# Axion–Sterile Neutrino Dark Matter

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The $a\nu $MSM and Generic Observational Bounds

## 3. Axion Dark Matter

## 4. Sterile Neutrino Dark Matter

#### 4.1. Non-Resonant Production

#### 4.2. Resonant Production

## 5. Sterile Neutrino Dark Matter in a CPT-Symmetric Universe

## 6. Primordial Black Holes as Dark Matter?

## 7. Axion–Sterile Neutrino Dark Matter

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Renormalization-Group Equations

## Notes

1 | The strong CP problem is the fine-tuning problem of explaining why the strong interactions do not break CP, while EW ones do. Addressing this fine-tuning problem through a symmetry without doing the same with the Higgs mass and cosmological constant fine-tuning problems appears to be a logical possibility, because the latter problems could be both addressed through anthropic arguments [8,9,10] (unlike the strong CP one). |

2 | |

3 | |

4 | As usual $h\equiv {H}_{0}/\left(100\mathrm{km}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}\right)$, where ${H}_{0}$ is the Hubble constant and ${\rho}_{\mathrm{cr}}$ is the critical energy density. |

5 | |

6 | See [73] for a study of this bound when sterile neutrinos account for the whole DM. |

7 | In this case, one neglects the dependence on ${m}_{s}/T$, where T is the photon temperature. This is justified as the resonant production of sterile neutrinos occurs at $T\sim 200$ MeV and ${m}_{s}\sim $ keV [93], so ${m}_{s}/T\sim {10}^{-5}$. |

8 | |

9 | |

10 | As usual, the expansion of the universe is nearly exponential for $\u03f5<1$ and becomes exactly exponential as $\u03f5\to 0$. |

11 | In that figure, we chose as an example the input values ${M}_{1}={10}^{11}\phantom{\rule{0.166667em}{0ex}}\mathrm{GeV}$, ${M}_{2}=6.4\times {10}^{13}\phantom{\rule{0.166667em}{0ex}}\mathrm{GeV}$, ${M}_{3}>{\overline{M}}_{\mathrm{Pl}}$, ${f}_{a}\simeq 2.5\times {10}^{10}\phantom{\rule{0.166667em}{0ex}}\mathrm{GeV}$, ${\lambda}_{A}\left({M}_{A}\right)\simeq 0.1$, $y\left({M}_{A}\right)\simeq 0.1$, ${\xi}_{H}\left({M}_{A}\right)\simeq 14$ and ${\xi}_{A}\left({M}_{A}\right)\simeq -2.6$. |

