Downloaded from http://rsta.royalsocietypublishing.org/ on December 4, 2017 The Higgs boson and cosmology rsta.royalsocietypublishing.org Mikhail Shaposhnikov
Institut de Théorie des Phénomènes Physiques, École Polytechnique Review Fédérale de Lausanne, 1015 Lausanne, Switzerland
Cite this article: Shaposhnikov M. 2015 The I will discuss how the Higgs field of the Standard Model may have played an important role in Higgs boson and cosmology. Phil.Trans.R. cosmology, leading to the homogeneity, isotropy and Soc. A 373: 20140038. flatness of the Universe; producing the quantum http://dx.doi.org/10.1098/rsta.2014.0038 fluctuations that seed structure formation; triggering the radiation-dominated era of the hot Big Bang; and contributing to the processes of baryogenesis and dark One contribution of 12 to a Discussion Meeting matter production. Issue ‘Before, behind and beyond the discovery of the Higgs boson’. 1. Introduction Subject Areas: A Higgs boson-like particle with mass 126 GeV has particle physics, relativity, high-energy physics recently been discovered at CERN [1,2], and thus we now have a theory of the strong, weak, electromagnetic and gravitational interactions that may be a self-consistent Keywords: effective field theory all the way up to the Planck Higgs boson, inflation, dark matter, ∼ × 18 scale E MP 2.44 10 GeV; see a recent discussion in baryogenesis [3–7]. Of course, this does not mean that the Standard Model (SM) is a correct theory of Nature up to these Author for correspondence: energies, as this would be an enormous extrapolation of known physics into a region that is not accessible to Mikhail Shaposhnikov present experiments. Nevertheless, since the assumption e-mail: [email protected] that the SM is valid up to the Planck scale is the most conservative option available to us, it has more predictive power than any other approach. The Higgs boson is a very special particle in the SM. It provides a mechanism for including weakly interacting massive vector bosons in the SM, and for ‘giving’ masses to quarks and leptons. As will be discussed in this paper, the Higgs field may also have had an important role in cosmology: it could have made the Universe flat, homogeneous and isotropic, it could have produced the fluctuations that led to structure formation and it could also have enabled the radiation-dominated epoch of the hot Big Bang to occur [8–10]. Moreover, in the modest extension of the SM by three relatively light Majorana fermions—heavy neutral leptons (HNLs)—the Higgs field is important for baryogenesis, leading to the charge asymmetric Universe, and for dark matter
2014 The Author(s) Published by the Royal Society. All rights reserved. Downloaded from http://rsta.royalsocietypublishing.org/ on December 4, 2017
U(c) 2
4 2 rsta.royalsocietypublishing.org lM /x /4 ......
Standard Model
l v4/4 lM4/x2/16 0 v Phil.Trans.R.Soc.A 0 c
Figure 1. Effective potential for a canonically normalized scalar field χ (related to the Higgs field in a way that is well understood) in the Einstein frame. For large h (or χ) the potential is flat.
