Neutrinoful Universe

The Standard Model of particle physics fails to explain the important pieces in the standard cosmology, such as inflation, baryogenesis, and dark matter of the Universe. We consider the possibility that the sector to generate small neutrino masses is responsible for all of them; the inflation is driven by the Higgs field to break $B-L$ gauge symmetry which provides the Majorana masses to the right-handed neutrinos, and the reheating process by the decay of the $B-L$ Higgs boson supplies the second lightest right-handed neutrinos whose CP violating decays produce $B-L$ asymmetry, a la, leptogenesis. The lightest right-handed neutrinos are also produced by the reheating process, and remain today as the dark matter of the Universe. In the minimal model of the inflaton potential, one can set the parameter of the potential by the data from CMB observations including the BICEP2 and the Planck experiments. In such a scenario, the mass of the dark matter particle is predicted to be of the order of PeV. We find that the decay of the PeV right-handed neutrinos can explain the high-energy neutrino flux observed at the IceCube experiments if the lifetime is of the order of $10^{28}$ s.


Introduction
Various cosmological observations are telling us that the Standard Model of particle physics needs some extension. The observation of the cosmic microwave background (CMB) and its anisotropy strongly supports the inflationary cosmology [1,2,3], which requires a process to generate the Standard Model particles after the inflation era. The particle-antiparticle asymmetry should also be generated after or during the reheating process. Also, the dark matter of the Universe must also be produced in the course of the cosmological history. The Standard Model should be extended to accommodate the inflation, baryogenesis and dark matter of the Universe.
One of the clues towards the mysteries of the Universe may be the finite neutrino masses, which are another evidence to go beyond the Standard Model. Once three kinds of righthanded neutrinos are introduced in the same way as other fermions, the global U (1) B−L symmetry becomes non-anomalous, and thus can be promoted to a Higgsed gauge symmetry.
It seems that all the ingredients to accommodate the realistic cosmology are present in this The inflation can be driven by the Higgs field to break U (1) B−L gauge symmetry [4,5] by assuming an appropriate form of the potential based on the idea of the chaotic inflation [6].
After the inflation, the B − L Higgs field oscillates about the minimum of the potential where U (1) B−L is broken. The spontaneous breaking of the B − L symmetry can give Majorana masses to the right-handed neutrinos through the Yukawa coupling, explaining the smallness of the neutrino masses by the seesaw mechanism [7]. The very same coupling allows the decay of the inflaton oscillation into the right-handed neutrinos to reheat the Universe. The subsequent decay of the right-handed neutrinos can provide the baryon asymmetry of the Universe by the leptogenesis mechanism [8]. The lightest right-handed neutrino should also be produced by the inflaton decay. If it is long-lived, this non-thermal component is a good candidate of the dark matter of the Universe.
There have been other minimalistic approaches to the connection between particle physics and cosmology. An realistic model with the minimal particle content has been constructed in Ref. [9], where the inflaton and the dark matter particle are both introduced as new scalar fields. The possibility of the inflaton as the Higgs-like field, thus playing important roles both in particle physics and cosmology, has been considered in Refs. [10,11,12]. The dark matter of the Universe as the right-handed neutrino has also been considered in Refs. [13,14,15,16] where the mass range of O(keV) are assumed.
In this paper, we consider the U (1) B−L extended Standard Model which covers the shortages in the Standard Model including the small neutrino masses as well as cosmological observations. We find that this minimalistic scenario is consistent with various observations such as tensor-to-scalar ratio, spectral index of the CMB fluctuations, the neutrino masses, baryon asymmetry of the Universe, and the energy density of the dark matter. We find, in the case where the reheating process is dominated by the decay of inflaton into the second lightest right-handed neutrinos, the mass of the dark matter particle is predicted to be of the order of PeV.
Since there is no reason to assume that the dark matter particle, the lightest righthanded neutrino, to be absolutely stable, we expect the decay of the dark matter to happen occasionally somewhere in the Universe. Through the dimension-four Yukawa interactions, the main decay mode would be into a lepton and a W boson, or a neutrino and a Z/h boson.
We demonstrate that the PeV neutrino events found at the IceCube experiment [17,18] can be explained by the decaying right-handed neutrinos if the lifetime is of the order of 10 28 s * .
In the following sections, based on the above scenario with U (1) B−L extended Standard Model, we discuss the neutrino flavor structure, an inflation model with the B − L Higgs, the non-thermal leptogenesis, the dark matter abundance produced by the decay of the inflaton, and the signals of decaying right-handed neutrinos at the IceCube experiment.

