# Constraints on the neutrino mass and mass hierarchy from cosmological observations

###### Abstract

Considering the mass splitting between three active neutrinos, we represent the new constraints on the sum of neutrino mass 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 Planck 2015 data of temperature anisotropy and polarizations of Cosmic Microwave Background (CMB), we find that the C.L. upper bounds on refer to eV for normal hierarchy (NH), eV for inverted hierarchy (IH) and eV for degenerate hierarchy (DH) respectively, and the normal hierarchy is slightly preferred than the inverted one (). In addition, the additional relativistic degrees of freedom and massive sterile neutrinos are neither favored at present.

###### pacs:

????## I Introduction

The phenomena of neutrino oscillation imply that there are mass splitting between three active neutrinos (see Lesgourgues:2006nd () 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 pdg ()

(1) | |||

(2) |

Thus we have two possible mass hierarchies, namely, a normal hierarchy (NH, ) and an inverted hierarchy (IH, ). Here () denote the mass eigenvalues of three neutrinos. The minimum sums of neutrino mass are eV for NH and 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 Hu:1997mj (); Hu:2001bc (); Komatsu:2008hk (); Jimenez:2010ev (); Reid:2009nq (); Thomas:2009ae (); Riemer-Sorensen:2013jsa (); Swanson:2010sk (); Cuesta:2015iho (); Palanque-Delabrouille:2015pga (); DiValentino:2015wba (); DiValentino:2015sam (); Rossi:2014nea (); Hou:2012xq (); Ade:2015xua (); Gerbino:2015ixa (); Zhang:2015rha (); Zhang:2015uhk (). 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 Hou:2012xq (). 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 Microwave Background (CMB), etc.

Planck collaboration Ade:2015xua () gave the C.L. upper bounds on the total mass of three active neutrinos by assuming a degenerate hierarchy (DH, where ) regardless of the mass splitting. The Planck TT+lowP constraint is eV and Planck TT,TE,EE+lowP constraint is eV for the CDM model. Here TT denotes the power spectrum of CMB temperature, EE denotes the power spectrum of CMB E-mode, and TE denotes the CMB temperature and E-mode cross correlation in the Planck 2015 data. “lowP” stands for Planck 2015 low- temperature-polarization data. Further adding the Planck 2015 CMB lensing data Ade:2015zua (), the constraints are slightly changed to eV and eV for two data combinations, respectively. However, by contrast, adding the Baryon Acoustic Oscillation (BAO) data including 6dFGS Beutler:2011hx (), MGS Ross:2014qpa (), BOSS DR11 CMASS Anderson:2013zyy () and LOWZ Anderson:2013zyy () can significantly improve the constraints to eV and 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 Cuesta:2015mqa () and power spectrum Gil-Marin:2015nqa () 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 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 and massive sterile neutrinos .

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 Cuesta:2015mqa () and power spectrum Gil-Marin:2015nqa () of CMASS and LOWZ galaxy samples from BOSS DR12 are listed in Tab. 1.

BOSS DR12 | ||||
---|---|---|---|---|

LOWZ | ||||

CMASS |

Only the consensus values Gil-Marin:2015nqa () are listed, which are used in this paper. Here denotes the effective redshift for CMASS and LOWZ samples, respectively, and are the Hubble parameter and angular diameter distance at reshift respectively, and is the comoving sound horizon at the redshift of baryon drag epoch. In addition, stands for the normalized correlation between and .

In this paper, we combine the BAO data including 6dFGS Beutler:2011hx (), MGS Ross:2014qpa (), BOSS DR12 CMASS Gil-Marin:2015nqa () and LOWZ Gil-Marin:2015nqa () with Planck 2015 likelihoods Aghanim:2015xee () 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) Heymans:2012gg (); Erben:2012zw (), redshift space distortion (RSD) Samushia:2013yga () and Planck cluster counts Ade:2013lmv (). 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 data Riess:2011yx (); Freedman:2012ny (); Efstathiou:2013via (). In addition, we do not use the data of supernovae of type Ia (SNe Ia), since the apparent magnitudes of SNe are insensitive to .

In the CDM model, there are six base cosmological parameters which are denoted by {,,,,}. Here is the physical density of baryons today and is the physical density of cold dark matter today. is the ratio between the sound horizon and the angular diameter distance at the decoupling epoch. is the Thomson scatter optical depth due to reionization. is the scalar spectrum index and is the amplitude of the power spectrum of primordial curvature perturbations at the pivot scale Mpc.

To constrain the neutrino sectors, we refer to the Markov Chain Monte Carlo sampler (CosmoMC) Lewis:2002ah () in the CDM 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 . The neutrino mass spectrum is

(3) |

and for NH, and

(4) |

and for IH. In addition, the neutrino mass spectrum is trivial for DH, namely

(5) |

Thus we can constrain the sum of neutrino mass via referring to the above three CDM model. It should be noted that there are lower cut-off values of which are eV for NH and 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 in Sec. III.1. In Sec. III.2, the relativistic degree of freedom is constrained. We simultaneously constrain and massive sterile neutrino in Sec. III.3.

### iii.1 Constraints on

In this subsection, we refer to two combinations of data sets, namely Planck TT,TE,EE+lowP+BAO and Planck TT+lowP+lensing+BAO, to constrain the sum of neutrino mass with NH, IH and DH, respectively. In the CDM model, the free cosmological parameters are given by

(6) |

where is the minimal eigenvalue of neutrino mass, and the total mass of neutrinos is a derived parameter, i.e. .

For three hierarchies, our constraints on as well as seven free parameters and can be found in Tab. 2.

Planck TT,TE,EE+lowP+BAO | PlanckTT+lowP+lensing+BAO | |||||

The likelihood distributions of and are depicted in Fig. 1.

