Cosmological constraints on neutrinos after BICEP2

Since the B-mode polarization of the cosmic microwave background (CMB) was detected by the BICEP2 experiment and an unexpectedly large tensor-to-scalar ratio, $r=0.20^{+0.07}_{-0.05}$, was found, the base standard cosmology should at least be extended to the 7-parameter $\Lambda$CDM+$r$ model. In this paper, we consider the extensions to this base $\Lambda$CDM+$r$ model by including additional base parameters relevant to neutrinos and/or other neutrino-like relativistic components. Four neutrino cosmological models are considered, i.e., the $\Lambda$CDM+$r$+$\sum m_\nu$, $\Lambda$CDM+$r$+$N_{\rm eff}$, $\Lambda$CDM+$r$+$\sum m_\nu$+$N_{\rm eff}$, and $\Lambda$CDM+$r$+$N_{\rm eff}$+$m_{\nu,{\rm sterile}}^{\rm eff}$ models. We combine the current data, including the Planck temperature data, the WMAP 9-year polarization data, the baryon acoustic oscillation data, the Hubble constant direct measurement data, the Planck Sunyaev-Zeldovich cluster counts data, the Planck CMB lensing data, the cosmic shear data, and the BICEP2 polarization data, to constrain these neutrino cosmological models. We focus on the constraints on the parameters $\sum m_\nu$, $N_{\rm eff}$, and $m_{\nu,{\rm sterile}}^{\rm eff}$. We also discuss whether the tension on $r$ between Planck and BICEP2 can be relieved in these neutrino cosmological models.

inflation models predict a negligible running (of order 10 −4 ).
To reduce the tension, more possibilities should be explored. One interesting suggestion is to consider additional sterile neutrino species in the universe [3,4]. Since the tensor-to-scalar ratio r is found to be around 0.2, the standard cosmology should at least be extended to ΛCDM+r model (now this is the base model with seven parameters). Thus, the model with sterile neutrino is naturally called ΛCDM+r+ν s model, in which two additional parameters, N eff and m eff ν,sterile , are included. It is shown that in the ΛCDM+r+ν s model the tension between Planck and BICEP2 can be greatly relieved at the expense of the increase of n s [3,4]. Moreover, actually, by including a sterile neutrino species in the universe, not only the tension between Planck and BICEP2 is relieved, but also the other tensions between Planck and other astrophysical observations, such as the H 0 direct measurement, the cluster counts, and the galaxy shear measurement, can all be significantly reduced. 1 Thus, the model with sterile neutrino seems to be an economical choice for the cosmology today. Furthermore, by combining the Planck+WP with the baryon acoustic oscillations (BAO), H 0 , Sunyaev-Zeldovich (SZ) cluster counts, CMB lensing, galaxy shear, and BICEP2 data, it is found that in the ΛCDM+r+ν s model the existing cosmological data prefer ∆N eff > 0 at the 2.7σ level and a nonzero mass of sterile neutrino at the 3.9σ level [3]. (See also Ref. [4] for a similar analysis.) Other proposals to address the large B modes include, e.g., foregrounds or some uncounted temperaturepolarization leakage [12], non-standard inflation models or more general early-universe scenarios [13][14][15][16][17][18][19][20][21], large-field excursions [22,23], primordial magnetic fields [24], topological defects [25,26], spatial variation of r [27], and so on. Obviously, the forthcoming new data from, e.g., Planck and Keck array are expected to improve the foreground model and provide more tight constraints on the B modes, resolving the current tension problem.
In this paper, we will consider neutrinos and extra relativistic components within the base ΛCDM+r model. We will use the current data to constrain the models with neutrinos. The models we consider in this paper include: (i) the active neutrinos with additional parameter m ν , (ii) the extra relativistic components with additional parameter N eff , (iii) the active neutrinos along with the extra relativistic components with additional parameters m ν and N eff , and (iv) the massive sterile neutrino with additional parameters N eff and m eff ν,sterile . The observational data we use in this paper are from Planck+WP+BAO, H 0 direct measurement, Planck SZ cluster counts, Planck CMB lensing, cosmic shear measurement, and BICEP2. This work will provide a detailed cosmological analysis on the models with neutrinos under the consideration of the BICEP2 data.
