GUT Scale Inflation, Non-Thermal Leptogenesis, and Atmospheric Neutrino Oscillations

Leptogenesis scenarios in supersymmetric hybrid inflation models are considered. Sufficient lepton asymmetry leading to successful baryogenesis can be obtained if the reheat temperature T_r>10^6 GeV and the superpotential coupling parameter kappa is in the range 10^-6<kappa<10^-2. For this range of kappa the scalar spectral index n_s=0.99+-0.01. Constraints from neutrino mixing further restrict the range of kappa that is allowed. We analyze in detail the case where the inflaton predominantly decays into the next-to-lightest right handed Majorana neutrino taking into account especially the constraints from atmospheric neutrino oscillations.


Introduction
Supersymmetric hybrid inflation models [1,2] provide a compelling framework for the understanding of the early universe. They account for the primordial density perturbations with a GUT scale symmetry breaking yet without any dimensionless parameters that are very small. As in any complete inflationary scenario, inflation in these models should be followed by a succesful reheating accounting for the observed baryon asymmetry of the universe.
In SUSY hybrid inflation it is generally preferable (and in many models necessary) to generate the baryon asymmetry via leptogenesis, which is then partially converted into baryon asymmetry by sphaleron effects [3]. If the gauge symmetry G = SO (10) or one of its subgroups (where inflation is associated with the breaking of a gauge symmetry G → H), the inflaton decays into the right handed neutrinos, whose subsequent out of equilibrium decay leads to the lepton asymmetry [4].
The right handed neutrinos could also be produced thermally, although it is difficult to reconcile the high reheat temperature required by thermal leptogenesis with the gravitino constraint [5].
In thermal leptogenesis [6] the lightest right handed Majorana neutrino N 1 washes away the previous asymmetry created by the heavier neutrinos. If, on the other hand, N 1 as well as the heavier neutrinos are out of equilibrium (T r < M 1 ), the lepton asymmetry could predominantly result from the inflaton χ decaying into the next-to-lightest neutrino N 2 . (χ → N 3 N 3 is ruled out by the gravitino constraint.) In this letter we focus on the latter scenario. It is easier to account for the observed baryon asymmetry in this case since the asymmetry per right handed neutrino decay is in general greater than the case where the inflaton decays into the lightest neutrino, and unlike thermal leptogenesis there is no washout factor.
The plan of the paper is as follows: In Section 2 we briefly review a class of supersymmetric hybrid inflation models. In Section 3 we qualitatively discuss leptogenesis scenarios for these models. In Section 4 we perform an analysis of the 'next-to-lightest' scenario, showing numerically that sufficient lepton asymmetry can be generated while satisfying, in particular, the constraints from atmospheric neutrino mixing.