The dashed lines denote constraints from Planck TT+lowP+lensing+BAO dataset while the solid lines denote constraints from Planck TT,TE,EE+lowP+BAO dataset. The red, black and blue lines denote constraints for the NH, IH and DH of neutrino mass spectrum, respectively. The grey dashed lines denote the minimum values for the total mass of three neutrinos for NH and IH, respectively.

For the DH, the C.L. upper limit on the total mass of three active neutrinos is eV for the data combination of Planck TT,TE,EE+lowP+BAO. The best-fit likelihoods is . Compared to Planck 2015 constraint eV in Ade:2015xua () from the Planck TT,TE,EE+lowP+BAO data where the BOSS DR11 CMASS and LOWZ data are used, there is about improvement on the uncertainty. The reason is that the total volume probed in BOSS DR12 has a increment and the experimental uncertainties are improved correspondingly. A conservative estimate is eV with the best-fit likelihood from the data combination of Planck TT+lowP+lensing+BAO.

For the NH, our constraint on is given by eV at C.L. from Planck TT,TE,EE+lowP+BAO dataset. It is around looser than the above constraint for the DH from the same dataset. The best-fit likelihoods for NH is which is slightly larger than that for DH. On the other hand, our constraint becomes eV at C.L. for Planck TT+lowP+lensing+BAO dataset, and the best-fit likelihood is . This constraint is similar to the constraint for DH by Planck TT+lowP+lensing+BAO dataset since this constraint is too loose to be sensitive to the neutrino mass hierarchy.

For the IH, our constraint on is given by eV at C.L. from Planck TT,TE,EE+lowP+BAO dataset. It is more than larger than that for the DH and about larger than that for the NH from the same dataset. The best-fit likelihoods is which is larger than that for NH by . It implies that the current data slightly prefers a normal hierarchy. On the other hand, our constraint becomes eV at C.L. for Planck TT+lowP+lensing+BAO dataset.

### iii.2 Constraints on

The total energy density of radiation in the Universe is given by

(7) |

where is the energy density of CMB photon and for counting the standard model neutrinos. 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 TT+lowP+lensing+BAO to constrain or equivalently the additional relativistic degree of freedom in the base CDM+ model. The free parameters include six base parameters and , while we fix eV with two massless and one massive active neutrinos. Our constraints on can be found in Tab. 3, where we also list constraints on other free parameters.

Planck TT,TE,EE+lowP+BAO | Planck TT+lowP+lensing+BAO | |||

CDM+ | CDM++ | CDM+ | CDM++ | |

– | – |

Our results are well consistent with the standard prediction . The constraints on the effective number of relativistic degrees of freedom are and at C.L. from Planck TT,TE,EE+lowP+BAO and Planck TT+lowP+lensing+BAO, respectively. , for example a fully thermalized sterile neutrino, is excluded at more than level by Planck TT,TE,EE+lowP+BAO data and at level by Planck TT+lowP+lensing+BAO data. A thermalized massless boson decoupled in the range MeV 100 MeV predicts which is disfavored at more than C.L. by these two data sets. If it decoupled at 100 MeV, which is slightly disfavored by Planck TT,TE,EE+lowP+BAO data but slightly favored by Planck TT+lowP+lensing+BAO data.

### iii.3 Simultaneous constraints on and

We can also consider extra one massive sterile neutrino whose effective mass is parametrized by eV. Assuming the sterile neutrino to be thermally distributed with an arbitrary temperature, is then given by

(8) |

where denotes the true mass. Here we consider the base CDM++ model, in which is a free parameter with a prior eV and has a flat prior with .

Our simultaneous constraints on and can be found in Tab. 3. From Planck TT,TE,EE+lowP+BAO data, we obtain constraints to be and eV at C.L.. From Planck TT+lowP+lensing+BAO data, we obtain and eV at C.L., which are similar to Planck 2015 results in Ade:2015xua (). can be excluded at much more than C.L.. One should note that the upper tail of 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 at C.L. is improved by about for the DH, comparing to Planck 2015 constraint at C.L. Ade:2015xua (). Meanwhile, we get updated C.L. upper limits for the NH and for the IH. For the NH (or the IH) and the DH, there is about (or ) 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 compared to the inverted hierarchy in our paper. Future precise observations might have potential to determine the neutrino mass and mass hierarchy Carbone:2010ik (); Wong:2011ip (); Hall:2012kg (); Hamann:2012fe (); Audren:2012vy (); Font-Ribera:2013rwa (); Abazajian:2013oma (); Wu:2014hta (); Mueller:2014dba (); Zhen:2015yba (); Villaescusa-Navarro:2015cca (); Errard:2015cxa (); Allison:2015qca (); Liu:2015txa (); Oyama:2015gma (); Zhao:2015gua ().

There are various tight constraints on in literatures. For instance, the combination of Lyman- absorption in the distant quasar spectra, BAO and Planck CMB data gave a constraint eV at C.L. in Palanque-Delabrouille:2015pga (). The combination of SDSS DR7 Luminous Red Galaxies (LRG), BAO and Planck CMB data gave an upper bound eV at C.L. in Cuesta:2015iho (). 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 Ade:2015xua (). 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.

Acknowledgments We acknowledge the use of HPC Cluster of SKLTP/ITP-CAS. This work is supported by Top-Notch Young Talents Program of China and grants from NSFC (grant NO. 11322545, 11335012 and 11575271). QGH would also like to thank the participants of the advanced workshop “Dark Energy and Fundamental Theory” supported by the Special Fund for Theoretical Physics from the National Natural Science Foundations of China (grant No. 11447613) for useful conversation.

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