% CL) and N eff < 3.4273 (95.4% CL). Thus, in this case, only upper limits on N eff and m eff ν,sterile (for the massive sterile neutrino) can be derived. Further including the other (H 0 +SZ+Lensing+WL) data significantly improves the constraints, and in this case we obtain m eff ν,sterile < 0.2417 eV (95.4% CL) and N eff = 3.3 +0. 12 −0.20 (68.3% CL). Thus, the current observations give a rather tight upper limit on m eff ν,sterile and favor ∆N eff > 0 at the 1.27σ level. This result seems to favor a massless sterile neutrino, in tension with the previous short-baseline neutrino oscillation experiments that prefer the mass of sterile neutrino at around 1 eV. But our result is consistent with the recent result of neutrino oscillation experiment done by the Daya Bay and MINOS collaborations [81], as well as the recent result of cosmic ray experiment done by the IceCube collaboration [82].