# A White Paper on keV Sterile Neutrino Dark Matter

M. Drewes, T. Lasserre, A. Merle, S. Mertens, R. Adhikari, M. Agostini, N. Anh Ky, T. Araki, M. Archidiacono, M. Bahr, et al.

► To cite this version:

M. Drewes, T. Lasserre, A. Merle, S. Mertens, R. Adhikari, et al. A White Paper on keV Sterile Neutrino Dark Matter. Journal of Cosmology and Astroparticle Physics, 2017, 01, pp.025. 10.1088/1475-7516/2017/01/025. [hal-01553987](https://hal.science/hal-01553987v1)

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<figure><img src="image_1.png" /></figure>
# A White Paper on keV sterile neutrino Dark Matter

To cite this article: R. Adhikari *et al* JCAP01(2017)025

View the [article online](#) for updates and enhancements.

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# A White Paper on keV sterile neutrino Dark Matter

Editors: M. Drewes, T. Lasserre, A. Merle and S. Mertens

E-mail: marcodrewes@gmail.com, thierry.lasserre@cea.fr, amerle@mpp.mpg.de, smertens@lbl.gov

Authors: R. Adhikari,⁶¹ M. Agostini,⁸⁴ N. Anh Ky,³⁹,⁷³ T. Araki,⁵⁷

M. Archidiacono,³⁴ M. Bahr,⁷⁰ J. Baur,² J. Behrens,⁶⁹

F. Bezrukov,⁶⁴ P.S. Bhupal Dev,³¹ D. Borah,³⁵ A. Boyarsky,⁴⁵

A. de Gouvea,⁶² C.A. de S. Pires,³⁷ H.J. de Vega,⁹,† A.G. Dias,³⁶

P. Di Bari,³² Z. Djurcic,²¹ K. Dolde,⁷ H. Dorrer,⁸¹ M. Durero,²

O. Dragoun,⁷¹ M. Drewes,¹ G. Drexlin,³⁰ Ch.E. Düllmann,⁸¹,⁸³

K. Eberhardt,⁸¹ S. Eliseev,⁸⁶ C. Enss,⁵⁰ N.W. Evans,⁵³

A. Faessler,⁸⁵ P. Filianin,⁸⁶ V. Fischer,² A. Fleischmann,⁵⁰

J.A. Formaggio,²⁰ J. Franse,¹⁶ F.M. Fraenkle,⁷ C.S. Frenk,⁶³

G. Fuller,⁷⁵ L. Gastaldo,⁵⁰ A. Garzilli,¹⁶ C. Giunti,²² F. Glück,⁷,⁶⁶

M.C. Goodman,²¹ M.C. Gonzalez-Garcia,¹⁹ D. Gorbunov,⁶⁵,⁷²

J. Hamann,⁴⁰ V. Hannen,⁶⁹ S. Hannestad,³⁴ S.H. Hansen,³³

C. Hassel,⁵⁰ J. Heeck,¹¹ F. Hofmann,⁸⁰ T. Houdy,²,⁴ A. Huber,⁷

D. Iakubovskyi,⁸⁹,⁴³ A. Ianni,²⁷ A. Ibarra,¹ R. Jacobsson,⁸⁷

T. Jeltema,⁷⁶ J. Jochum,¹²,¹³ S. Kempf,⁵⁰ T. Kieck,⁸¹,⁸²

M. Korzeczek,⁷,² V. Kornoukhov,⁴² T. Lachenmaier,¹³ M. Laine,⁷⁴

P. Langacker,⁶⁶,⁶⁷ T. Lasserre,¹,²,³,⁴ J. Lesgourgues,¹⁵ D. Lhuillier,²

Y.F. Li,⁷⁷ W. Liao,⁷⁹ A.W. Long,⁹⁰ M. Maltoni,²⁶ G. Mangano,²⁴

N.E. Mavromatos,⁴⁴ N. Menci,⁵⁸ A. Merle,⁵ S. Mertens,⁶,⁷

A. Mirizzi,²⁵,⁴⁶ B. Monreal,⁷⁰ A. Nozik,⁶⁵,⁷² A. Neronov,⁴⁹ V. Niro,²⁶

Y. Novikov,⁵² L. Oberauer,¹ E. Otten,⁸² N. Palanque-Delabrouille,²

M. Pallavicini,²³ V.S. Pantuev,⁶⁵ E. Papastergis,⁵¹ S. Parke,⁷⁸

S. Pascoli,⁵⁵ S. Pastor,²⁸ A. Patwardhan,⁷⁵ A. Pilaftsis,⁵⁴

D.C. Radford,⁹¹ P.C.-O. Ranitzsch,⁶⁹ O. Rest,⁶⁹ D.J. Robinson,¹⁷

P.S. Rodrigues da Silva,³⁷ O. Ruchayskiy,⁸⁹,¹⁰ N.G. Sanchez,⁸

M. Sasaki,¹² N. Saviano,¹⁴,⁵⁵ A. Schneider,⁶⁰ F. Schneider,⁸¹,⁸²
T. Schwetz,³⁰ S. Schönert,<fnref n="1" /> S. Scholl,<fnref n="12" /> F. Shankar,³² R. Shrock,<fnref n="19" />

N. Steinbrink,⁶⁹ L. Strigari,⁵⁶ F. Suekane,⁴¹ B. Suerfu,⁶⁸

R. Takahashi,³⁸ N. Thi Hong Van,³⁹ I. Tkachev,⁶⁵ M. Totzauer,<fnref n="5" />

Y. Tsai,<fnref n="18" /> C.G. Tully,⁶⁸ K. Valerius,<fnref n="7" /> J.W.F. Valle,²⁸ D. Venos,⁷¹

M. Viel,⁴⁷,⁴⁸ M. Vivier,<fnref n="2" /> M.Y. Wang,⁵⁹ C. Weinheimer,⁶⁹

K. Wendt,⁸² L. Winslow,<fnref n="20" /> J. Wolf,<fnref n="7" /> M. Wurm,<fnref n="14" /> Z. Xing,⁷⁷

S. Zhou⁷⁷ and K. Zuber⁸⁸

<footnote n="1">¹Physik-Department and Excellence Cluster Universe, Technische Universität München, James-Franck-Str. 1, 85748 Garching, Germany</footnote>

<footnote n="2">²Commissariat à l'énergie atomique et aux énergies alternatives, Centre de Saclay, DSM/IRFU, 91191 Gif-sur-Yvette, France</footnote>

<footnote n="3">³Institute for Advance Study, Technische Universität München, James-Franck-Str. 1, 85748 Garching, Germany</footnote>

<footnote n="4">⁴AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/IRFU, Observatoire de Paris, Sorbonne Paris Cité, 75205 Paris Cedex 13, France</footnote>

<footnote n="5">⁵Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Foehringer Ring 6, 80805 München, Germany</footnote>

<footnote n="6">⁶Institute for Nuclear and Particle Astrophysics, Lawrence Berkeley Laboratory, Berkeley, CA 94720, U.S.A.</footnote>

<footnote n="7">⁷KCETA, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany</footnote>

<footnote n="8">⁸CNRS LERMA Observatoire de Paris, PSL, UPMC Sorbonne Universités, France</footnote>

<footnote n="9">⁹CNRS LPTHE UPMC Univ P. et M. Curie Paris VI, France</footnote>

<footnote n="10">¹⁰Ecole Polytechnique Federale de Lausanne, FSB/ITP/LPPC, BSP, CH-1015, Lausanne, Switzerland</footnote>

<footnote n="11">¹¹Service de Physique Théorique, Université Libre de Bruxelles, Boulevard du Triomphe, CP225, 1050 Brussels, Belgium</footnote>

<footnote n="12">¹²Institute for Astronomy and Astrophysics, Kepler Center for Astro and Particle Physics, University of Tübingen, Germany</footnote>

<footnote n="13">¹³Eberhard Karls Universität Tübingen, Physikalisches Institut, 72076 Tübingen, Germany</footnote>

<footnote n="14">¹⁴Institute of Physics and EC PRISMA, University of Mainz, Mainz Germany</footnote>

<footnote n="15">¹⁵Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen University, D-52056 Aachen, Germany</footnote>

<footnote n="16">¹⁶Lorentz Institute, Leiden University, Niels Bohrweg 2, Leiden, NL-2333 CA, The Netherlands</footnote>

<footnote n="17">¹⁷Department of Physics, University of California, Berkeley, CA 94720, U.S.A.</footnote>

<footnote n="18">¹⁸Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742, U.S.A.</footnote>

<footnote n="19">¹⁹C.N.Yang Institute for Theoretical Physics, SUNY at Stony Brook, Stony Brook, NY 11794-3840, U.S.A.</footnote>

<footnote n="20">²⁰Massachusetts Institute of Technology, U.S.A.</footnote>

<footnote n="21">²¹Argonne National Laboratory, Argonne, Illinois 60439, U.S.A.</footnote>

<footnote n="22">²²INFN, Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy</footnote>
²³Dipartimento di Fisica dell'Università di Genova,
via Dodecaneso 33, 16146 Genova, Italy

²⁴INFN, Sezione di Napoli, Monte S.Angelo, Via Cintia I-80126, Napoli, Italy

²⁵Dipartimento Interateneo di Fisica Michelangelo Merlin,
Via Amendola 173, 70126 Bari, Italy

²⁶Departamento de Física Teórica, Universidad Autónoma de Madrid and
Instituto de Física Teórica UAM/CSIC, Calle Nicolás Cabrera 13-15,
Cantoblanco, E-28049 Madrid, Spain

²⁷Laboratorio Subterráneo de Canfranc Paseo de los Ayerbe,
S/N 22880 Canfranc Estacion Huesca Spain

²⁸Instituto de Física Corpuscular (CSIC-Universitat de Valencia), Valencia, Spain

²⁹Instituto de Física Teórica UAM/CSIC, Calle de Nicolás Cabrera 13-15,
Universidad Autónoma de Madrid, Cantoblanco, E-28049 Madrid, Spain

³⁰Institut für Kernphysik, Karlsruher Institut für Technologie (KIT),
D-76021 Karlsruhe, Germany

³¹Consortium for Fundamental Physics, School of Physics and Astronomy,
University of Manchester, Manchester M13 9PL, U.K.

³²School of Physics and Astronomy, University of Southampton,
Southampton, SO17 1BJ, U.K.

³³Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen,
Juliane Maries Vej 30, 2100 Copenhagen, Denmark

³⁴Department of Physics and Astronomy, University of Aarhus,
DK-8000 Aarhus C, Denmark

³⁵Department of Physics, Indian Institute of Technology Guwahati,
Assam-781039, India

³⁶Centro de Ciências Naturais e Humanas, UFABC,
Av. dos Estados, 5001, 09210-580, Santo André, SP, Brazil

³⁷Departamento de Física, UFPB,
Caixa Postal 5008, 58051-970, João Pessoa, PB, Brazil

³⁸Graduate School of Science and Engineering, Shimane University, Japan

³⁹Institute of physics, Vietnam academy of science and technology,
10 Dao Tan, Ba Dinh, Hanoi, Vietnam

⁴⁰Sydney Institute for Astronomy, School of Physics, The University of Sydney,
NSW 2006, Australia

⁴¹RCNS, Tohoku University, Japan

⁴²ITEP, ul. Bol. Cheremushkinskaya, 25, 117218 Moscow, Russia

⁴³Bogolyubov Institute for Theoretical Physics,
14-b Metrologichna str., 03680 Kyiv, Ukraine

⁴⁴King's College London, Physics Department, Strand, London WC2R 2LS, U.K.

⁴⁵Universiteit Leiden - Instituut Lorentz for Theoretical Physics,
P.O. Box 9506, NL-2300 RA Leiden, Netherlands, Netherlands

⁴⁶Istituto Nazionale di Fisica Nucleare - Sezione di Bari,
Via Amendola 173, 70126 Bari, Italy

⁴⁷INAF/OATs Osservatorio Astronomico di Trieste, Via Tiepolo 11, 34143 Trieste, Italy

⁴⁸INFN / National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, Italy
<sup>49</sup>ISDC, Astronomy Department, University of Geneva,
Ch. d'Ecogia 16, Versoix, 1290, Switzerland

<sup>50</sup>Kirchhoff-Institute for Physics, Heidelberg University,
Im Neuenheimer Feld 227 D-69120 Heidelberg, Germany

<sup>51</sup>Kapteyn Astronomical Institute, University of Groningen,
Landleven 12, Groningen NL-9747AD, The Netherlands

<sup>52</sup>Petersburg Nuclear Physics Institute, 188300, Gatchina, Russia and
St. Petersburg State University, 199034 St. Petersburg, Russia

<sup>53</sup>Institute of Astronomy, Madingley Rd, Cambridge, CB3 0HA, U.K.

<sup>54</sup>School of Physics and Astronomy, University of Manchester,
Manchester M13 9PL, U.K.

<sup>55</sup>Institute for Particle Physics Phenomenology, Department of Physics,
Durham University, Durham DH1 3LE, U.K.

<sup>56</sup>Mitchell Institute for Fundamental Physics and Astronomy,
Texas A et M University, U.S.A.

<sup>57</sup>Department of physics, Saitama University,
Shimo-Okubo 255, 338-8570 Saitama, Sakura-ku, Japan

<sup>58</sup>INAF-Osservatorio Astronomico di Roma,
via di Frascati 33, 00040 Monte Porzio Catone, Italy

<sup>59</sup>Department of Physics and Astronomy, Mitchell Institute for Fundamental Physics and Astronomy, Texas A et M University, College Station, TX 77843-4242, U.S.A.

<sup>60</sup>Institute for Computational Science, University of Zurich, 8057 Zurich, Switzerland

<sup>61</sup>Centre for Theoretical Physics, Jamia Millia Islamia (Central University),
New Delhi-110025, India

<sup>62</sup>Northwestern University, U.S.A.

<sup>63</sup>Institute for Computational Cosmology, Durham University, U.K.

<sup>64</sup>University of Connecticut, U.S.A.

<sup>65</sup>Institute for Nuclear Research, Russian Academy of Sciences, Moscow, 117312, Russia

<sup>66</sup>Wigner Research Center for Physics, Budapest, Hungary

<sup>67</sup>Institute for Advanced Study, Princeton, NJ 08540, U.S.A.

<sup>68</sup>Princeton University, Princeton, NJ 08542, U.S.A.

<sup>69</sup>Westfälische Wilhelms Universität Münster, Institut für Kernphysik,
Wilhelm Klemm-Str.9, D-48149 Münster, Germany

<sup>70</sup>University of California, Santa Barbara, U.S.A.

<sup>71</sup>Nuclear Physics Institute, ASCR, CZ-25068 Rez near Prague, Czech Republic

<sup>72</sup>MIPT, Institutskiy per. 9, Dolgoprudny, Moscow Region, 141700, Russia

<sup>73</sup>Laboratory for high energy physics and cosmology, Faculty of physics,
VNU university of science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

<sup>74</sup>ITP, AEC, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland

<sup>75</sup>Department of Physics, University of California, San Diego,
La Jolla, California 92093-0319, U.S.A.

<sup>76</sup>University of California, Santa Cruz, U.S.A.

<sup>77</sup>Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China
⁷⁸Fermi National Accelerator Laboratory, U.S.A.

⁷⁹Institute of Modern Physics, School of Sciences, East China University of Science and Technology, 130 Meilong Road, Shanghai, China

⁸⁰Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstrasse, 85748 Garching, Germany

⁸¹Institut für Kernchemie, Johannes Gutenberg-Universität, 55099 Mainz, Germany

⁸²Institut für Physik, Johannes Gutenberg-Universität, 55099 Mainz, Germany

⁸³GSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany

⁸⁴Gran Sasso Science Institute (INFN), L’Aquila, Italy

⁸⁵Institute of Theoretical Physics, University of Tübingen, 72076 Tübingen, Germany

⁸⁶Max-Planck-Institut für Kernphysik, 69117 Heidelberg, Germany

⁸⁷European Organization for Nuclear Research (CERN), Geneva, Switzerland

⁸⁸Institut für Kern- und Teilchenphysik, TU Dresden, Germany

⁸⁹Discovery Center, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark

⁹⁰Kavli Institute for Cosmological Physics, The University of Chicago, Chicago, Illinois 60637, U.S.A.

⁹¹Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A.

†Deceased.

Received April 26, 2016

Revised October 20, 2016

Accepted October 26, 2016

Published January 13, 2017

**Abstract.** We present a comprehensive review of keV-scale sterile neutrino Dark Matter, collecting views and insights from all disciplines involved — cosmology, astrophysics, nuclear, and particle physics — in each case viewed from both theoretical and experimental/observational perspectives. After reviewing the role of active neutrinos in particle physics, astrophysics, and cosmology, we focus on sterile neutrinos in the context of the Dark Matter puzzle. Here, we first review the physics motivation for sterile neutrino Dark Matter, based on challenges and tensions in purely cold Dark Matter scenarios. We then round out the discussion by critically summarizing all known constraints on sterile neutrino Dark Matter arising from astrophysical observations, laboratory experiments, and theoretical considerations. In this context, we provide a balanced discourse on the possibly positive signal from X-ray observations. Another focus of the paper concerns the construction of particle physics models, aiming to explain how sterile neutrinos of keV-scale masses could arise in concrete settings beyond the Standard Model of elementary particle physics. The paper ends with an extensive review of current and future astrophysical and laboratory searches, highlighting new ideas and their experimental challenges, as well as future perspectives for the discovery of sterile neutrinos.

**Keywords:** cosmological neutrinos, dark matter experiments, dark matter theory, particle physics - cosmology connection

**ArXiv ePrint:** 1602.04816
# Contents

1 Executive summary 1

2 Neutrinos in the Standard Model of particle physics and beyond 4

2.1 Introduction: massive neutrinos and lepton mixing (author: S. Parke) 4

2.2 Current status of three-neutrino masses and mixings (authors: M.C. Gonzalez-Garcia, M. Maltoni, T. Schwetz, R. Shrock) 6

2.2.1 Neutrino oscillations 6

2.2.2 Absolute neutrino mass measurements 11

2.3 Open questions in neutrino physics (author: A. de Gouvêa) 13

2.4 Sterile neutrinos — general introduction (author: P. Langacker) 15

2.5 The seesaw mechanism (author: M. Drewes) 19

2.5.1 Possible origins of neutrino mass 19

2.5.2 The seesaw mechanism 20

3 Neutrinos in the Standard Model of cosmology and beyond 23

3.1 The standard model of cosmology (author: J. Hamann) 23

3.1.1 Geometry 23

3.1.2 Energy content 24

3.1.3 Parameters of base ΛCDM 24

3.1.4 The cosmological standard model vs. observations 25

3.1.5 Internal consistency 25

3.1.6 External consistency 26

3.1.7 Occam's razor 27

3.2 Active neutrinos in cosmology (authors: J. Lesgourgues, S. Pastor) 27

3.2.1 The cosmic neutrino background 27

3.2.2 The effective number of neutrinos 28

3.2.3 Massive neutrinos as Dark Matter 28

3.2.4 Effects of standard neutrinos on cosmology 29

3.2.5 Current cosmological bounds on standard neutrinos 30

3.3 Big bang nucleosynthesis (author: G. Mangano) 32

3.3.1 What it is and how it works 32

3.3.2 Constraints on the baryon density and $N_{\text{eff}}$ 33

3.4 Sterile neutrinos in cosmology 34

3.4.1 eV-scale (authors: M. Archidiacono, N. Saviano) 34

3.4.2 keV-scale (authors: A. Boyarsky, O. Ruchaisky) 38

3.4.3 MeV-scale (authors: S. Pascoli, N. Saviano) 41

3.4.4 GeV–TeV-scale (authors: A. Ibarra) 43

3.4.5 Leptogenesis (author: P. Di Bari) 44

4 Dark Matter at galactic scales: observational constraints and simulations 47

4.1 Astrophysical clues to the identity of the Dark Matter (author: C. Frenk) 47

4.2 Missing dwarf galaxies (author: N. Menci) 49

4.3 Inner density profiles of small galaxies and the cusp-core problem (author: W. Evans) 52
<table><tr><td>4.4</td><td>Too-big-to-fail (author: E. Papastergis)</td><td>54</td></tr><tr><td>4.4.1</td><td>Introduction and background</td><td>54</td></tr><tr><td>4.4.2</td><td>Possible solutions of the TBTF problem within ΛCDM</td><td>57</td></tr><tr><td>4.4.3</td><td>Can Warm Dark Matter solve the TBTF problem?</td><td>58</td></tr><tr><td>4.5</td><td>The kinematics and formation of subhaloes in Warm Dark Matter simulations (authors: M. Wang, L. Strigari)</td><td>59</td></tr></table>

## 5 Observables related to keV neutrino Dark Matter 62

5.1 Phase space analysis (author: D. Gorbunov) 62

5.2 Lyman-α forest constraints (author: M. Viel) 65

5.3 X-ray observations (authors: O. Ruchayskiy, T. Jeltema, A. Neronov, D. Iakubovskyi) 68

5.3.1 X-ray signals — overview 68

5.3.2 3.5 keV line 72

5.3.3 Other line candidates in keV range 78

5.4 Laboratory constraints (author: O. Dragoun) 78

## 6 Constraining keV neutrino production mechanisms 83

6.1 Thermal production: overview (authors: M. Drewes, G. Fuller, A.V. Patwardhan) 83

6.1.1 Motivation 83

6.1.2 Active-sterile neutrino oscillations 84

6.1.3 De-cohering scatterings 86

6.1.4 MSW-effect and resonant conversion 86

6.2 Thermal production: state of the art (authors: M. Drewes, M. Laine) 90

6.2.1 Examples of complete frameworks 90

6.2.2 Matter potentials and active neutrino interaction rate 94

6.2.3 Open questions 96

6.3 Production by particle decays (authors: F. Bezrukov, A. Merle, M. Totzauer) 97

6.3.1 Decay in thermal equilibrium 98

6.3.2 Production from generic scalar singlet decays 101

6.4 Dilution of thermally produced DM (author: F. Bezrukov) 104

## 7 keV neutrino theory and model building (particle physics) 107

7.1 General principles of keV neutrino model building (authors: A. Merle, V. Niro) 107

7.2 Models based on suppression mechanisms 108

7.2.1 The split seesaw mechanism and its extensions (author: R. Takahashi) 108

7.2.2 Suppressions based on the Froggatt-Nielsen mechanism (authors: A. Merle, V. Niro) 111

7.2.3 The minimal radiative inverse seesaw mechanism (authors: A. Pilaftsis, B. Dev) 113

7.2.4 Models based on loop-suppressions (authors: D. Borah, R. Adhikari) 115

7.3 Models based on symmetry breaking 118

7.3.1 $L_e - L_\mu - L_\tau$ symmetry (authors: A. Merle, V. Niro) 118

7.3.2 $Q_6$ symmetry (author: T. Araki) 120

7.3.3 $A_4$ symmetry (author: A. Merle) 122

7.4 Models based on other principles 124
7.4.1 Extended seesaw (author: J. Heeck) 124

7.4.2 Dynamical mass generation and composite neutrinos (authors: D. Robinson, Y. Tsai) 126

7.4.3 3-3-1-models (authors: A.G. Dias, N. Anh Ky, C.A. de S. Pires, P.S. Rodrigues da Silva, N. Thi Hong Van) 129

7.4.4 Anomalous Majorana neutrino masses from torsionful quantum gravity (authors: N. Mavromatos, A. Pilaftsis) 131

8 Current and future keV neutrino search with astrophysical experiments 135

8.1 Previous bounds (authors: A. Boyarsky, J. Franse, A. Garzilli, D. Iakubovskyi) 135

8.2 X-ray telescopes and observation of the 3.55 keV line (authors: A. Boyarsky, J. Franse, A. Garzilli, D. Iakubovskyi) 136

8.3 Lyman-α methods for keV-scale Dark Matter (authors: A. Boyarsky, J. Franse, A. Garzilli, D. Iakubovskyi) 139

8.4 Pulsar kicks (author: S. Hansen) 139

8.5 Supernovae (authors: S. Hansen and S. Zhou) 140

8.5.1 The vacuum limit 141

8.5.2 Matter effects 142

9 Current and future keV neutrino search with laboratory experiment 144

9.1 Introduction 144

9.2 Tritium beta decay experiments (author: S. Mertens) 144

9.2.1 The Troitsk experiment (authors: V.S. Pantuev, I.I. Tkachev, A.A. Nozik) 145

9.2.2 The KATRIN experiment (authors: S. Mertens, J. Behrens, K. Dolde, V. Hannen, A. Huber, M. Korcekzek, T. Lasserre, D. Radford, P.C.-O. Ranitzsch, O. Rest, N. Steinbrink, C. Weinheimer) 148

9.2.3 The project 8 experiment (author: B. Monreal) 153

9.2.4 PTOLEMY experiment (authors: B. Suerfu, C.G. Tully) 154

9.2.5 Full kinematic reconstruction of the beta decay (authors: F. Bezrukov, E. Otten) 159

9.3 Electron capture experiments (author: L. Gastaldo) 163

9.3.1 The electron capture in ¹⁶³Ho experiment ECHo (authors: L. Gastaldo, T. Lasserre, A. Faessler) 166

9.3.2 Other nuclides from the electron capture sector (authors: L. Gastaldo, Y. Novikov) 171

9.4 Direct detection 175

9.4.1 Direct detection via inverse β decay (authors: Y. Li, W. Liao, and Z. Xing) 176

9.4.2 Prospects for sterile neutrino dark matter direct detection (author: A.J. Long) 178

9.5 Search for heavy sterile neutrinos with SHiP (author: R. Jacobsson on behalf of SHiP) 180

10 Discussion — pro and cons for keV neutrino as Dark Matter and perspectives 184
# 1 Executive summary

Despite decades of searching, the nature and origin of Dark Matter (DM) remains one of the biggest mysteries in modern physics. Astrophysical observations over a vast range of physical scales and epochs clearly show that the movement of celestial bodies, the gravitational distortion of light and the formation of structures in the Universe cannot be explained by the known laws of gravity and observed matter distribution [1-7]. They can, however, be brought into very good agreement if one postulates the presence of large amounts of non-luminous DM in and between the galaxies, a substance which is much more abundant in the Universe than ordinary matter [1]. Generic ideas for what could be behind DM, such as Massive Compact Halo Objects (MACHOs) [8-11] are largely ruled out [12, 13] or at least disfavored [14, 15]. Alternative explanations based on a modification of the law of gravity [16] have not been able to match the observations on various different scales. Thus, the existence of one or several new elementary particles appears to be the most attractive explanation.