The paper is organized as follows. In Sec. II, we briefly describe the cosmological models with neutrinos and the observational data. In Sec. III, we present the fit results and discuss these results in detail.
Conclusion is given in Sec. IV.

A. Cosmological models involving neutrinos
The cosmology with neutrinos has been described in detail and reviewed by the WMAP Collaboration [28][29][30] and the Planck Collaboration [2]. In this paper, our conventions are consistent with those adopted by the Planck Collaboration [2], i.e., those used in the camb Boltzmann code. So, we will not describe in detail the equations but only specify the models with different parameters; for the details about the cosmology with neutrinos we refer the reader to Refs. [2,[28][29][30].
Under the current situation that the large PGWs have been discovered, the base cosmology should be extended to the 7-parameter ΛCDM+r model. The base parameters for this model are: 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 approximation (used in CosmoMC) to the angular size of the sound horizon at the time of lastscattering r s (z * )/D A (z * ), τ is the Thomson scattering optical depth due to reionization, n s and A s are the spectral index and amplitude of the primordial curvature perturbations, respectively, and r 0.05 is the tensorto-scalar ratio at k 0 = 0.05 Mpc −1 . Other parameters, such as Ω Λ , Ω m , σ 8 , H 0 , r 0.002 , and so on, are the derived parameters.
In this base cosmology, there are three active neutrino species. Due to non-instantaneous decoupling corrections and other subtle corrections, the effective number of relativistic species in the standard cosmology is N eff = 3.046. A minimal-mass normal hierarchy for the neutrino mass is assumed in the base cosmology, i.e., only one massive eigenstate with m ν = 0.06 eV (Ω ν h 2 ≈ m ν /93.04 eV ≈ 0.0006).
In this paper, we consider the extensions to this base cosmology. Neutrinos and extra relativistic components bring additional base parameters to the model.
• Consider the total mass of active neutrinos. In this case, a degenerate model is assumed in which the three active neutrino species are degenerate in mass and the total mass m ν is a free parameter.
Thus, in this extension, one additional base parameter, m ν , is introduced.
• Consider the extra neutrino-like radiation. In this case, the extra relativistic degrees of freedom are effectively massless. The total mass of active neutrinos m ν is kept fixed at 0.06 eV, but the parameter N eff is free. Thus, in this extension, one additional base parameter, N eff , is introduced.
• Simultaneously consider the active neutrino mass and extra radiation. In this case, the parameters N eff and m ν are both free. So, two additional parameters, N eff and m ν , are introduced.
• Consider the massive sterile neutrino. In this case, the total mass of active neutrinos m ν is kept fixed at 0.06 eV, but we add one massive sterile neutrino in the model. Thus, two additional parameters, N eff and m eff ν,sterile , are introduced.
We use flat priors for the base parameters. When the base parameters are varied, the prior ranges are chosen to be much wider than the posterior so that the results of parameter estimation are not affected.
The priors are set following the Planck Collaboration [2]. In addition to these priors, a "hard" prior on the Hubble constant H 0 of [20, 100] km s −1 Mpc −1 is imposed.

B. Observational data
We consider the following data sets: • Planck+WP: the CMB TT angular power spectrum data from Planck [2], in combination with the large-scale EE and TE polarization power spectrum data from 9-year WMAP [30].  2 There are also other BAO datasets, e.g., the 6dF with one point, r s /D V (0.1) = 0.336 ± 0.015 [32], and the WiggleZ with three points, r s /D V (0.44) = 0.0916 ± 0.0071, r s /D V (0.60) = 0.0726 ± 0.0034, and r s /D V (0.73) = 0.0592 ± 0.0032 [33]. The inverse covariance matrix for the WiggleZ data is given in Ref. [33] (see also Ref. [30]). In this work, we choose to only use the latest two most accurate BAO measurements from BOSS DR11 [31], which is sufficient for the purpose of this work in breaking the CMB parameter degeneracies. 3 There is also another accurate H 0 measurement given by the Carnegie Hubble Program, H 0 = [74.3 ± 1.5 (statistical) ± 2.1 (systematic)] km s −1 Mpc −1 [35]. This result agrees well with that of Ref. [34]. In this work, we do not consider the measurement result of Ref. [35], but only use the one of Ref. [34].