Model
We extend the gauge group of the Standard Model into, and introduce the right-handed neutrinos, N i (i = 1, 2, 3), and the U (1) B−L Higgs field φ B−L which is neutral under the Standard Model gauge group and has charge −2 under U (1) B−L .
The U (1) B−L symmetry is gauged, and thus the spontaneous breaking of U (1) B−L would not leave the massless Nambu-Goldstone boson. The following interaction terms are added to the Standard Model: where ℓ i and N i are four-component Weyl fermions, i.e., P L ℓ i = ℓ i , P R N i = N i . The coupling constant λ i can be taken to be real and positive without loss of generality, and the components of y ij ν , in general, are complex valued. The potential terms for φ B−L field can be written as, There can also be an interaction term such as, For v B−L 5M Pl which we assume later, the coupling constant κ ′ is extremely small if we demand this term would not contribute significantly to the Higgs potential.
The spontaneous breaking of U (1) B−L by φ B−L = v B−L generates masses of N i : The neutrino masses are, in turn, generated by the seesaw mechanism: We assume that the lightest right-handed neutrino, N 1 , to be long-lived, and it serves as the dark matter of the Universe. That means, |y 1i ν | ≪ 1.
As we will see in Sec. 5, in the scenario where the PeV neutrino events at the IceCube experiment to be explained by the decay of N 1 , the lifetime of N 1 has to be around 10 28 s.
This lifetime corresponds to y 1i ν ∼ 10 −29 . In fact, this model has various unexplained small numbers such as the Higgs mass parameter, the θ parameter in QCD, the cosmological constant, κ ′ , κ as well as y 1i ν . Although we do not look for particular reasons for such small numbers here, a very small y 1i ν is somewhat special since it can be protected by a Z 2 symmetry, N 1 ↔ −N 1 . If such a symmetry is only violated by some non-perturbative effects of gauge or gravity interactions at high scales, the size may be understood as a natural value † .
In such a scenario, it is likely that the non-perturbative effects respect the flavor symmetry, and thus the effective operator to break the Z 2 symmetry, for example, takes the form of The quantum theory of gravity may give natural ground for such considerations [21,22]. Here, Λ is expected to be the scale which characterizes the non-perturbative effects such as, µe −8π 2 /g 2 (µ) , in the case of a gauge theory. This is analogous to the interaction considered in QCD [23]. Together with the Yukawa interactions of the charged lepton sector y ij e in the Standard Model and y αi ν (α = 2, 3) in Eq. (1), y 1i ν is generated as in the diagram in Fig. 1: One can also consider interactions such as From this operator, we obtain y 1k ν ∝ (det y d )ǫ ijk y 2i ν y 3j ν . In any case, the flavor symmetry implies an interesting proportionality: We will see in Sec. 5 that if this type of contribution is dominated, the branching ratio of the N 1 decay is directly related to the neutrino mixing parameters. By introducing a small parameter c, Eq. (9) is explicitly written as Because of tiny y 1ℓ ν 's, N 1 provides very little contribution to the neutrino masses. In this case, the neutrino sector is essentially that of the model with only two right-handed neutrinos [24,25]. Here, we define the following Yukawa matrixỹ and mass matrixM : Neutrino masses are given by, where U MNS is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [26,27]: Eq. (12) tells us that the lightest neutrino is massless (up to O((y 1i ν ) 2 ) contributions) because the rank ofỹ andM is two. There is only one Majorana phase in Eq. (13) in this effectively two-generation model. We can parametrizeỹ which satisfies Eq. (12) by using a 3×2 complex matrix R [28,29]:ỹ where R can be expressed in terms of a complex parameter z, for normal hierarchy, and, for inverted hierarchy.
By using the above parametrization and Eqs. (10,14), we can determine the structure of the Yukawa coupling y ν . For normal hierarchy, we obtain, For inverted hierarchy, Here, we used the unitarity of U PMNS for the calculation of y 1ℓ ν . These structures are important for the discussion of the flavor of the decay products of N 1 , We will discuss their effects on the energy spectrum of the neutrino flux from the decay of N 1 in Sec. 5.