Supersymmetric Hybrid Inflation
In a class of realistic supersymmetric models, inflation is associated with the breaking of either a grand unified symmetry or one of its subgroups. Here we will limit ourselves to supersymmetric hybrid inflation models [2]. The simplest such model [1] is realized by the renormalizable potential (consistent with a U(1) R-symmetry) [7] where φ(φ) denote a conjugate pair of superfields transforming as nontrivial representations of some gauge group G, S is a gauge singlet superfield, and κ (> 0) is a dimensionless coupling. In the absence of supersymmetry breaking, the potential energy minimum corresponds to non-zero (and equal in magnitude) vevs (= M) for the scalar components in φ and φ, while the vev of S is zero. (We use the same notation for superfields and their scalar components.) Thus, G is broken to some subgroup H.
In order to realize inflation, the scalar fields φ, φ, S must be displayed from their present minima. For |S| > M, the φ, φ vevs both vanish so that the gauge symmetry is restored, and the tree level potential energy density κ 2 M 4 dominates the universe.
With supersymmetry thus broken, there are radiative corrections from the φ − φ supermultiplets that provide logarithmic corrections to the potential which drives inflation.
The temperature fluctuations δT /T turn out to be proportional to (M/M P ) 2 , where M denotes the symmetry breaking scale of G, and M P = 1.2 × 10 19 GeV is the Planck mass [1,2]. Comparison with the δT /T measurements by COBE [8] and WMAP [9] shows that the gauge symmetry breaking scale M is naturally of order The inflationary scenario based on the superpotential W 1 in Eq. (1) has the characteristic feature that the end of inflation essentially coincides with the gauge symmetry breaking. Thus, modifications should be made to W 1 if the breaking of G to H leads to the appearance of topological defects such as monopoles, strings or domain walls.
For instance, the breaking of G P S ≡ SU(4) c × SU(2) L × SU(2) R [10] to the MSSM by fields belonging to φ(4, 1, 2), φ(4, 1, 2) produces magnetic monopoles that carry two quanta of Dirac magnetic charge [11]. As shown in [12], one simple resolution of the monopole problem is achieved by supplementing W 1 with a non-renormalizable term: where µ is comparable to the GUT scale, M S ∼ 5 × 10 17 GeV is a superheavy cutoff scale, and the dimensionless coefficient β is of order unity. The presence of the non-renormalizable term enables an inflationary trajectory along which the gauge symmetry is broken. Thus, in this 'shifted' hybrid inflation model the magnetic monopoles are inflated away.
A variation on these inflationary scenarios is obtained by imposing a Z 2 symmetry on the superpotential, so that only even powers of the combination φφ are allowed [13]: where the dimensionless parameters κ and β (see Eq. (2)) are absorbed in µ and M S . The resulting scalar potential possesses two (symmetric) valleys of local minima which are suitable for inflation and along which the GUT symmetry is broken. The inclination of these valleys is already non-zero at the classical level and the end of inflation is smooth, in contrast to inflation based on the superpotential W 1 (Eq. (1)).
An important consequence is that, as in the case of shifted hybrid inflation, potential problems associated with topological defects are avoided.
In all these models, for the symmetry breaking scale M ∼ 10 16 GeV, one predicts an essentially scale invariant spectrum (0.98 n s 1 depending on the value of κ or of M and |dn s /d ln k| < 10 −3 [14]) which is consistent with a variety of CMB measurements including the recent WMAP results [9,15].
After the end of inflation, the system falls toward the SUSY vacuum and performs damped oscillations about it. The inflaton, which we collectively denote as χ, consists of the two complex scalar fields equal mass m χ . In the presence of N = 1 supergravity, SUSY breaking is induced by the soft SUSY violating terms in the tree level potential and S acquires a vev comparable to the gravitino mass m 3/2 (∼ TeV). This (mass) 2 term provides an extra force driving S to the minimum, but its effect is negligible for κ 10 −6 .
More often than not, SUGRA corrections tend to derail an otherwise succesful inflationary scenario by giving rise to scalar (mass) 2 terms of order H 2 , where H denotes the Hubble constant. Remarkably, it turns out that for a canonical SUGRA potential (with minimal Kähler potential |S| 2 + |φ| 2 + |φ| 2 ), the problematic (mass) 2 term cancels out for the superpotential W 1 in Eq. (1) [7]. This property also persists when non-renormalizable terms that are permitted by the U(1) R symmetry are included in the superpotential. 2 As noted in [19,14], for large values of κ the presence of SUGRA corrections due to the minimal Kähler potential can give rise to n s values that exceed unity by an amount that is not favored by the data on smaller scales. SUGRA corrections also become important for tiny values of κ. Nevertheless, they remain ineffective for a wide range of κ (10 −6 κ 10 −2 ). As we shall discuss below, leptogenesis consistent with the observed baryon asymmetry generally constrains κ to a similar range.