production [11]. In addition, active neutrino masses and mixing are induced via Yukawa 20140038 : 373 couplings of both HNLs and left-handed neutrinos to the Higgs field. 2. Higgs field and gravity In order to embed the SM in the cosmological framework, one has to fix the coupling of the Higgs field to gravity. In addition to the replacement of the Minkowski metric ημν by the generic curved space metric gμν , one should add to the Lagrangian a non-minimal coupling of the Higgs field to gravity (e.g. [12]): M2 S = d4x −g − P R − ξH†H R. (2.1) G 2 Here R is the scalar curvature, g is the determinant of the metric, H is the Higgs field and ξ is a new dimensionless coupling constant of the SM. As is the case with other couplings of the SM, the value of ξ cannot be fixed from within the model, but instead can be determined by specific experiments (cosmological observations in our case). ξ To elucidate√ the role of the non-minimal coupling , let us consider large Higgs fields > / ξ 2 = † / h MP ,(h H H 2), which may have existed in the early Universe. In this regime, the Higgs field not only gives masses ∝ h to fermions and vector bosons, but also determines the gravity interaction strength, which is simply the inverse coefficient in front of the scalar curvature R: eff = 2 + ξ 2 ∝ MP MP h h. In the limit of large Higgs fields, physical observables do not depend on / eff h, as in all dimensionless ratios the magnitude of h cancels out; MW MP is one such example. In particular, the physical effective potential (obtained by transforming the theory from the so-called Jordan frame to the Einstein frame) does not depend on the Higgs field, as depicted in figure 1. It has been shown [13] that the form of the potential is not changed by perturbative higher order corrections, provided the mass of the Higgs boson obeys the requirement (see also [14]): ξ M > M − 0.1 log ± 1 GeV. (2.2) H crit 1000 Here yt(μt) − 0.9361 αs(M ) − 0.1184 M = 129.1 + × 2.0 − Z × 0.5 GeV, (2.3) crit 0.0058 0.0007 where yt(μt) is the top Yukawa coupling in the MS scheme taken at μt = 173.2 GeV, and αs(MZ) μ = is the strong coupling at the scale MZ. The theoretical uncertainty in Mcrit is very small– approximately 70 MeV (see [6] and the discussion in [4,7]). The comparison of Mcrit with experiment for ξ ∼ 1 is presented in figure 2. There is 1–2 s.d. tension between the experimental Downloaded from http://rsta.royalsocietypublishing.org/ on December 4, 2017
Higgs mass M = 125.3 ± 0.6 GeV H 3 0.121 rsta.royalsocietypublishing.org ...... ) z 0.120 ( M s 0.119
0.118
strong coupling a 0.117 Phil.Trans.R.Soc.A 0.116 170 171 172 176175174173 pole top mass Mt (GeV)
Figure 2. The shaded regions account for 1 and 2 s.d. experimental uncertainties in αs (αs = 0.1184 ± 0.0007) and the pole top quark mass Mt (Tevatron: Mt = 173.2 ± 0.51 ± 0.71 GeV), and also include theoretical errors in extraction of yt from 20140038 : 373 experiment. The thick straight line (blue in the online version) marks the relation between αs and the pole top mass following from equation (2.3)ifMH is identified with Mcrit. The enclosing shaded regions correspond to 1 and 2 s.d. experimental errors in the Higgs mass. Small ellipses (red in the online version) correspond to the accuracy achievable at the e+e− collider [15]. (Online version in colour.) values of the top and Higgs masses and the bound (2.2), with the main uncertainty coming from + − Mt; it is therefore imperative to obtain more precise measurements at the future e e collider. In what follows I will assume that (2.2) is satisfied. 3. Higgs boson, cosmological inflation and the hot Big Bang It is well known that a number of important cosmological problems, such as the flatness, isotropy and homogeneity of the Universe, can be solved simultaneously by the accelerated expansion of the Universe in the distant past. This epoch of inflation is expected to be driven by some scalar field called the inflaton. The Higgs field is the only scalar field included within the framework of the SM, and in [8] it was shown that the Higgs field is a good inflaton candidate. The evolution of the Universe with the Higgs field playing the role of the inflaton (Higgs inflation) proceeds as follows [9,10]. At the first stage, as is√ the case in any chaotic inflation > / ξ scenario [16], the value of the Higgs field is large (h MP ), and it rolls slowly towards the minimum of the potential in figure 1. The potential energy of the Higgs field leads to the exponential expansion of the Universe, which then becomes flat, homogeneous