As a first step, the suitability of known particles within the well-tested Standard Model (SM) has been examined. Indeed, the neutral, weakly interacting, massive neutrino could in principle be a DM candidate. However, neutrinos are so light that even with the upper limit for their mass [17, 18] they could not make up all of the DM energy density [19]. Moreover, neutrinos are produced with such large (relativistic) velocities that they would act as hot DM (HDM), preventing the formation of structures such as galaxies or galaxy clusters [20].

Consequently, explaining DM in terms of a new elementary particle clearly requires physics beyond the SM. There are multiple suggested extensions to the SM, providing a variety of suitable DM candidates, but to date there is no clear evidence telling us which of these is correct. Typically, extensions of the SM are sought at high energies, resulting in DM candidates with masses above the electroweak scale. In fact, there is a class of good DM candidates available at those scales, which are called Weakly Interacting Massive Particles (WIMPs). If these particles couple with a strength comparable to the SM weak interaction, they would have been produced in the early Universe via thermal freeze-out in suitable amounts [21]<fnref n="1" /> WIMPs generically avoid the structure formation problem associated with SM neutrinos, as they are much more massive and therefore non-relativistic at the time of galaxy formation, because their velocities have been considerably redshifted.<fnref n="2" /> That is, WIMPs act as cold DM (CDM). Typical examples for WIMPs are neutralinos as predicted by supersymmetry [22-25] or Kaluza-Klein bosons as predicted by models based on extra spatial dimensions [26-29]. More minimal extensions of the SM also predict WIMPs, e.g. an inert scalar doublet [30, 31].

One of the advantages of WIMPs is that there is a variety of ways to test their existence. WIMPs could annihilate in regions of sufficiently high density, such as the center of a galaxy, thereby producing detectable (indirect) signals [32] in e.g. photons, antimatter, or neutrinos. The same interactions that are responsible for the annihilation of two WIMPs in outer space can also be responsible for their production at colliders [33] or their scattering with ordinary matter in direct search experiments [34].<fnref n="3" /> While a lot of experiments are currently taking data, no conclusive evidence for WIMPs has yet been found. Direct searches keep on pushing the limit on DM-matter cross sections towards smaller and smaller values [35-37], indirect

<footnote n="1">¹Note that this is true independently of the WIMP mass — up to logarithmic corrections — as long as they freeze out cold, since the main dependence on the mass drops out in the formula for the DM abundance [22].</footnote>

<footnote n="2">²Note that, of course, also other reasons can be responsible for DM having non-relativistic speed such as, e.g., strong interactions of condensation.</footnote>

<footnote n="3">³At the level of amplitudes, this relation between “break it”, “make it” and “shake it” can be visualized by rotating the Feynman diagram in steps of 90 degrees.</footnote>
searches yield some interesting but still inconclusive hints [38–40], and as of today the LHC has not discovered a hint of a DM-like particle [41–44]. WIMPs are certainly not yet excluded, nevertheless the current experimental results suggest the thorough exploration of alternative DM candidates.

A seemingly unrelated issue arose recently in N-body simulations of cosmological structure formation. Advanced simulations [45] revealed some discrepancies between purely CDM scenarios and observations at small scales (a few 10 kpc or smaller). For example, there seem to be too few dwarf satellite galaxies observed compared to simulations (the missing satellite problem) [46, 47]; the density profile of galaxies is observed to be cored, whereas simulations predict a cusp profile (the cusp-core problem) [48, 49] and, finally, the observed dwarf satellite galaxies seem to be less dense than expected. This could possibly be explained if larger and very dense galaxies exist but are invisible due to a suppression of star formation [50–52]. However, no mechanism is known to suppress star formation in these types of galaxies: they are too big to fail producing enough stars (too-big-to-fail problem) [53, 54].

While the discrepancy between simulation and observation is apparent, its origin is not so clear. A natural possibility would be that earlier simulations did not include baryons, although we clearly know they exist. The full inclusion of baryons and their interactions is highly non-trivial and only recently has it been attempted [55, 56]. Another source for the discrepancy could arise from astrophysical feedback effects [50, 51]. These include, for example, relatively large supernova rates in dwarf galaxies which could wipe out all the visible material so that many dwarfs are simply invisible [52]. Finally, it could also be that the DM velocity spectrum is not as cold as assumed [57]. It has been shown that a *warm* DM (WDM) spectrum can significantly affect structure formation and strongly reduce the build-up of small objects [58]. Even more generally, the DM spectrum need not be thermal at all. It could have various shapes depending on the production mechanism (see section 5) and thereby influence structure formation in non-trivial ways. Thus, DM may be not simply *cold*, *warm*, or *hot*, but the spectra could be more complicated resembling, e.g., mixed scenarios [59]. In any case, resolving the small-scale structure problem by modifying the DM spectrum would require a new DM candidate.

The candidate particle discussed in this White Paper is a *sterile neutrino with a keV-scale mass*. A sterile neutrino is a hypothetical particle which, however, is connected to and can mix with the known active neutrinos. In SM language, sterile neutrinos are right-handed fermions with zero hypercharge and no color, i.e., they are total singlets under the SM gauge group and thus perfectly neutral. These properties allow sterile neutrinos to have a mass that does not depend on the Higgs mechanism. This so-called Majorana mass [60] can exist independently of electroweak symmetry breaking, unlike the fermion masses in the SM. In particular, the Majorana mass can have an arbitrary scale that is very different from all other fermion masses. Typically, it is assumed to be very large, but in fact it is just unrelated to the electroweak scale and could also be comparatively small. Observationally and experimentally the magnitude of the Majorana mass is almost unconstrained [61–74].

Depending on the choice of the Majorana mass, the implications for particle physics and cosmology are very different, see e.g. [67]. Two reasons motivate a keV mass scale for a sterile neutrino DM candidate. First, fermionic DM can not have an arbitrarily small mass, since in dense regions (e.g. in galaxy cores) it cannot be packed within an infinitely small volume, due to the Pauli principle. This results in a lower bound on the mass, the so-called Tremaine-Gunn bound [75]. Second, sterile neutrinos typically have a small mixing with the active neutrinos, which would enable a DM particle to decay into an active neutrino and a
mono-energetic photon. Since the decay rate scales with the fifth power of the initial state mass, a non-observation of the corresponding X-ray peak leads to an upper bound of a few tens of keV.<fnref n="4" /> It is these two constraints, the phase space and X-ray bounds, which enforce keV-scale masses for sterile neutrinos acting as DM.

This White Paper attempts to shed light on sterile neutrino DM from *all* perspectives: astrophysics, cosmology, nuclear, and particle physics, as well as experiments, observations, and theory. Progress in the question of sterile neutrino DM requires expertise from all these different areas. The goal of this document is thus to advance the field by stimulating fruitful discussions between these communities. Furthermore, it should provide a comprehensive compendium of the current knowledge of the topic, and serve as a future reference.<fnref n="5" /> The list of authors indicates that there is great interest in the subject among scientists from many areas of physics.

This White Paper is laid out as follows. First, sterile neutrinos are introduced from the particle physics (section 1) and cosmology/astrophysics (section 2) perspectives. section 3 reviews the current tensions of CDM simulations with small-scale structure observations, and discusses attempts to tackle them. section 4 gives a comprehensive summary of current constraints on keV sterile neutrino DM, arising from all accessible observables. The different sterile neutrino DM production mechanisms in the early Universe, and how they are constrained by astrophysical observations, are treated in section 5. section 6 turns to particle physics by reviewing attempts to explain or motivate the keV mass scale in various scenarios of physics beyond the SM. Current and future astrophysical and laboratory searches are discussed in sections 7 and 8, respectively, highlighting new ideas, their experimental challenges, and future perspectives for the discovery or exclusion of sterile neutrino DM. We end by giving an overall conclusion, involving all the viewpoints discussed in this paper.

Let us now start our journey into the fascinating world of keV sterile neutrino DM and address one of the biggest questions in modern science:

*What is Dark Matter and where did it come from?*

**Note added.** Several sections in the White Paper make reference to Japan's space-borne Astro-H/Hitomi X-ray observatory. The Japanese Space Agency (JAXA) successfully launched the Astro-H satellite from Tanegashima Space Center in Japan on the 16th of February 2016, but after an apparent break off of bigger parts of the satellite occuring on March 26th, it was finally decided on April 28th to give up on the spacecraft. We give a short wrap-up of the events in the paragraph right before section 8.3. More detailed information can be found on the JAXA webpage, http://global.jaxa.jp/projects/sat/astro_h/.

<footnote n="4">⁴This only holds if active-sterile mixing is not switched off or forbidden, which may be the case in certain scenarios, see section 6.</footnote>

<footnote n="5">⁵The reader should be warned that the texts contributed to this work by the different authors cannot treat the various topics in full detail. They should, however, serve as possible overview and we made a great effort to ensure that they do contain all the relevant references, so that the present White Paper can guide the inclined reader to more specific information.</footnote>
## 2 Neutrinos in the Standard Model of particle physics and beyond

Section Editors:
Carlo Giunti, André de Gouvea

The existence of sterile neutrinos is an exciting possible manifestation of new physics beyond the standard scenario of three-neutrino mixing, which has been established by the observation of neutrino flavor oscillations in many solar, reactor, and accelerator experiments (see the recent reviews in refs. [76, 77]). Sterile neutrinos [78] are observable through their mixing with the active neutrinos. In this section we present a brief introduction to the standard theory of three-neutrino mixing in subsection 2.1 and a summary of its current phenomenological status in subsection 2.2. In subsection 2.3 we summarize the open questions in neutrino physics and in subsection 2.4 we present a general introduction to sterile neutrinos.

### 2.1 Introduction: massive neutrinos and lepton mixing (author: S. Parke)

In the Standard Model (SM), as constructed around 1970, the neutrinos, ($\nu_e, \nu_\mu, \nu_\tau$), are massless and interact diagonally in flavor, as follows

$$ 
\begin{align} 
W^+ \to e^+ + \nu_e, & \qquad W^- \to e^- + \bar{\nu}_e, & Z & \to \nu_e + \bar{\nu}_e, \nonumber \\ 
W^+ \to \mu^+ + \nu_\mu, & \qquad W^- \to \mu^- + \bar{\nu}_\mu, & Z & \to \nu_\mu + \bar{\nu}_\mu, \tag{2.1} \\ 
W^+ \to \tau^+ + \nu_\tau, & \qquad W^- \to \tau^- + \bar{\nu}_\tau, & Z & \to \nu_\tau + \bar{\nu}_\tau. \nonumber 
\end{align} 
$$

Since they travel at the speed of light, their character cannot change from production to detection. Therefore, in flavor terms, massless neutrinos are relatively uninteresting compared to quarks.

Since then many experiments have seen neutrino flavor transitions, therefore neutrinos must have a mass and, like the quarks, there is a mixing matrix relating the neutrino flavor states, $\nu_e, \nu_\mu, \nu_\tau$, with the mass eigenstates, $\nu_1, \nu_2, \nu_3$:

$$ |\nu_{\alpha}\rangle = \sum_{j=1}^{3} U_{\alpha j} |\nu_{j}\rangle \quad (\alpha = e, \mu, \tau), \qquad (2.2) $$

where the mixing matrix $U$ is unitary and referred to as the PMNS$^6$ matrix. By convention, the mass eigenstates are labeled such that $|U_{e1}|^2 > |U_{e2}|^2 > |U_{e3}|^2$, which implies that

$$ \nu_1 \text{ component of } \nu_e > \nu_2 \text{ component of } \nu_e > \nu_3 \text{ component of } \nu_e. $$

With this choice of labeling of the neutrino mass eigenstates, the solar neutrino oscillations/transformations are governed by $\Delta m^2_{21} \equiv m^2_2 - m^2_1$, as these two are electron neutrino rich, and the atmospheric neutrino oscillations by $\Delta m^2_{31}$ and $\Delta m^2_{32}$. The SNO experiment [83] determined the mass ordering of the solar pair, $\nu_1$ and $\nu_2$, such that $m^2_2 > m^2_1$, i.e. $\Delta m^2_{21} > 0$. The atmospheric neutrino mass ordering,

$$ m_3^2 > m_2^2 \text{ or } m_3^2 < m_1^2, \qquad (2.3) $$

is still to be determined, see figure 1. If $m_3^2 > m_2^2$, the ordering is known as the normal ordering (NO), whereas if $m_3^2 < m_1^2$ the ordering is known as the inverted ordering (IO).

<footnote>$^6$Pontecorvo-Maki-Nakagawa-Sakata [79–82].</footnote>
<figure><img src="image_2.png" /><figcaption>**Figure 1.** What is known about the square of the neutrino masses for the two atmospheric mass orderings.</figcaption></figure>

The mass splittings of the neutrinos are approximately [84]:

$$ \Delta m_{32}^2 \approx \pm 2.5 \times 10^{-3} \, \text{eV}^2 \quad \text{and} \quad \Delta m_{21}^2 \approx +7.5 \times 10^{-5} \, \text{eV}^2, \qquad (2.4) $$

and the sum of the masses of the neutrinos satisfies

$$ \sqrt{\delta m_A^2} \approx 0.05 \, \text{eV} < \sum_{i=1}^{3} m_i < 0.5 \, \text{eV}. \qquad (2.5) $$

So the sum of neutrino masses ranges from $10^{-7}$ to $10^{-6}$ times $m_e$, however the mass of the lightest neutrino, $m$, could be very small. If $m \ll \sqrt{\delta m_\odot^2} \sim 0.01 \, \text{eV}^2$, then this is an additional scale to be explained by a theory of neutrino masses and mixings.

The standard representation [85] of the PMNS mixing matrix is given as follows:

$$ \begin{aligned} U &= \begin{pmatrix} U_{e1} & U_{e2} & U_{e3} \\ U_{\mu1} & U_{\mu2} & U_{\mu3} \\ U_{\tau1} & U_{\tau2} & U_{\tau3} \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & c_{23} & s_{23} \\ 0 & -s_{23} & c_{23} \end{pmatrix} \begin{pmatrix} c_{13} & 0 & s_{13}e^{-i\delta} \\ 0 & 1 & 0 \\ -s_{13}e^{i\delta} & 0 & c_{13} \end{pmatrix} \begin{pmatrix} c_{12} & s_{12} & 0 \\ -s_{12} & c_{12} & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} e^{i\alpha_1} & 0 & 0 \\ 0 & e^{i\alpha_2} & 0 \\ 0 & 0 & 1 \end{pmatrix} \\ &= \begin{pmatrix} c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta} \\ -s_{12}c_{23} - c_{12}s_{13}s_{23}e^{i\delta} & c_{12}c_{23} - s_{12}s_{13}s_{23}e^{i\delta} & c_{13}s_{23} \\ s_{12}s_{23} - c_{12}s_{13}c_{23}e^{i\delta} & -c_{12}s_{23} - s_{12}s_{13}c_{23}e^{i\delta} & c_{13}c_{23} \end{pmatrix} \begin{pmatrix} e^{i\alpha_1} & 0 & 0 \\ 0 & e^{i\alpha_2} & 0 \\ 0 & 0 & 1 \end{pmatrix}, \end{aligned} \qquad (2.6) $$

where $s_{ij} = \sin\theta_{ij}$ and $c_{ij} = \cos\theta_{ij}$. The Dirac phase, $\delta$, allows for the possibility of CP violation in the neutrino oscillation appearance channels. The Majorana phases $\alpha_1$ and $\alpha_2$ are unobservable in oscillations since oscillations depend on $U_{\alpha i}^* U_{\beta i}$ but they have observable, CP conserving effects, in neutrinoless double beta decay. If the neutrinos are Dirac, then neutrinoless double beta decay will be absent and the Majorana phases in the PMNS matrix
<figure><img src="image_3.png" /><figcaption>**Figure 2.** The flavor content of the neutrino mass eigenstates (figure similar to figure 1 in ref. [87]). The width of the lines is used to show how these fractions change as $\cos \delta$ varies from -1 to +1. Of course, this figure must be the same for neutrinos and anti-neutrinos, if CPT is conserved.</figcaption></figure>

are non-physical and can be set to zero. Note that there is some arbitrariness involved in which parameter combinations are called the physical phases, which is the reason why the “distribution” of the phases in eq. (2.6) looks a little asymmetric. This can be avoided when using the symmetric parametrization instead [86].

The approximate values of the mixing parameters are as follows:

$$ \sin^2 \theta_{13} \equiv |U_{e3}|^2 \approx 0.02, \qquad (2.7) $$

$$ \sin^2 \theta_{12} \equiv |U_{e2}|^2/(1 - |U_{e3}|^2) \approx 1/3, \qquad (2.8) $$

$$ \sin^2 \theta_{23} \equiv |U_{\mu3}|^2/(1 - |U_{e3}|^2) \approx 1/2, \qquad (2.9) $$

$$ 0 \le \delta < 2\pi. \qquad (2.10) $$

More precise values will be given in the next section. These mixing angles and mass splittings are summarized in figure 2, which also shows the dependence of the flavor fractions on the CP violating Dirac phase $\delta$.

## 2.2 Current status of three-neutrino masses and mixings (authors: M.C. Gonzalez-Garcia, M. Maltoni, T. Schwetz, R. Shrock)

### 2.2.1 Neutrino oscillations

Thanks to remarkable discoveries by a number of neutrino oscillation experiments it is now an established fact that neutrinos have mass and that leptonic flavors are not symmetries of Nature [78, 88]. Historically neutrino oscillations were first observed in the disappearance of solar $\nu_e$'s and atmospheric $\nu_\mu$'s which could be interpreted as flavor oscillations with two very different wavelengths. Over the last 15 years, these effects were confirmed also by terrestrial experiments using man-made beams from accelerators and nuclear reactors (see ref. [89] for an overview). In brief, at present we have observed neutrino oscillation effects in:
* atmospheric neutrinos, in particular in the high-statistics results of Super-Kamiokande [90];

* event rates of solar neutrino radiochemical experiments Chlorine [91], GALLEX/GNO [92], and SAGE [93], as well as time- and energy-dependent rates from the four phases in Super-Kamiokande [94–97], the three phases of SNO [98], and Borexino [99, 100];

* disappearance results from accelerator long-baseline (LBL) experiments in the form of the energy distribution of $\nu_\mu$ and $\bar{\nu}_\mu$ events in MINOS [101] and T2K [102];

* LBL $\nu_e$ appearance results for both neutrino and antineutrino events in MINOS [103] and $\nu_e$ appearance in T2K [104];

* reactor $\bar{\nu}_e$ disappearance at medium baselines in the form of the energy distribution of the near/far ratio of events at Daya Bay [105] and RENO [106] and the energy distribution of events in the near Daya Bay [107] and RENO [108] detectors and in the far Daya Bay [107], RENO [108] and Double Chooz [109, 110] detectors.

* the energy spectrum of reactor $\bar{\nu}_e$ disappearance at LBL in KamLAND [111].

This wealth of data can be consistently described by assuming mixing among the three known neutrinos ($\nu_e$, $\nu_\mu$, $\nu_\tau$), which can be expressed as quantum superpositions of three massive states $\nu_i$ ($i = 1, 2, 3$) with masses $m_i$. As explained in the previous section this implies the presence of a leptonic mixing matrix in the weak charged current interactions which can be parametrized in the standard representation, see eq. (2.6).

In this convention, disappearance of solar $\nu_e$'s and long-baseline reactor $\bar{\nu}_e$'s dominantly proceed via oscillations with wavelength $\propto E/\Delta m_{21}^2$ ($\Delta m_{ij}^2 \equiv m_i^2 - m_j^2$ and $\Delta m_{21}^2 \ge 0$ by convention) and amplitudes controlled by $\theta_{12}$, while disappearance of atmospheric and LBL accelerator $\nu_\mu$'s dominantly proceed via oscillations with wavelength $\propto E/|\Delta m_{31}^2|$ $\ll E/\Delta m_{21}^2$ and amplitudes controlled by $\theta_{23}$. Generically $\theta_{13}$ controls the amplitude of oscillations involving $\nu_e$ flavor with $E/|\Delta m_{31}^2|$ wavelengths. So, given the observed hierarchy between the solar and atmospheric wavelengths, there are two possible non-equivalent orderings for the mass eigenvalues, which are conventionally chosen as:

$$m_1 < m_2 < m_3 \quad \text{with} \quad \Delta m_{21}^2 \ll (\Delta m_{32}^2 \approx \Delta m_{31}^2 > 0), \qquad (2.11)$$

$$m_3 < m_1 < m_2 \quad \text{with} \quad \Delta m_{21}^2 \ll -(\Delta m_{31}^2 \approx \Delta m_{32}^2 < 0). \qquad (2.12)$$

As it is customary, we refer to the first option, eq. (2.11), as normal ordering (NO), and to the second one, eq. (2.12), as inverted ordering (IO); in this form they correspond to the two possible choices of the sign of $\Delta m_{31}^2$. In this convention the angles $\theta_{ij}$ can be taken without loss of generality to lie in the first quadrant, $\theta_{ij} \in [0, \pi/2]$, and the CP phase $\delta \in [0, 2\pi]$. In the following we adopt the (arbitrary) convention of reporting results for $\Delta m_{31}^2$ for NO and $\Delta m_{32}^2$ for IO, i.e., we always use the one which has the larger absolute value. Sometimes we will generically denote such quantity as $\Delta m_{3\ell}^2$, with $\ell = 1$ for NO and $\ell = 2$ for IO.

