Constraints on the neutrino mass and mass hierarchy from cosmological observations

Considering the mass splitting between three active neutrinos, we represent the new constraints on the sum of neutrino mass $\sum m_\nu$ by updating the anisotropic analysis of Baryon Acoustic Oscillation (BAO) scale in the CMASS and LOWZ galaxy samples from Data Release 12 of the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS DR12). Combining the BAO data of 6dFGS, MGS, LOWZ and CMASS with $\textit{Planck}$~2015 data of temperature anisotropy and polarizations of Cosmic Microwave Background (CMB), we find that the $95\%$ C.L. upper bounds on $\sum m_\nu$ refer to $\sum m_{\nu,\rm NH}<0.18$ eV for normal hierarchy (NH), $\sum m_{\nu,\rm IH}<0.20$ eV for inverted hierarchy (IH) and $\sum m_{\nu,\rm DH}<0.15$ eV for degenerate hierarchy (DH) respectively, and the normal hierarchy is slightly preferred than the inverted one ($\Delta \chi^2\equiv \chi^2_{\rm NH}-\chi^2_{\rm IH} \simeq -3.4$). In addition, the additional relativistic degrees of freedom and massive sterile neutrinos are neither favored at present.


I. INTRODUCTION
The phenomena of neutrino oscillation imply that there are mass splitting between three active neutrinos (see [1] for a review). Currently only two independent mass squared differences have been determined by neutrino oscillation experiments. Regardless of experimental uncertainties, they are given by [2] ∆m 2 21 ≡ m 2 2 − m 2 1 = 7.5 × 10 −5 eV 2 , Thus we have two possible mass hierarchies, namely, a normal hierarchy (NH, m 1 < m 2 < m 3 ) and an inverted hierarchy (IH, m 3 < m 1 < m 2 ). Here m i (i = 1, 2, 3) denote the mass eigenvalues of three neutrinos. The minimum sums of neutrino mass are 0.06 eV for NH and 0.10 eV for IH.
Up to now, the absolute neutrino mass and mass hierarchy are still unknown.
Cosmology provides possibilities to measure the neutrino mass or the sum of neutrino mass m ν [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Massive neutrinos are initially relativistic and become non-relativistic today. They can impact on the cosmic expansion since they evolves differently from pure radiations and pure cold dark matter. They can influence the evolution of cosmological perturbations at early times and affect the CMB temperature anisotropies via the early-time Integrated Sachs-Wolfe (ISW) effect [14]. In addition, relativistic neutrinos suppress the clustering of matter and then modify the growth of structure. Thus one might extract useful signals of cosmic neutrinos from cosmological observations such as the matter clustering and the anisotropies and polarizations of Cosmic  [21], the constraints are slightly changed to m ν < 0.68 eV and m ν < 0.59 eV for two data combinations, respectively. However, by contrast, adding the Baryon Acoustic Oscillation (BAO) data including 6dFGS [22], MGS [23], BOSS DR11 CMASS [24] and LOWZ [24] can significantly improve the constraints to m ν < 0.21 eV and m ν < 0.17 eV, respectively. The reason is that the BAO data can significantly break the acoustic scale degeneracy.
Recently the BAO distance scale measurements were updated via an anisotropic analysis of BAO scale in the correlation function [25] and power spectrum [26] of the CMASS and LOWZ galaxy samples from Data Release 12 of the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS DR12). The total volume probed in DR12 has a 10% increment from DR11 and the experimental uncertainty has been reduced correspondingly. Thus in this paper we update the constraints on the total mass of three active neutrinos by using BOSS DR12 CMASS and LOWZ data, which are combined with other cosmological observations such as Planck 2015 CMB data. In this paper, we also consider the mass splitting between three neutrinos implied by the neutrino oscillations between three generations. We estimate whether the current data sets can distinguish the neutrino mass hierarchy. In addition, we also update constraints on additional relativistic degree of freedom ∆N eff ≡ N eff − 3.046 and massive sterile neutrinos m eff ν,sterile . The rest of the paper is arranged as follows. In Sec. II, we reveal our methodology and cosmological data sets used in this paper. In Sec. III, we demonstrate our constraints on the sum of neutrino mass, additional relativistic degree of freedom and massive sterile neutrinos, respectively.
Our conclusions are listed in Sec. IV.