Leptogenesis in SUSY Hybrid Inflation Models
The observed baryon asymmetry of the universe can be naturally explained via leptogenesis in SUSY hybrid inflation models. If inflation is associated with the breaking of the gauge symmetry G = SO(10) [20] or one of its subgroups such as , the inflaton decays into right handed neutrino superfields [4]. Their subsequent out of equilibrium decay to lepton and Higgs superfields leads to the observed baryon asymmetry via sphaleron effects [3].
Before discussing the constraints on κ from leptogenesis, we note that an important constraint that is independent of the details of the seesaw parameters already arises from considering the reheat temperature T r after inflation, taking into account the gravitino problem which requires that T r 10 10 GeV [5]. We expect the heaviest right handed neutrino to have a mass of around 10 14 GeV, which is in the right ball park to provide via the seesaw a mass scale of about .05 eV to explain the atmospheric neutrino anomaly through oscillations. Comparing this with [2] T r = 45 (where m P ≃ 2.4 × 10 18 GeV is the reduced Planck mass, and the decay rate of the 2 In general, K is expanded as K = |S| 2 + |φ| 2 + |φ| 2 + α|S| 4 /M 2 P + . . ., and only the |S| 4 term in K generates a mass 2 for S, which would spoil inflation for α ∼ 1 [16,17]. From the requirement |S| < M P , one obtains an upper bound on α ( 10 −3 ) [18]. Since smaller values of α do not effect the dynamics of inflation significantly and other terms in K are supressed, we take the Kähler potential to be minimal for simplicity.
, we see that for m χ 10 5 GeV, M i should not be identified with the heaviest right handed neutrino, otherwise T r would be too high [21]. Here we have assumed that the right handed neutrinos N i acquire mass from a The gravitino constraint expressed by Eq. (5)  We now consider the case where the inflaton χ predominantly decays into a right handed neutrino that is heavy compared to the reheat temperature T r . The ratio of the number density of the right-handed (s)neutrino n N to the entropy density s is given by where B r denotes the branching ratio into the right handed neutrino channel. The resulting lepton asymmetry is where ǫ is the lepton asymmetry produced per right handed neutrino decay.
Note that unlike thermal leptogenesis, there is no washout factor in non-thermal leptogenesis since lepton number violating 2-body scatterings mediated by right handed neutrinos are out of equilibrium as long as the lightest right handed neutrino mass M 1 ≫ T r [23]. More precisely, the washout factor is proportional to e −z where z = M 1 /T r [6], and can be neglected for z 10.
An alternative scenario [27] is the case where and M 2 < m χ /2. Since the decay width of the inflaton is proportional to M 2 i , the branching ratios to 2N 1 and 2N 2 are (M 1 /M 2 ) 2 and 1 − (M 1 /M 2 ) 2 respectively. Thus, provided ǫ 1 ǫ 2 , the contribution to the lepton asymmetry from N 1 is negligible.
From Eq. (9) (with permuted indices) and Eq. (10) (16) or, since the first term is negligible for hierarchical Dirac neutrino masses We can also write this as Eq. (11) with permuted indices and recover Eq. (12) with M 1 replaced by M 2 (see [24], section 2.1). This indicates that ǫ 2 can easily attain values ǫ 1 , so that the dominant contribution to the lepton asymmetry is from N 2 .
Qualitatively, Eq. (12) shows that lepton asymmetry sufficient to meet the observational constraint |n L /s| ≃ 2.4 × 10 −10 can be generated with reasonable values for the phases.
We conclude this section by summarizing the various constraints on κ and the symmetry breaking scale M. As noted in the previous section, for large κ (or M) the SUGRA contribution gives rise to n s values that exceed unity by an amount that is not favored by the data on smaller scales (n s ≤ 1 at k = 0.05 Mpc −1 [9]). This provides an upper bound on κ and M for the shifted and smooth hybrid inflation models [14]. For SUSY hybrid inflation with the renormalizable potential Eq. (1), the gravitino constraint (Eq. (5)) provides a more stringent upper bound.
For small values of κ, the SUGRA correction and the soft SUSY breaking (mass) 2 term become important. We find by numerical calculation that the primordial density perturbations are too small for κ 10 −6 for SUSY hybrid inflation and κ 10 −7 for the shifted model. Sufficient leptogenesis requires Eqs. (13,15), and these are satisfied