In summary, in total the $3\nu$ oscillation analysis of the existing data involves six parameters: 2 mass square differences (one of which can be positive or negative), 3 mixing angles, and the Dirac CP phase $\delta$. For the sake of clarity we summarize in table 1 which experiment contribute dominantly to the present determination of the different parameters.
<table><caption>**Table 1.** Experiments contributing to the present determination of the oscillation parameters.</caption><thead><tr><th>Experiment</th><th>Dominant</th><th>Important</th></tr></thead><tbody><tr><td>Solar Experiments</td><td>θ<sub>12</sub></td><td>Δm<sup>2</sup><sub>21</sub>, θ<sub>13</sub></td></tr><tr><td>Reactor LBL (KamLAND)</td><td>Δm<sup>2</sup><sub>21</sub></td><td>θ<sub>12</sub>, θ<sub>13</sub></td></tr><tr><td>Reactor MBL (Daya-Bay, Reno, D-Chooz)</td><td>θ<sub>13</sub></td><td>|Δm<sup>2</sup><sub>3ℓ</sub>|</td></tr><tr><td>Atmospheric Experiments</td><td>θ<sub>23</sub></td><td>|Δm<sup>2</sup><sub>3ℓ</sub>|, θ<sub>13</sub>,δ</td></tr><tr><td>Accelerator LBL ν<sub>μ</sub> Disapp. (Minos, T2K)</td><td>|Δm<sup>2</sup><sub>3ℓ</sub>|, θ<sub>23</sub></td><td></td></tr><tr><td>Accelerator LBL ν<sub>e</sub> App. (Minos, T2K)</td><td>δ</td><td>θ<sub>13</sub>, θ<sub>23</sub>, sign(Δm<sup>2</sup><sub>3ℓ</sub>)</td></tr></tbody></table>

The consistent determination of these leptonic parameters requires a global analysis of the data described above which, at present, is in the hands of a few phenomenological groups [84, 112, 113]. Here we summarize the results from ref. [84]. We show in figure 3 the one-dimensional projections of the $\Delta\chi^2$ of the global analysis as a function of each of the six parameters. The corresponding best-fit values and the derived ranges for the six parameters at the $1\sigma$ ($3\sigma$) level are given in table 2. For each parameter the curves and ranges are obtained after marginalizing with respect to the other five parameters. The ranges presented in the table are shown for three scenarios. In the first and second columns we assume that the ordering of the neutrino mass states is known “a priori” to be normal or inverted, respectively, so that the ranges of all parameters are defined with respect to the minimum in the given scenario. In the third column we make no assumptions on the ordering, so in this case the ranges of the parameters are defined with respect to the global minimum (which corresponds to IO) and are obtained by marginalizing also over the ordering. For this third case we only give the $3\sigma$ ranges. Of course in this case the range of $\Delta m^2_{3\ell}$ is composed of two disconnected intervals, one one containing the absolute minimum (IO) and the other the secondary local minimum (NO).

As mentioned, all the data described above can be consistently interpreted as oscillations of the three known active neutrinos. However, together with this data, several anomalies at short baselines (SBL) have been observed which *cannot* be explained as oscillations in this framework but could be interpreted as oscillations involving an *O*($eV$) mass sterile state. They will be discussed in section 2.4. In what respect the results presented here the only SBL effect which has to be treated in some form is the so-called *reactor anomaly* by which the most recent reactor flux calculations [114–116], fall short at describing the results from reactor experiments at baselines $\lesssim 100$ m from Bugey4 [117], ROVNO4 [118], Bugey3 [119], Krasnoyarsk [120, 121], ILL [122], Gösgen [123], SRP [124], and ROVNO88 [125], to which we refer as reactor short-baseline experiments (RSBL). We notice that these RSBL do not contribute to oscillation physics in the $3\nu$ framework, but they play an important role in constraining the unoscillated reactor neutrino flux if they are to be used instead of the theoretically calculated reactor fluxes. Thus, to account for the possible effect of the reactor anomaly in the determined ranges of neutrino parameters in the framework of $3\nu$ oscillations, the results in figure 3 are shown for two extreme choices. The first option is to leave the normalization of reactor fluxes free and include the RSBL data, experiments (labeled “Free+RSBL”) The second option is not to include short-baseline reactor data but assume reactor fluxes and uncertainties as predicted in [115] (labeled “Huber”).
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Global $3\nu$ oscillation analysis. The orange (violet) curves are for NO (IO). For solid curves the normalization of reactor fluxes is left free and data from short-baseline (less than 100 m) reactor experiments are included. For dashed curves, short-baseline data are not included but reactor fluxes as predicted in [115] are assumed. Note that we use $\Delta m^2_{31}$ for NO and $\Delta m^2_{32}$ for IO to denote the mass square differences. (Figure similar to figure 2 in ref. [84].)</figcaption></figure>

From the results in the figure and table we conclude that:

1. if we define the $3\sigma$ relative precision of a parameter $x$ by $2(x^{\text{up}} - x^{\text{low}})/(x^{\text{up}} + x^{\text{low}})$, where $x^{\text{up}}$ ($x^{\text{low}}$) is the upper (lower) bound on $x$ at the $3\sigma$ level, from the numbers in the table we find $3\sigma$ relative precision of 14% ($\theta_{12}$), 32% ($\theta_{23}$), 15% ($\theta_{13}$), 14% ($\Delta m^2_{21}$), and 11% ($|\Delta m^2_{3\ell}|$) for the various oscillation parameters;

2. for either choice of the reactor fluxes the global best-fit corresponds to IO with $\sin^2 \theta_{23} > 0.5$, while the second local minimum is for NO and with $\sin^2 \theta_{23} < 0.5$;

3. the statistical significance of the preference for IO versus NO is quite small, $\Delta\chi^2 \lesssim 1$;

4. the present global analysis disfavors $\theta_{13} = 0$ with $\Delta\chi^2 \approx 500$. Such impressive result is mostly driven by the reactor data from Daya Bay with secondary contributions from RENO and Double Chooz;
<table><caption>**Table 2.** Three-flavor oscillation parameters from our fit to global data after the NOW 2014 conference. The results are presented for the “Free Fluxes + RSBL” in which reactor fluxes have been left free in the fit and short-baseline reactor data (RSBL) with $L \lesssim 100$ m are included. The numbers in the 1st (2nd) column are obtained assuming NO (IO), i.e., relative to the respective local minimum, whereas in the 3rd column we minimize also with respect to the ordering. Note that $\Delta m_{3\ell}^2 \equiv \Delta m_{31}^2 > 0$ for NO and $\Delta m_{3\ell}^2 \equiv \Delta m_{32}^2 < 0$ for IO.</caption><thead><tr><th rowspan="2"></th><th colspan="2">Normal ordering (Δχ<sup>2</sup> = 0.97)</th><th colspan="2">Inverted ordering (best-fit)</th><th rowspan="2">Any ordering<br>3σ range</th></tr><tr><th>bfp ±1σ</th><th>3σ range</th><th>bfp ±1σ</th><th>3σ range</th></tr></thead><tbody><tr><td>sin<sup>2</sup> θ<sub>12</sub></td><td>0.304<sup>+0.013</sup><sub>-0.012</sub></td><td>0.270 → 0.344</td><td>0.304<sup>+0.013</sup><sub>-0.012</sub></td><td>0.270 → 0.344</td><td>0.270 → 0.344</td></tr><tr><td>θ<sub>12</sub>/°</td><td>33.48<sup>+0.78</sup><sub>-0.75</sub></td><td>31.29 → 35.91</td><td>33.48<sup>+0.78</sup><sub>-0.75</sub></td><td>31.29 → 35.91</td><td>31.29 → 35.91</td></tr><tr><td>sin<sup>2</sup> θ<sub>23</sub></td><td>0.452<sup>+0.052</sup><sub>-0.028</sub></td><td>0.382 → 0.643</td><td>0.579<sup>+0.025</sup><sub>-0.037</sub></td><td>0.389 → 0.644</td><td>0.385 → 0.644</td></tr><tr><td>θ<sub>23</sub>/°</td><td>42.3<sup>+3.0</sup><sub>-1.6</sub></td><td>38.2 → 53.3</td><td>49.5<sup>+1.5</sup><sub>-2.2</sub></td><td>38.6 → 53.3</td><td>38.3 → 53.3</td></tr><tr><td>sin<sup>2</sup> θ<sub>13</sub></td><td>0.0218<sup>+0.0010</sup><sub>-0.0010</sub></td><td>0.0186 → 0.0250</td><td>0.0219<sup>+0.0011</sup><sub>-0.0010</sub></td><td>0.0188 → 0.0251</td><td>0.0188 → 0.0251</td></tr><tr><td>θ<sub>13</sub>/°</td><td>8.50<sup>+0.20</sup><sub>-0.21</sub></td><td>7.85 → 9.10</td><td>8.51<sup>+0.20</sup><sub>-0.21</sub></td><td>7.87 → 9.11</td><td>7.87 → 9.11</td></tr><tr><td>δ/°</td><td>306<sup>+39</sup><sub>-70</sub></td><td>0 → 360</td><td>254<sup>+63</sup><sub>-62</sub></td><td>0 → 360</td><td>0 → 360</td></tr><tr><td>Δm<sup>2</sup><sub>21</sub>/10<sup>-5</sup> eV<sup>2</sup></td><td>7.50<sup>+0.19</sup><sub>-0.17</sub></td><td>7.02 → 8.09</td><td>7.50<sup>+0.19</sup><sub>-0.17</sub></td><td>7.02 → 8.09</td><td>7.02 → 8.09</td></tr><tr><td>Δm<sup>2</sup><sub>3ℓ</sub>/10<sup>-3</sup> eV<sup>2</sup></td><td>+2.457<sup>+0.047</sup><sub>-0.047</sub></td><td>+2.317 → +2.607</td><td>-2.449<sup>+0.048</sup><sub>-0.047</sub></td><td>-2.590 → -2.307</td><td>+2.325 → +2.599<br>-2.590 → -2.307</td></tr></tbody></table>

5. the uncertainty on $\theta_{13}$ associated with the choice of reactor fluxes is at the level of $0.5\sigma$ in the global analysis. This is so because the most precise results from Daya Bay, and RENO are reactor flux normalization independent;

6. a non-maximal value of the $\theta_{23}$ mixing is slightly favored, at the level of $\sim 1.4\sigma$ for IO at of $\sim 1.0\sigma$ for NO;

7. the statistical significance of the preference of the fit for the second (first) octant of $\theta_{23}$ is $\le 1.4\sigma$ ($\le 1.0\sigma$) for IO (NO);

8. the best-fit for $\delta$ for all analyses and orderings occurs for $\delta \simeq 3\pi/2$, and values around $\pi/2$ are disfavored with $\Delta\chi^2 \simeq 6$. Assigning a confidence level to this $\Delta\chi^2$ is non-trivial, due to the non-Gaussian behavior of the involved $\chi^2$ function, see ref. [84] for a discussion and a Monte Carlo study.

From this global analysis one can also derive the $3\sigma$ ranges on the magnitude of the elements of the leptonic mixing matrix to be:

$$ |U| = \begin{pmatrix} 0.801 & 0.845 & 0.514 & 0.580 & 0.137 & 0.158 \\ 0.225 & 0.517 & 0.441 & 0.699 & 0.614 & 0.793 \\ 0.246 & 0.529 & 0.464 & 0.713 & 0.590 & 0.776 \end{pmatrix}. \quad (2.13) $$

The present status of the determination of leptonic CP violation is further illustrated in figure 4. On the left panel we show the dependence of the $\Delta\chi^2$ of the global analysis on
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Left: dependence of the global $\Delta\chi^2$ function on the Jarlskog invariant. The orange (violet) curves are for NO (IO). (Figure similar to figure 3b in [84].) Right: leptonic unitarity triangle. After scaling and rotating so that two of its vertices always coincide with (0, 0) and (1, 0) we plot the $1\sigma$, $2\sigma$, $3\sigma$ CL (2 dof) allowed regions of the third vertex. (Figure similar to figure 4d in [84].)</figcaption></figure>

the Jarlskog invariant which gives a convention-independent measure of CP violation [126], defined as:

$$ \mathrm{Im}[U_{\alpha i} U_{\alpha j}^* U_{\beta i} U_{\beta j}] = \cos \theta_{12} \sin \theta_{12} \cos \theta_{23} \sin \theta_{23} \cos^2 \theta_{13} \sin \theta_{13} \sin \delta = J_{\mathrm{CP}}^{\mathrm{max}} \sin \delta, \quad (2.14) $$

where in the second equality we have used the parametrization in eq. (2.6). Thus the determination of the mixing angles yields at present a maximum allowed CP violation

$$ J_{\mathrm{CP}}^{\mathrm{max}} = 0.0329 \pm 0.0009 (\pm 0.0027) \qquad (2.15) $$

at $1\sigma$ ($3\sigma$) for both orderings. The preference of the present data for non-zero $\delta$ implies a best-fit of $J_{\mathrm{CP}}^{\mathrm{best}} = -0.032$, which is favored over CP conservation at the $\sim 1.2\sigma$ level. These numbers can be compared to the size of the Jarlskog invariant in the quark sector, which is determined to be $J_{\mathrm{CP}}^{\mathrm{quarks}} = (2.96^{+0.20}_{-0.16}) \times 10^{-5}$ [85].

On the right panel of figure 4 we recast the allowed regions for the leptonic mixing matrix in terms of one leptonic unitarity triangle. Since in the analysis $U$ is unitary by construction, any given pair of rows or columns can be used to define a triangle in the complex plane. In the figure we show the triangle corresponding to the unitarity conditions on the first and third columns which is the equivalent to the one usually shown for the quark sector. In this figure the absence of CP violation implies a flat triangle, i.e., $\mathrm{Im}(z) = 0$. As can be seen, the horizontal axis marginally crosses the $1\sigma$ allowed region, which for 2 dof corresponds to $\Delta\chi^2 \simeq 2.3$. This is consistent with the present preference for CP violation, $\chi^2(J_{\mathrm{CP}} = 0) - \chi^2(J_{\mathrm{CP}} \text{ free}) = 1.5$.

### 2.2.2 Absolute neutrino mass measurements

Oscillation experiments provide information on $\Delta m_{ij}^2$ and on the leptonic mixing angles $\theta_{ij}$. But they are insensitive to the absolute mass scale for the neutrinos. Of course, the results of an oscillation experiment do provide a lower bound on the heavier mass in $\Delta m_{ij}^2$,
<figure><img src="image_6.png" /><figcaption>Figure 5. 95% allowed regions (for 2 d.o.f.) in the planes $(m_{\nu_e}, \sum m_\nu)$ and $(m_{ee}, \sum m_\nu)$ obtain from projecting the results of the global analysis of oscillation data.</figcaption></figure>

$|m_i| \ge \sqrt{\Delta m_{ij}^2}$ for $\Delta m_{ij}^2 > 0$, but there is no upper bound on this mass. In particular, the corresponding neutrinos could be approximately degenerate at a mass scale that is much higher than $\sqrt{\Delta m_{ij}^2}$. Moreover, there is neither an upper nor a lower bound on the lighter mass $m_j$.

Information on the absolute neutrino masses, rather than mass differences, can be extracted from kinematic studies of reactions in which a neutrino or an anti-neutrino is involved. In the presence of mixing the most relevant constraint comes from the study of the end point $(E \sim E_0)$ of the electron spectrum in tritium beta decay $^{3}\text{H} \rightarrow ^{3}\text{He} + e^{-} + \bar{\nu}_{e}$. This spectrum can be effectively described by a single parameter $m_{\nu_e}$ [127], if for all neutrino states $E_0 - E \gg m_i$. In this case:

$$ \frac{dN}{dE} \approx R(E) \sum_i |U_{ei}|^2 \sqrt{(E_0 - E)^2 - m_{\nu_e}^2}, \qquad (2.16) $$

where $R(E)$ contains all the $m_\nu$-independent factors, and

$$ m_{\nu_e}^2 = \frac{\sum_i m_i^2 |U_{ei}|^2}{\sum_i |U_{ei}|^2} = \sum_i m_i^2 |U_{ei}|^2 = c_{13}^2 c_{12}^2 m_1^2 + c_{13}^2 s_{12}^2 m_2^2 + s_{13}^2 m_3^2, \qquad (2.17) $$

where the second equality holds if unitarity is assumed. At present we only have a bound $m_{\nu_e} \le 2.2$ eV at 95% CL [17, 128] which is expected to be superseded soon by KATRIN [129] with about one order of magnitude improvement in sensitivity.

Direct information on neutrino masses can also be obtained from neutrinoless double beta decay $(A, Z) \rightarrow (A, Z + 2) + e^{-} + e^{-}$. This process violates lepton number by two units. Hence, in order to induce $0\nu\beta\beta$ decay, $\nu$'s must be Majorana particles [130, 131]. In particular, for the case in which the only effective lepton number violation at low energies is induced by the Majorana mass term for the neutrinos, the rate of $0\nu\beta\beta$ decay is proportional
to the square of the effective Majorana mass of $\nu_e$ [132, 133]:

$$m_{ee} = \left| \sum_{i} m_i U_{ei}^2 \right| = \left| m_1 c_{13}^2 c_{12}^2 e^{i2\alpha_1} + m_2 c_{13}^2 s_{12}^2 e^{i2\alpha_2} + m_3 s_{13}^2 e^{-i2\delta} \right| \quad (2.18)$$

which, unlike eq. (2.17), also depends on all three CP violating phases.

Recent searches carried out with $^{76}$Ge (GERDA experiment [134]) and $^{136}$Xe (KamLAND-Zen [135] and EXO-200 [136] experiments) have established the lifetime of this decay to be longer than $10^{25}$ yr, corresponding to a limit on the neutrino mass of $m_{ee} \le 0.2 - 0.4$ eV at 90% C.L. A series of new experiments is planned with sensitivity of up to $m_{ee} \sim 0.01$ eV [137].

Neutrino masses have also interesting cosmological effects and, as we will see in more detail in section 2, cosmological data mostly gives information on the sum of the neutrino masses $\Sigma = \sum_i m_i$. However, in particular when trying to derive conclusions on neutrino parameters, it is very important to have all systematics under control [138].

Correlated information on these three probes of the neutrino mass scale can be obtained by mapping the results from the global analysis of oscillations presented previously. We show in figure 5 the present status of this exercise. The relatively large width of the regions in the right panel are due to the unknown Majorana phases. Thus from a positive determination of two of these probes information can be obtained on the value of the Majorana phases and/or the mass ordering.

## 2.3 Open questions in neutrino physics (author: A. de Gouvêa)

The discovery of non-zero neutrino masses and mixing in the lepton sector invites several fundamental particle physics questions. Searches for the answers to these open questions are among the key driving forces behind current and future research in neutrino physics, both experimentally and theoretically.

Some of these questions are related to the neutrino masses themselves. First and foremost, we do not know the exact values of the neutrino masses. While the oscillation data measure, sometimes very precisely, the neutrino mass-squared differences, they are not at all sensitive to the values of the masses themselves. As mentioned above, cosmological surveys and precision measurements of the $\beta$ spectrum of weak nuclear decays place complementary upper bounds on different combinations of the neutrino masses. Some information on the neutrino masses can also be extracted from searches for neutrinoless double beta decay if one assumes that neutrinos are Majorana fermions.

The neutrino mass ordering — inverted or normal — is also unknown, as discussed earlier. This is a question that can, and very likely will, be addressed by neutrino oscillation experiments, including the current generation of long-baseline accelerator-based experiments, NOvA [139] and T2K [140], along with near-future measurements of the atmospheric neutrino flux (see, for example, ref. [141]). Very ambitious long-baseline reactor antineutrino experiments, like JUNO [142], may also be able to determine the neutrino mass ordering by measuring very precisely $\Delta m_{31}^2$ and $\Delta m_{32}^2$ (see, for example, refs. [143, 144]).

The answers to the two questions above will reveal not only the values and ordering of the masses, but they will also provide crucial information regarding the dynamical mechanism behind the neutrino masses. Current data do not allow one to distinguish scenarios where the lightest neutrino mass is zero from scenarios where all three neutrino masses are almost degenerate, $|m_i - m_j| \ll m_i, m_j$. None of the electrically charged fermions — charged leptons and quarks — have almost degenerate masses. Note also that, if the neutrino mass ordering
is inverted, $m_1$ and $m_2$ are guaranteed to be almost degenerate even for that case where the lightest neutrino mass, $m_3$, is very small.

The fact that the neutrinos are massive and neutral invites another question: are neutrinos Majorana fermions? While all known fundamental fermions are massive Dirac fermions, i.e., the particles and antiparticles are distinct objects and all contain four degrees of freedom, the massive neutrinos might be Majorana fermions, i.e., the particle and antiparticle are related and containing only two degrees of freedom. The answer to this question is intimately tied to whether or not baryon number minus lepton number, $B-L$, is an exact symmetry. Majorana neutrinos imply that $B-L$ is explicitly broken and one should be able to observe that experimentally (see, for example, refs. [145, 146], for recent overviews). The most precise probes of $B-L$ violation are searches for neutrinoless double beta decay, which are currently being pursued in earnest. In the next ten years we hope to learn much more about the nature of the neutrino, especially if the neutrino mass ordering turns out to be inverted [132].

Neutrino oscillations allow one to explore another important fundamental question: is there CP violation in the leptonic sector? More concretely, we hope to address whether neutrinos and antineutrinos oscillate differently. While the survival probabilities for neutrinos and antineutrinos are guaranteed to be the same, the transition probabilities need not be. If $P(\nu_\alpha \to \nu_\beta) \neq P(\bar{\nu}_\alpha \to \bar{\nu}_\beta)$, $\alpha, \beta = e, \mu, \tau$, $\alpha \neq \beta$, CP is violated. In the three-flavor paradigm discussed above, CP violation is governed by the CP-odd parameter $\delta$. If $\delta \neq 0, \pi$, CP-invariance is violated in neutrino oscillations. Given what is currently known about the neutrino oscillation parameters, it is clear that the current generation of neutrino experiments is not very sensitive to potential CP-invariance violation in neutrino oscillations. Next-generation experiments, especially the long-baseline proposals LBNF/DUNE (see, for example, ref. [147]) and T2HK [148], are designed to significantly challenge CP-invariance in the leptonic sector. We currently understand very little about the phenomenon of CP violation. We do know that it is necessary in order to dynamically generate a matter-antimatter asymmetry in the early Universe [149] — a mechanism dubbed baryogenesis, see e.g. [150] for a general discussion. Since we know that the CP violation observed in the quark sector is insufficient to explain the baryon-asymmetry, the community is eagerly awaiting information on leptonic CP violation so that it can make progress on understanding baryogenesis.

Precision neutrino experiments, especially precision neutrino oscillation experiments, allow one to ask other fundamental physics questions. For example, are there more than three neutrinos? While we know that there are three families of SM fermions, including the three active neutrinos $\nu_e, \nu_\mu, \nu_\tau$, there could be more neutral fermionic states. Collider data, especially those from LEP, restrict the number of active neutrinos to three [85]. New fermions that do not couple to the W- and Z-bosons with SM strength — usually referred to as sterile neutrinos — are not severely constrained and their existence might be revealed via neutrino oscillation experiments. While these sterile neutrinos do not necessarily couple to SM particles, they mix with the active neutrinos in such a way that neutrino oscillation experiments might depend on more mixing parameters and more oscillation frequencies. If the new neutrinos are light enough, they can only be probed experimentally in neutrino oscillation experiments. There are many current neutrino oscillation experiments dedicated to the search for sterile neutrinos.

Neutrino oscillation experiments are also sensitive to new “weaker-than-weak” neutrino interactions with matter. These modify neutrinos production and detection and, often more significantly, they lead to non-standard matter effects. Long-baseline experiments, especially NOνA [139] and LBNF/DUNE (see, for example, ref. [147]), are particularly sensitive to non-standard neutrino interactions.
Finally, neutrino oscillations are very special phenomena. They are a consequence of quantum mechanical interference, and the associated coherence lengths are of the order of several kilometers (or significantly more). This provides a unique sensitivity to more exotic physics, including the violation of Lorentz invariance and tests of the CPT-theorem, or departures from the basic laws of quantum mechanics.