Inflation with the B − L Higgs field
In this section, we consider an inflation model with the B − L Higgs field. The potential for where µ 2 = 2v 2 B−L and Λ 4 = κv 4 B−L /4, and we define µ > 0. The phase direction can be gauged away.
The inflaton field can slow roll when µ ≫ M Pl , either from the |φ| > µ or |φ| < µ region towards the minimum at φ = µ. In both cases, the slow-roll parameters at φ = φ 0 are given by [30], The field value φ 0 at the pivot scale k 0 = 0.002 Mpc −1 is expressed in terms of the number of e-folds N : where the field value at the end of the inflation, φ end , is obtained from, The tensor-to-scalar ratio, r, and the spectral index, n s , is expressed in terms of the slow-roll parameters as, The Planck normalization sets the overall scale [31],  The dark blue region corresponds to the region consistent with the BICEP2 at 1σ level whereas the light blue region does to that at 2σ level. The left figure is consistent with the result obtained in Ref. [5]. and the observed spectral index is given by, The results from the BICEP2 experiment prefer, r = 0.20 +0.07 −0.05 , V 1/4 = 2.0 × 10 16 GeV · r 0.16 when one combines the data from the Planck experiment. ‡ Here, the preferred range of r will be modified to r = 0.16 +0.06 −0.05 after subtracting the best available estimate for foreground dust [32].
The predictions for r and n s is shown in Fig. 2 with varying µ. The region favored by the CMB observations are also shown. We see that for N = 60, |φ| > µ and µ 5M Pl is favored. The inflaton mass, m φ = 2 √ 2Λ 2 /µ, as a function of µ/M Pl is also shown in Fig. 2.
For |φ| > µ and µ 5M Pl , we find, m φ ∼ 10 13 GeV. (31) ‡ We will not consider the tension between the data from the Planck satellite (r < 0.11) [31] and that from the BICEP2 experiment (r ∼ 0.2). The tension can be relaxed if one considers a running spectral index, an extra relativistic component, non-zero neutrino mass [32,33], an anti-correlation between tensor and scalar modes [34] or between tensor and isocurvature modes [35]. See also [36,37] for other solutions.
This value corresponds to a very small value of κ such as κ ∼ 10 −12 for µ ∼ 5M Pl . In the following discussion, we fix the inflaton mass at this value, and will see that the correct amount of the baryon asymmetry and the dark matter can be obtained after the decay of the inflaton fields.

Reheating by the inflaton decay
After the inflation, the decay of φ can produce the Standard Model particles. The dominant decay mode can either be into two right-handed neutrinos via the interaction term in Eq. (1) or two Higgs fields (including the Goldstone modes) via the term in Eq. (3).
In the case where the φ → N i N i mode is dominated and for λ 1 ≪ λ 2 and M 3 > m φ which are justified later, the decay width is given by, By equating Γ φ with the Hubble parameter H(T R ) at the reheating temperature, T R , we obtain, Here we used T R = (90/π 2 g * (T R )) 1/4 Γ φ M Pl and g * (T R ) = 106.75, where g * (T R ) is the relativistic degrees of freedom in plasma at the temperature T R . If the Higgs mode φ → hh, W W, ZZ is the dominant decay channel through Eq. (3), the reheating temperature can be arbitrarily higher than the above estimate. If T R is higher than m φ , the perturbative analysis of the reheating process becomes unreliable [38]. Therefore, we restrict ourselves to the region of T R < m φ ∼ 10 13 GeV.