and isotropic. The small-scale quantum fluctuations of the Higgs field are inflated and seed structure formation, thus leading to the creation of galaxies and clusters√ of galaxies. / ξ After the Higgs field reaches the value h MP , the slow roll ends, and the Higgs field starts to oscillate. The exponential expansion of the Universe becomes a power law, corresponding to matter domination. The Higgs field oscillations lead to the creation of the particles of the SM that couple most strongly to H, namely to intermediate vector bosons W and Z and the top quark. Eventually, W and Z thermalize through decays and inverse decays into fermions of the SM. Owing to all these processes the decay of the scalar field is completed sometime before the /ξ amplitude of the Higgs field reaches the value h MP . As a result, the Universe is heated up to ∼ 14 the temperature Treh 10 GeV [9,10]. This is the start of the hot Big Bang stage of the Universe evolution, when the Universe is dominated by radiation. The cosmological predictions of Higgs inflation can be compared with observations performed by the Planck satellite. The Higgs-inflaton potential depends on one unknown parameter, ξ.Itcan be fixed by the amplitude of the cosmic microwave background temperature fluctuations δT/T Downloaded from http://rsta.royalsocietypublishing.org/ on December 4, 2017
(a) 0.25 4 Planck + WP rsta.royalsocietypublishing.org ...... ) Planck + WP + highL 0.20 Planck + WP + BAO 0.002
predictions of 0.15 inflationary models:
0.10 Higgs inflation
0.05 tensor-to-scalar ratio ( r tensor-to-scalar Phil.Trans.R.Soc.A
0
(b) 0.25 Planck + WP 20140038 : 373 ) Planck + WP + highL 0.20 Planck + WP + BAO
0.002 convex natural inflation concave 0.15 power-law inflation low-scale SSB SUSY R2 inflation 0.10 V µ f 2/3 V µ f 0.05 V µ f 2 tensor-to-scalar ratio ( r tensor-to-scalar V µ f 3
N* =50 0 0.94 0.96 0.98 1.00 N* =60 primordial tilt (ns)
Figure3. (a)Predictionsfortheinflationaryindexesns andr forHiggsinflation.(b)Predictionsfordifferentinflationarymodels contrasted with the Planck results (adapted from [17]). The definitions of different models and measurements can be foundin [17]. (Online version in colour.) at the Wilkinson Microwave Anisotropy Probe normalization scale approximately 500 Mpc, with the use of precise knowledge of the top quark and Higgs masses, and with αs. In general, ξ>600 [13]. The relatively large value of the non-minimal coupling ξ plays an essential role in the self- consistency of the Higgs inflation (see [14] and references therein). Since the Higgs mass lies near ξ Mcrit, the actual value of may be close to the lower bound. In addition, the value of the spectral index, ns, of scalar density perturbations and the amplitude of tensor perturbations r = δρs/δρt can be determined. The predictions, together with the Planck results, are presented in figure 3 and are well inside the 1 s.d. experimental contour. Moreover, as is the case for most single-field inflationary models, the perturbations are Gaussian, and therefore in excellent agreement with Planck [17]. 4. Minimal physics beyond the Standard Model In order to continue this discussion, we need to consider physics beyond that described by the SM, since the SM cannot explain the matter–antimatter asymmetry of the Universe or the nature of dark matter. Towards this end, we refer to the minimal extension of the SM by three Majorana leptons: the νMSM (for neutrino minimal SM). It is based on exactly the same gauge group as the SM and does not introduce any new particle physics scale (figure 4). These three new particles, the Downloaded from http://rsta.royalsocietypublishing.org/ on December 4, 2017
three generations 5 of matter (fermions) spin 1/2 rsta.royalsocietypublishing.org I II III ...... mass 2.4 MeV 1.27 GeV 173.2 GeV 0 charge 2/3 uct2/3 2/3 0 g left left left right right name up right charm top gluon
4.8 MeV 104 MeV 4.2 GeV 0 –1/3 d –1/3 sb–1/3 0 Y quarks left left left down right down right down right photon Phil.Trans.R.Soc.A ~10 keV ~GeV ~GeV 91.2 GeV 0 126 GeV 0 0 0 ve N1 00vm N2 vt N3 H Z 0 muon tau left electron left left weak Higgs neutrino neutrino neutrino force boson 105.7 MeV 1.777 GeV 80.4 GeV spin 0 0.511 MeV ± 20140038 : 373 –1e –1m –1 t ±1 W bosons (forces) spin 1 leptons
left left left weak electron right muon right tau right force
Figure4. ParticlecontentoftheνMSM.IntheSMtheright-handedpartnersofneutrinosareabsent.IntheνMSM,allfermions have both left- and right-handed components and masses below the Fermi scale. (Online version in colour.)