## 2.4 Sterile neutrinos — general introduction (author: P. Langacker)

Sterile neutrinos, also known as singlet or right-handed neutrinos, are SU(2) × U(1)-singlet leptons. They therefore have no ordinary charged or neutral current weak interactions except those induced by mixing. Most extensions of the original standard model involve one or more sterile neutrinos, with model-dependent masses which can vary from zero to extremely large. One usually defines the right-chiral component of a sterile neutrino field as $\nu_R$, i.e., $\nu_R$ annihilates a right-chiral state, where chirality coincides with helicity in the massless limit. The CP-conjugate field is then

$$ (\nu_R)^c \equiv C \overline{\nu_R}^T, \qquad (2.19) $$

where we are following the notation in ref. [67]. In eq. (2.19), C is the charge conjugation matrix, given by $= i\gamma_2\gamma_0$ in the Weyl representation, and $\overline{\nu_R} \equiv (\nu_R)^\dagger\gamma^0$ is the Dirac adjoint. Note that the CP conjugate in eq. (2.19) is always well-defined, independent of whether CP is violated, and that $(\nu_R)^c$ is the field which annihilates a left-chiral antineutrino.<fnref n="7" />

In contrast, an active (or doublet or ordinary) neutrino is in an SU(2) doublet with a charged lepton, and it has conventional weak interactions. There are three known left-chiral active neutrinos $\nu_{L,\alpha}$, where the flavor index $\alpha = e, \mu, \tau$ denotes the associated charged lepton. The CP-conjugate $(\nu_L)^c \equiv C \overline{\nu_L}^T$ (suppressing the flavor index) is the field associated with a right-chiral antineutrino. The number $n$ of right chiral neutrinos is unknown (and could even be zero, as there are alternative explanations of neutrino masses, see section 2.5.1). In the remainder of this subsections we use an illustrative toy model with only one LH and one RH neutrino flavour.

$\nu_L \leftrightarrow (\nu_L)^c$ and $\nu_R \leftrightarrow (\nu_R)^c$ each describe two degrees of freedom and are known as Weyl spinors. Fermion mass terms describe transitions between left and right-chiral states. There are two possible types for neutrinos. A Dirac mass term connects the left and right components of two different Weyl spinors. These are typically active and sterile, such as

$$ \mathcal{L}_D = -m_D (\overline{\nu_L} \nu_R + \overline{\nu_R} \nu_L), \qquad (2.20) $$

where we have chosen the phases of the fields so that $m_D$ is real. $\mathcal{L}_D$ allows a conserved lepton number $L$, but violates weak isospin by 1/2 unit. It can be generated by the Higgs mechanism, as in figure 6, and it is analogous to the quark and charged lepton masses. That is, $m_D = y_D v$,<fnref n="8" /> where $v = 174$ GeV is the expectation value of the neutral Higgs field. If eq. (2.20) is the only neutrino mass term, then $\nu_L$ and $\nu_R$ can be combined to form a four-component Dirac spinor $\nu_D \equiv \nu_L + \nu_R$, with CP conjugate $(\nu_D)^c \equiv (\nu_L)^c + (\nu_R)^c$.

Unlike quarks and charged leptons, neutrinos are not charged under any unbroken gauge symmetries. They may therefore have Majorana mass terms, which connect a Weyl spinor with its own CP conjugate. These could be present for either active or sterile neutrinos,

$$ \mathcal{L}_M = -\frac{1}{2} m_T (\overline{\nu_L} \nu_L^c + \overline{\nu_L}^c \nu_L) - \frac{1}{2} m_S (\overline{\nu_R} \nu_R^c + \overline{\nu_R}^c \nu_R) \equiv -\frac{1}{2} m_T (\overline{\nu}_a \nu_a) - \frac{1}{2} m_S (\overline{\nu}_s \nu_s), \quad (2.21) $$

<footnote n="7">⁷Some authors use alternative notations, such as $\nu_{R,L}^c$ for $C \overline{\nu_{R,L}}^T$.</footnote>

<footnote n="8">⁸In the minimal seesaw model, (2.25), the number $y_D$ is to be identified with the matrix F.</footnote>
<figure><img src="image_7.png" /><figcaption>**Figure 6.** Left: Dirac mass from the vacuum expectation value of the neutral component $\phi^0$ of a Higgs doublet. $y_D$ is a Yukawa coupling. Center: a Majorana mass for an active neutrino due to a Higgs triplet $\phi_T^0$ with the Yukawa coupling $y_T$. An analogous diagram can generate a Majorana mass for a sterile neutrino, with $\phi_T^0$ replaced by a Higgs singlet (or bare mass). Right: Majorana mass for $\nu_L$ generated by a higher-dimensional operator involving two Higgs doublets. (Figure similar to figure 1 in ref. [151].)</figcaption></figure>

where $\nu_a \equiv \nu_L + (\nu_L)^c$ and $\nu_s \equiv \nu_R + (\nu_R)^c$ are active (a) and sterile (s) Majorana two-component spinors. They are self-conjugate, i.e., $\nu_a = C \bar{\nu}_a^T$ and $\nu_s = C \bar{\nu}_s^T$. Both $m_T$ and $m_s$ violate lepton number by two units. The mass $m_T$ also violates weak isospin by one unit. It can be generated by the expectation value of a Higgs triplet $\phi_T = (\phi_T^0, \bar{\phi}_T^-, \bar{\phi}_T^{--})^T$, i.e., $m_T = y_T \langle \phi_T^0 \rangle$, where $y_T$ is a Yukawa coupling, as illustrated in figure 6. It can also be due to a higher-dimensional operator involving two Higgs doublets with coefficient $C/M$, with $C$ a dimensionless constant and $M$ a new physics scale. For mass dimension 5, the only operator of this kind is given by (2.26). The singlet mass $m_S$ does not violate weak isospin and could in principle be due to a bare mass. However, in many extensions of the standard model a bare mass is forbidden by new broken symmetries, so that $m_S = y_S \langle \phi_S \rangle$, where $y_S$ is a Yukawa coupling and $\phi_S$ is a standard model singlet.

Dirac and Majorana mass terms<fnref n="9" /> can be present simultaneously and can be generalized to three or more active neutrinos, to an arbitrary number of sterile neutrinos, and to non-standard assignments such as additional sterile neutrinos in which the *left*-chiral component carries lepton number $L = +1$.

Sterile neutrino masses can have almost any value, but here we mention a number of important special cases. For simplicity we focus on $m_T = 0$ except when it induced by the seesaw mechanism, but generalization is straightforward. We also take $m_D$ and $m_S$ real and nonnegative.

* $m_S = 0$: this is the pure Dirac case. $\nu_L$ and $\nu_R$ are the left and right-chiral components of a Dirac neutrino with conserved lepton number $L$. $\nu_R$ has no standard model interactions except for the extremely weak Higgs-Yukawa coupling $y_D \sim 3 \times 10^{-13} m_D/(0.05 \text{ eV})$. This small value could occur via fine-tuning, or more likely because it is strongly suppressed by new symmetries or extra-dimensional effects, see e.g. ref. [151].

* $m_D = 0$: this is the pure Majorana case. The sterile Majorana neutrino can have arbitrary mass, but it has no standard model interactions, and there is no active-sterile mixing.

<footnote n="9">⁹Off-diagonal mass terms are generally defined as Majorana if the mass eigenstates are.</footnote>
* $m_S \ll m_D$: this pseudo-Dirac limit is a perturbation on the Dirac case. The four components of the Dirac neutrino split into two Majorana neutrinos, with masses $|m_{1,2}| \sim m_D \pm m_S/2$ and left-chiral components $\sim [\nu_L \pm (\nu_R)^c] / \sqrt{2}$. The non-observation of solar oscillations into the sterile state requires $m_S < O(10^{-9} \text{ eV})$, see ref. [152].

* $m_S \gg m_D$ with $m_T = 0$: this is the (minimal type-I) seesaw limit [153–158]. There are two Majorana mass eigenstates, with one light mainly-active neutrino and a heavier mainly-sterile state, with respective masses:

$$|m_1| \sim \frac{m_D^2}{m_S}, \quad m_2 \sim m_S, \qquad (2.22)$$

and the active-sterile mixing angle

$$|\theta| \sim \frac{m_D}{m_S} \sim \sqrt{\frac{m_1}{m_2}} \sim 3 \times 10^{-3} \left( \frac{m_1}{0.05 \text{ eV}} \right)^{1/2} \left( \frac{6 \text{ keV}}{m_S} \right)^{1/2}. \qquad (2.23)$$

The Higgs Yukawa coupling required is

$$y_D = \frac{\sqrt{m_1 m_S}}{v} \sim 10^{-10} \left( \frac{m_1}{0.05 \text{ eV}} \right) \left( \frac{m_S}{6 \text{ keV}} \right). \qquad (2.24)$$

The mass $m_S$ is often directly associated with a new physics scale, especially for $m_S \gg v$. In that limit, the sterile state is often integrated out, so that $m_1$ plays the role of $m_T$ in the low-energy effective theory. These results are generalized to three families in section 2.5.2. An important phenomenological difference in the case of several families is that there may be cancellations in the neutrino mass matrix that allow individual entries of the mixing matrix $\theta$ to be much bigger than the estimate (2.23). This is in particular the case in models with an approximate $B-L$ symmetry.

We now mention several special cases:

- $m_S = O(\text{eV})$: oscillations of active into eV-scale sterile neutrinos have been suggested by the LSND and MiniBooNE $\bar{\nu}_\mu \to \bar{\nu}_e$ (and to a lesser extent $\nu_\mu \to \nu_e$) results, by possible $\bar{\nu}_e$ disappearance in short-baseline reactor experiments, and by radioactive source calibrations carried out by the SAGE and GALLEX experiments, e.g., ref. [159–161]. There is, however, tension with a number of other $\nu_\mu$, $\bar{\nu}_\mu$, $\nu_e$, and $\bar{\nu}_e$ disappearance or appearance results in longer-baseline reactor and accelerator experiments and in atmospheric and solar neutrinos, especially for a single eV-scale sterile state, ref. [161–163]. There was some suggestion of additional radiation in the early Universe from CMB experiments, e.g., ref. [164], but the number density of even one eV-scale neutrino with large mixing would have to be suppressed compared to the expectations of an oscillation-induced thermalized species [165].

The LSND and other positive results present a challenge to theory, because they would require mixing between active and sterile states with the same chirality. This does not occur for the pure Dirac or pure Majorana cases, or in the conventional seesaw with large $m_S$. They would require that both Dirac and Majorana mass terms are tiny.<fnref n="10" /> A promising possibility is the minimal

<footnote n="10">¹⁰More generally, two distinct types of mass terms, such as active-sterile Dirac, and sterile-sterile Dirac in the presence of extra sterile states, would be necessary.</footnote>
mini-seesaw ($m_T = 0$ and $m_D \ll m_S \sim \mathcal{O}(\text{eV})$, e.g., ref. [166, 167]), which yields a reasonable relation between mixing angles and masses, as seen in eq. (2.23). The tiny values for the masses would suggest that they are suppressed by some new symmetry, ref. [168, 169].

Sterile neutrinos in the eV range (and higher, up to kinematic limits) could lead to observable kinks or lines in $\beta$ decay and other weak decay spectra, ref. [127, 170-173]. Both eV-scale, ref. [174], and heavier sterile neutrinos up to the TeV range, ref. [175], could also significantly affect neutrinoless double $\beta$ decay [176].

- $m_S = \mathcal{O}(\text{keV})$: the keV range is especially interesting because the sterile neutrino would be a viable DM candidate. It could be warm or cold depending on the cosmological production mechanisms, e.g., produced by oscillations between active and sterile neutrinos, ref. [177]. In particular, warm or moderately cold dark matter could describe the low-scale structure observations better than cold dark matter candidates such as WIMPs or axions (see section 4 and e.g. [178-185]). The keV neutrinos would not be absolutely stable, but could decay by mixing into a photon and light neutrino with a cosmologically-long lifetime. There are stringent astrophysical and cosmological constraints on sterile neutrinos in the keV and other mass ranges, as reviewed in refs. [66, 171, 186]. The observational limits on X-ray emission by the radiative decays are especially stringent, but they are consistent with sterile DM<fnref n="11" /> for $m_S = \mathcal{O}(\text{keV})$. There is some tentative evidence for a 3.5 keV X-ray line, which could possibly be due to a decaying sterile neutrino, see refs. [190, 191]. A keV-scale sterile could also explain the observed pulsar velocities, ref. [171]. However, they are very difficult to detect in a lab, see e.g. refs. [192-194] for investigations of neutrinoless double beta decay and section 8 for direct detection attempts.

The neutrino minimal standard model (νMSM) [188, 189] is a minimal extension of the SM, with no new physics other than three sterile neutrinos up to the Planck scale. One is at the keV scale, to account for DM. Two heavier sterile neutrinos can account for the observed light neutrino masses by a seesaw mechanism. If they are nearly degenerate and in the range 150 MeV–100 GeV they can account for the baryon asymmetry through oscillation-induced leptogenesis. General theoretical models for keV steriles are reviewed in ref. [66].

- $m_S = \mathcal{O}(\text{MeV}-\text{TeV})$: there are a variety of constraints from astrophysics, ref. [171], from weak decays (e.g., spectral lines or from lepton violation such as $\ell_2^+ \to \ell_1^- M^+ M^+$ or $M_1^+ \to M_2^- \ell^+ \ell^+$, where $\ell$ and $M$ represent leptons and mesons, respectively), and from same-sign dilepton production at the LHC or Tevatron, e.g., ref [71, 170]. These results are typically sensitive to mixings larger than those suggested by eq. (2.23).

- $m_S \gg \text{TeV}$: this is the canonical minimal type-I seesaw range, in which $y_D$ is comparable to the quark and charged-lepton Yukawas (i.e., $m_D \sim m_e$ for $m_S \sim$ several TeV, while $m_D \sim m_t$ for $m_S \sim 10^{14}$ GeV). The only direct observational consequence other than the mixing-induced active neutrino masses is the possibility of leptogenesis, usually by heavy sterile decay. There may also be model-dependent

<footnote n="11">¹¹The relevant mixing angle, $|\theta| = \mathcal{O}(10^{-5} - 10^{-4})$ would lead to a very small $m_1$ using the seesaw formula eq. (2.23), which however would still be applicable [187]. The observed solar and atmospheric scales could, however, be associated with heavier steriles, as in the νMSM [188, 189].</footnote>
associated signatures, such as $\mu \to e\gamma$ by sneutrino exchange for $m_S = O(\text{TeV})$ in supersymmetric models. A very large $m_S$ may be associated with a grand unified theory, with implications for proton decay. For example, $m_S = 10^{14}$ GeV is about two orders of magnitude below the scale of supersymmetric gauge unification.

- $m_S \sim M_P$: for completeness we mention the possibility that $m_S$ is comparable to the Planck scale, $M_P \sim 10^{19}$ GeV. This would imply $m_1 < 10^{-5}$ eV for $y_D \sim 1$. This is the typical maximum value of $m_1$ expected from gravitational mechanisms,$^{12}$ unless there are large extra dimensions or multiple contributions.

## 2.5 The seesaw mechanism (author: M. Drewes)

### 2.5.1 Possible origins of neutrino mass

Any model of neutrino masses should explain the fact that they are orders of magnitude smaller than any other fermion masses in the SM ("mass puzzle"). It is convenient to classify the most popular models according to the way how this mass hierarchy is realized.

* **Small coupling constant:** if the light neutrino masses $m_i$ are generated via spontaneous symmetry breaking, then their smallness could simply be due to a tiny coupling constant. For instance, in the model (2.25), Dirac masses can be generated via the standard Higgs mechanism with $F \sim 10^{-12}$ and $M_M = 0$.

* **Seesaw mechanism:** if the neutrino masses $m_i$ are generated at classical level, they may be suppressed by the new heavy scale $M$. The most studied version is the type-I seesaw [153, 155–158, 195] discussed below in 2.5.2, the two other possibilities [196] are the type-II [158, 197–200] and type-III [201] seesaw.

* **Flavor ("horizontal") symmetry:** individual entries of the light neutrino mass matrix $m_\nu$ may be much larger than the $m_i$ if there is a symmetry that leads to approximate lepton number conservation and cancellations in $m_\nu m_\nu^\dagger$. Prominent realizations of this idea are Froggatt-Nielsen type models [202] and models with approximate lepton number conservation, e.g. [203–209].

* **Radiative masses:** neutrinos could be massless in the classical limit, with their masses being generated by quantum corrections. The suppression due to the "loop factor" $(4\pi)^2$ is usually not sufficient to explain the smallness of the $m_i$, but the new particles in the loop couple to $\nu_L$ with some small coupling constants that lead to an additional suppression. Flavor symmetries or an additional seesaw-like suppression can help to make such models more "natural", see e.g. [210–218].

Of course, any combination of these ideas may be realized in nature. In addition to the smallness of the neutrino masses, it is desirable to find an explanation for the observed mixing pattern of neutrinos ("flavor puzzle"). While the quark mass matrix is approximately diagonal in the weak interaction basis, there is no obvious symmetry or structure in $m_\nu$. Numerous attempts have been made to identify discrete or continuous symmetries, see e.g. [219] and section 6.13 The basic problem is that the reservoir of possible symmetries is practically unlimited; for any possible observed pattern of neutrino masses and mixings one could find a symmetry that "predicts" it. In this situation, models can only be convincing if

<footnote>$^{12}$In this case, $m_S$ might correspond, e.g., to a string excitation rather than a particle.</footnote>

<footnote>$^{13}$See refs. [220–222] for some seesaw implementations not covered in section 6.</footnote>
they either predict observables that have not been measured, such as mass [219, 223–226] or mixing [219, 227–229] sum rules, or are “simple” and aesthetically appealing from some viewpoint. Prior to the measurement of $\theta_{13}$, models predicting $\theta_{13} = 0$ seemed well-motivated, such as those leading to tri/bi-maximal mixing [230–234]. The observed $\theta_{13} \neq 0$, however, requires an additional symmetry-breaking sector in these models. That makes it difficult to explain $m_\nu$ in terms of a simple symmetry and a small number of parameters, and interest in anarchic models [235] with random values has grown. A more detailed overview of models that incorporate sterile neutrinos with keV masses that could act as DM candidates is given in section 6. In the following we briefly review the main features of the probably minimal and most studied mechanism for neutrino masses, the (type I) seesaw mechanism. Is particularly important in the present context because i) it predicts the existence of heavy sterile neutrinos, ii) these heavy particles mix with ordinary neutrinos (which is the basis for many experimental and astrophysical searches) ans iii) the type I seesaw is implemented in many theories of particle physics, such as grand unified theories based on SO(10) or any other theories that involve a (spontaneously broken) gauged $B-L$ symmetry.

### 2.5.2 The seesaw mechanism

The type-I seesaw model is defined by adding $n$ RH neutrinos $\nu_R$ to the SM, i.e., singlet fermions with RH chirality to the SM that couple to the SM neutrinos $\nu_L$ in the same way as the RH and LH components of the charged leptons. The Lagrangian reads

$$ \mathcal{L} = \mathcal{L}_{\text{SM}} + i \bar{\nu}_R \partial \nu_R - \bar{\ell}_L F \nu_R \tilde{\Phi} - \tilde{\Phi}^\dagger \bar{\nu}_R F^\dagger \ell_L - \frac{1}{2} (\bar{\nu}_R^c M_M \nu_R + \bar{\nu}_R M_M^\dagger \nu_R^c). \quad (2.25) $$

Here, flavor and isospin indices have been suppressed. $\mathcal{L}_{\text{SM}}$ is the Lagrangian of the SM, $\ell_L = (\nu_L, e_L)^T$ are the LH lepton doublets, $\Phi$ is the Higgs doublet and $\tilde{\Phi} = \epsilon\Phi^*$, where $\epsilon$ is the antisymmetric SU(2)-invariant tensor, and $F$ is a matrix of Yukawa interactions. An explicit Majorana mass term $M_M$ is allowed for $\nu_R$ because the $\nu_R$ are gauge singlets. This is a specific realization of the term $m_S$ in (2.21). It is often assumed that the eigenvalues $M_I$ of $M_M$ are far above the electroweak scale. Then the $\nu_R$ are experimentally unobservable. The only effect they have on low energy physics is mediated by the dimension-5 operator [236]:

$$ \mathcal{L}_{\text{eff}} = \mathcal{L}_{\text{SM}} + \frac{1}{2} \bar{\ell}_L \tilde{\Phi} F M_M^{-1} F^T \tilde{\Phi}^T \ell_L^c, \quad (2.26) $$

as obtained by integrating out the fields $\nu_R$ instead of (2.25). The Higgs mechanism generates a Majorana mass term $\bar{\nu}_L m_\nu \nu_L^c$, with $m_\nu$ is given by

$$ m_\nu = -v^2 F M_M^{-1} F^T, \quad (2.27) $$

where $v = 174$ GeV is the Higgs field expectation value. This case is not interesting in the context of this review because superheavy $\nu_R$ are too short-lived to be DM candidates. However, experimentally the magnitude of the $M_I$ is almost unconstrained, and different choices have very different implications for particle physics, cosmology and astrophysics, see e.g. [67, 159]. There are several scenarios that predict eigenvalues of $M_M$ at or below the electroweak scale, including the inverse seesaw [237] and linear seesaw [238], the $\nu$MSM [239, 240], low-scale seesaw (e.g. [220–222]), or Coleman-Weinberg type models [216], see also section 6. The full neutrino mass term after electroweak symmetry breaking reads

$$ \frac{1}{2} (\bar{\nu}_L \overline{\nu_R^c}) \mathfrak{M} \begin{pmatrix} \nu_L^c \\ \nu_R \end{pmatrix} + \text{h.c.} \equiv \frac{1}{2} (\bar{\nu}_L \overline{\nu_R^c}) \begin{pmatrix} 0 & m_D \\ m_D^T & M_M \end{pmatrix} \begin{pmatrix} \nu_L^c \\ \nu_R \end{pmatrix} + \text{h.c.}, \quad (2.28) $$
where $m_D \equiv Fv$ and $v = 174$ is the Higgs field vacuum expectation value. For $M_I \gg 1$ eV there is a hierarchy $m_D \ll M_M$, and one observes two distinct sets of mass eigenstates: one set of “active” light neutrinos that are mostly SU(2) doublets and one set of “sterile” heavy neutrinos that are mostly gauge singlets. Mixing between the active and sterile neutrinos is suppressed by elements of the mixing matrix

$$ \theta \equiv m_D M_M^{-1}. \qquad (2.29) $$

This allows to rewrite (2.27) as

$$ m_{\nu} = -v^2 F M_M^{-1} F^T = -m_D M_M^{-1} m_D^T = -\theta M_M \theta^T. \qquad (2.30) $$

The fact that heavier $M_I$ imply lighter $m_i$ motivates the name “seesaw mechanism”. An important implication of this relation is that one RH neutrino with non-vanishing mixing is needed for each non-zero light neutrino mass. That is, if the minimal seesaw mechanism is the only source of light neutrino masses, there must be at least $n = 2$ RH neutrinos because two mass splittings (“solar” and “atmospheric”) have been observed. If the lightest neutrino is massive, i.e. $m_{\text{lightest}} \equiv \min(m_1, m_2, m_3) \neq 0$, then $n \ge 3$ is required. A heavy neutrino that is a DM candidate would not count in this context - to ensure its longevity, the three mixing angles $\theta_{\alpha I}$ must be so tiny that the effect of the DM candidate on light neutrino masses is negligible. This has an interesting consequence for the scenario with $n = 3$, in which the number of sterile flavors equals that of active flavors: if one of the heavy neutrinos composes the DM, then the lightest neutrino is effectively massless ($m_{\text{lightest}} \simeq 0$). If, on the other hand, the absolute neutrino mass scale $m_{\text{lightest}}$ is larger than $10^{-3}$ eV, then all $M_I$ are expected to be larger than 100 MeV due to cosmological constraints [68]. These conclusions can of course be avoided for $n > 3$ or if there is another source of neutrino mass. The matrix

$$ U = \begin{bmatrix} 1 - \frac{1}{2}\theta\theta^{\dagger} & \theta \\ -\theta^{\dagger} & 1 - \frac{1}{2}\theta^{\dagger}\theta \end{bmatrix} + \mathcal{O}[\theta^3] \begin{bmatrix} U_{\nu} \\ U_N^* \end{bmatrix}. \qquad (2.31) $$

diagonalizes the full $6 \times 6$ neutrino mass matrix $\mathfrak{M}$ to second order in the small expansion parameters $|\theta_{\alpha I}|$:

$$ U^{\dagger} \mathfrak{M} U^* = \begin{pmatrix} U_{\nu}^{\dagger} m_{\nu} U_{\nu} & \\ & U_{N}^{T} M_{N} U_{N}^{*} \end{pmatrix}. \qquad (2.32) $$

The unitary matrices $U_{\nu}$ and $U_N$ diagonalize the mass matrices $m_{\nu}$ defined in (2.30) and $M_N \equiv M_M + \frac{1}{2}(\theta^{\dagger}\theta M_M + M_M^T\theta^T\theta^*)$,

$$ m_{\nu} = U_{\nu} m_{\nu}^{\text{diag}} U_{\nu}^{T}, \qquad M_{N} = U_{N}^{*} M_{N}^{\text{diag}} U_{N}^{\dagger}, \qquad (2.33) $$

$$ M_N^{\text{diag}} = \text{diag}(M_1, M_2, M_3), \qquad m_{\nu}^{\text{diag}} = \text{diag}(m_1, m_2, m_3). \qquad (2.34) $$

The eigenvalues of $M_M$ and $M_N$ coincide in very good approximation; we do not distinguish them in what follows and use the notation $M_I$ for both. The free (quadratic) part of the neutrino Lagrangian reads $\frac{1}{2}\bar{N}(i\not{\partial} - M_N^{\text{diag}})N + \frac{1}{2}\bar{\nu}(i\not{\partial} - m_{\nu}^{\text{diag}})\nu$. These relations hold at tree level. Phenomenological implications of quantum corrections are e.g. discussed in [69, 71], but are not relevant if the sterile neutrinos are DM because $\theta_{\alpha I}$ are too small. The mass terms $m_{\nu}$ and $M_N$ are not exactly identical to the terms $m_T$ and $m_S$ in (2.21) because the latter are mass terms for the chiral fields $\nu_L$ and $\nu_R$, while the former involve the mass eigenstates $\nu$ and
N. These can only be identified with each other if there is no Dirac mass term (2.20), which in the minimal seesaw model (2.25) means $F = \Theta = 0$ and massless active neutrinos. In other models, however, $m_T$ can be generated without a Dirac mass term and mixing with $\nu_R$.