12 |

## References

- Salvio, A. A Simple Motivated Completion of the Standard Model below the Planck Scale: Axions and Right-Handed Neutrinos. Phys. Lett. B
**2015**, 743, 428. [Google Scholar] [CrossRef] [Green Version] - Kim, J.E. Weak interaction singlet and strong CP invariance. Phys. Rev. Lett.
**1979**, 43, 103. [Google Scholar] [CrossRef] - Shifman, M.A.; Vainshtein, A.I.; Zakharov, V.I. Can confinement ensure natural CP invariance of strong interactions? Nucl. Phys. B
**1980**, 166, 493. [Google Scholar] [CrossRef] - Bezrukov, F.L.; Shaposhnikov, M. The Standard Model Higgs boson as the inflaton. Phys. Lett. B
**2008**, 659, 703. [Google Scholar] [CrossRef] [Green Version] - Bezrukov, F.L.; Magnin, A.; Shaposhnikov, M. Standard Model Higgs boson mass from inflation. Phys. Lett. B
**2009**, 675, 88. [Google Scholar] [CrossRef] [Green Version] - Bezrukov, F.; Shaposhnikov, M. Standard Model Higgs boson mass from inflation: Two loop analysis. J. High Energy Phys.
**2009**, 907, 89. [Google Scholar] [CrossRef] - Salvio, A. Higgs Inflation at NNLO after the Boson Discovery. Phys. Lett. B
**2013**, 727, 234. [Google Scholar] [CrossRef] [Green Version] - Weinberg, S. Anthropic Bound on the Cosmological Constant. Phys. Rev. Lett.
**1987**, 59, 2607. [Google Scholar] [CrossRef] - Agrawal, V.; Barr, S.M.; Donoghue, J.F.; Seckel, D. The Anthropic principle and the mass scale of the standard model. Phys. Rev. D
**1998**, 57, 5480. [Google Scholar] [CrossRef] [Green Version] - D’Amico, G.; Strumia, A.; Urbano, A.; Xue, W. Direct anthropic bound on the weak scale from supernovæ explosions. Phys. Rev. D
**2019**, 100, 083013. [Google Scholar] [CrossRef] [Green Version] - Peccei, R.D.; Quinn, H.R. CP Conservation in the Presence of Instantons. Phys. Rev. Lett.
**1977**, 38, 1440. [Google Scholar] [CrossRef] [Green Version] - Peccei, R.D.; Quinn, H.R. Constraints Imposed by CP Conservation in the Presence of Instantons. Phys. Rev. D
**1977**, 16, 1791. [Google Scholar] [CrossRef] - Salvio, A. Critical Higgs inflation in a Viable Motivated Model. Phys. Rev. D
**2019**, 99, 015037. [Google Scholar] [CrossRef] [Green Version] - Hamada, Y.; Kawai, H.; Oda, K.y.; Park, S.C. Higgs Inflation is Still Alive after the Results from BICEP2. Phys. Rev. Lett.
**2014**, 112, 241301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bezrukov, F.; Shaposhnikov, M. Higgs inflation at the critical point. Phys. Lett. B
**2014**, 734, 249. [Google Scholar] [CrossRef] - Hamada, Y.; Kawai, H.; Oda, K.y.; Park, S.C. Higgs inflation from Standard Model criticality. Phys. Rev. D
**2015**, 91, 053008. [Google Scholar] [CrossRef] [Green Version] - Buttazzo, D.; Degrassi, G.; Giardino, P.P.; Giudice, G.F.; Sala, F.; Salvio, A.; Strumia, A. Investigating the near-criticality of the Higgs boson. J. High Energy Phys.
**2013**, 12, 1–49. [Google Scholar] [CrossRef] [Green Version] - Burgess, C.P.; Lee, H.M.; Trott, M. Power-counting and the Validity of the Classical Approximation During Inflation. J. High Energy Phys.
**2009**, 2009, 103. [Google Scholar] [CrossRef] [Green Version] - Barbon, J.L.F.; Espinosa, J.R. On the Naturalness of Higgs Inflation. Phys. Rev. D
**2009**, 79, 081302. [Google Scholar] [CrossRef] [Green Version] - Hertzberg, M.P. On Inflation with Non-minimal Coupling. J. High Energy Phys.