HNLs or simply ‘Ns’—with masses in the range keV to GeV can explain simultaneously neutrino masses and oscillations, dark matter and the baryon asymmetry of the Universe [11] (for a review, see [18]). The HNLs interact with the Higgs boson via Yukawa interactions in exactly the same way ¯ α T c α α the other fermions do: F IH NI L (cf. with the name HNL), where the L are left leptonic doublets, ‘c’ is the sign of the charge conjugation and ‘T’ is the sign of the matrix transpose. These interactions lead to active neutrino masses via the GeV scale see-saw mechanism, the creation of matter–antimatter asymmetry at temperatures T ∼ 100 GeV, and finally dark matter production at T ∼ 100 MeV, as outlined below and in figure 5.
(a) Baryogenesis In the νMSM, nothing essentially interesting happens between temperatures 103 and 1013 GeV, as all the SM elementary particles are nearly in thermal equilibrium. At the same time, the HNLs N2,3 are out of equilibrium due to the tiny values of their Yukawa couplings. HNLs are created in interactions with real or virtual Higgs bosons in processes like H ↔ Nν, tt¯ ↔ Nν, and come into thermal equilibrium at T ∼ 100 GeV. The charge–parity (CP) violation in these reactions leads to lepton asymmetry [11,19], which is then converted to baryon asymmetry of the Universe by SM sphalerons. The baryon number violating processes freezes out at T 140 GeV.
(b) Electroweak crossover Yet another event, though without observable consequences, happens around temperatures T ∼ 100 GeV. The Higgs field expectation value grows from small values up to the zero temperature value 250 GeV. For the experimentally determined value of the Higgs mass, there Downloaded from http://rsta.royalsocietypublishing.org/ on December 4, 2017
a ~ eh t 6 a ~ t2/3 rsta.royalsocietypublishing.org a ~ t1/2 ...... matter dark energy domination domination 2/3 radiation
Universe size a ~ t domination
a ~ eH t Phil.Trans.R.Soc.A hot Big Bang recombination baryogenesis dark matter production Universe age Higgs inflation s 10−43 10−36 10−32 10−12 10−4 1012 1017 temperature 1013 103 10−1 GeV 20140038 : 373
Figure 5. A possible history of the Universe from inflation until the present time. (Online version in colour.)
T T
critical point symmetric phase A vapour critical point B water
ice Higgs phase
P MH
Figure6. Thephasediagramoftheelectroweaktheoryandofthevapour–liquidsystem.Thecriticalpoint(endpointoftheline Tcrit = ± Mcrit = ± of the first-order phase transition) has coordinates 109.2 0.8 GeV, H 72.3 0.7 GeV. (Online version in colour.) is no phase transition [20], but instead a smooth crossover with the temperature given in the lowest order of perturbation theory by 1/2 M2 T v H , (4.1) c 2 + 2 + 2 / + 2 MH MW MZ 2 mt and shifted by non-perturbative effects to Tc 160 GeV [21]. The phase diagram of the electroweak theory on temperature versus Higgs mass plane is similar to the phase diagram of the vapour–liquid system on the pressure–temperature plane: one can transfer from the symmetric ‘phase’ (vapour phase) to the Higgs ‘phase’ (liquid phase) continuously, without crossing any phase transition boundary (figure 6).