All six mass eigenstates are Majorana fermions and can be represented by the elements of the flavor vectors

$$ \nu = V_{\nu}^{\dagger}\nu_{L} - U_{\nu}^{\dagger}\theta\nu_{R}^{c} + \text{c.c.} \quad \text{and} \quad N = V_{N}^{\dagger}\nu_{R} + \Theta^{T}\nu_{L}^{c} + \text{c.c.} \qquad (2.35) $$

Here, “c.c.” stands for the charge conjugation defined in eq. (2.19). The observed light mass eigenstates $\nu_i$ are related to the active flavor eigenstates by the matrix

$$ V_{\nu} = \left(1 - \frac{1}{2}\theta\theta^{\dagger}\right) U_{\nu}. \qquad (2.36) $$

$V_\nu$ is the usual light neutrino neutrino mixing matrix and $U_\nu$ its unitary part, $V_N$ and $U_N$ are their equivalents in the sterile sector. Unfortunately, unitarity violation due to heavy neutrinos does not appear to be able to resolve the long-standing issues of short-baseline anomalies [161]. The active-sterile mixing is determined by the matrix

$$ \Theta \equiv \theta U_N^*. \qquad (2.37) $$

In summary, one can express the interaction eigenstates (singlets $\nu_R$ and doublet component $\nu_L$) as

$$ \begin{pmatrix} \nu_L \\ \nu_R^c \end{pmatrix} = P_L \mathcal{U} \begin{pmatrix} \nu \\ N \end{pmatrix}. \qquad (2.38) $$

The interactions of neutrinos in the SM are described by the Lagrangian term

$$ -\frac{g}{\sqrt{2}} \overline{\nu_L} \gamma^\mu e_L W_\mu^+ - \frac{g}{\sqrt{2}} \overline{e_L} \gamma^\mu \nu_L W_\mu^- - \frac{g}{2 \cos \theta_W} \overline{\nu_L} \gamma^\mu \nu_L Z_\mu, \qquad (2.39) $$

The interactions of light and heavy neutrinos can be determined by inserting $\nu_L = P_L(V_\nu \nu + \Theta N)$ from eq. (2.38) into eq. (2.39). The unitarity violation in (2.36) implies a flavor-dependent suppression of the weak interactions of the light mass eigenstates $\nu_i$ in eq. (2.35). The doublet component $\Theta_{I\alpha}^T \nu_{L\alpha}^c + \Theta_{I\alpha}^\dagger \nu_{L\alpha}$ of $N_I$ in (2.35) implies that the heavy states have $\theta$-suppressed weak interactions: if kinematically allowed, they appear in any process in which $\nu_{L\alpha}$ appears, but with an amplitude suppressed by $\Theta_{\alpha I}$. The $N_I$ also interact with Higgs particles and $\nu_L$ directly via the Yukawa coupling. This makes various different direct and indirect searches for heavy neutrinos possible [127, 172, 173], see e.g. [67, 71, 170, 241] and references therein for a recent summary.
# 3 Neutrinos in the Standard Model of cosmology and beyond

Section Editors:
Julien Lesgourgues, Alessandro Mirizzi

## 3.1 The standard model of cosmology (author: J. Hamann)

Over the past fifteen years, we have seen the emergence of a theoretical description of the Universe often dubbed the “standard model of cosmology”.<fnref n="14" />

Unlike its namesake and distant relative, the SM, the cosmological standard model is of a more empirical nature. Many of its aspects are not so much fixed by the cold hard requirements imposed by fundamental symmetries, but rather driven by the desire to describe the measurements in a maximally economical way. While this approach might imply a certain degree of arbitrariness and lack of predictivity, it is even more remarkable that it has up to now defied countless attempts to challenge its position and has been left virtually untouched by the avalanche of new information that has accompanied the evolution of cosmology into a precision science.

In this section we will briefly summarize the broad features of this model. During the history of the Universe, all known interactions were relevant, at least during a given range of time. The basic framework of the cosmological standard model is thus given by the theory of General Relativity for gravity, and by the SM for the weak, strong and electromagnetic interactions. The properties of the model can roughly be categorised in terms of its assumptions regarding the geometry and the energy content of the Universe.

### 3.1.1 Geometry

The cosmological standard model does have one symmetry condition, known as the *cosmological principle*, which stipulates that at leading order, the Universe is homogeneous and isotropic. This assumption is very constraining: the most general metric satisfying it is the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, in spherical coordinates given by $ds^2 = dt^2 - a^2(t) (\frac{dr^2}{1-\kappa r^2} + d\Omega^2)$, with a freedom of a one-parameter function, the scale factor $a(t)$, and one discrete *curvature* parameter $\kappa \in \{-1, 0, 1\}$, corresponding to a spatially open, flat, and closed geometry, respectively. In base $\Lambda$CDM however, space is taken to be flat (i.e., $\kappa = 0$). The time evolution of the scale factor is governed by Einstein's equations and determined by the energy content up to a boundary condition, e.g., today's expansion rate, the Hubble parameter $H_0 \equiv \frac{1}{a} \frac{da}{dt}\bigg|_{t=t_0}$.

The fact that today's Universe contains objects such as stars, galaxies, and galaxy clusters requires small deviations from exact isotropy and homogeneity at early times. Actually, initial density fluctuations of the order of $\delta\rho/\rho \sim 10^{-5}$, which eventually start growing under the influence of gravity, are sufficient to seed the rich phenomenology of structures we observe in our neighborhood today. The FLRW metric thus ought to be interpreted as a background solution, with perturbations around it described by *statistically isotropic and homogeneous random fields*.

In the cosmological standard model, the state of these initial perturbations is a particularly simple one: they are taken to be Gaussian and adiabatic curvature perturbations, with

<footnote n="14">¹⁴In the following we shall also refer to it as the “base $\Lambda$CDM” model.</footnote>
an amplitude only weakly dependent on their wavenumber $k$, and described by a power-law power spectrum,

$$P_R(k) = A_s \left(\frac{k}{k_*}\right)^{n_s-1}, \qquad (3.1)$$

where $k_*$ is an arbitrary pivot scale, $A_s$ the amplitude of the spectrum at $k = k_*$, and $n_s$ the spectral index. Coincidentally, perturbations with these properties are a generic prediction of the simplest models of slow-roll single-field inflation — which also explain the flat spatial geometry.

### 3.1.2 Energy content

Whereas there is no evidence for interactions not described by the SM or by general relativity, the particle content of the SM turns out to be insufficient to explain cosmological observations. The base $\Lambda$CDM model requires the presence of five constituents:

* **Baryonic matter** (also including charged leptons) with an energy density parameterized by $\omega_b$. During the process of Big Bang Nucleosynthesis (discussed in section 3.3), light elements are formed; astrophysical measurements of their abundances have traditionally been one of the observational cornerstones in support of the cosmological standard model.

* **Photons** making up the relic radiation of the Big Bang, the Cosmic Microwave Background (CMB). This is easily the most well-measured ingredient of the cosmic recipe. Not only has its blackbody nature been impressively confirmed [242] and its mean temperature been determined to high precision [243], but the measurement of its temperature fluctuations and polarization [244] makes it by far the most powerful source of information for cosmology to date.

* **Neutrinos**: the Cosmic Neutrino Background (CNB) is the neutrino equivalent of the CMB, but has not yet been directly detected. In base $\Lambda$CDM, it is assumed to consist of the three standard model neutrino flavors. The physics of the CNB is covered in more detail in section 3.2.1.

* **Dark Matter (DM)**: this component with energy density parameter $\omega_c$ cannot be identified with any standard model particle, and is characterized by its lack of electromagnetic interaction. In base $\Lambda$CDM, its velocity dispersion is zero, thereby making it cold Dark Matter (CDM).

* **Dark energy**: in base $\Lambda$CDM, this takes the form of a cosmological constant $\Lambda$, i.e., a non-vanishing vacuum energy, whose value is fixed by the requirement of spatial flatness.

### 3.1.3 Parameters of base $\Lambda$CDM

On top of the five free parameters introduced above, a sixth one needs to be taken into account: the optical depth to reionization, $\tau$, which should be derivable from first principles. In practice, however, such a calculation would prove to be extremely complicated, due to the highly non-linear nature of the problem. In any case this simple parametrization is sufficient, since current CMB data are not very sensitive to the exact details of the reionization process.
<table><caption>**Table 3.** Constraints on the base ΛCDM parameters from *Planck* temperature and polarization data [1].</caption><tr><td>Ω<sub>b</sub>h<sup>2</sup></td><td>0.02225 ± 0.00016</td></tr><tr><td>Ω<sub>c</sub>h<sup>2</sup></td><td>0.1198 ± 0.0015</td></tr><tr><td>H<sub>0</sub></td><td>67.27 ± 0.66</td></tr><tr><td>τ</td><td>0.079 ± 0.017</td></tr><tr><td>ln(10<sup>10</sup>A<sub>s</sub>)</td><td>3.094 ± 0.034</td></tr><tr><td>n<sub>s</sub></td><td>0.9645 ± 0.0049</td></tr></table>

This completes the parameter space of the base ΛCDM model:

$$ \Theta_{\Lambda\text{CDM}} = \{\omega_b, \omega_c, H_0, A_s, n_s, \tau\}. \qquad (3.2) $$

Note that this is only one possible parametrization; there exist equivalent ones that are related by parameter transformations. Also, when inferring the values of these parameters from data, one may in addition have to vary extra “nuisance” parameters, depending on the data sets being considered.

### 3.1.4 The cosmological standard model vs. observations

In terms of cosmological observations, a special role is played by the anisotropies of the CMB. The physics of CMB anisotropies is very well understood, so they can be predicted very accurately, leaving no significant theoretical ambiguity. In addition, CMB data are the only single observable able to constrain all parameters of the base ΛCDM model simultaneously. The state-of-the-art is provided by the *Planck* collaboration, who recently released their full mission temperature and polarization data [244], with a further re-analysis that has been announced and is to be expected by 2017.

When faced with observational data, any model claiming the title of a “standard model” needs at the very least fulfill the following criteria: firstly, it ought to give a reasonably good fit to individual observations (internal consistency). Secondly, the parameter values inferred from different observables should be consistent with each other (external consistency). And finally, it should pass the Occam’s razor test, i.e., not be unnecessarily complex. Let us now turn to addressing these three criteria.

### 3.1.5 Internal consistency

Does the base ΛCDM model fit current CMB data well? For the *Planck* data, the goodness-of-fit with respect to base ΛCDM is discussed in ref. [1]: for the best fit of the ΛCDM model to the *Planck* temperature plus large-scale polarization data, the $\ell \ge 30$ TT-data have a reduced $\chi^2$ of 1.03, corresponding to a $p$-value of $p \approx 0.17$. The analogous $p$-values for the TE- and EE-data are $p \approx 0.10$ and $p \approx 0.37$, respectively. In other words, if the true underlying model is indeed the best-fit of base ΛCDM, the observed goodness-of-fit lies well within the expected range and the effective $\chi^2$ is not anomalously large. In addition, the analysis of higher order correlations of the *Planck* temperature shows no signs of departure from Gaussianity [245]. With internal consistency thus established, it makes sense to infer estimates of the free parameters from the data; we list current constraints in table 3.
### 3.1.6 External consistency

Besides the CMB, there are a number of other observables sensitive to subsets of $\Theta_{\Lambda\text{CDM}}$. If these observables happened to prefer different parameter regions within base $\Lambda$CDM, such discrepancies could be a sign for new physics, but they might as well indicate insufficient understanding of experimental or theoretical systematic uncertainties in any of the data sets.

Most cosmologically interesting observables broadly fall into one of two classes: those sensitive to properties of the homogeneous Universe (geometric observables or measurements of primordial element abundances, the latter of which will be discussed in section 3.3), and those constraining properties of its inhomogeneities (shape measurements).

**Geometric observables.** These observables make use of standard rulers or standard candles to relate a known distance to an observed scale. Examples include the baryon acoustic oscillation (BAO) scale, the luminosity distances of high-redshift type Ia supernovae (SNe Ia), or the determination of the Hubble parameter from low-redshift SNe Ia. Combining these measurements with CMB data can help breaking the geometric degeneracy [246] which, although not relevant anymore for the base $\Lambda$CDM, can reappear in extended models (with, e.g., a free curvature parameter, additional radiation, or hot Dark Matter (HDM) components).

While all of the recent galaxy-based BAO scale measurements, as well as the latest SN Ia data show excellent agreement with *Planck* data, there do remain two occurrences of slight discrepancies of ~2.5σ with geometric observables. (i) The BAO scale extracted from measurements of the Lyman-α forest at high redshifts [247]. (ii) The Hubble parameter reported by Riess et al. [248] based on SNe Ia measurements.

Whereas the reason for the discrepancy in (i) remains unclear, a re-analysis of the SNe Ia data [249] finds a value within one standard deviation of the *Planck* value, pointing to systematic issues as a more likely explanation for this tension. On the other hand, even in a more recent analysis, the tension seems to remain present [250].

**Shape measurements.** Even though the CMB power spectra are exquisitely sensitive to the structure of the Universe and the physics around the time of recombination (at a redshift $z \approx 1100$), much of the later evolution of cosmic structures only weakly affects the temperature and polarization power spectra. It may thus be helpful to combine CMB data with lower-redshift tracers of the cosmic structure, particularly with regard to models in which the growth of perturbations at late times is affected (e.g., HDM or dark energy models). Information about the late-time structure of the Universe can for instance be extracted from galaxy surveys (2-/3-dimensional power spectra of the galaxy distribution, redshift space distortions, weak gravitational lensing of galaxies), quasar spectra (1-dimensional power spectrum of hydrogen gas from measurements of the Lyman-α forest), and even from the CMB maps (weak lensing of the CMB, cluster counts). Compared to the CMB power spectra, the interpretation of most of these observations tends not to be as straightforward, due to complications introduced by, e.g., non-linear structure growth and complicated astrophysical processes.

Of particular note here is a mild tension concerning the amplitude of matter perturbations on small scales. Assuming base $\Lambda$CDM, *Planck* CMB data prefer somewhat more power than has been inferred from galaxy weak lensing [251], cluster counts [252], and redshift space distortions [253]. It has been suggested that this could potentially be a sign of an additional HDM component [162, 254–256], but this interpretation is not fully compelling either, as it is not statistically required by the data [257]. Lyman-α data, on the other hand, appear perfectly consistent with CMB data [258] in base $\Lambda$CDM.
### 3.1.7 Occam's razor

Following the detection of the CMB's B-polarization on intermediate scales by the BICEP2 collaboration [259], it seemed for a while that the base $\Lambda$CDM model might have to be extended to include primordial tensor perturbations. However, a more thorough analysis using Planck polarization data to estimate the contribution of polarized emission from galactic dust showed that the signal is consistent with foreground emission [260].

The Planck data themselves have been fitted to many classes of extended models, in a large number of combinations with other data sets.<fnref n="15" /> None of these extended models improve the fit by a sufficient amount to warrant the introduction of extra parameters though. This in turn implies that for the time being, there does not seem to be any convincing evidence against the base $\Lambda$CDM model. With ongoing experimental efforts on many fronts of observational cosmology however, this situation may eventually change.

## 3.2 Active neutrinos in cosmology (authors: J. Lesgourgues, S. Pastor)

### 3.2.1 The cosmic neutrino background

The existence of a relic sea of neutrinos is a generic feature of the standard hot Big Bang model, in number only slightly below that of relic photons. This cosmic neutrino background (CNB) has not been detected yet, but its presence is indirectly established by the accurate agreement between the calculated and observed primordial abundances of light elements, as well as from the analysis of other cosmological observables, such as CMB anisotropies.

Produced at large temperatures by frequent weak interactions, flavor neutrinos ($\nu_{e,\mu,\tau}$) were kept in equilibrium with the rest of the primeval relativistic plasma (electrons, positrons, and photons). Neutrinos had a momentum spectrum with an equilibrium Fermi-Dirac phase-space distribution with temperature $T$: $f_{\text{eq}}(p, T) = [\exp(p/T) - 1]^{-1}$, where we included a neutrino chemical potential $\mu_\nu$ that would exist in the presence of a neutrino-antineutrino asymmetry. It has been shown that, even if $\mu_\nu$ is non-zero, the contribution of the chemical potential cannot be very relevant [261].

As the Universe cools, the weak interaction rate falls below the expansion rate of the Universe and neutrinos decouple from the rest of the plasma. An estimate of the decoupling temperature $T_{\text{dec}}$ can be found by equating the thermally averaged value of the weak interaction rate $\Gamma_\nu = \langle\sigma_\nu n_\nu\rangle$, where $\sigma_\nu \propto G_F^2$ is the cross section of the electron-neutrino processes and $n_\nu$ is the neutrino number density, to the expansion rate given by the Hubble parameter $H = (8\pi\rho/3M_P^2)^{1/2}$. If we approximate the numerical factors to unity, with $\Gamma_\nu \approx G_F^2 T^5$ and $H \approx T^2/M_P$, we obtain the rough estimate $T_{\text{dec}} \approx 1$ MeV. Neutrinos were ultra-relativistic at decoupling, because they cannot possess masses much larger than 1 eV.

Although neutrino decoupling is not described by a unique $T_{\text{dec}}$, it can be approximated as an instantaneous process. In this simple but reasonably accurate approximation, the spectrum is preserved after decoupling, because the number density of non-interacting neutrinos has remained constant in any co-moving volume since decoupling. Shortly after neutrino decoupling the temperature drops below the electron mass, favoring $e^\pm$ annihilations into photons over their production. If one assumes that this entropy transfer did not affect the neutrinos because they were already completely decoupled, it is easy to find that the ratio between the temperatures of relic photons and neutrinos is $T_\gamma/T_\nu = (11/4)^{1/3} \approx 1.40102$. It turns out that neutrino decoupling and $e^\pm$ annihilations are sufficiently close in time so that some relic electron-neutrino interactions exist. These processes are more efficient for

<footnote n="15">¹⁵http://www.cosmos.esa.int/web/planck/pla.</footnote>
larger neutrino energies, leading to non-thermal distortions in the neutrino spectra at the per cent level, and to a smaller increase of the co-moving photon temperature [262]. These changes slightly enhance the contribution of relic neutrinos to the total energy density. In practice, these distortions only have small consequences on the evolution of cosmological perturbations, and for many purposes they can be safely neglected.

The neutrino number density, inferred from the Fermi-Dirac spectrum and $T_ν$, is equal today to 113 neutrinos and antineutrinos of each flavor per cm$^3$. The energy density for massive neutrinos must be calculated numerically at each time, because they contribute initially to radiation, but they behave as matter after the non-relativistic transition. In units of the critical value, the present energy density of neutrinos is $\Omega_ν = \rho_ν/\rho_c^0 = \sum m_ν/93.14\,h^2\,\text{eV}$ (i.e. of order unity for eV masses), where $\sum m_ν$ includes the mass of all the neutrino states that are non-relativistic today.$^{16}$ Taking into account the results of neutrino oscillation experiments, as well as the bounds from neutrinoless double beta decay and tritium beta decay experiments, the possible present values of $\Omega_ν$ are restricted to the approximate range $0.0013 (0.0022) \lesssim \Omega_ν \lesssim 0.13$ for NO (IO), where we already included that $h \approx 0.7$.

### 3.2.2 The effective number of neutrinos

In the base $\Lambda$CDM model, together with photons, neutrinos fix the expansion rate while the Universe is dominated by radiation. Their contribution to the total radiation content can be parametrized in terms of the effective number of neutrinos $N_{\text{eff}}$, defined as

$$ \rho_r = \rho_\gamma + \rho_\nu = \left[ 1 + \frac{7}{8} \left( \frac{4}{11} \right)^{4/3} N_{\text{eff}} \right] \rho_\gamma. \qquad (3.3) $$

Here, we have normalized $\rho_r$ to the photon energy density because its value today is known from the measurement of the CMB temperature. This equation is valid when neutrino decoupling is complete and holds as long as all neutrinos are relativistic.

We know that the number of light neutrinos sensitive to weak interactions (i.e., flavor or active neutrinos) equals three from the analysis of the invisible Z-boson width at LEP, $N_ν = 2.984 \pm 0.008$ [85]. The effect of the small distortions in the CNB spectra from relic $e^±$ annihilations leads to $N_{\text{eff}} \simeq 3.046$ [262] (disregarding neutrino oscillations, one may obtain the slightly higher value of $N_{\text{eff}} \simeq 3.052$ [263]). Any departure of $N_{\text{eff}}$ from this last value would be due to non-standard neutrino features or additional relativistic relics. A detailed discussion of cosmological scenarios where $N_{\text{eff}}$ is not fixed to three can be found in [264–266]. We will see in section 3.3 that the theory of Big Bang Nucleosynthesis (BBN) and the measurement of light element abundances provide interesting bounds on $N_{\text{eff}}$.

### 3.2.3 Massive neutrinos as Dark Matter

A priori, massive neutrinos are excellent DM candidates, in particular because we are certain that they exist, in contrast to other candidate particles. DM particles with a large velocity dispersion such as that of neutrinos are called HDM.