**2010**, 2010, 1–14. [Google Scholar] [CrossRef] [Green Version] - Burgess, C.P.; Patil, S.P.; Trott, M. On the Predictiveness of Single-Field Inflationary Models. J. High Energy Phys.
**2014**, 2014, 1–31. [Google Scholar] [CrossRef] [Green Version] - Burgess, C.P.; Lee, H.M.; Trott, M. Comment on Higgs Inflation and Naturalness. J. High Energy Phys.
**2010**, 1007, 7. [Google Scholar] [CrossRef] [Green Version] - Salvio, A. Initial Conditions for Critical Higgs Inflation. Phys. Lett. B
**2018**, 780, 111–117. [Google Scholar] [CrossRef] - Salvio, A.; Mazumdar, A. Classical and Quantum Initial Conditions for Higgs Inflation. Phys. Lett. B
**2015**, 750, 194. [Google Scholar] [CrossRef] [Green Version] - Salvio, A. Hearing Higgs with gravitational wave detectors. J. Cosmol. Astropart. Phys.
**2021**, 6, 40. [Google Scholar] [CrossRef] - Preskill, J.; Wise, M.; Wilczek, F. Cosmology of the invisible axion. Phys. Lett. B
**1983**, 120, 127. [Google Scholar] [CrossRef] [Green Version] - Abbott, L.; Sikivie, P. A cosmological bound on the invisible axion. Phys. Lett. B
**1983**, 120, 133. [Google Scholar] [CrossRef] - Dine, M.; Fischler, W. The not so harmless axion. Phys. Lett. B
**1983**, 120, 137. [Google Scholar] [CrossRef] - Davis, R.L. Cosmic Axions from Cosmic Strings. Phys. Lett. B
**1986**, 180, 225–230. [Google Scholar] [CrossRef] - Harari, D.; Sikivie, P. On the Evolution of Global Strings in the Early Universe. Phys. Lett. B
**1987**, 195, 361–365. [Google Scholar] [CrossRef] - Davis, R.L.; Shellard, E.P.S. Do Axions Need Inflation? Nucl. Phys. B
**1989**, 324, 167–186. [Google Scholar] [CrossRef] - Battye, R.A.; Shellard, E.P.S. Global string radiation. Nucl. Phys. B
**1994**, 423, 260–304. [Google Scholar] [CrossRef] [Green Version] - Nagasawa, M.; Kawasaki, M. Collapse of axionic domain wall and axion emission. Phys. Rev. D
**1994**, 50, 4821–4826. [Google Scholar] [CrossRef] [Green Version] - Hiramatsu, T.; Kawasaki, M.; Saikawa, K.; Sekiguchi, T. Production of dark matter axions from collapse of string-wall systems. Phys. Rev. D
**2012**, 85, 105020. [Google Scholar] [CrossRef] [Green Version] - Gorghetto, M.; Hardy, E.; Villadoro, G. Axions from Strings: The Attractive Solution. J. High Energy Phys.
**2018**, 7, 151. [Google Scholar] [CrossRef] [Green Version] - Gorghetto, M.; Hardy, E.; Villadoro, G. More Axions from Strings. SciPost Phys.
**2021**, 10, 50. [Google Scholar] [CrossRef] - Ballesteros, G.; Redondo, J.; Ringwald, A.; Tamarit, C. Standard Model-axion-seesaw-Higgs portal inflation. Five problems of particle physics and cosmology solved in one stroke. J. Cosmol. Astropart. Phys.
**2017**, 1708, 1. [Google Scholar] [CrossRef] [Green Version] - Dodelson, S.; Widrow, L.M. Sterile-neutrinos as dark matter. Phys. Rev. Lett.
**1994**, 72, 17–20. [Google Scholar] [CrossRef] [Green Version] - Shi, X.D.; Fuller, G.M. A New dark matter candidate: Nonthermal sterile neutrinos. Phys. Rev. Lett.
**1999**, 82, 2832–2835. [Google Scholar] [CrossRef] [Green Version] - Kusenko, A. Sterile neutrinos: The Dark side of the light fermions. Phys. Rep.
**2009**, 481, 1–28. [Google Scholar] [CrossRef] [Green Version] - Drewes, M.; Lasserre, T.; Merle, A.; Mertens, S.; Adhikari, R.; Agostini, M.; Ky, N.A.; Araki, T.; Archidiacono, M.; Bahr, M.; et al. A White Paper on keV Sterile Neutrino Dark Matter. J. Cosmol. Astropart. Phys.