(c) Dark matter
The N1 HNL can be sufficiently stable to play the role of the dark matter particle [22–29]. It is ¯→ ν produced in the early Universe via processes like ll N1 (figure 7) at a temperature in the region Downloaded from http://rsta.royalsocietypublishing.org/ on December 4, 2017
H N1 7 rsta.royalsocietypublishing.org ......
n Z
n n n Phil.Trans.R.Soc.A Figure 7. Production of dark matter particles via process ll¯→ Nν. The Higgs vacuum expectation value leads to N − ν mixing. (Online version in colour.)
of: 20140038 : 373 / M 1 3 T ∼ 130 I MeV, (4.2) 1keV and never come into thermal equilibrium since their interaction strength is very small. The Higgs field is essential for these processes, as it induces the mixing between HNLs and active neutrinos via Yukawa couplings. For a review of the various constraints on N1 properties, see [18].
5. Experiments Of course, the picture of the Universe evolution presented above is based on the extrapolation of known physics up to high-energy scales and on specific hypotheses about physics beyond the SM. Since the model of new physics—the νMSM—is extremely minimalistic, it is highly predictive and thus can be tested in future experiments. In order to check the self-consistency of the SM and of the νMSM up to the inflationary scale, we should determine the mass of the Higgs boson, the top Yukawa coupling and strong coupling constant with the highest possible accuracy (figure 2). This is one of the arguments in favour + − of the future e e collider: a top quark factory in which a precision measurement of the top quark mass would be possible. In order to verify the cosmological predictions of Higgs inflation, and to distinguish it from other models, such as Starobinsky R2 inflation [30], we need to have −3 measurements of the spectral index ns of scalar perturbations at the level of 10 (figure 8). In addition, the determination of the tensor-to-scalar ratio should be performed down to values −4 of r 0.003 and that of the running of the spectral index dns/d log k down to 5 × 10 .These measurements are thought to be possible at COrE (http://www.core-mission.org/science.php), PRISM (http://www.prism-mission.org/)andSKA(http://www.skatelescope.org/). In order to verify the mechanism for baryogenesis and neutrino mass generation, one should experimentally look for new particles—the HNLs N2,3. This is a challenging task, since they must be very weakly coupled to satisfy the Sakharov condition of out of equilibrium. The expression of interest to search for HNLs [31] has recently been submitted to the CERN SPS Committee; see also http://ship.web.cern.ch/ship/. The nature of the dark matter HNL, N1, is very different from that of weakly interacting massive particles (WIMPs). This dark matter candidate can be observed with the help of high- → νγ resolution X-ray telescopes (for a review, see [32]), which can look for HNL decays N1 by detecting monochromatic photons from regions where dark matter is concentrated. Some indications for the existence of an unidentified X-ray line which may correspond to the HNL dark matter particle with mass 7.1 keV have recently been reported in [33,34]. Downloaded from http://rsta.royalsocietypublishing.org/ on December 4, 2017
1 Planck + WP + H 8 x rsta.royalsocietypublishing.org =0 PRISM precision ...... x = 0.001 10–1 x (f2/2 )R + l (f4/4 ) x = 0.01 Higgs inflation R2 inflation –2 10 x = 0.1 tensor-to-scalar ratio(r) x =1 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00
primordial tilt (ns) Phil.Trans.R.Soc.A
Figure 8. Predictions of the Higgs inflation versus other models. The accuracy of the Polarized Radiation Imaging and Spectroscopy Mission (PRISM) is indicated. (Online version in colour.)
6. Conclusion 20140038 : 373 The SM Higgs field could play an important role in cosmology:
— it could help generate a universe that is flat, homogeneous and isotropic; — quantum fluctuations of the Higgs field could lead to structure formation; — coherent oscillations of the Higgs field could make the hot Big Bang and produce all the matter in the Universe; — real and virtual Higgs bosons could play a crucial role in baryogenesis, leading to a charge asymmetric Universe; — dark matter production may come about as an effect of mixing between neutrinos and HNLs, induced by the Higgs field; and — several new experiments are needed to reveal the ‘secret’ couplings of the Higgs boson.