Leptogenesis
For the case where φ → N 2 N 2 is the dominant decay mode, the decay of N 2 can generate B−L asymmetry by leptogenesis. The baryon-to-entropy ratio obtained from the non-thermal leptogenesis is [39], where (3/2)T R /m φ is the number density n N 2 ≃ n φ /2 divided by the entropy density produced by the decay of φ. The ǫ factor is the magnitude of the CP violation [40]: for M 2 ≪ M 3 . It is bounded by [25,41], 3 16π Here, we take ∆m 2 ⊙ = (0.0086 eV) 2 and ∆m 2 A = (0.048 eV) 2 [42]. Therefore, For normal hierarchy, compared with the observed baryon-to-entropy ratio, n B /s ≃ 10 −10 [43], we need M 2 10 12 GeV. On the other hand, for inverted hierarchy, we need M 2 10 13 GeV which is on the border of the constraint: m φ > 2M 2 . In any case, these result justify M 3 > m φ which we assumed before.
If the Higgs mode is important, the branching ratio into M 2 is suppressed, and thus nonthermal leptogenesis becomes difficult. With fixed m φ from the CMB observations, there is no freedom to make M 2 larger since the decay into N 2 becomes kinematically forbidden.
Instead, if the reheating temperature is high enough, it is possible to produce N 2 thermally.

Dark matter abundance
The inflaton also decays into two N 1 's. The assumption that N 1 is long-lived makes it possible to identify this component to be the dark matter of the Universe.
The partial decay width is given by, By using the relation H(T R ) ∼ Γ φ ∼ T 2 R /M Pl , and n N 1 /s ≃ (3/2)(T R /m φ )Br(φ → N 1 N 1 ), we find, Here, we used Ω NT N 1 = (M 1 n N 1 /s)/(ρ c /s) 0 , where (ρ c /s) 0 ≃ 1.8 × 10 −9 GeV is the critical density divided by the entropy density today. The contribution from the thermal production from the scattering processes by the U (1) B−L gauge interaction is much smaller such as [16], This is estimated with the interaction between N 1 and the Standard Model fermions in plasma through the s-channel exchange of the U (1) B−L gauge boson. We summarize the allowed regions in Fig. 3. We see that one obtains the correct amount of the baryon asymmetry and the dark matter abundance at M 1 ∼ 1 PeV and M 2 ∼ 10 12 GeV within the region consistent with the BICEP2 results at the 1σ level. The PeV dark matter opens up an interesting possibility that the high energy neutrinos observed at the IceCube experiment [17,18] are explained by the decay of N 1 , which will be studied in the next section. For a heavier N 1 , we see a region where the thermal leptogenesis works. There, a high enough reheating temperature is realized by the inflaton decay into Higgs fields through the coupling in Eq. (3).

PeV neutrinos as a signal of decaying N 1
In this section, we discuss observational signatures of the right-handed neutrino dark matter.
As discussed so far, the inflation, the baryon asymmetry and the correct amount of dark matter can be explained for M 1 = O(1) PeV. Since there is no reason to expect that N 1 is absolutely stable, we have a chance to see high energy cosmic rays produced from the decay of N 1 . It is interesting that the PeV is indeed the energy region where an excess of high energy neutrinos events are observed at the IceCube experiment. In this section, we discuss the possibility that neutrino excess which is observed at IceCube experiment [17,18] is explained by the decay products of N 1 .  Figure 3: Consistent regions with neutrino masses and cosmological observations. The two shaded regions (green and light orange) are consistent with the BICEP2 at 1σ level respectively, and imply that the non-thermal leptogenesis works (green) and the dark matter is explained via the inflaton decay (light orange). Here, we assume normal hierarchy. We also show the mass range of N 1 favored by the IceCube experiment (pink shaded region). In the dark orange region where thermal leptogenesis is viable, the reheating temperature is treated as a free parameter satisfying M 2 ≤ T R ≤ m φ = 10 13 GeV with 5M Pl ≤ v B−L . A high reheating temperature is realized by the decay into hh, W W and ZZ via the coupling in Eq. (3).
On the other hand, the branching fractions for inverted hierarchy is completely determined independent of δ.