The role of neutrinos as HDM particles has been widely discussed since the 1970s (see e.g. [267]). It was realized in the mid-1980s that HDM affects the evolution of cosmological perturbations in a particular way: it erases the density contrasts on wavelengths smaller

<footnote>$^{16}$Alternatively, one could state $\Omega_ν = \sum m_ν/94.10\,h^2\,\text{eV}$ for exactly $N_ν = 3$ neutrino flavours, which may be more appropriate given that the true value of $N_{\text{eff}}$, to be defined in the next subsection, is not unknown. See ref. [263] for more detailed explanations.</footnote>
than a mass-dependent free-streaming scale. As we will see, in a Universe dominated by HDM, this suppression contradicts various observations, so the attention then turned to CDM candidates, i.e. particles which were non-relativistic at the epoch when the Universe became matter-dominated. Still in the mid-1990s, it appeared that a small admixture of HDM in a Universe dominated by CDM fitted the observational data on density fluctuations at small scales better than a pure CDM model. However, within the presently favored ΛCDM model, dominated at late times by a cosmological constant (or some form of dark energy), a significant contribution of HDM is not needed. Instead, one can use the available cosmological data to find how large the neutrino contribution can be to constrain the possible values of neutrino masses.

### 3.2.4 Effects of standard neutrinos on cosmology

The physical effect of standard neutrinos on cosmological observables is described briefly in [268], and in further details in sections 5.3.3 and 6.1.4 of [264]. In few words, the most crucial point is that at least two out of the three neutrino masses are such that neutrinos become non-relativistic at some time between photon decoupling and today. This has two important consequences, explaining why standard neutrinos can be probed with cosmology. First, they provide a non-trivial background density and pressure, behaving initially as radiation and later as non-relativistic matter. Second, since neutrinos entered the non-relativistic regime only very recently, they still have considerable velocities compared to CDM particles, which prevents them from clustering on small scales: their velocity is greater than the escape velocity from gravitational potential wells on those scales. These two points have important implications for the matter and CMB power spectra in the Universe.

**Matter power spectrum.** The quantity $P(k, z) = \langle |\delta\rho_m/\rho_m|^2 \rangle$, measurable through several types of astrophysical observations, characterizes the amount of clustering of total matter on different wavenumbers $k$ and redshifts $z$. Since total matter includes non-relativistic neutrinos at late times, the absence of neutrino clustering on small scales is expected to reduce the power spectrum by $(1-f_\nu)^2$, where $f_\nu$ is the fraction of total matter in the form of neutrinos, proportional to the total neutrino mass $\sum m_\nu$. In reality, however, the effect is larger than that, because the presence of a non-clustering component contributing nevertheless to the background expansion rate also changes the growth rate of the CDM fluctuations themselves. Hence, on small scales, the matter power spectrum suppression is close to $1-8f_\nu$ according to linear perturbation theory. N-body simulations show that the effect is even further enhanced to about $1-10f_\nu$ through the non-linear clustering happening at a slightly later epoch [269].

While $f_\nu$ only depends on the sum $\sum m_\nu$ of all light neutrino masses, second-order effects on the growth of structure do depend on individual neutrino masses (which govern the individual value of free-streaming scales at a given time). Mass-splitting effects have been shown to be so small that it is unlikely that they can be measured one day (see e.g. [264, 268, 270]).

**CMB anisotropies.** For masses smaller than 0.6 eV, neutrinos are still relativistic at the time of photon decoupling, and their mass cannot impact the evolution of acoustic oscillations. However, the presence of relativistic neutrinos is felt by the CMB perturbations: (i) through the neutrino contribution to the background radiation density and expansion history, that depends on $N_{\text{eff}}$; (ii) through gravitational forces between photon/baryon clumps and neutrino overdensities, important on Hubble-crossing scales during radiation domination: the existence of such forces can be tested by measuring effective parameters with the CMB, like the sound speed and viscosity speed ($c_{\text{eff}}^2, c_{\text{vis}}^2$) of non-photon radiation.
An effect of neutrino masses on the CMB can only appear at two levels: that of the background evolution around and after the non-relativistic transition; and that of secondary anisotropies (related to the behavior of photons after decoupling: integrated Sachs Wolfe (ISW) effect, weak lensing by large scale structure, etc. [264]).

Since the CMB is very sensitive to the redshift $z_{\text{eq}}$ of equality between matter and radiation, the effect of neutrino masses is best discussed for a fixed $z_{\text{eq}}$, and in particular when the density of baryons and CDM in the early Universe is kept fixed. In that case, playing with the neutrino mass still changes a bit the expansion history of the Universe at late times, when neutrinos become non-relativistic. This shows up in the angular distance to recombination (controlling the angle under which we see CMB peaks), and in the time of matter-to-cosmological-constant equality (controlling the late ISW effect). While the second effect is hard to observe due to cosmic variance, the first effect is important. It explains in particular why the CMB tends to constrain the particular combination of $\sum m_{\nu}$ and $H_0$ that fixes the angular peak scale.

Soon after the photon decoupling, at least one neutrino eigenstate is expected to become non-relativistic. This impacts the evolution of metric fluctuations at high redshift. Through the early ISW effect, this non-relativistic transition produces a depletion in the CMB power spectrum of the order of $(\Delta C_l/C_l) \sim -\langle \sum m_{\nu}/0.3 \, \text{eV} \rangle\%$ on multipoles $20 < l < 500$. This effect is roughly thirty times smaller than the depletion in the small-scale matter power spectrum, $\Delta P(k)/P(k) \sim -\langle \sum m_{\nu}/0.01 \, \text{eV} \rangle\%$, and hence difficult to observe.

Finally, since neutrino masses reduce the amplitude of matter fluctuations on small scales, they also result in a weaker lensing of CMB anisotropy patterns. This shows up directly in the temperature and polarization spectra: less lensing means less smoothing and sharper contrasts between maxima and minima, and also a larger spectrum tail at high multipoles. This effect can be seen even better in the reconstruction of the CMB lensing spectrum, that can be inferred from high-order statistics in CMB maps.

### 3.2.5 Current cosmological bounds on standard neutrinos

Recent results from the *Planck* satellite strongly support the existence of the standard neutrino background, through measurements of the background density and of gravitational effects of neutrinos. The previously defined parameters ($N_{\text{eff}}$, $c_{\text{eff}}^2$, $c_{\text{vis}}^2$) are expected to be equal to (3.046, 1/3, 1/3) in a Universe with no relativistic relics beyond standard neutrinos. The *Planck* 2015 release [1] gives (2.99 ± 0.20, 0.3240 ± 0.0060, 0.327 ± 0.037) at 68% C.L., assuming the minimal cosmological model with neutrino masses close to their smallest possible values in the normal hierarchy scenario. These results are well compatible with theoretical expectations.

The neutrino mass bound coming from CMB temperature/polarization spectra is $\sum m_{\nu} < 0.49 \, \text{eV}$ (95% C.L.) for the *Planck* 2015 data [1]. We recall that this bound comes mainly from the angular diameter distance and lensing effects. The first effect can be better constrained by adding information on the Baryon Acoustic Oscillation (BAO) scale at redshifts $0.1 < z < 0.8$, in order to reduce parameter degeneracies. This gives $\sum m_{\nu} < 0.17 \, \text{eV}$ (95% C.L.) for *Planck* 2015 + BAO data [1].

It is important to know that the observed CMB temperature/polarization spectra present a low contrast between maxima and minima, already *slightly* smaller than what would be expected in the base $\Lambda$CDM model with negligible neutrino masses. Any parameter predicting less lensing (like $\sum m_{\nu}$) is hence disfavored by the data. This “slight lensing excess” is statistically compatible with the minimal $\Lambda$CDM model at the $2\sigma$ level, and it could be due to
a statistical fluke. However one cannot totally exclude that it may be induced by unknown systematics or by new physics; in that case, the previous upper bound on $\sum m_\nu$ would be relaxed.

Instead of looking at the smoothing of the temperature/polarization spectra, one can extract from CMB maps the lensing potential spectrum, to probe CMB lensing more directly. In that case, no “slight lensing excess” is observed: on the contrary, the lensing potential spectrum is a bit low on small scales. Hence lensing extraction does not strengthen the neutrino mass bound, and even makes it slightly weaker: $\sum m_\nu < 0.23$ eV (95% C.L.) for Planck 2015 + BAO + lensing [1].

To improve these bounds, one needs to use information from Large Scale Structure (LSS), to probe the neutrino free-streaming effect and the reduction of the matter power spectrum on small scales. These measurements usually suffer from high systematics and modeling uncertainties. They actually give results on neutrino masses that are in moderate tension with each other. The problem is that any LSS data returning a high amplitude of fluctuations on inter-galactic scales will automatically lead to a strong neutrino mass bound, and vice-versa. However a high/low amplitude could easily be the result of unknown systematics. This problem should be solved with future LSS surveys, which will have much better sensitivity, better control on systematics, and — even more importantly — a wide coverage of scales and redshifts, allowing to probe the actual scale-dependent and time-dependent effect of neutrino masses on the growth of structures, without trivial degeneracies with the global amplitude of fluctuations.

Hence the following list of neutrino mass bounds from current LSS data should be taken with a grain of salt. The measurement of the one-dimensional flux power spectrum from Lyman-α forests in BOSS quasars gives $\sum m_\nu < 0.15$ eV (95% C.L.) for Planck 2015 + Lyman-α [258]. A slightly tighter bound, $\sum m_\nu < 0.13$ eV (95% C.L.), is found by combining Planck 2015 with a compilation of baryon acoustic oscillation data, and with the galaxy correlation spectrum of the Sloan Digital Sky Survey (SDSS) [271]. The measurement of the structure formation rate through redshift space distortions (RSD) in the BOSS galaxy survey gives $\sum m_\nu < 0.23$ eV (95% C.L.) for Planck 2015 + RSD [272]. Instead, various measurements of galaxy weak lensing (GWL) and of the cluster mass function (CMF) are in tension with Planck: they would prefer a higher neutrino mass, of the order of 0.4 eV (see, e.g., refs. [1, 273]), but at the cost of adopting values of the Hubble rate in tension with Planck. It is not clear at the moment whether these tensions will be resolved by a better understanding of systematics, or by a change of the physical paradigm. What is clear is that the tension between CMB data and GWL/CMF data cannot be resolved by simply playing with neutrino parameters [1]: extra ingredients would be needed (like for instance a departure from Einstein gravity).

In the future, we expect a sensitivity $\sigma(\sum m_\nu)$ close to 0.02 eV around the year 2025 for a LSS survey like Euclid combined with Planck [274]; or 0.015 eV for a CMB satellite of next generation like Core+ combined with Euclid [275]; or, under very optimistic assumptions, even 0.0075 eV for a futuristic 21cm survey like FFFT [276]. Possibilities to detect the CMB polarization, such as CMB-S4 [277], and observations aiming at dark energy, like DESI [278], will further improve the situation. Hence there are very good prospects to detect the absolute neutrino mass scale with cosmology, and even to exclude the IO scenario if the range $\sum m_\nu \ge 0.12$ eV is found to be incompatible with the data at several $\sigma$'s. If instead the total mass is found close to $\sum m_\nu \sim 0.12$ eV, cosmological data will probably never reach the required sensitivity to discriminate between NO and IO.
## 3.3 Big bang nucleosynthesis (author: G. Mangano)

### 3.3.1 What it is and how it works

Big Bang Nucleosynthesis (BBN) is the epoch in the evolution history of the Universe when light nuclei, basically $^2$H, $^3$H, $^3$He, $^4$He, and $^7$Li, were produced out of the initial plasma of protons and neutrons. This happened via weak, electromagnetic, and strong interactions. The whole phase started about one second after the Bang and lasted for approximately 15 minutes. In terms of the photon temperature this corresponds to the range from 1 MeV (10$^{10}$ K) to 20-30 keV(10$^8$ K).

The basic picture of BBN was understood in the decade after the seminal $\alpha\beta\gamma$ paper [279], and since then it became one of the main observational pillars of the hot Big Bang cosmological scenario. Furthermore, since the abundances produced of nuclear species are quite sensitive to the expansion rate during BBN, as well as to other key parameters such as the baryon density, BBN is a robust probe of the early Universe, and of exotic physics that could be relevant at these stages (extra particle species, new interactions, etc.).

The dynamics of BBN can be summarized as follows:

* Initial conditions. At high temperatures ($T > \text{MeV}$) the Universe is filled by free protons and neutrons, which are maintained in mutual chemical equilibrium by charged current weak interactions with $e^{\pm}$ and electron neutrinos/antineutrinos. These enforce their number density ratio to follow the standard equilibrium result, usually referred to as the Saha equation:

$$ \frac{n_n}{n_p} = \exp \left( -\frac{\Delta m}{T} \right), \qquad (3.4) $$

with $\Delta m \simeq 1.29 \text{ MeV}$ being the neutron-proton mass difference. In this early phase there is no room for formation of bound nuclei, since there are too many photons per baryon. As soon as, say, $^2$H is formed, it is immediately photodissociated by the inverse process,

$$ \gamma + {}^2\text{H} \rightarrow n + p. \qquad (3.5) $$

This is again the effect of chemical equilibrium now applied to the electromagnetic process of eq. (3.5). Defining the baryon to photon number density ratio $\eta_b = n_b/n_\gamma$, the Saha equation gives

$$ \frac{n_{2\text{H}}}{n_{\text{H}}} \sim \eta_{\text{b}} \left( \frac{T}{m_{\text{N}}} \right)^{3/2} \exp \left( \frac{B_{2\text{H}}}{T} \right), \qquad (3.6) $$

with $B_{2\text{H}} \sim 2.2 \text{ MeV}$ being the deuterium binding energy and $m_N$ the mean nucleon mass.

* Weak interaction freeze-out. As it has been already discussed, at $T_{\text{dec}} \sim \text{MeV}$ weak interaction rates become slower than expansion rate of the Universe. At this scale, charged current processes freeze out, and the number of neutrons per co-moving volume is only reduced via neutron $\beta$-decays. Using eq. (3.4), for $T \le T_{\text{dec}}$ we have

$$ \frac{n_n}{n_p} = \exp \left( -\frac{\Delta m}{T_{\text{dec}}} \right) \exp \left( -\frac{t(T)}{\tau_n} \right), \qquad (3.7) $$

where $t(T) \propto (\frac{\text{MeV}}{T})^2$ is the time-temperature relation during radiation domination, and $\tau_n = (880.3 \pm 1.1) \text{ sec}$ is the neutron lifetime [85].
Notice that the time-temperature relation is sensitive to the total radiation energy density, which means that the number of neutrons available at the onset of BBN, which eventually get almost entirely bound in $^4$He nuclei, strongly depends on $N_{\text{eff}}$.

* Deuterium formation. The baryon to photon number density ratio is related to the current energy density of baryons through $\eta_b \sim 274 \cdot 10^{-10} \omega_b$. The latter is accurately measured by CMB anisotropy data, leading to $\eta_b \sim 6 \cdot 10^{-10}$. From eq. (3.6), we see that for $n_{2\text{H}}/n_{\text{H}}$ to become of order one it is necessary to wait until the photon temperature decreases well below $B_{2\text{H}}$. In particular, at $T_{\text{BBN}} \sim 70$ keV, the photodissociation process become ineffective and the deuterium abundance starts raising. This is the *deuterium bottleneck*.

* The nuclear chain. After $^2$H formation, a complicated network of nuclear processes starts. They take place in a rapidly cooling environment with low nucleon density, resulting in a peculiar out-of-equilibrium nuclear chain. Deuterium is rapidly burned into $^4$He, which is the main BBN outcome, since it has the highest binding energy per nucleon among light nuclei. Its final density fraction, $Y_p = 4n_{4\text{He}}/n_b$, is weakly sensitive to the details of the nuclear network, and can be computed in good approximation under the assumption that all neutrons are eventually bound in helium nuclei

$$Y_p \sim 4 \frac{n_n}{2} \frac{1}{n_n + n_p} \Big|_{T_{\text{BBN}}} = \frac{2}{1 + \exp(\Delta m/T_{\text{dec}}) \exp(t(T_{\text{BBN}})/\tau_n)} \sim 0.25, \quad (3.8)$$

which is in very good agreement with experimental results, see the next section. A more accurate determination of $Y_p$ and in particular of other nuclides, which are much less abundant,<fnref n="17" /> can only be obtained numerically, by solving a set of coupled kinetic equations along with the Einstein equations, the constraint of conservation of baryon and electric charge, and the covariant conservation of the total stress-energy tensor, as in [280–282]. The reader interested in this issue or in the details of the nuclear reaction network can refer e.g. to [264, 283, 284].

### 3.3.2 Constraints on the baryon density and $N_{\text{eff}}$

In the standard (minimal) model of BBN, the only free parameter is the baryon-to-photon number density $\eta_b$, or equivalently the present baryon energy density parameter $\omega_b$. In fact, already in the late sixties it was understood that comparing experimental observations of primordial light nuclei abundances with theoretical BBN results was a powerful method to constrain the cosmological baryon density [280]. Today the value of $\omega_b$ is measured with an exquisite precision by CMB temperature and polarization maps, and found to be in the same range required by BBN. Let us discuss this point in more details. The most recent and precise determination by the Planck satellite, reported in table 3, is in agreement with previous results of several earlier CMB experiments. With this value and a standard relativistic energy density at BBN, $N_{\text{eff}} = 3.046$, one obtains from the numerical code of [282]:<fnref n="18" />

$$Y_p = 0.2467 \pm 0.0006, \quad (3.9)$$

$$^2\text{H/H} = (2.61 \pm 0.13) \cdot 10^{-5}, \quad (3.10)$$

<footnote n="17">¹⁷For instance, $^2\text{H/H}$ and $^3\text{He/H}$ are expected to be of order $10^{-5}$, while $^7\text{Li/H}$ is of order $10^{-10}$.</footnote>

<footnote n="18">¹⁸For the still controversial primordial origin of $^7\text{Li}$ observations see e.g. the BIG-BANG NUCLEOSYN-THESIS review in [85].</footnote>
where the 95% C.L. errors are mainly due to uncertainties in the neutron lifetime (for $^4$He) and in nuclear rates like $d(p, \gamma)^3$He (for deuterium). The error coming from uncertainties on $\omega_b$ is strongly sub-dominant for $^4$He, and slight sub-dominant for deuterium. These values are in very good agreement with the most recent experimental determinations of these primordial nuclei ratio from [285, 286] (with 68% C.L.),

$$Y_p = 0.2465 \pm 0.0097, \qquad (3.11)$$

$$^{2}\text{H/H} = (2.53 \pm 0.04) \cdot 10^{-5}. \qquad (3.12)$$

As Planck scientists comment in [1]: “This is a remarkable success for the standard theory of BBN.”. Indeed, it is a remarkable success of the standard cosmological model.

Because the baryon density is basically fixed by the CMB, while BBN can be exploited to constrain non-minimal cosmological models. For instance, nuclei abundances could be affected by a non-standard value of $N_{\text{eff}}$, active neutrino chemical potentials, extra sterile neutrino states, exotic properties of neutrinos and of their phase space distribution, or non-standard neutrino interactions. For a review on these issues see e.g. [264, 284]. Here we only briefly consider how $N_{\text{eff}}$ can be bounded by combining the Planck results with $Y_p$ and $^{2}\text{H}$ experimental determinations, still assuming standard BBN, but with an arbitrary number of relativistic degrees of freedom.

As we mentioned, relativistic particles, such as neutrinos, fix the expansion rate during BBN, which in turn fixes the produced abundances of light elements, and in particular that of $^4$He. This is why BBN gave the first constraints on the number of neutrino species, even before accelerators. The Planck collaboration [1] found that the joint constraints on $(\omega_b, N_{\text{eff}})$ obtained either from CMB data alone or from $^4$He and deuterium abundances nicely agree with each other, showing once more the remarkable consistency of the simplest cosmological models. After marginalizing over the baryon density, the CMB+BBN joint constraint on $N_{\text{eff}}$ at 95% C.L. is

$$N_{\text{eff}} = 2.99 \pm 0.39, \qquad (3.13)$$

using the $^4$He abundance estimation of [285], and

$$N_{\text{eff}} = 2.91 \pm 0.37 \qquad (3.14)$$

if the $^{2}\text{H/H}$ value of [286] is assumed.

## 3.4 Sterile neutrinos in cosmology

The role of sterile neutrinos in cosmology strongly depends on the magnitude of their mass. For different choices of the mass scale, they can be responsible for various phenomena. In this White Paper we are primarily interested in sterile neutrinos with keV masses. However, in the following we briefly sketch other well-motivated mass scales and their phenomenological implications. More details are e.g. given in the reviews [67, 159].

### 3.4.1 eV-scale (authors: M. Archidiacono, N. Saviano)

In recent years, light sterile neutrinos ($m \sim 1$ eV) have been introduced in order to explain intriguing but controversial anomalies in laboratory oscillation experiments. Global analyses [161, 162, 287], based on different scenarios with one (dubbed “3+1”) or two (“3+2”) eV sterile neutrinos, have been performed; however there is still no consensus on a preferred model and the search for sterile neutrinos in laboratory experiments is presently open.
Given the systematic uncertainties of any experimental measurement, it is important to use as many observations as possible to corner sterile neutrinos. In this context, cosmological observations represent a valid complementary tool to probe these scenarios, being sensitive to the number of neutrinos and to the sum of their masses. Indeed, sterile neutrinos can be produced by oscillations with the active neutrinos in the early Universe, contributing — if sufficiently light — to the radiation content (parametrized in terms of the effective number of relativistic degrees of freedom $N_{\text{eff}}$) in addition to photons and ordinary neutrinos and leaving possible traces on different cosmological observables. Moreover, once sterile neutrinos become non-relativistic, they contribute to the matter density and, as well as active neutrinos, they damp density perturbations on scales smaller than the characteristic mass-dependent free-streaming scale, leaving a peculiar imprint on structure formation. Therefore cosmology can reveal sterile neutrinos in two ways: constraining both $N_{\text{eff}}$ (if the steriles are still relativistic) and the effective sterile neutrino mass, $m_{\text{eff}}$, which is defined as the physical mass weighted by the sterile abundance.

The first cosmological observable able to constrain $N_{\text{eff}}$ is Big Bang Nucleosynthesis (BBN). Indeed if these additional states are produced before the active-neutrino decoupling ($T_d \sim 1$ MeV), they would acquire quasi-thermal distributions (depending on their temperature) and behave as extra degrees of freedom at the time of primordial nucleosynthesis. This would anticipate weak interaction decoupling leading to a larger neutron-to-proton ratio and eventually resulting into a larger $^4$He fraction. Furthermore, sterile neutrinos, can distort the $\nu_e$ phase space distribution via flavor oscillations with the active ones, leading to a possible effect on the helium and deuterium abundances. However several and independent investigations [286, 288, 289] showed that one fully thermalized sterile neutrino is still marginally allowed.

Concerning Cosmic Microwave Background (CMB), the basic effect induced by a variation in $N_{\text{eff}}$ is related to the impact on the expansion rate of the Universe at recombination and it is detectable at small scales, as it is explained in refs. [290, 291]. Furthermore the CMB angular power spectrum is also affected both by the background variation due to the sterile neutrino mass and by the early Integrated Sachs-Wolfe effect due to their non-relativistic transition close to the recombination epoch. The combined analysis of $N_{\text{eff}}$ and $m_{\text{eff}}$ performed by the Planck collaboration, using the up-to-date cosmological data [1] including Planck temperature power spectrum, lensing data, and baryonic acoustic oscillations, lead to the following 95% C.L. upper bounds (for massive sterile neutrinos):

$$N_{\text{eff}} < 3.7 \quad \text{and} \quad m_{\text{eff}} < 0.52 \text{ eV}, \qquad (3.15)$$

$$N_{\text{eff}} < 3.7 \quad \text{and} \quad m_{\text{eff}} < 0.38 \text{ eV}, \qquad (3.16)$$

where the former set is computed in presence of the prior cutoff of $m_{\text{ster}}^{\text{ther}} < 10$ eV, while the latter restricting to the region where $m_{\text{ster}}^{\text{ther}} < 2$ eV.