**2017**, 1, 25. [Google Scholar] - Boyarsky, A.; Drewes, M.; Lasserre, T.; Mertens, S.; Ruchayskiy, O. Sterile neutrino Dark Matter. Prog. Part. Nucl. Phys.
**2019**, 104, 1–45. [Google Scholar] [CrossRef] [Green Version] - Boyle, L.; Finn, K.; Turok, N. CPT-Symmetric Universe. Phys. Rev. Lett.
**2018**, 121, 251301. [Google Scholar] [CrossRef] [Green Version] - Boyle, L.; Finn, K.; Turok, N. The Big Bang, CPT, and neutrino dark matter. arXiv
**2018**, arXiv:1803.08930. [Google Scholar] - Randjbar-Daemi, S.; Salvio, A.; Shaposhnikov, M. On the decoupling of heavy modes in Kaluza-Klein theories. Nucl. Phys. B
**2006**, 741, 236–268. [Google Scholar] [CrossRef] [Green Version] - Elias-Miro, J.; Espinosa, J.R.; Giudice, G.F.; Lee, H.M.; Strumia, A. Stabilization of the Electroweak Vacuum by a Scalar Threshold Effect. J. High Energy Phys.
**2012**, 6, 031. [Google Scholar] [CrossRef] [Green Version] - Asaka, T.; Blanchet, S.; Shaposhnikov, M. The nuMSM, dark matter and neutrino masses. Phys. Lett. B
**2005**, 631, 151–156. [Google Scholar] [CrossRef] [Green Version] - Asaka, T.; Shaposhnikov, M. The nuMSM, dark matter and baryon asymmetry of the universe. Phys. Lett. B
**2005**, 620, 17. [Google Scholar] [CrossRef] [Green Version] - Asaka, T.; Shaposhnikov, M.; Kusenko, A. Opening a new window for warm dark matter. Phys. Lett. B
**2006**, 638, 401–406. [Google Scholar] [CrossRef] [Green Version] - Asaka, T.; Laine, M.; Shaposhnikov, M. Lightest sterile neutrino abundance within the nuMSM. J. High Energy Phys.
**2015**, 1, 91. [Google Scholar] - Canetti, L.; Drewes, M.; Shaposhnikov, M. Sterile Neutrinos as the Origin of Dark and Baryonic Matter. Phys. Rev. Lett.
**2013**, 110, 061801. [Google Scholar] [CrossRef] - Abazajian, K.N.; Kusenko, A. Hidden treasures: Sterile neutrinos as dark matter with miraculous abundance, structure formation for different production mechanisms, and a solution to the σ
_{8}problem. Phys. Rev. D**2019**, 100, 103513. [Google Scholar] [CrossRef] [Green Version] - Perez, K.; Ng, K.C.Y.; Beacom, J.F.; Hersh, C.; Horiuchi, S.; Krivonos, R. Almost closing the νMSM sterile neutrino dark matter window with NuSTAR. Phys. Rev. D
**2017**, 95, 123002. [Google Scholar] [CrossRef] [Green Version] - Garcia-Bellido, J.; Morales, E.R. Primordial black holes from single field models of inflation. Phys. Dark Univ.
**2017**, 18, 47–54. [Google Scholar] [CrossRef] [Green Version] - Ezquiaga, J.M.; Garcia-Bellido, J.; Morales, E.R. Primordial Black Hole production in Critical Higgs Inflation. Phys. Lett. B
**2018**, 776, 345–349. [Google Scholar] [CrossRef] - Ballesteros, G.; Taoso, M. Primordial black hole dark matter from single field inflation. Phys. Rev. D
**2018**, 97, 023501. [Google Scholar] [CrossRef] [Green Version] - Motohashi, H.; Hu, W. Primordial Black Holes and Slow-Roll Violation. Phys. Rev. D
**2017**, 96, 063503. [Google Scholar] [CrossRef] [Green Version] - Hertzberg, M.P.; Yamada, M. Primordial Black Holes from Polynomial Potentials in Single Field Inflation. Phys. Rev. D
**2018**, 97, 083509. [Google Scholar] [CrossRef] [Green Version] - Carr, B.; Kohri, K.; Sendouda, Y.; Yokoyama, J. Constraints on Primordial Black Holes. arXiv
**2020**, arXiv:2002.12778. [Google Scholar] - Luzio, L.D.; Giannotti, M.; Nardi, E.; Visinelli, L. The landscape of QCD axion models. Phys. Rep.