Leptogenesis and Atmospheric Neutrino Oscillations
Two-family numerical calculations for SUSY hybrid inflation models discussed here have been carried out previously in refs. [27,22,12,28]. Here we update and extend these calculations using recent measurements of neutrino oscillation parameters.
A comment is in order whether two-family calculations are physically relevant.
We consider the case where the χ → N 2 N 2 branch is dominant, and Eq. (17) approx-imates the lepton asymmetry in terms of two families only. Since the Dirac masses are assumed to be hierarchical, the µτ block is dominant. Furthermore, the gauge symmetries suggest a Dirac mixing matrix close to the CKM matrix V CKM , which is close to the unit matrix especially in the µτ sector. Under these conditions the neutrino mixing matrix U MNS is approximately obtained by rotating the charged lepton and neutral Dirac sectors only in the µτ sector with respect to the weak basis and diagonalizing the resulting light neutrino mass matrix [27].
Note that the mixing angle obtained this way can only be identified with the atmospheric neutrino mixing angle if the mixing angles θ 13 and θ 12 are both small. While the solar mixing angle at weak scale is not small [29], its RG evolution can lead to a small angle at the reheat temperature [30]. This occurs for a wide range of CP phases for large tan β (≃ m t /m b ∼ 50 for G ⊃ SU(4) c ) and degenerate neutrino masses of ≃ 0.1 eV [31]. For hierarchical neutrino masses radiative effects on the mixing are in general small [32,33]. The solar mixing in this case could be accounted for by non-diagonal Majorana masses of ∼ 10 −3 eV that can arise from higher dimensional operators [34].
Thus, we can ignore the first family only if we consider the special case of a small solar mixing angle at large energy scales. For this special case the lepton asymmetry and the atmospheric neutrino mixing angle can be calculated without assuming any particular ansatz for the Dirac and Majorana mass matrices.
The lepton asymmetry in this case is given by [27] n L s Here H u = 174 sin β GeV (≈ 174 GeV for large tan β), where β = H u / H d . The light neutrino mass matrix is given by the seesaw formula: where m D is the Dirac neutrino mass matrix and M ν c the right handed Majorana mass matrix. The atmospheric neutrino mixing angle θ 23 lies [27] in the range where ϕ is the rotation angle which diagonalizes the light neutrino mass matrix in the basis where the Dirac mass matrix is diagonal and θ D is the Dirac mixing angle.
In our analysis we will assume θ D ≈ 0 and so take ϕ = θ 23 . (From SU(4) c symmetry, θ D ≃ |V cb | ≃ 0.03 [35].) We take m D 3 compatible with G, i.e. m D 3 = m τ × tan β for G LR , and m D 3 = m t for SO(10) or G P S . These relations hold at M GUT , while the relevant values of the parameters are those at the leptogenesis scale. We estimate the Dirac masses by using the above relations as approximations, with the values for the quark and lepton masses at T r = 10 9 GeV given in [36].
The light neutrino masses are assumed to be either hierarchical with m ν2 = 8.5 × Using these Dirac and light neutrino masses, we have numerically calculated the range of κ, the symmetry breaking scale M and the reheat temperature T r consistent with the observed baryon asymmetry n B /s ≃ 8.7 × 10 −11 [9] and the near maximal atmospheric mixing sin 2 2θ 23 0.95 [39,37,38]. 5 For the allowed range of κ we also GeV (Fig. 5).

Smooth hybrid inflation with G P S
a. Hierarchical neutrinos: As in case 3a, we take m D 2 = 2 GeV to allow solutions. The baryogenesis and neutrino mixing constraints can be satisfied with T r ∼ 10 10 GeV, and the heavy Majorana masses are M 2 ∼ 10 11 GeV and M 3 ∼ 10 15 GeV (Fig. 7).
b. Degenerate neutrinos: Taking m D 2 = 2 GeV to allow solutions, the baryogenesis and neutrino mixing constraints can only be satisfied for a narrow range of masses, with the symmetry breaking scale M 10 16 GeV and T r 10 10 GeV.
Note that in our calculations we have assumed M 1 ≫ T r so that washout effects are negligible. Since M 2 /T r turns out to be in the range 10−100, this assumption conflicts with a strong hierarchy between M 1 and M 2 . Eqs. (17) and (18) have to be suitably modified for M 2 ∼ M 1 . However, the resulting lepton asymmetry does not change significantly, unless the right handed neutrinos are quasi degenerate ((M 2 − M 1 ) ≪ M 1 ). On the other hand, if we drop the assumption M 1 ≫ T r , the constraints to generate sufficient lepton asymmetry become more stringent [6,40].

Conclusion
We have reviewed non-thermal leptogenesis in SUSY hybrid inflation models. For the simplest SUSY hybrid inflation model, sufficient lepton asymmetry can be generated provided that the dimensionless coupling constant appearing in the superpotential We conclude that SUSY hybrid inflation models can satisfactorily meet the gravitino and baryogenesis constraints, consistent with the observed neutrino (mass) 2 differences and near maximal atmospheric neutrino mixing.