The tension with the eV sterile neutrinos, as hinted by laboratory experiments, is twofold [165, 292, 293]: on one hand the constraint on $N_{\text{eff}}$ rules out one fully thermalized sterile neutrino at high significance (more than 99% C.L.), on the other hand eV sterile neutrinos are too heavy to be consistent with the cosmological mass bounds. However, it has been noticed in [254] that massive sterile neutrinos might partially alleviate the tension between Planck and local Universe observations. Compared to the base-ΛCDM model, adding sterile neutrinos increases the value of the Hubble constant, making it compatible with the measurements of the Hubble Space Telescope; at the same time additional massive
neutrinos lower $\sigma_8$ (the r.m.s. of density fluctuations on a sphere of $8 \text{ h}^{-1}\text{Mpc}$), consistently with the value preferred by lensing and clusters measurements. Finally we note here that lensing and clusters measurements may provide a detection of a non-zero effective mass $m_{\text{eff}} = (0.67 \pm 0.18) \text{ eV}$ [273].

Given this partially contradictory situation and the existence of the laboratory anomalies, it is necessary to study the physical conditions under which the sterile neutrino production can occur in the early Universe. As already mentioned, sterile neutrinos are produced in the primordial plasma by mixing with the active species. Therefore, in order to assess their abundance, it is necessary to solve the quantum kinetic equations for the flavor evolution of the active-sterile neutrino ensemble.

**Flavor evolution for the active-sterile system.** It is commonly assumed that sterile neutrinos $\nu_s$ are produced in the early Universe via mixing with the other neutrino species $\nu_e, \nu_\mu, \nu_\tau$ in presence of collisions. Considering only one additional sterile state, the flavor eigenstates $\nu_\alpha$ are related to the mass eigenstates $\nu_i$ via a unitary matrix $U = U(\theta_{12}, \theta_{13}, \theta_{23}, \theta_{14}, \theta_{24}, \theta_{34})$ [294].

The neutrino (antineutrino) ensemble, simultaneously mixing and scattering in the universal plasma, is described in terms of a $4 \times 4$ momentum-dependent density matrix $\varrho_{\mathbf{p}}(\bar{\varrho}_{\mathbf{p}})$, in which the diagonal components represent the flavor contents while the off-diagonal ones encode the phase information and vanish in the absence of mixing. The general evolution equation for the density matrix $\varrho_{\mathbf{p}}(\bar{\varrho}_{\mathbf{p}})$ is the following [295–298]:

$$i \frac{d\rho}{dt} = [\Omega, \rho] + C[\rho], \qquad (3.17)$$

and a similar expression holds for the antineutrino matrix $\bar{\rho}$. The evolution in terms of the comoving observer proper time $t$ can be easily recast in function of the temperature $T$ (see [297] for a detailed treatment). The first term on the right-hand side of eq. (3.17) describes the flavor oscillations Hamiltonian,

$$\Omega = \frac{M^2}{2} \frac{1}{p} + \sqrt{2} G_F \left[ -\frac{8p}{3} \left( \frac{E_\ell}{m_W^2} + \frac{E_\nu}{m_Z^2} \right) + N_\nu \right], \qquad (3.18)$$

where $M^2 = U^\dagger M^2 U$ is the neutrino mass matrix, while the terms proportional to the Fermi constant $G_F$ encode the matter effects in the neutrino oscillations. At the temperatures of interest, the primordial plasma is composed almost only by $e^\pm$ pairs. The $E_\ell$ term is related to the energy density of $e^\pm$ pairs and describes their refractive effect. Moreover, since the neutrino gas is very dense, the neutrinos themselves form a background medium; as a consequence, we must consider neutrino self-interactions, which make the flavor evolution equations non-linear. Neutrino self-interactions are described by a first order term $N_\nu$, proportional to the difference of the density matrices for $\nu$ and $\bar{\nu}$, namely a primordial $\nu$ asymmetry, which has a negligible contribution in a standard scenario, and by a second order term $E_\nu$, which is proportional to the energy density of $\nu$ and $\bar{\nu}$. Finally, the last term on the right-hand side of eq. (3.17) is the collisional term, which is proportional to $G_F^2$ and describes pair processes, annihilations, and any momentum exchange process. Concerning the mass and mixing parameters involved in the equations of motion, they can be fixed at the best-fit values obtained from the global analysis in the active sector [112] and in the sterile sector [162]. In particular the active-sterile mixing angle turns out to be $\mathcal{O}(10^\circ)$, while the active-sterile mass splitting is $\Delta m_{\text{st}}^2 \sim \mathcal{O}(1) \text{ eV}^2$, much larger than the solar and atmospheric ones.
<figure><img src="image_8.png" /><figcaption>Figure 7. Evolution of the sterile abundance $\rho_{ss}$ in function of the temperature $T$.</figcaption></figure>

**Production of sterile neutrinos and cosmological implications.** For the mass and mixing parameters preferred by laboratory anomalies, sterile neutrinos are copiously produced by oscillations with the active species [165] and they represent one extra degree of freedom. Figure 7 shows the behavior of the sterile component of the density matrix as a function of the temperature of the Universe $T$: at $T \sim 30$ MeV the active-sterile flavor conversions start and the sterile neutrino production reaches $\rho_{ss} = 1$ (which translates into $\Delta N_{\text{eff}} = 1$), in tension with the cosmological bounds, see eqs. (3.15)–(3.16). Furthermore, fixing $\Delta N_{\text{eff}}$ to 1 leads to a strong cosmological bound on $m_{\text{eff}}$ (see figure 2 of ref. [255]), which results in a severe tension with the mass range suggested by laboratory experiments. However cosmological analyses show a negative correlation between $N_{\text{eff}}$ and $m_{\text{eff}}$. This anti-correlation suggests the possibility of reconciling eV sterile neutrinos with cosmology. Indeed, if the production of sterile neutrinos is suppressed in the early Universe, they do not thermalize and their physical mass is weighted by a lower abundance. Several mechanisms have been proposed in order to achieve this suppression. The first one is represented by the introduction of a primordial neutrino-antineutrino asymmetry term [299, 300] in the equations for the flavor evolution (see eqs. (3.17) and (3.18)). This new term, acting as a type of matter effect, would suppress the flavor conversions and by that the sterile production. However, in order to accommodate the cosmological bounds, a very large primordial asymmetry is required, $L \sim \mathcal{O}(10^{-2})$. In the presence of such a large asymmetry, the flavor conversions occur at lower temperature (a few MeV) around the neutrino decoupling time. Therefore active neutrinos are not repopulated anymore by collisions and their abundances are depleted. This causes distortions in the electron (anti-)neutrino spectra which directly participate in the weak interactions ruling the neutron-proton interconversion. The observable consequence is an alteration to the primordial abundances of light nuclei predicted by BBN. Recently a different model has been proposed by several authors [301, 302] in order to
achieve an analogous suppression of the flavor evolution through a new matter term in the equations of motion. This model is based on new secret interactions among sterile neutrinos only,<fnref n="19" /> mediated by a new gauge boson. In the presence of the new interactions, flavor conversions are shifted at lower temperature and sterile neutrinos are produced resonantly by collisions. BBN and mass bounds strongly constrain the model, disfavoring values of the mass of the new gauge boson $M_X > 1$ MeV [304, 305]. Interestingly, if the new light boson ($M_X < 1$ MeV) also couples to DM [302, 306–308], secret interactions will be able to solve the small-scale problems of the CDM scenario. Finally, we mention that new interactions mediated by a very light (or even massless) pseudoscalar might represent an escape route to the cosmological mass bounds [309].

### 3.4.2 keV-scale (authors: A. Boyarsky, O. Ruchaisky)

It is the keV mass scale sterile neutrinos that could play a very prominent role in cosmology. A sterile neutrino is a neutral, massive particle and its lifetime can be very long (see the discussion below). Therefore it is a viable DM candidate. However, to constitute 100% of the DM *its mass should be above* 0.4 keV. Indeed, as sterile neutrinos are fermions, they satisfy the so-called Tremaine-Gunn bound [75], i.e., their phase space distribution in a galaxy cannot exceed that of the degenerate Fermi gas. This bound is very robust as it makes no assumption about possible distribution of DM particles inside a galaxy or even in the early Universe. One can simply estimate the total mass of a galaxy, its size, and the volume in velocity space, defined by escape velocity and then define the average phase space density within the object. These quantities are very robust and close to the direct observables. The phase space density appears to be the largest for dwarf satellite galaxies of the Milky Way (for review see e.g. [310]), so that these objects provide the strongest bounds. At the same time, dwarf galaxies are the most DM dominated objects which also reduces the uncertainty in the determination of the DM mass. Still, there are appreciable astronomical uncertainties in deducing phase-space density from astronomical observations, and under reasonable assumptions they can be changed by a factor ~2, which has already been taken into account when quoting the above bound, see e.g. [311–313].

To be a DM candidate, keV sterile neutrinos need to be produced efficiently in the early Universe. Since they cannot thermalize easily, the simplest production mechanism is via mixing with the active neutrinos in the primordial plasma [177]. What is important, however, is that sterile neutrino DM is practically always produced out of thermal equilibrium, and therefore its primordial momentum distribution is in general *not* given by a Fermi-Dirac distribution. Indeed, sterile neutrinos in equilibrium have the same number density as ordinary neutrinos, i.e., $112 \text{ cm}^{-3}$. With the lower bound on the sterile neutrino mass being 0.4 keV, this would lead to the energy density today $\rho_{\text{sterile,eq}} \simeq 45 \text{ keV/cm}^3$, which significantly exceeds the critical density of the Universe $\rho_{\text{crit}} = 10.5 \text{ h}^2 \text{ keV/cm}^3$. Therefore, sterile neutrino DM cannot be a thermal relic (unless entropy dilution is exploited, see section 5), and its primordial properties are in general different from such a particle.

Assuming the validity of Big Bang theory already below $T \sim 1$ GeV,<fnref n="20" /> one can relate the sterile neutrino mass $m_N$ and the mixing angle $\theta^2$ needed to produce the correct DM abundance. This relation depends on one additional parameter, namely the level of *lepton-antilepton asymmetry* present in the plasma at the epoch of production. If the lepton asymmetry

<footnote n="19">¹⁹The cosmological impact of secret interactions among active neutrinos has been discussed, see e.g. [303].</footnote>

<footnote n="20">²⁰This hypothesis has never been rigorously proven by any observation. Strictly speaking, the hot Big Bang model is only valid for temperatures below few MeV, corresponding to Big Bang Nucleosynthesis.</footnote>
is significantly larger than the baryon asymmetry, i.e. if $\eta_L \equiv n_L/n_\gamma \gtrsim 10^6 \eta_b$ ($\eta_b = n_b/n_\gamma = 6 \times 10^{-10}$ is the baryon asymmetry of the Universe),<fnref n="21" /> the mixing of sterile neutrino with active neutrinos can occur resonantly [314–317] (similar to the MSW effect [318, 319]). Otherwise, non-resonant production [177, 320] takes place. The non-resonant production defines a smallest amount of DM that can be produced for given $m_N$ and $\theta^2$, while the resonance may (significantly) enhance it depending on the lepton asymmetry present in plasma.

The lower bound on the DM mass becomes stronger once a particular production mechanism is specified and the corresponding primordial distribution of sterile neutrinos is taken into account. As the DM particles obey dissipationless Liouville dynamics (neglecting the baryonic feedback for a moment), the maximum of their phase-space distribution remains constant. The coarse-grained distribution (i.e., the only one that can be inferred from the astronomical observations) can only decrease this maximum, see section 4.1. This gives a bound on the mass of the sterile neutrino DM that is stronger by a factor of few than that of a degenerate Fermi gas, bringing the lower bound to $\sim 1$ keV or above.

A keV sterile neutrino mixes with ordinary neutrinos. In the presence of sterile neutrinos, the leptonic weak neutral current is nondiagonal in mass eigenstates [321], so the $N$ can decay at tree-level via Z-exchange, as $N \to \nu_i\bar{\nu}_j\nu_j$, where $\nu_i$ and $\nu_j$ are mass eigenstates. The necessary and sufficient condition for the leptonic weak neutral current to be diagonal in mass eigenstates is that all leptons of the same chirality and charge must have the same weak $T$ and $T_3$, which is violated in the presence of sterile neutrinos [321]. Its keV-scale mass makes the decay $N \to \nu_\alpha\nu_\beta\bar{\nu}_\beta$ possible (here $\alpha, \beta$ are neutrino flavors and all possible combinations of flavors are allowed, including charge conjugated decay channels as $N$ is a Majorana particle). The total decay width for $N \to 3\nu$ is given by [321–323]:

$$ \Gamma_{N \to 3\nu} = \frac{G_F^2 m_N^5}{96\pi^3} \sin^2 \theta = \frac{1}{4.7 \times 10^{10} \text{ sec}} \left( \frac{m_N}{50 \text{ keV}} \right)^5 \sin^2 \theta, \quad (3.19) $$

and one requires that the corresponding lifetime should be much longer than the age of the Universe. This imposes a bound on the mixing angle $\theta^2$ [324]:

$$ \theta^2 < 1.1 \times 10^{-7} \left( \frac{50 \text{ keV}}{m_N} \right)^5 \quad \text{— lifetime longer than the age of the Universe.} \quad (3.20) $$

Already this bound tells us that the contribution of the DM sterile neutrino to neutrino oscillations driven by $\delta m_\nu \sim m_N \theta^2$ becomes much smaller than the solar neutrino mass difference $\sqrt{\Delta m_\odot^2} \sim 10^{-4}$ eV) [189, 325]. Therefore, at least two more sterile neutrinos are required to explain to observed mass differences if the neutrino masses are due to exactly two sterile neutrinos, and the lightest active mass is (almost) zero — a non-trivial prediction that can be checked by the Euclid space mission [274], by CMB-S4 [277], or by DESI [278].

A further bound on the mixing angle, much stronger than (3.20) comes from a one loop mediated radiative decay $N \to \nu+\gamma$ that leads to a monochromatic X-ray line signal. If sterile neutrinos are produced via mixing (whether resonantly, or not) with ordinary neutrinos, their mass should be below about 50 keV [186, 316], both to yield the correct DM abundance and not to produce a too strong decay line. Therefore, in the minimal setting, the mass of DM sterile neutrinos should be in the range of 0.4–50 keV. Note that the lower bound is fairly

<footnote n="21">²¹Usually $\eta_b$ and $\eta_L$ are thought to be of the same order of magnitude. However, the observational constraints on $\eta_L$ are very weak (see e.g. [150] and references therein) and allow for asymmetries that are many orders of magnitude larger than $\eta_b$.</footnote>
<figure><img src="image_9.png" /><figcaption>**Figure 8.** Left panel: characteristic form of the resonantly produced (RP) sterile neutrino distribution function for $m_N = 3$ keV and various values of the lepton asymmetry parameter $L_6 \equiv 10^6(n_\nu - \bar{n}_\nu)/s$. The spectrum for $L_6 = 16$ (red solid line) is shown together with its resonant (dashed) and non-resonant (dot-dashed) components. All these spectra have the same shape for $q \gtrsim 3$. Right panel: transfer functions for the spectrum with $m_N = 3$ keV and $L_6 = 16$ together with a CWDM spectrum for $m_N = 3$ keV and $F_{\text{WDM}} \simeq 0.2$ (blue dashed-dotted line). Vertical lines mark the free-streaming horizon of non-resonant ($k_{\text{FSH}}$, left line) and resonant ($k_{\text{RES}}^{\text{res}}$, right line) components. The green dashed line shows the transfer function for non-resonantly produced sterile neutrino with $m_N = 3$ keV. Reproduced from [330].</figcaption></figure>

universal (if 100% of DM is made of this particle) and does not depend on the production mechanism, however, taking into account more specific information about the details of the DM production may be used to derive much stronger bounds from structure formation data.

If (part of) the DM particles are born relativistic, they stream out of the smallest overdensities, erasing structure on scales below their *free-streaming horizon* $\lambda_{\text{FS}}$ (see e.g. [59] for a definition). This means that the power spectrum of sterile neutrino DM is suppressed at scales below $\lambda_{\text{FS}}$ (as compared to a CDM power spectrum with the same cosmological parameters). For example, the suppression of the power spectrum of density perturbations due to the non-zero masses of active neutrinos (that were relativistic and cooled down during the matter-dominated epoch) is at the few per cent level (see e.g. [268]). The situation is different if particles were born relativistically, but cooled down and became non-relativistic while deeply in the radiation dominated. Such DM candidates are called *warm* DM (WDM), although their spectrum may not even necessarily be thermal. The free-streaming of WDM particles modifies the primordial spectrum of density perturbations in the epoch when these perturbations are still linear. Once the WDM particles became non-relativistic, their evolution is governed by the same dynamics as that of conventional CDM. The suppression of power for WDM particles however occurs on smaller scales than for the massive active neutrinos (below a comoving Mpc or similar). These scales are significantly less constrained by the current cosmological data. Currently two major methods are used: (1) the Lyman-$\alpha$ forest (see e.g. [59, 326–332]) and (2) studying the properties of sub-structures in the Local Group [58, 313, 333–342]. These methods are subject to large statistical and systematic uncertainties [59]. Based on the current observational limits, the scales of interest for WDM range from 1 Mpc (comoving scale) to tens of kpc. At the Mpc scales the suppression of the power spectrum as compared to the standard (ΛCDM) prediction cannot be more than tens of per cent, but at smaller scales (order 100 kpc or below) the suppression can be stronger. Current constraints are discussed in a recent review [343].

The production of the DM sterile neutrino via mixing becomes most efficient at temperatures $T \sim 150–500$ MeV [177, 315, 320, 324] resulting in the population of “warm”
DM particles. Resonant production, in turn, results into an efficient conversion of an excess of $\nu_e(\bar{\nu}_e)$ into DM neutrinos $N$ [314, 316]. While non-resonant (NRP) oscillations result in (quasi-) thermal momentum distribution [320, 344], resonantly produced (RP) sterile neutrinos are typically much colder and the dispersion of their momentum distribution is also much smaller than thermal (see figure 8, left panel). Therefore, resonantly produced sterile neutrinos behave in some sense like a *mixture of a cold and warm DM (CWDM)* over some range of scales (see figure 8, right panel). As resonant production does practically not depend on the coupling constant between active and sterile neutrinos (that is why it is so efficient), for DM particles produced in that way this coupling is *not* fixed by the DM abundance (as opposed to the non-resonant case) and it could even be smaller. The expected flux of X-ray photons, produced by decays of DM particles in galaxies and clusters is, on the contrary, always proportional to this coupling constant. Therefore, the bounds from X-ray observations are relaxed for resonantly produced sterile neutrinos. It has been demonstrated in [58, 59, 330, 345-347] that resonantly produced sterile neutrino DM is fully compatible with all existing astrophysical and cosmological observations but, at the same time, it is “warm” enough to suppresses sub-structures in Milky-Way-size galaxies, consistent with observations [58]. Therefore, the value of lepton asymmetry present in the plasma at these temperatures is an important parameter for such type of DM. It defines the fraction of CDM (when the spectra are approximated as CWDM mixture) and also the lifetime of the DM particles.

### 3.4.3 MeV-scale (authors: S. Pascoli, N. Saviano)

Heavy sterile neutrinos $N_i$, $i = 4, 5, \dots$, with masses $M_i$ in the range $O(1-100)$ MeV and mixing with active neutrinos (with $|U_{ai}| \ll 1$) emerge in low scale seesaw models in order to explain neutrino masses, as for instance in the $\nu$MSM (Neutrino Minimal Standard Model — see [186] for a review). They have many possible experimental signatures depending on their masses and mixing angles.

Bounds on these heavy sterile neutrinos are obtained from several terrestrial experiments (as first obtained in [127, 172, 173, 348]; for detailed reviews see ref. [170] and also [64, 71, 349]). For masses up to $\sim 3$ MeV, kink searches [127] pose model-independent bounds on $U_{e4}$ down to $|U_{e4}|^2 \lesssim 10^{-3}$ and for heavier masses the strongest limits are due to peak searches, looking for the spectrum of electrons in pion and kaon decays, as strong as $|U_{e4}|^2 \lesssim \text{few} \times 10^{-8}$ around (60–100) MeV [85, 127, 172, 173, 348, 350]. For heavier masses, constraints are obtained from looking for the decay products of these sterile neutrinos in accelerator experiments. If of Majorana nature, they would also induce neutrinoless double beta decay which currently provides the strongest bound, $|U_{e4}|^2 < 10^{-8}$ at 100 MeV. It should be pointed out that this latter limit could be evaded if a cancellation of the sterile neutrino contributions to the decay takes places, for instance due to the interplay among different sterile neutrinos or due to additional lepton number violating processes. For mixing with muons, peak searches [127, 172, 173] provide the most stringent bounds on $|U_{\mu4}|^2$ using pion decays, $|U_{\mu4}|^2 \lesssim \text{few} \times 10^{-5}$, and kaon decays, $|U_{\mu4}|^2 \lesssim 10^{-4}-10^{-9}$, depending on the heavy neutrino mass [85, 351]. Mixing with tau neutrinos is significantly less known, with bounds coming only from searches of decays of heavy neutrinos, $|U_{\tau4}|^2 < 10^{-4}$ at 100 MeV. Furthermore, since $N_i$ mix with active neutrinos, they can, in principle be emitted by core-collpase supernovae. In this context limits have been placed from the SN 1987A observation, requiring that the SN core should not emit too much energy in the $N_i$ channels, since this additional energy-loss would shorten the observed neutrino burst [352, 353]. Furthermore, in [354] the possible role of heavy sterile neutrinos in enhancing supernova explosions has been discussed.
In this section, we focus on the constraints on $N_i$ that can be placed from cosmological arguments. The basic idea is that $N_i$ can be produced in the Early Universe by the mixing with active neutrinos, and then decay into lighter species. For sufficiently large mixing angles, they would reach equilibrium and decouple at a temperature higher than that of active neutrinos by a factor $|U_{\alpha i}|^{2/3}$. For smaller mixing, they would be produced by decoherence in the active neutrino processes, and even a small fraction could lead to very significant effects. In particular, for masses in the range $1 \text{ MeV} < M_i \lesssim M_\pi \sim 140 \text{ MeV}$, the main decay channels are $N_i \to 3\nu$ and $N_i \to \nu e^+ e^-$ with a decay rate of [322]:

$$ \Gamma_i \approx \frac{1 + (g_L^a)^2 + g_R^2}{768\pi^3} G_F^2 M_i^5 |U_{\alpha i}|^2, \qquad (3.21) $$

where we have considered the neutral current contribution, and $g_L^a = 1/2 + \sin^2 \theta_W$, $g^{\mu,\tau} = -1/2+\sin^2 \theta_W$, $g_R = \sin^2 \theta_W$. If $N_i$ mix with electron neutrinos, the decay $N_i \to \nu e^+ e^-$ would receive an additional contribution from the charged current interactions. The decay products of the sterile neutrinos are injected into the primordial plasma, increasing its temperature and shifting the chemical equilibrium. Depending on their mass and mixing, the $N_i$ would decay in different cosmological epochs affecting different observables. For masses heavier than that of the pion additional decay channels are kinematically allowed, e.g. $N_i \to \pi^0 \nu$, $N_i \to \pi^\pm e(\mu)$. They dominate with a much faster decay rate. Here, however, we restrict our discussion to heavy neutrino masses below the pion threshold.

The decay of $N_i$ into active neutrinos and other particles, occurring during or shortly before BBN, would influence the abundances of light elements through indirect and direct effects. The decays into electrons and positrons will inject more energy into the electromagnetic part of the primeval plasma, thereby slowing down the cooling of the Universe and affecting the neutron-to-proton ratio which rules the abundance of the primordial yields [62, 71, 352, 353]. Concerning the decay into neutrinos, the injection can modify the spectra of the active species which directly participate in the charged current interaction governing the proton-neutron interconversion, and consequently the BBN products. Moreover, the decay products could directly dissociate nuclei that have already formed. The most updated bounds from BBN have been discussed in [62] where it has been shown that, in the region of masses $1 \text{ MeV} < M_i \lesssim 140 \text{ MeV}$, primordial nucleosynthesis restricts the lifetime of sterile neutrinos to be well below 1 sec. For higher values of the mass of $N_i$ the bounds would relax since sterile neutrinos can decay also into heavier species, and the branching ratio into $3\nu$ becomes sub-dominant.