Funding statement. This work was supported by the Swiss National Science Foundation. I am grateful to Mark Lovell for careful reading of the manuscript and important suggestions.
References 1. Aad G et al. 2012 Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Phys. Lett. B 716,1– 29. (doi:10.1016/j.physletb.2012.08.020) (ATLAS collaboration: http://cds.cern.ch/record/ 1523727) 2. Chatrchyan S et al. 2012 Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. Phys. Lett. B 716, 30. (doi:10.1016/j.physletb.2012.08.021)(CMS collaboration: http://cds.cern.ch/record/1542387) 3. Ellis J, Espinosa JR, Giudice GF, Hoecker A, Riotto A. 2009 The probable fate of the Standard Model. Phys. Lett. B 679, 369–375. (doi:10.1016/j.physletb.2009.07.054) 4. Bezrukov F, Kalmykov MY, Kniehl BA, Shaposhnikov M. 2012 Higgs boson mass and new physics. J. High Energy Phys. 10, 140. (doi:10.1007/JHEP10(2012)140) 5. Degrassi G, Di Vita S, Elias-Miro J, Espinosa JR, Giudice GF, Isidori G, Strumia A. 2012 Higgs mass and vacuum stability in the Standard Model at NNLO. J. High Energy Phys. 1208, 098. (doi:10.1007/JHEP08(2012)098) 6. Buttazzo D, Degrassi G, Giardino PP, Giudice GF, Sala F, Salvio A, Strumia A. 2013 Investigating the near-criticality of the Higgs boson. J. High Energy Phys. 1312, 089. (doi:10.1007/JHEP12(2013)089) 7. Shaposhnikov M. 2013 Cosmology: theory. (http://arxiv.org/abs/1311.4979) Downloaded from http://rsta.royalsocietypublishing.org/ on December 4, 2017
8. Bezrukov FL, Shaposhnikov M. 2008 The Standard Model Higgs boson as the inflaton. Phys. 9 Lett. B 659, 703–706. (doi:10.1016/j.physletb.2007.11.072) rsta.royalsocietypublishing.org 9. Bezrukov F, Gorbunov D, Shaposhnikov M. 2009 On initial conditions for the hot big bang. J...... Cosmol. Astropart. Phys. 0906, 029. (doi:10.1088/1475-7516/2009/06/029) 10. Garcia-Bellido J, Figueroa DG, Rubio J. 2009 Preheating in the Standard Model with the Higgs- inflaton coupled to gravity. Phys. Rev. D 79, 063531. (doi:10.1103/PhysRevD.79.063531) 11. Asaka T, Shaposhnikov M. 2005 The nuMSM, dark matter and baryon asymmetry of the Universe. Phys. Lett. B620, 17–26. (doi:10.1016/j.physletb.2005.06.020) 12. Birrell ND, Davies PCW. 1982 Quantum fields in curved space. Cambridge Monographs on Mathematical Physics. Cambridge, UK: Cambridge University Press. 13. Bezrukov F, Shaposhnikov M. 2009 Standard model Higgs boson mass from inflation: two loop analysis. J. High Energy Phys. 0907, 089. (doi:10.1088/1126-6708/2009/07/089)