II. DATA AND METHOD
The recent distance measurements from the anisotropic analysis of BAO scale in the correlation function [25] and power spectrum [26] of CMASS and LOWZ galaxy samples from BOSS DR12 are listed in Tab. I. Only the consensus values [26] are listed, which are used in this paper. Here z  denotes the effective redshift for CMASS and LOWZ samples, respectively, H(z) and D A (z) are the Hubble parameter and angular diameter distance at reshift z respectively, and r d is the comoving sound horizon at the redshift of baryon drag epoch. In addition, ρ D A ,H stands for the normalized correlation between D A (z) and H(z).
In this paper, we combine the BAO data including 6dFGS [22], MGS [23], BOSS DR12 CMASS [26] and LOWZ [26] with Planck 2015 likelihoods [27] of CMB temperature and polarizations as well as CMB lensing. In fact, we employ two combinations of observational data, namely Planck TT,TE,EE+lowP+BAO and Planck TT+lowP+lensing+BAO. The latter one is expected to give conservative constraints on the neutrino sectors while the former one gives more severe constraints.
There are tensions on the amplitude of fluctuation spectrum between Planck CMB data and other astrophysical data such as weak lensing (WL) [28,29], redshift space distortion (RSD) [30] and Planck cluster counts [31]. Thus we do not take them into consideration in this paper. We neither consider the direct measurements of cosmic expansion, since there are certain debates on the H 0 data [32][33][34]. In addition, we do not use the data of supernovae of type Ia (SNe Ia), since the apparent magnitudes of SNe are insensitive to m ν .
In the ΛCDM model, there are six base cosmological parameters which are denoted by Here ω b is the physical density of baryons today and ω c is the physical density of cold dark matter today. θ MC is the ratio between the sound horizon and the angular diameter distance at the decoupling epoch. τ is the Thomson scatter optical depth due to reionization. n s is the scalar spectrum index and A s is the amplitude of the power spectrum of primordial curvature perturbations at the pivot scale k p = 0.05 Mpc −1 .
To constrain the neutrino sectors, we refer to the Markov Chain Monte Carlo sampler (Cos-moMC) [35] in the νΛCDM model. By considering the mass splitting in Eq. (1) and Eq. (2), we can express the neutrino mass spectrum by two independent mass squared differences and one minimum mass eigenvalue m ν,min . The neutrino mass spectrum is and m ν,min = m 1 for NH, and and m ν,min = m 3 for IH. In addition, the neutrino mass spectrum is trivial for DH, namely Thus we can constrain the sum of neutrino mass m ν via referring to the above three νΛCDM model. It should be noted that there are lower cut-off values of m ν which are 0.06 eV for NH and 0.10 eV for IH, respectively.

III. RESULTS
In this section, we represent the constraints on the neutrino sectors by updating cosmological data. To be specific, we give an updated upper bound on the sum of neutrino mass m ν in Sec. III A. In Sec. III B, the relativistic degree of freedom N eff is constrained. We simultaneously constrain N eff and massive sterile neutrino m eff ν,sterile in Sec. III C.
A. Constraints on m ν In this subsection, we refer to two combinations of data sets, namely Planck where m ν,min is the minimal eigenvalue of neutrino mass, and the total mass of neutrinos is a derived parameter, i.e. m ν = m 1 + m 2 + m 3 .
For three hierarchies, our constraints on m ν as well as seven free parameters and χ 2 can be found in Tab. II. The likelihood distributions of m ν and m ν,min are depicted in Fig. 1