**2020**, 870, 1–117. [Google Scholar] [CrossRef] - Esteban, I.; Gonzalez-Garcia, M.C.; Maltoni, M.; Schwetz, T.; Zhou, A. The fate of hints: Updated global analysis of three-flavor neutrino oscillations. J. High Energy Phys.
**2020**, 9, 178. [Google Scholar] [CrossRef] - de Salas, P.F.; Forero, D.V.; Gariazzo, S.; Martínez-Miravé, P.; Mena, O.; Ternes, C.A.; Tórtola, M.; Valle, J.W.F. 2020 Global reassessment of the neutrino oscillation picture. arXiv
**2020**, arXiv:2006.11237. [Google Scholar] - Zyla, P.A.; et al.; Particle Data Group Particle Data Group. Prog. Theor. Exp. Phys.
**2020**, 083C01. Available online: https://pdg.lbl.gov/2020/tables/rpp2020-sum-quarks.pdf (accessed on 22 September 2021). - Hagiwara, K.; Hikasa, K.; Nakamura, K.; Tanabashi, M.; Aguilar-Benitez, M.; Amsler, C.; Barnett, R.M.; Burchat, P.R.; Carone, C.D.; Lugovsky, V.S.; et al. Particle Data Group. Phys. Rev. D
**2018**, 98, 030001. Available online: http://pdg.lbl.gov/2018/tables/rpp2018-sum-gauge-higgs-bosons.pdf (accessed on 22 September 2021). - Bethke, S. World Summary of α
_{s}(2012). Nucl. Phys. Proc. Suppl.**2013**, 234, 229. [Google Scholar] [CrossRef] [Green Version] - Petreczky, P.; Schadler, H.P.; Sharma, S. The topological susceptibility in finite temperature QCD and axion cosmology. Phys. Lett. B
**2016**, 762, 498–505. [Google Scholar] [CrossRef] [Green Version] - Borsanyi, S.; Fodor, Z.; Guenther, J.; Kampert, K.H.; Katz, S.D.; Kawanai, T.; Kovacs, T.G.; Mages, S.W.; Pasztor, A.; Pittler, F.; et al. Calculation of the axion mass based on high-temperature lattice quantum chromodynamics. Nature
**2016**, 539, 69–71. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Alion, T.; Back, J.J.; Bashyal, A.; Bass, M.; Bishai, M.; Cherdack, D.; Diwan, M.; Djurcic, Z.; Evans, J.; Fernandez-Martinez, E.; et al. Particle Data Group. Review of Particle Physics. Phys. Rev. D. 2016. Available online: https://arxiv.org/pdf/1606.09550.pdf (accessed on 22 September 2021).
- Bezrukov, F.; Gorbunov, D.; Shaposhnikov, M. On initial conditions for the Hot Big Bang. J. Cosmol. Astropart. Phys.
**2009**, 906, 29. [Google Scholar] [CrossRef] [Green Version] - Garcia-Bellido, J.; Figueroa, D.G.; Rubio, J. Preheating in the Standard Model with the Higgs-Inflaton coupled to gravity. Phys. Rev. D
**2009**, 79, 063531. [Google Scholar] [CrossRef] [Green Version] - Kawasaki, M.; Saikawa, K.; Sekiguchi, T. Axion dark matter from topological defects. Phys. Rev. D
**2015**, 91, 065014. [Google Scholar] [CrossRef] [Green Version] - Tremaine, S.; Gunn, J.E. Dynamical Role of Light Neutral Leptons in Cosmology. Phys. Rev. Lett.
**1979**, 42, 407–410. [Google Scholar] [CrossRef] - Gorbunov, D.; Khmelnitsky, A.; Rubakov, V. Constraining sterile neutrino dark matter by phase-space density observations. J. Cosmol. Astropart. Phys.
**2008**, 10, 41. [Google Scholar] [CrossRef] - Boyanovsky, D.; de Vega, H.J.; Sanchez, N. Constraints on dark matter particles from theory, galaxy observations and N-body simulations. Phys. Rev. D
**2008**, 77, 043518. [Google Scholar] [CrossRef] [Green Version] - de Vega, H.J.; Sanchez, N.G. Model independent analysis of dark matter points to a particle mass at the keV scale. Mon. Not. R. Astron. Soc.
**2010**, 404, 885. [Google Scholar] [CrossRef] [Green Version] - Savchenko, D.; Rudakovskyi, A. New mass bound on fermionic dark matter from a combined analysis of classical dSphs. Mon. Not. R. Astron. Soc.