14. Bezrukov F, Magnin A, Shaposhnikov M, Sibiryakov S. 2011 Higgs inflation: consistency and Phil.Trans.R.Soc.A generalisations. J. High Energy Phys. 1101, 016. (doi:10.1007/JHEP01(2011)016) 15. Alekhin S, Djouadi A, Moch S. 2012 The top quark and Higgs boson masses and the stability of the electroweak vacuum. Phys. Lett. B 716, 214–219. (doi:10.1016/j.physletb.2012.08.024) 16. Linde AD. 1983 Chaotic inflation. Phys. Lett. B 129, 177–181. (doi:10.1016/0370-2693(83) 90837-7) 17. Ade PAR et al. 2013 Planck 2013 results. XXII. Constraints on inflation. (http://arxiv. org/abs/1303.5082) 20140038 : 373 18. Boyarsky A, Ruchayskiy O, Shaposhnikov M. 2009 The role of sterile neutrinos in cosmology and astrophysics. Annu. Rev. Nucl. Part. Sci. 59, 191–214. (doi:10.1146/annurev.nucl. 010909.083654) 19. Akhmedov EK, Rubakov VA, Smirnov AY. 1998 Baryogenesis via neutrino oscillations. Phys. Rev. Lett. 81, 1359–1362. (doi:10.1103/PhysRevLett.81.1359) 20. Kajantie K, Laine M, Rummukainen K, Shaposhnikov ME. 1996 Is there a hot electroweak phase transition at mH ≥ mW? Phys. Rev. Lett. 77, 2887–2890. (doi:10.1103/PhysRevLett. 77.2887) 21. D’Onofrio M, Rummukainen K, Tranberg A. 2012 The sphaleron rate at the electroweak crossover with 125 GeV Higgs mass. PoS LATTICE 2012, 055. (http://arxiv.org/abs/1212. 3206) 22. Dodelson S, Widrow LM. 1994 Sterile-neutrinos as dark matter. Phys. Rev. Lett. 72, 17–20. (doi:10.1103/PhysRevLett.72.17) 23. Shi X-D, Fuller GM. 1999 A new dark matter candidate: nonthermal sterile neutrinos. Phys. Rev. Lett. 82, 2832–2835. (doi:10.1103/PhysRevLett.82.2832) 24. Dolgov AD, Hansen SH. 2002 Massive sterile neutrinos as warm dark matter. Astropart. Phys. 16, 339–344. (doi:10.1016/S0927-6505(01)00115-3) 25. Abazajian K, Fuller GM, Patel M. 2001 Sterile neutrino hot, warm, and cold dark matter. Phys. Rev. D 64, 023501. (doi:10.1103/PhysRevD.64.023501) 26. Abazajian K, Fuller GM, Tucker WH. 2001 Direct detection of warm dark matter in the X-ray. Astrophys. J. 562, 593–604. (doi:10.1086/323867) 27. Asaka T, Laine M, Shaposhnikov M. 2007 Lightest sterile neutrino abundance within the nuMSM. J. High Energy Phys. 0701, 091. (doi:10.1088/1126-6708/2007/01/091) 28. Shaposhnikov M. 2008 The νMSM, leptonic asymmetries, and properties of singlet fermions. J. High Energy Phys. 0808, 008. (doi:10.1088/1126-6708/2008/08/008) 29. Laine M, Shaposhnikov M. 2008 Sterile neutrino dark matter as a consequence of νMSM-induced lepton asymmetry. J. Cosmol. Astropart. Phys. 0806, 031. (doi:10.1088/1475- 7516/2008/06/031) 30. Starobinsky AA. 1980 A new type of isotropic cosmological models without singularity. Phys. Lett. B 91, 99–102. (doi:10.1016/0370-2693(80)90670-X) 31. Bonivento W et al. Proposal to search for heavy neutral leptons at the SPS. (http:// arxiv.org/abs/1310.1762) 32. Boyarsky A, Iakubovskyi D, Ruchayskiy O. 2012 Next decade of sterile neutrino studies. Phys. Dark Univ. 1, 136–154. (doi:10.1016/j.dark.2012.11.001) 33. Bulbul E, Markevitch M, Foster A, Smith RK, Loewenstein M, Randall SW. Detection of an unidentified emission line in the stacked X-ray spectrum of galaxy clusters. (http://arxiv.org/abs/1402.2301) 34. Boyarsky A, Ruchayskiy O, Iakubovskyi D, Franse J. An unidentified line in X-ray spectra of the Andromeda galaxy and Perseus galaxy cluster. (http://arxiv.org/abs/1402.4119)