Actually, the Planck data are in tension with several astrophysical observations, as discussed by the  [2], which are in tension with the H 0 direct measurement [34], the cluster counts [7], 6 and the cosmic shear measurement [39] at the 2-3σ level. In addition, Planck is also in mild tension with the SNLS type Ia supernova compilation (at about the 2σ level).
Due to the complexity of these astrophysical data, these tensions can possibly be interpreted in terms of that some sources of systematic errors are not completely understood in these astrophysical measurement.
An alternative explanation is that the base ΛCDM model is incorrect or should be extended.
The possibilities that the tensions between Planck and these astrophysical data might imply new physics have been explored. For example, the tension between Planck and the H 0 direct measurement might hint that dark energy is not the cosmological constant but is some dynamical field (or fluid). It is shown in Ref. [44] that in a dynamical dark energy model, such as the constant w model or the holographic dark energy model, 4 The SZ cluster counts result quoted here is based on the use of the mass function from Ref. [36]. A different mass function from Ref. [37] leads to a slightly different value of σ 8 (Ω m /0.27) 0.3 = 0.802 ± 0.014. In addition, the result also depends, more or less, on the bias (1 − b) that is assumed to account for all possible observational biases including departure from hydrostatic equilibrium, absolute instrument calibration, temperature inhomogeneities, residual selection bias, etc. Numerical simulations based on the consideration of several ingredients of the gas physics of clusters give (1 − b) = 0.8 +0.  Table 2 of Ref. [7]. 5 We note that CMB lensing power spectrum and CFHTLens survey are two absolutely different, physically and observationally independent datasets. 6 Actually, for the cluster counts, besides the Planck SZ-selected cluster sample [7], there are also several other accurate datasets, including the SZ clusters from SPT [40] and ACT [41], and X-ray [42] and optical richness [43] selected cluster samples. Constraints from these cluster samples on σ 8 (Ω m /0.27) 0.3 can be found in Table 3 (and Fig. 10) of Ref. [7]. In this paper, we only focus on the Planck SZ cluster counts [7]. the tension between Planck and H 0 is greatly reduced. But the mild tension between the Planck data and the SNLS type Ia supernova data may come from the systematic error, which could be greatly eliminated by considering the new effects of supernova, such as the evolution of the color-luminosity parameter β, as analyzed in Refs. [45,46].
Sterile neutrino can also play a very significant role in relieving the tensions between Planck and the astrophysical observations. Involving sterile neutrino can increase the early-time Hubble expansion rate and the free-streaming damping, leading to the changes of the acoustic scale and the growth of cosmic structure, thus the tensions between Planck and H 0 , cluster counts, and cosmic shear can simultaneously be greatly reduced when the massive sterile neutrino is considered [8][9][10]. Furthermore, very recently, it was shown that the tension between Planck and BICEP2 can also be significantly relieved when the sterile neutrino is involved in the model [3,4]. Therefore, in the ΛCDM+r+ν s model, almost all the tensions between Planck and other astrophysical observations can be simultaneously alleviated.
In this paper, we use the latest observational data to place constraints on the neutrino cosmological models. Since we use the uniform data sets, we actually can make a direct comparison for these models.
We do not use the type Ia supernova data in this analysis because dynamical dark energy is not considered and also the systematic errors in the supernova data cannot be well quantified [45,46]. But we assume that other astrophysical data sets, such as H 0 , SZ cluster counts, and cosmic shear, have accurately quantified estimates of systematic errors. Since there is no tension between Planck and BAO, we can always safely use the Planck+WP+BAO data combination. In order to measure the impacts from the other astrophysical observations on the neutrino physics, we can further combine the H 0 +SZ+Lensing data in the analysis. Furthermore, to see the role of the BICEP2 data play in constraining the neutrino cosmological models, we finally use an all data combination involving the BICEP2 data. Thus, in our analysis, we use the data combinations: (i) Planck+WP+BAO, (ii) Planck+WP+BAO+H 0 +SZ+Lensing, and (iii) Planck+WP+BAO+H 0 +SZ+Lensing+BICEP2. In the next section, we will report and discuss the fitting results of the neutrino cosmological models in the light of these data sets.