**2019**, 487, 5711–5720. [Google Scholar] [CrossRef] [Green Version] - de Vega, H.J.; Sanchez, N.G. Galaxy phase-space density data exclude Bose-Einstein condensate Axion Dark Matter. arXiv
**2014**, arXiv:1401.1214. [Google Scholar] - Pal, P.B.; Wolfenstein, L. Radiative Decays of Massive Neutrinos. Phys. Rev. D
**1982**, 25, 766. [Google Scholar] [CrossRef] - Barger, V.D.; Phillips, R.J.N.; Sarkar, S. Remarks on the KARMEN anomaly. Phys. Lett. B
**1995**, 352, 365–371. [Google Scholar] [CrossRef] [Green Version] - Benso, C.; Brdar, V.; Lindner, M.; Rodejohann, W. Prospects for Finding Sterile Neutrino Dark Matter at KATRIN. Phys. Rev. D
**2019**, 100, 115035. [Google Scholar] [CrossRef] [Green Version] - Ng, K.C.Y.; Roach, B.M.; Perez, K.; Beacom, J.F.; Horiuchi, S.; Krivonos, R.; Wik, D.R. New Constraints on Sterile Neutrino Dark Matter from NuSTAR M31 Observations. Phys. Rev. D
**2019**, 99, 083005. [Google Scholar] [CrossRef] [Green Version] - Abazajian, K. Production and evolution of perturbations of sterile neutrino dark matter. Phys. Rev. D
**2006**, 73, 063506. [Google Scholar] [CrossRef] [Green Version] - Abazajian, K.N. Sterile neutrinos in cosmology. Phys. Rep.
**2017**, 711–712, 1–28. [Google Scholar] [CrossRef] [Green Version] - Palazzo, A.; Cumberbatch, D.; Slosar, A.; Silk, J. Sterile neutrinos as subdominant warm dark matter. Phys. Rev. D
**2007**, 76, 103511. [Google Scholar] [CrossRef] [Green Version] - Abazajian, K.; Fuller, G.M.; Patel, M. Sterile neutrino hot, warm, and cold dark matter. Phys. Rev. D
**2001**, 64, 023501. [Google Scholar] [CrossRef] [Green Version] - Wolfenstein, L. Neutrino Oscillations in Matter. Phys. Rev. D
**1978**, 17, 2369–2374. [Google Scholar] [CrossRef] - Mikheyev, S.P.; Smirnov, A.Y. Resonance Amplification of Oscillations in Matter and Spectroscopy of Solar Neutrinos. Sov. J. Nucl. Phys.
**1985**, 42, 913–917. [Google Scholar] - Ghiglieri, J.; Laine, M. Improved determination of sterile neutrino dark matter spectrum. J. High Energy Phys.
**2015**, 11, 171. [Google Scholar] [CrossRef] [Green Version] - Venumadhav, T.; Cyr-Racine, F.Y.; Abazajian, K.N.; Hirata, C.M. Sterile neutrino dark matter: Weak interactions in the strong coupling epoch. Phys. Rev. D
**2016**, 94, 043515. [Google Scholar] [CrossRef] [Green Version] - Bodeker, D.; Klaus, A. Sterile neutrino dark matter: Impact of active-neutrino opacities. J. High Energy Phys.
**2020**, 7, 218. [Google Scholar] [CrossRef] - Laine, M.; Shaposhnikov, M. Sterile neutrino dark matter as a consequence of nuMSM-induced lepton asymmetry. J. Cosmol. Astropart. Phys.
**2008**, 6, 31. [Google Scholar] [CrossRef] [Green Version] - Canetti, L.; Drewes, M.; Frossard, T.; Shaposhnikov, M. Dark Matter, Baryogenesis and Neutrino Oscillations from Right Handed Neutrinos. Phys. Rev. D
**2013**, 87, 093006. [Google Scholar] [CrossRef] [Green Version] - Eijima, S.; Shaposhnikov, M.; Timiryasov, I. Freeze-in generation of lepton asymmetries after baryogenesis in the νMSM. arXiv
**2020**, arXiv:2011.12637. [Google Scholar] - Akhmedov, E.K.; Rubakov, V.A.; Smirnov, A.Y. Baryogenesis via neutrino oscillations. Phys. Rev. Lett.
**1998**, 81, 1359. [Google Scholar] [CrossRef] [Green Version] - Drewes, M.; Garbrecht, B. Leptogenesis from a GeV Seesaw without Mass Degeneracy. J. High Energy Phys.