Neutrino flux from decay of N 1
We have calculated the energy spectrum of neutrinos dN ν /dE ν from decay of N 1 by using PYTHIA 8.1 [46]. The neutrino spectrum for M 1 = 2.3 PeV is shown in Fig. 4. We have a sharp peak in the neutrino energy spectrum at E ν = M 1 /2. In the case of inverted hierarchy, since the fractions of muon and tau are large compared to the normal hierarchy, the number of neutrinos is slightly larger around E ν ∼ 10 5−6 GeV due to the decay products of the muons and taus.
As the neutrino travels towards the Earth, the neutrinos change their flavors by the neutrino oscillation according to the following probabilities:  In both figure, we take δ = 0. Red, green and blue lines show the spectrum of ν e +ν e , ν µ +ν µ and ν τ +ν τ , respectively.
In Fig. 5, we show the energy spectrum of the neutrinos after the oscillation.
We estimate the observed flux of neutrinos on the Earth in the following way. We have two classes of contribution from the decaying dark matter; one is from halo of our galaxy, and another is from extra-galactic. For a review of the calculation of the neutrino flux, e.g., see Ref. [47]. The halo contribution which is averaged over the full sky is proportional to where D halo is determined by the halo density profile ρ halo (r), The parameter s in the integral of Eq. (51) is the distance from the Earth, and it is related to the distance r from the galactic center as, r(s, θ, φ) = s 2 + R 2 ⊙ − 2sR ⊙ cos θ cos φ. Here, R ⊙ is the distance of the Sun to the galactic center, and we take its value as 8.0 kpc [48].
For the calculation of D halo , we adopt the Navarro-Frenk-White (NFW) density profile [49], and take r c = 20 kpc and ρ ⊙ = 0.4 GeV cm −3 [50]. Then, D halo is calculated as, Extra galactic contribution is redshifted because of the expansion of the Universe. Their contribution is written by, where we estimate the integrand just from z = 0 to z eq for simplicity and hence also neglect the contribution from the dark matter which had decayed at the radiation dominated era, because we assume the dark matter mass is around PeV and the energy of neutrino from early universe is too low to explain the IceCube excess. In Eq. (54), is the Hubble expansion rate at the redshift z. c = 3.0 × 10 10 cm s −1 is the speed of light.
Finally, the expected number of events at the IceCube detector per 662 days with given energy is calculated as, where σ (ν ℓ ) eff is the neutrino effective area for each flavor which is given in Refs. [18,52]. The IceCube experiment observed 28 events with deposited energies between 30 and 1200 TeV, and the expected number of events from atmospheric muons and neutrinos is 10.6 +5.0 −3.6 [18]. For 2.3 PeV dark matter, the total expected number of events for each pattern of the neutrino mass hierarchy is, From this estimate, we see that the total excess can be explained for τ N 1 ≃ 1 × 10 28 s for both normal and inverted hierarchy. We also show the energy distribution of the neutrinos in Fig. 6. In this figure, we take M 1 = 2.3 PeV and τ N 1 = 10 28 s.
The IceCube experiment provides the data of the event rate per the deposited energies in the detector in Fig. 4 in Ref. [18]. Note that our results in Fig. 6 are, in contrast, those for incoming neutrino energies, and thus the deposited ones should be smaller due to escaping neutrinos and muons. One needs to take into account the correction when the shape of the distribution is compared. For M N 1 = 2.3 PeV, the expected number of neutrinos with the energy higher than 1 PeV is, for both normal and inverted hierarchy. Thus, by assuming that the deposited energy is equal to that of incoming neutrinos, the two observed neutrino events around PeV energies can be explained for τ N 1 ≃ 2× 10 28 s. We can expect more sub-PeV events for the inverted hierarchy than the normal hierarchy. Implications from the IceCube experiment will be important to distinguish neutrino models.

Summary
In this paper, we considered a minimalistic cosmological scenario based on the U If the coupling between the Standard Model Higgs and the B − L Higgs field is significant, the reheating temperature can be higher than the second lightest right-handed neutrino N 2 , depending on the coupling. In such a case, thermal leptogenesis is possible whereas the dark matter should be heavier than O(10) PeV to explain the abundance by the inflaton decay.