III. RESULTS AND DISCUSSIONS
For convenience, the four models considered in this paper are called: (i) ΛCDM+r+ m ν , (ii)  Tables I-IV. In the tables, we quote the ±1σ errors, but for the parameters that cannot be well constrained, we quote the 95%

CL upper limits.
A. Constraints on the total mass of active neutrinos m ν Figure 1 and Table I    But, H 0 , n s and r 0.002 are not affected evidently.
The posterior distribution is shown by the red curve in Fig. 1. Further including the BICEP2 data does not improve the constraint on the neutrino mass, m ν = 0.28 +0.07 −0.08 eV (68% CL; Planck+WP+BAO+H 0 +SZ+Lensing+BICEP2).
The posterior distribution is shown in Fig. 1 by the blue curve which is nearly coincident with the red one.
Thus, we find that in the ΛCDM+r+ m ν model the combined cosmological data prefer a nonzero total mass of active neutrinos at about the 4σ significance. The BICEP2 does not affect other parameters, either, except for the tensor-to-scalar ratio r. In the ΛCDM+r+ m ν model, it is shown from Fig. 1 and Table I that the tension between Planck and BICEP2 cannot be effectively reduced. The Planck+WP+BAO data combination gives r 0.002 < 0.12 (95% CL), and further adding H 0 +SZ+Lensing data weakens the limit to r 0.002 < 0.15 (95% CL). Including the BICEP2 data could improve the constraint on r to r 0.002 = 0.18 +0.03 −0.04 (68% CL).
B. Constraints on the effective number of relativistic species N eff Figure 2 and Table II   The addition of the parameter N eff can slightly relieve the tension between Planck and H 0 . The Planck+WP+BAO data combination gives H 0 = 70.4 +1.8 −1.9 km s −1 Mpc −1 . In the same case we also find a high amplitude for the present-day matter fluctuations, σ 8 = 0.849 ± 0.020. When the H 0 +SZ+Lensing data are added, the value of H 0 is not affected significantly, H 0 = 69.1 ± 1.4 km s −1 Mpc −1 , but the value of σ 8 becomes much smaller, σ 8 = 0.792 ± 0.009 (with the error also shrinking significantly).
In the ΛCDM+r+N eff model, the constraint results for the parameter N eff are: We also find that in the ΛCDM+r+N eff model the upper limit for the tensor-to-scalar ratio becomes a little bit higher, r 0.002 < 0.15, from the Planck+WP+BAO data, and this limit does not change when the H 0 +SZ+Lensing data are added. So, this model cannot effectively alleviate the tension between Planck and BICEP2. When the BICEP2 data are included, the constraint on r becomes r 0.002 = 0.18 ± 0.04. The Planck+WP+BICEP2 constraints on the ΛCDM+r+ m ν and ΛCDM+r+N eff models were also discussed recently in Ref. [47].
C. Simultaneous constraints on N eff and m ν Figure 3 and Table III    In the ΛCDM+r+ m ν +N eff model, the constraint results for the parameters N eff and m ν are: We find that with the basic data combination Planck+WP+BAO, only an upper limit for the total mass of active neutrinos can be given, but the weak preference for N eff > 3.046 at about the 1.6σ level is shown.
Combining the H 0 +SZ+Lensing data can tightly constrain both m ν and N eff , giving the evidence for nonzero mass of active neutrinos and ∆N eff ≡ N eff − 3.046 > 0 at the 3.9σ and 2.9σ, respectively. Further adding the BICEP2 data can improve the results to some extent, favoring m ν > 0 and ∆N eff > 0 at the 4.0σ and 3.6σ levels, respectively.
It is interesting to compare the current results with those derived from data before Planck and BICEP2.