**2013**, 3, 96. [Google Scholar] [CrossRef] [Green Version] - Serpico, P.D.; Raffelt, G.G. Lepton asymmetry and primordial nucleosynthesis in the era of precision cosmology. Phys. Rev. D
**2005**, 71, 127301. [Google Scholar] [CrossRef] [Green Version] - Duran, A.; Morrison, L.; Profumo, S. Sterile Neutrino Dark Matter from Generalized CPT-Symmetric Early-Universe Cosmologies. arXiv
**2021**, arXiv:2103.08626. [Google Scholar] - Kohri, K.; Lyth, D.H.; Melchiorri, A. Black hole formation and slow-roll inflation. J. Cosmol. Astropart. Phys.
**2008**, 4, 38. [Google Scholar] [CrossRef] [Green Version] - Kohri, K.; Lin, C.M.; Matsuda, T. Primordial black holes from the inflating curvaton. Phys. Rev. D
**2013**, 87, 103527. [Google Scholar] [CrossRef] [Green Version] - Schwarz, D.J.; Terrero-Escalante, C.A.; Garcia, A.A. Higher order corrections to primordial spectra from cosmological inflation. Phys. Lett. B
**2001**, 517, 243–249. [Google Scholar] [CrossRef] - Bezrukov, F.; Rubio, J.; Shaposhnikov, M. Living beyond the edge: Higgs inflation and vacuum metastability. Phys. Rev. D
**2015**, 92, 083512. [Google Scholar] [CrossRef] [Green Version] - Bezrukov, F.; Pauly, M.; Rubio, J. On the robustness of the primordial power spectrum in renormalized Higgs inflation. J. Cosmol. Astropart. Phys.
**2018**, 1802, 40. [Google Scholar] [CrossRef] [Green Version] - Salvio, A.; Strumia, A. Agravity. J. High Energy Phys.
**2014**, 1406, 80. [Google Scholar] [CrossRef] - Salvio, A. Solving the Standard Model Problems in Softened Gravity. Phys. Rev. D
**2016**, 94, 096007. [Google Scholar] [CrossRef] [Green Version] - Salvio, A.; Strumia, A. Agravity up to infinite energy. Eur. Phys. J. C
**2018**, 78, 124. [Google Scholar] [CrossRef] [PubMed] - Salvio, A. Metastability in Quadratic Gravity. Phys. Rev. D
**2019**, 99, 103507. [Google Scholar] [CrossRef] [Green Version] - Salvio, A. Quasi-Conformal Models and the Early Universe. Eur. Phys. J. C
**2019**, 79, 750. [Google Scholar] [CrossRef] [Green Version] - Salvio, A. Quadratic Gravity. Front. Phys.
**2018**, 6, 77. [Google Scholar] [CrossRef] [Green Version] - Salvio, A. Dimensional Transmutation in Gravity and Cosmology. Int. J. Mod. Phys. A
**2021**, 36, 2130006. [Google Scholar] [CrossRef] - Liddle, A.R.; Leach, S.M. How long before the end of inflation were observable perturbations produced? Phys. Rev. D
**2003**, 68, 103503. [Google Scholar] [CrossRef] [Green Version] - Ade, P.A.R.; Aghanim, N.; Arnaud, M.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; et al. Planck 2015 results. XX. Constraints on inflation. Astron. Astrophys.
**2016**, 594, A20. [Google Scholar] - Akrami, Y.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; Bartolo, N.; Basak, S.; et al. Planck 2018 results. X. Constraints on inflation. Astron. Astrophys.
**2020**, 641, A10. [Google Scholar] - Machacek, M.E.; Vaughn, M.T. Two Loop Renormalization Group Equations in a General Quantum Field Theory. 1. Wave Function Renormalization. Nucl. Phys.
**1983**, B222, 83. [Google Scholar] [CrossRef] - Machacek, M.E.; Vaughn, M.T. Two Loop Renormalization Group Equations in a General Quantum Field Theory. 2. Yukawa Couplings. Nucl. Phys.
**1984**, B236, 221. [Google Scholar] [CrossRef] - Machacek, M.E.; Vaughn, M.T. Two Loop Renormalization Group Equations in a General Quantum Field Theory. 3. Scalar Quartic Couplings. Nucl. Phys.
**1985**, B249, 70. [Google Scholar] [CrossRef]