B. Constraints on N eff
The total energy density of radiation in the Universe is given by where ρ γ is the energy density of CMB photon and N eff = 3.046 for counting the standard model neutrinos. N eff > 3.046 will indicate that there are some unknown relativistic degrees of freedom in the Universe.
In this subsection, we use two data combinations of Planck TT,TE,EE+lowP+BAO and Planck ∆N eff = 1, for example a fully thermalized sterile neutrino, is excluded at more than 5σ level by Planck TT,TE,EE+lowP+BAO data and at 4σ level by Planck TT+lowP+lensing+BAO data. A thermalized massless boson decoupled in the range 0.5 MeV < T < 100 MeV predicts ∆N eff = 4/7 0.57 which is disfavored at more than 95% C.L. by these two data sets. If it decoupled at  We can also consider extra one massive sterile neutrino whose effective mass is parametrized by m eff ν,sterile ≡ 94.1Ω ν,sterile h 2 eV. Assuming the sterile neutrino to be thermally distributed with an arbitrary temperature, m eff ν,sterile is then given by where m thermal sterile denotes the true mass. Here we consider the base ΛCDM+N eff +m eff ν,sterile model, in which m thermal sterile is a free parameter with a prior m thermal sterile < 10 eV and N eff has a flat prior with N eff > 3.046.
Our simultaneous constraints on N eff and m eff ν,sterile can be found in Tab. III. From Planck TT,TE,EE+lowP+BAO data, we obtain constraints to be N eff < 3.39 and m eff ν,sterile < 0.60 eV at 95% C.L.. From Planck TT+lowP+lensing+BAO data, we obtain N eff < 3.69 and m eff ν,sterile < 0.48 eV at 95% C.L., which are similar to Planck 2015 results in [17]. ∆N eff = 1 can be excluded at much more than 95% C.L.. One should note that the upper tail of m eff ν,sterile is closely related to high physical masses near to the prior cutoff.

IV. CONCLUSIONS
In this paper, we updated cosmological constraints on the total mass of three active neutrinos by updating BOSS DR11 to DR12 of CMASS and LOWZ samples. We considered the mass splitting between three neutrinos and then considered the neutrino mass spectrum with the NH, IH and DH, respectively. When the Planck TT,TE,EE+lowP+BAO combination is updated, our constraint m ν < 0.15 eV at 95% C.L. is improved by about 10% for the DH, comparing to Planck 2015 constraint m ν < 0.17 eV at 95% C.L. [17]. Meanwhile, we get updated 95% C.L.
upper limits m ν < 0.18 eV for the NH and m ν < 0.20 eV for the IH. For the NH (or the IH) and the DH, there is about 20% (or 27%) difference between their upper limits on the absolute neutrino mass. Thus it is meaningful to take into consideration the data of neutrino mass splitting obtained from the experimental particle physics. Although the current cosmological data may be not good enough to distinguish different neutrino mass hierarchies, the normal hierarchy is slightly preferred by ∆χ 2 −3.4 compared to the inverted hierarchy in our paper. Future precise observations might have potential to determine the neutrino mass and mass hierarchy [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52].
There are various tight constraints on m ν in literatures. For instance, the combination of Lyman-α absorption in the distant quasar spectra, BAO and Planck CMB data gave a constraint m ν < 0.12 eV at 95% C.L. in [15]. The combination of SDSS DR7 Luminous Red Galaxies (LRG), BAO and Planck CMB data gave an upper bound m ν < 0.11 eV at 95% C.L. in [16].
Both constraints, close to the lower cut-off values of 0.10 eV for the IH, are tighter than ours obtained in this paper. Thus it is interesting to include the observational data sets regarding to the matter power spectrum into our exploration, besides the lensed-CMB and BAO data. We will remain these considerations as our future work.
In addition, we also updated the constraints on the relativistic degree of freedom and massive sterile neutrinos. Our results are similar to Planck 2015 constraints in [17]. We found no significant evidence for additional relativistic degree of freedom and fully thermalized massive sterile neutrinos by using current data sets in this paper. Nevertheless, a significant density of additional radiations is still allowed by considering uncertainties of the data.