It is also important to show that this model is very helpful in reconciling the tension between Planck and BICEP2. With only the Planck+WP+BAO data, we find that the upper limit on the tensor-to-scalar ratio r is weakened to r 0.002 < 0.19 (95% CL). Once the H 0 +SZ+Lensing data are included, the limit on r is further weakened to r 0.002 < 0.27 (95% CL), which is well compatible with the BICEP2 result, r 0.002 = 0.20 +0.07 −0.05 [1]. Combining with the BICEP2 data, the r constraint is tightened to r 0.002 = 0.22 +0.04 −0.05 . We also notice that due to the positive correlation between n s and r (see the n s -r 0.002 contours in grey and red in Fig. 3), once the tensor-to-scalar ratio r is increased, the scalar spectral index n s is also enlarged.
According to the fitting results, the exact scale-invariant perturbation spectrum cannot be excluded but actually is favored in this model. The ΛCDM+r+N eff +m eff ν,sterile model has been discussed in Refs. [3,4]. In Ref. [3], this model is also called ΛCDM+r+ν s model, with ν s denoting the sterile neutrino with two extra parameters N eff and m eff ν,sterile . In this paper, we duplicate the calculations in Ref. [3], but we will provide more information about the fit results. Figure 4 and Table IV summarize the fit results for the ΛCDM+r+N eff +m eff ν,sterile model. It has been discussed in Refs. [3,4] (see also Refs. [8][9][10]) that the sterile neutrino can reconcile the tensions between Planck and other astrophysical observations such as the direct measurement of H 0 [34], the Planck SZ cluster counts [7], and the cosmic shear measurement [39]. Here, we can see from We now show the constraint results for the parameters N eff and m eff ν,sterile in this model:  We find that the mass of sterile neutrino cannot be well constrained using only the basic data combination Planck+WP+BAO, but the addition of H 0 , SZ, and Lending data significantly improves the constraint on the mass, strongly favoring a nonzero mass of sterile neutrino at the 3.6σ statistical significance. The posterior distributions of m eff ν,sterile for the two cases are shown as grey and red curves, respectively, in Fig. 4, and the direct comparison of the two curves is very impressive. This shows that the SZ cluster data (as well as the H 0 and Lensing data) play an important role in constraining the mass of sterile neutrino, as discussed in Refs. [3,4]. Further including the BICEP2 data improves the evidence for nonzero mass of sterile neutrino to be at the 3.9σ significance. For the N eff constraints, the basic combination Planck+WP+BAO shows the preference for ∆N eff > 0 at the 1.7σ level, and the inclusion of H 0 +SZ+Lensing data improves slightly the preference for ∆N eff > 0 to be at the 1.9σ level. The BICEP2 data play a significant role in improving the constraint on N eff , which can be seen directly from the posterior distribution curves in Fig. 4. Further adding the BICEP2 data favors the ∆N eff > 0 result at the 2.7σ level.
The sterile neutrino can help reconcile the tension between Planck and BICEP2, as analyzed in Refs. [3,4]. Using only Planck+WP+BAO can lead to r 0.002 < 0.20 (95% CL), and including H 0 +SZ+Lensing can give r 0.002 < 0.23 (95% CL), consistent with the BICEP2 result. Further adding the BICEP2 data, we obtain the tightly constrained result, r 0.002 = 0.21 +0.04 −0.05 . As pointed out by Refs. [3,4], the increase of r is at the expense of the increase of n s , due to the positive correlation between n s and r 0.002 (as shown by the grey and red contours in the n s -r 0.002 plane in Fig. 4). Hence, as the same as the ΛCDM+r+ m ν +N eff model discussed in the last subsection, this model can resolve the tension between Planck and BICEP2, but at the same time cannot exclude the exact scale-invariant primordial perturbation spectrum.
The light massive sterile neutrino is motivated to explain the anomalies appearing in the short-baseline neutrino oscillation experiments [48][49][50][51][52][53]. It is of great interest to see that the evidence of the existence of the light sterile neutrino can be found in the existing cosmological data with high statistical significance (see also Refs. [3,4,[8][9][10]). Moreover, in this model almost all the tensions of Planck with other astrophysical observations can be simultaneously relieved.