**Figure 2.**The typical shape of the effective potential as a function of the canonically normalized Higgs field close to criticality (in this plot, we approach the critical regime by varying ${\lambda}_{HA}$).

**Figure 3.**The curvature power spectrum and the canonically normalized (Higgs) field close to criticality (which we approach by varying ${\lambda}_{HA}$). The corresponding values of $\u03f5$ and $\delta $ are shown as well in an inset of the right plot. The parameters are set as in Figure 30.

**Figure 4.**Sterile neutrino production range as the axion decay constant changes. For the first value of the axion decay constant, ${f}_{a}$, we only take into account the misalignment mechanism for axion production. For the second value, ${f}_{a}^{\mathrm{mis}+\mathrm{string}}$, we take into account both the misalignment mechanism and the decay of topological defects setting ${\lambda}_{A}=0.1$. The upper line corresponds to the non-resonant production and the lower line is the BBN bound discussed in Section 4.2.

**Figure 5.**The upper and lower lines of Figure 4 (here depicted in solid black and orange, respectively) compared with the X-ray and phase-space bounds discussed in Section 4 (dashed lines). The X-ray bounds are the upper ones in blue [42] and red [81], while the phase-space ones are the lower ones in black. In this figure, we also provide the corresponding plot for ${X}_{s}=1$ (the bottom one, see Ref. [42] for a review).

**Figure 6.**The non-resonant sterile neutrino (black solid lines) and the X-ray and phase-space bounds of Figure 5 together with the structure formation bounds of Section 4.1 (dash-dotted lines).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Salvio, A.; Scollo, S.
Axion–Sterile Neutrino Dark Matter. *Universe* **2021**, *7*, 354.
https://doi.org/10.3390/universe7100354

**AMA Style**

Salvio A, Scollo S.
Axion–Sterile Neutrino Dark Matter. *Universe*. 2021; 7(10):354.
https://doi.org/10.3390/universe7100354

**Chicago/Turabian Style**

Salvio, Alberto, and Simone Scollo.
2021. "Axion–Sterile Neutrino Dark Matter" *Universe* 7, no. 10: 354.
https://doi.org/10.3390/universe7100354