If $N_i$ decay after BBN, one can make use of the Hubble constant, of Supernovae Ia luminosity distances, of the Cosmic Microwave Background shift parameter, as well as of measurements of the Baryon Acoustic Oscillation scale to set bounds on the mixing parameters. Indeed, the decay of additional heavy neutrino species can modify the Universe expansion rate between BBN and recombination, thereby causing entropy production diluting the density of ordinary neutrinos and producing extra radiation affecting the value of $N_{\text{eff}}$. Specifically, they would produce a highly non-thermal component of active neutrinos whose distribution is dictated by the kinematics of the heavy neutrino decay and which is severely constrained by the amount of dark radiation allowed at recombination. Information on the early expansion rate, e.g. from Baryon Acoustic Oscillations, essentially imposes that the decay time needs to be faster than 0.1 s. A very detailed recent analysis in [355] finds that the bounds are as strong as $|U_{\alpha 4}|^2 \lesssim 10^{-14}-10^{-17}$ for $M_i \sim 1 \text{ MeV}-100 \text{ MeV}$.
These bounds are much stronger than those which can be obtained in terrestrial experiments, discussed briefly above. However, they can be evaded in nonstandard particle physics and/or cosmological scenarios. A possibility would be to consider a low-reheating model in which the temperature at the end of inflation, the so-called reheating temperature $T_{RH}$, is low enough $\ll 100$ MeV [356]. The abundance of sterile neutrinos becomes very suppressed with respect to that obtained within the standard framework, allowing to relax the previous bounds. For $M_i \gg 1$ MeV, another option is to consider heavy sterile neutrinos which have additional interactions so that they decay much faster than in the standard case, $\Gamma_i^{-1} \ll 1$ s, and their decay products become thermalised before BBN.

### 3.4.4 GeV–TeV-scale (authors: A. Ibarra)

The stringent limits on the mixing angle of sterile neutrinos with mass in the keV–MeV range reviewed in the previous section from Big Bang Nucleosynthesis or from the Cosmic Microwave Background totally disappear when the mass is larger than $\sim 1$ GeV. Nevertheless, heavy sterile neutrinos can still play an important role in cosmology. If the sterile neutrino is short lived on cosmological time scales, its out-of-equilibrium decay generates a cosmic lepton asymmetry provided C and CP are violated which requires the existence of more than one sterile neutrino state in Nature. Moreover, if the decays occur earlier than $\sim 10^{-12}$ s after the Big Bang, when the sphalerons are still in thermal equilibrium, the lepton asymmetry can be efficiently converted into a baryon asymmetry, thus providing an explanation for the matter-antimatter asymmetry observed in our Universe [150]. This is the renowned leptogenesis mechanism [357], which will be reviewed in the next subsection. On the other hand, if the sterile neutrino is very long-lived on cosmological time scales, it contributes to the DM of our Universe. The longevity of the DM particle implies that the mixing angle with the active neutrinos must be tiny, which in turn implies that DM thermal production via active-sterile oscillations is highly inefficient. Thus, other mechanisms must then be at work in order to generate the observed DM abundance $\Omega_{DM}h^2 \simeq 0.12$, as measured by the Planck satellite [164]. Regardless of how the DM particle was produced, this scenario has phenomenological interest since the decay of the DM particles could lead to experimental signatures through the observation of an excess in the fluxes of cosmic antimatter, gamma-rays, or neutrinos with respect to the expected astrophysical backgrounds (for a review, see [358]).

The most distinctive signal of sterile neutrino decay is the gamma-ray line at $E = M_N/2$ produced in the one-loop decay $N \to \gamma\nu$. The predicted partial decay width reads [322, 359]:

$$ \Gamma_{\gamma\nu} = \frac{9 \alpha_{\text{EM}} G_F^2}{256 \pi^4} \sin^2 \theta m_N^5 = \frac{1}{6 \times 10^{29} \text{s}} \left( \frac{\sin \theta}{10^{-26}} \right)^2 \left( \frac{m_N}{500 \text{ GeV}} \right)^5, \quad (3.22) $$

where $\alpha_{\text{EM}} = 1/137$ and $G_F = 1.166 \times 10^{-5} \text{ GeV}^{-2}$. Gamma-ray lines have been searched for using data collected by the Fermi-LAT [39], as well as by the H.E.S.S. [360] and MAGIC telescopes [361]. The strongest limits on the inverse decay width read $\Gamma_{\gamma\nu}^{-1} \gtrsim 10^{29}$ s for a DM mass in the range of 10-600 GeV [39] and $\Gamma_{\gamma\nu}^{-1} \gtrsim 4 \times 10^{27}$ s for $m_N = 1-30$ TeV [358], with practically no sensitivity to the choice of the DM halo profile. These limits in turn translate, respectively, into $\sin\theta \lesssim 1.4 \times 10^{-24} (M_N/100 \text{ GeV})^{-5/2}$ and $\sin\theta \lesssim 2.2 \times 10^{-26} (M_N/1 \text{ TeV})^{-5/2}$.

Complementary limits on the mixing angle can be obtained from the tree-level decays $N \to W^{\pm}\ell^{\mp}$, $Z\nu$, when kinematically accessible, which produce a continuum flux of gamma-
rays, antimatter particles, and neutrinos... The decay widths of these processes read [362]:

$$ \Gamma_{W^{+}\ell^{-}} = \Gamma_{W^{-}\ell^{+}} = \frac{G_F}{8\sqrt{2}\pi} \sin^2\theta m_N^3 \left(1 + 2\frac{M_W^2}{m_N^2}\right) \left(1 - \frac{M_W^2}{m_N^2}\right)^2, \quad (3.23) $$

$$ \Gamma_{Z\nu} = \frac{G_F}{8\sqrt{2}\pi} \sin^2\theta m_N^3 \left(1 + 2\frac{M_Z^4}{m_N^4}\right) \left(1 - \frac{M_Z^2}{m_N^2}\right)^2. \quad (3.24) $$

Numerically, for $M_N \gg M_W, M_Z$, we have:

$$ \Gamma_{W^{+}\ell^{-}} = \Gamma_{W^{-}\ell^{+}} \approx \Gamma_{Z\nu} \approx \frac{1}{2 \times 10^{26} \text{ s}} \left( \frac{\sin \theta}{10^{-26}} \right)^2 \left( \frac{m_N}{500 \text{ GeV}} \right)^3. \quad (3.25) $$

Limits on the DM decay width with gauge bosons in the final state were derived in [363] from an analysis of the diffuse gamma-ray emission using Fermi-LAT data, in [361] from observations of the dwarf galaxy Segue 1 using the MAGIC telescopes, and in [364] from Fermi-LAT observations of several nearby galaxies and clusters. Besides, limits from antiprotons were derived in [365, 366] from the non-observation of an excess in the antiproton-to-proton fraction measured by the PAMELA experiment [367]. Lastly, limits from positrons were derived in [368] from the AMS-02 measurements of the positron fraction [369] and the positron flux [370]. All these different observations typically require $\Gamma_{Z\nu}^{-1} \gtrsim 10^{26}$ s for $m_N \le 10$ TeV, which translates into an upper limit on the mixing angle $\sin\theta \lesssim 1.2 \times 10^{-26}$ ($M_N/500$ GeV)$^{-3/2}$. Tree level decays into weak gauge bosons then provide, for $M_N \lesssim 5$ TeV, the strongest limits on this scenario, due to the larger branching fraction.

For larger masses, the strongest limits stem from the non-observation of the neutrinos produced in the two body decay $N \to Z\nu$. More specifically, limits on the high energy neutrino flux from AMANDA between 16 TeV and $2.5 \times 10^3$ TeV [371], from IceCube between 340 TeV and $6.3 \times 10^6$ TeV [372, 373], from Auger between $10^5$ TeV and $10^8$ TeV [374] and from ANITA between $10^6$ TeV and $3.2 \times 10^{11}$ TeV [375] typically require $\Gamma_{W^{\ell,Z\nu}}^{-1} \gtrsim 10^{26} - 10^{27}$ s for masses between $\sim 10$ TeV and the Grand Unification scale [376], which translates into $\sin\theta \lesssim 1.4 \times 10^{-28} \left(\frac{M_N}{10 \text{ TeV}}\right)^{-3/2}$. Clearly, this scenario requires a very strong suppression of the mixing angle, assuming that the sterile neutrino constitutes to totality of the DM of the Universe. If the sterile neutrino contributes only to a fraction $f$ of the total DM density, the previous limits on the mixing angle should be scaled by a factor $f^{1/2}$.

### 3.4.5 Leptogenesis (author: P. Di Bari)

The same minimal extension of the SM that can account for neutrino masses and mixing, the (type I) seesaw mechanism [153–156, 377], can also elegantly explain the observed matter-antimatter asymmetry of the Universe [150] with leptogenesis [357]. The out-of-equilibrium decays of heavy RH neutrinos at temperatures above the electroweak symmetry breaking temperature $T_{EW} \sim 100$ GeV, can produce a sizeable $B-L$ asymmetry that, first injected in the form of a lepton asymmetry, is then partly reprocessed by sphaleron processes into a baryon asymmetry able to reproduce the observed baryon-to-photon number ratio $\eta_b \simeq (6.1 \pm 0.1) \times 10^{-10}$ [164]. In the absence of strong mass degeneracies or fine tuned cancellations in the seesaw formula, the mechanism requires very high masses for the RH neutrinos producing the asymmetry, $M_i \gtrsim 10^9$ GeV [378]. These can be either the lightest RH neutrinos, as in the original proposal [357], or the next-to-lightest RH neutrinos [379–381], or both [382–385]. In any case this lower bound, even with flavor effects [386, 387] included [388], also implies
a similar lower bound on the reheating temperature $T_{RH} \gtrsim 10^9$ GeV. It is also interesting that there is an upper bound on the lightest neutrino mass $m_1 \lesssim (0.1)$ eV [381, 389–391] on an upper bound that now cosmological observations, in particular the new results from the Planck satellite, indicate to be respected.

Within a minimal extension of the SM the scale of leptogenesis can be lowered to the TeV scale thanks to a resonant enhancement of the *CP* asymmetries if at least two RH neutrino masses are quasi-degenerate [392]. In recent years, in the LHC era, much attention has been devoted to scenarios of resonant leptogenesis that might produce signals at colliders (e.g. [393]) or large lepton flavor violating rates [394]. These scenarios can be motivated and embedded within extensions of the SM beyond the minimal type I seesaw scenario such as the inverse seesaw [237].

An interesting alternative scenario of leptogenesis is offered by the possibility that the matter-antimatter asymmetry is generated by RH neutrino oscillations [395] instead of decays. This is realized within a different region of the seesaw parameter space with RH neutrino masses of a few GeV. The key point of leptogenesis from RH neutrino oscillations is that RH neutrinos also carry (chiral) asymmetries. These can be generated by RH neutrino oscillations and then transferred to the active sector by the Yukawa couplings in a way that the total asymmetry the sum of those in the RH and LH sector vanishes. The violation of total $B-L$ (involving both chiralities) is suppressed because the asymmetries are generated at temperatures that are much larger than the $B-L$-violating Majorana masses, but sphalerons only “see” the left chiral charges and therefore generate a net $B$-charge. In this regard, the scenario is similar to Dirac leptogenesis [396]. In the original proposal [395] it is essential that one of the two oscillating RH neutrinos thermalizes after sphaleron processes become inoperative. In this way, RH neutrino mixing can generate opposite asymmetries in the two RH neutrino species but only that in the thermalized RH neutrino is communicated to the active sector and then reprocessed into a baryon asymmetry. *CP* violation in the RH neutrino mixing, requiring a non-vanishing value of the Jarlskog invariant for RH neutrino mixing, is a necessary ingredient.

However, in [188] it was noticed that the same coupling between the LH and the RH neutrino sectors provides an additional source of *CP* violation that has to be taken into account. This source — and not *CP* violation in the RH neutrino mixing — can be responsible for a generation of an asymmetry able to explain the observed one within the so called $\nu$MSM [189], where the mixing between the two heavier RH neutrinos generate the asymmetry and the lightest RH neutrino, with mass $M_1 \sim$ keV, can play the role of (warm) DM. In this way the model provided a picture where neutrino masses, mixing, DM and matter-antimatter asymmetry are all explained within just the minimal type I seesaw extension of the SM [397, 398].

The scenario requires RH neutrino masses $M_{2,3} \sim (1–80)$ GeV and highly degenerate ($|M_3 - M_2| \sim 10^{-6} M_2$). Leptogenesis occurs when the oscillating RH neutrinos are ultra-relativistic at $T_L \sim 10^4$ GeV. In the $\nu$MSM, the Dirac neutrino masses are below the keV scale and, therefore, neutrino Yukawa couplings are all smaller than $\sim 10^{-8}$, reintroducing somehow the problem of the smallness of neutrino Yukawa couplings that the seesaw mechanism was supposed to solve. However, despite this price to be paid, the intriguing feature is that the seesaw formula and the neutrino data enforce a mixing of the RH with LH neutrinos that is exactly what is needed in order to produce the correct amount of DM in the early Universe. This would happen at temperatures above MeV, and thus prior to the onset of Big Bang Nucleosynthesis. On the other hand a sufficient production also requires the presence of a large lepton asymmetry resonantly enhancing the active-sterile neutrino mixing. This can
be generated by the leptogenesis mechanism itself, though in this case even more fine tuned conditions are required in order to have a much smaller baryon asymmetry reproducing the observed one [399].

If one does not require the lightest RH neutrino to be the DM particle, then the possibility to reproduce the correct asymmetry with leptogenesis from RH neutrino oscillations (together with correct light neutrino masses and mixing parameters) occurs for a larger parameter space. In particular RH neutrino masses can be much heavier than a few GeV [400] and there is no need for a degeneracy in the heavy neutrino masses [401]. This opens up the possibility to find the heavy neutrinos that generated the BAU at existing experiments [401] or at the proposed SHiP experiment [402]. In principle, even the relevant source of CP-violation may be probed [398, 403], allowing to unveil the origin of matter in experiments. However, it has also been noticed that in this case a fine tuning in the seesaw formula is then necessarily required anyway [404].

An alternative possibility to combine DM and Leptogenesis is to fully decouple one of the RH neutrinos that would then play the role of DM. This has to be induced through physics beyond a minimal type I seesaw extension, for example from inflation or introducing the 5-dim effective operator [405, 406]:

$$ \hat{O} = \frac{\lambda_{AB}}{\Lambda_{\text{eff}}} |\Phi|^2 \bar{N}_A^c N_b, \qquad (3.26) $$

where $\Phi$ is the SM Higgs field, $\lambda_{AB}$ are dimensionless couplings, and $\Lambda_{\text{eff}}$ is the scale of new physics, a simple example being a Higgs portal operator. Leptogenesis could then proceed within a 2 RH neutrino mixing scenario [385].

Many leptogenesis models relying on extensions of the minimal type I scenarios have been also proposed (for a recent review see [407]). In some cases these models also offer new possibilities to explain simultaneously DM and matter-antimatter asymmetry.

As we discussed, leptogenesis scenarios at low scale, as in the inverse seesaw or in the $\nu$MSM, have the advantage that they can be probed with direct experimental tests that are already producing important constraints on the parameters of the models. In this respect the LHC14 stage will be extremely exciting.

It should be said, however, that also high energy scale leptogenesis scenarios might be testable with low energy neutrino experiments within some specific frameworks. An interesting example is provided by SO(10)-inspired leptogenesis [380, 408–411], which produces interesting testable constraints on low energy neutrino parameters, especially when a condition of independence of the initial conditions is imposed (strong thermal leptogenesis) [412–414].

In conclusion we have fully entered an exciting stage where finally we are starting to test phenomenological consequences from leptogenesis scenarios.
# 4 Dark Matter at galactic scales: observational constraints and simulations

Section Editors:
Aurel Schneider, Francesco Shankar, Oleg Ruchayskiy

Dark Matter particles were created (or have decoupled from the primordial plasma) deeply in the radiation-dominated era. Current constraints extracted from cosmological observations of, e.g., galaxy clustering or the Cosmic Microwave Background (CMB) anisotropies, rule out the possibility that a significant fraction of Dark Matter particles remained relativistic up until the matter-radiation equality time. These observations, however, still allow for Dark Matter to possibly retain significant primordial velocities deeply in the radiation-dominated epoch. Such Dark Matter models, often labeled as *Warm Dark Matter*, WDM as opposed to its non-relativistic counterpart “cold” Dark Matter CDM, are still allowed by current data on CMB at least at large scales, where they are about indistinguishable.

The difference between cold and warm Dark Matter particles would appear at approximately galactic scales (co-moving Mpc or below) and it is only recently that such small-scale effects are starting to be resolvable both observationally and numerically.

The power spectrum at (sub)-Mpc scales can be probed indirectly by the Lyman-α forest method (see section 5.2). A number of other astronomical observables of both the local and the high redshift Universe can be sensitive to modifications of the primordial power spectrum. Examples are the abundance and internal structure of dwarf galaxies, the formation of first stars, or the process of reionization. Some of these observations have been claimed to be in tension with the ΛCDM “concordance” model, and WDM scenarios have been proposed as promising alternatives.

In this section we will critically discuss some of the main issues around the nature of Dark Matter and how this can be probed via a variety of astrophysical and cosmological tests. Our aim is to go through the main “challenger” of small scale structure formation, providing an overview of the current state of research and discussing how sterile neutrino DM could alter this picture.

In particular, we will first provide in section 4.1 an overview of the state-of-the-art in this field, giving a brief but complete framework of the main features of warm versus cold DM and their validation via galactic observables. We will then move on to discuss the specifics of each of the topics of current interest to the community. We will start in section 4.2 by analyzing the so-called “missing satellite problem”, in light of the most recent discoveries and theoretical approaches. We will continue in section 4.3 by emphasizing the role of CDM and WDM in shaping the inner regions of galactic and host halo structures. Section 4.4 will instead be specifically devoted to a thorough examination of one of the most recent and puzzling problems, the so-called “Too-big-to-fail” problem, also in light of the most recent available data on the rotation curves and number densities of dwarf galaxies in different environments. We then conclude our journey in section 4.5 in which we consider the kinematics and formation of subhaloes in WDM cosmologies.

## 4.1 Astrophysical clues to the identity of the Dark Matter (author: C. Frenk)

Cold, collisionless particles are the most popular candidates for the DM. There are sound reasons for this, both from the points of view of astrophysics and particle physics. The current standard model of cosmology, ΛCDM (where Λ stands for Einstein's cosmological
constant and CDM for cold DM), is based on this idea. ΛCDM accounts for an impressive array of data on the structure of the Universe on large scales, from a few gigaparsecs down to a few megaparsecs, where the cosmic microwave background (CMB) radiation and the clustering of galaxies provide clean and well-understood diagnostics. Particle physics provides well-motivated CDM candidates, most famously the lightest supersymmetric particle, which would have the relic abundance required to provide the measured DM content of our Universe.

In recent years, an altogether different kind of elementary particle, a sterile neutrino, has emerged as alternative plausible candidate for the Dark Matter. These particles are predicted in a simple extension of the Standard Model of particle physics and, if they occur as three, could explain the observed neutrino oscillation rates as well as baryogenesis. In the neutrino Minimal Standard Model one of them would have a mass in the keV range and behave as warm WDM. These particles would be relativistic at the time of decoupling, and subsequent free streaming would damp primordial density fluctuations below some critical cutoff scale which varies inversely with the particle mass and, for keV-mass thermal relics, corresponds to a dwarf galaxy mass.

CDM and WDM models produce very similar large-scale structure and are thus indistinguishable by standard CMB and galaxy clustering tests. However, as is clearly shown in figure 9, they produce completely different structures on scales smaller than dwarf galaxies, where the WDM primordial power spectrum is cut off whereas the CDM power spectrum continues to increase (logarithmically).

The search for the identity of the Dark Matter is currently at a very exciting stage, with tentative, mutually exclusive, claims for detections of CDM [415] and WDM [190]. While eagerly awaiting developments on the Dark Matter particle detection front, it is incumbent upon astrophysicists to develop tests that could distinguish between the two frontrunner candidates. A glance at the figure suggests that such a test should not be too difficult: the two simulated galactic Dark Matter haloes look completely different. In CDM, there is a very large number of small subhaloes whereas in WDM only a handful of the most massive ones are present. Surely, it cannot be beyond the ingenuity of observational astronomers to tell us whether the Milky Way resides in a halo like that on the left or in one like that on the right of the figure!

It turns out that distinguishing CDM from WDM merely by counting the number of small satellites orbiting the Milky Way (or similar galaxies) is not possible once galaxy formation is taken into account. This is because there are two physical processes that inhibit the formation of galaxies in small haloes: the reionization of hydrogen in the early Universe heats up gas to a temperature higher than the virial temperature of small haloes (a few kilometers per second) and, in slightly larger haloes where gas can cool, the first supernovae explosions expel the remaining gas. Models of galaxy formation going back over 10 years, see e.g. [416, 417] and more recent simulations, confirm this conclusion. In fact, if anything, requiring that a galactic halo should contain at least as many subhaloes as there are observed satellites in the Milky Way sets a lower limit on the mass of WDM particle candidates [340]. A similar limit can be placed from the degree of inhomogeneity in the density distribution at early times as measured by the clustering of Lyman-α clouds [331].

Other tests based on the observed properties of Milky Way satellites also fail to distinguish CDM from WDM. In CDM, haloes and subhaloes have cuspy “NFW” [418] density profiles, while in WDM one would expect cores to form in their inner regions. However, detailed calculations show that such cores would be much too small to be astrophysically relevant [342]. Another test is offered by the “too-big-to-fail” argument [53] that CDM N-body
<figure><img src="image_10.png" /><figcaption>**Figure 9.** N-body simulations of galactic haloes in universes dominated by CDM (left) and WDM (right; for a particle mass of 2 keV). Intensity indicates the line-of-sight projected square of the density, and hue the projected density-weighted velocity dispersion. (Figure similar to figure 3 in ref. [58].)</figcaption></figure>

simulations predict a population of massive, dense subhaloes whose structure is incompatible with the observed kinematics of stars in the brightest Milky Way satellites. WDM models neatly avoid this problem because the inner densities of subhaloes are slightly less dense than those of their cold Dark Matter counterparts, on account of their more recent formation epoch [58] (which has also shown in connection to the 3.5 keV hint [419]). However, baryon physics produce similar effects in CDM subhaloes [420].

At present, the best strategy for distinguishing CDM from WDM would seem to be the use of gravitational lensing. There are two complementary techniques that could reveal the presence of the myriad small subhaloes predicted to exist in the haloes of galaxies by CDM simulations: flux ratio anomalies and distortions of giant arcs. Whether these techniques will work in practice is still an open question.

## 4.2 Missing dwarf galaxies (author: N. Menci)

The abundance and the properties of low-mass galaxies and in particular of satellites have long been constituting a challenging issue for CDM galaxy formation models. The latter connect the physical processes involving gas and star formation to the merging histories (and hence to the power spectrum of density perturbations) of Dark Matter haloes [421–428].

The large amplitude of the CDM spectrum on small mass scales $M \lesssim 5 \times 10^9 M_\odot$, a characteristic of dwarf galaxies, results into the following critical issues for galaxy formation models:

(a) The predicted abundance of local low-luminosity dwarfs exceeds the observed values unless a strong energy feedback — heating the gas and suppressing star formation — is assumed, see e.g. [429–431]. However, at high redshifts the larger escape velocities of galaxies (due to their larger densities) make the feedback inefficient in suppressing star formation even in low-mass objects. This results into an over-prediction (of factors $\sim 5$) of faint galaxies increasing with redshift when compared, e.g., to the evolution of the K-band luminosity function for $1 \lesssim z \lesssim 3$ [see 432, and figure 10] or with the faint