The best-fit results, ∆N eff ≈ 1 and m thermal sterile ≈ m eff ν,sterile ≈ 0.5 eV, derived in this paper and Refs. [3,4], indicate a fully thermalized sterile neutrino with sub-eV mass. However, the short baseline neutrino oscillation experiments prefer the mass of sterile neutrino at around 1 eV. So, there is still a tension on the sterile neutrino mass between the cosmological data and the short-baseline neutrino oscillation data.
The implication of this tension for cosmology deserves further investigations. See Ref. [54] for a recent discussion.

IV. CONCLUSION
After the detection of the PGWs by the BICEP2 experiment, the base standard cosmology should at least be extended to the 7-parameter ΛCDM+r model. In this paper, we consider the extensions to this base ΛCDM+r model by including additional base parameters relevant to neutrinos and/or other neutrinolike relativistic components. Four neutrino cosmological models are considered, i.e., the ΛCDM+r+ m ν , ΛCDM+r+N eff , ΛCDM+r+ m ν +N eff , and ΛCDM+r+N eff +m eff ν,sterile models. We use the current observational data to constrain these models. The cosmological data used in this paper include: Planck+WP, BAO, H 0 , Planck SZ cluster, Planck CMB lensing, cosmic shear, and BICEP2 data. The main results of this paper are shown in Figs. 1-4 and Tables I-IV. Here, we summarize the findings from our analysis.
• The ΛCDM+r+ m ν model. With the Planck+WP+BAO data, we find a limit on the active neutrino mass, m ν < 0.28 eV (95% CL). Including the H 0 +SZ+Lensing data leads to a strikingly tight constraint: m ν = 0.28 ± 0.07 eV, preferring a nonzero mass of active neutrinos at about the 4σ level. Further adding the BICEP2 data does not improve the constraint on the mass. We also find that this model cannot alleviate the tension on r between Planck and BICEP2.
• The ΛCDM+r+N eff model. Using only the Planck+WP+BAO data gives N eff = 3.52 +0. 31 −0.32 , and further adding the H 0 +SZ+Lensing data gives N eff = 2.97 +0.20 −0.22 , and combination of all data (including BICEP2) leads to N eff = 3.07 ± 0.20. These results are consistent with the standard value of 3.046.
We also find that this model cannot effectively alleviate the tension on r between Planck and BICEP2.
• The ΛCDM+r+ m ν +N eff model. With the Planck+WP+BAO data, we obtain m ν < 0.50 eV (95% CL) and N eff = 3.69 +0. 33 −0.40 , so in this case only an upper limit on the total mass of active neutrinos can be given, but the weak preference for N eff > 3.046 at about the 1.6σ level is shown.
Combining with the H 0 +SZ+Lensing data can lead to tight constraints, m ν = 0.58 +0.14 −0.15 eV and N eff = 4.04 +0. 35 −0.34 , giving the evidence for nonzero mass of active neutrinos and ∆N eff > 0 at the 3.9σ and 2.9σ, respectively. Further adding the BICEP2 data can improve the results to m ν = 0.63 +0.13 −0.16 eV and N eff = 4.20 ± 0.32, favoring m ν > 0 and ∆N eff > 0 at the 4.0σ and 3.6σ levels, respectively.
We also show that this model is very helpful in relieving the tension between Planck and BICEP2.
The increase of r is at the cost of the increase of n s , and consequently the exact scale-invariant spectrum cannot be excluded.
• The ΛCDM+r+N eff +m eff ν,sterile model. With the Planck+WP+BAO data, we obtain m eff ν,sterile < 0.51 eV (95% CL) and N eff = 3.72 +0. 32 −0.40 , thus in this case only an upper limit on the sterile neutrino mass can be derived and the preference for ∆N eff > 0 at the 1.7σ level is shown. Further including the H 0 +SZ+Lensing data significantly improves the constraints, m eff ν,sterile = 0.48 +0.11 −0.13 eV and N eff = 3.75 +0. 34 −0.37 , favoring a nonzero mass of sterile neutrino and ∆N eff > 0 at the 3.6σ and 1.9σ levels, respectively. Finally, further adding the BICEP2 data improves the constraints to m eff ν,sterile = 0.51 +0.12