Non-Minimal Quartic Inflation in Supersymmetric SO(10)

We describe how quartic ($\lambda \phi^4$) inflation with non-minimal coupling to gravity is realized in realistic supersymmetric $SO(10)$ models. In a well-motivated example the $16-\overline{16}$ Higgs multiplets, which break $SO(10)$ to $SU(5)$ and yield masses for the right-handed neutrinos, provide the inflaton field $\phi$. Thus, leptogenesis is a natural outcome in this class of $SO(10)$ models. Moreover, the adjoint (45-plet) Higgs also acquires a GUT scale value during inflation so that the monopole problem is evaded. The scalar spectral index $n_s$ is in good agreement with the observations and $r$, the tensor to scalar ratio, is predicted for realistic values of GUT parameters to be of order $10^{-3}-10^{-2}$.

By incorporating a single right-handed neutrino per generation to cancel new anomalies from gauging the accidental global U(1) B−L symmetry of the Standard Model (SM), both SU (4) × SU (2) L × SU (2) R [1] and SO(10) [2] provide particularly compelling examples of unifying the strong and electroweak forces. A non-supersymmetric model of SO (10) inflation [3], based on an earlier SU (5) model [4], was proposed a longtime ago. In this class of SO(10) inflation models, driven by a gauge singlet field with minimal coupling to gravity and utilizing the Coleman-Weinberg potential [5], the scalar to tensor ratio r, a canonical measure of gravity waves generated during inflation, is estimated to be 0.02, for n s = 0.96 − 0.97 [6]. Depending on the SO(10) symmetry breaking pattern, an observable number density of intermediate mass magnetic monopoles may be present in our galaxy [7].
In this letter we propose to implement primordial inflation in realistic supersymmetric SO(10) models [8]. We do this with a supergravity generalization of non-minimal λφ 4 inflation [9]. Recall that λφ 4 inflation with a minimal coupling to gravity predicts an r value close to 0.25 − 0.3, depending on the number of e-foldings (N 0 = 60 − 50). This prediction for r lies well outside the 2-σ range allowed by Planck [10] and WMAP 9 [11]. In contrast, λφ 4 inflation with a suitable non-minimal coupling to gravity is in good agreement with the data regarding the key parameters n s and r. The quantity r, in particular, can be as low as 0.003 or so, for n s = 0.96 − 0.97. The discussion closely follows a previous model [12] based on supersymmetric SU (5).
In order to retain perturbative unification of the MSSM gauge couplings in supersymmetric SO(10) we prefer to work with lower dimensional SO(10) representations. We employ 16 − 16 Higgs to break SO(10) to SU (5) while keeping supersymmetry unbroken. The 16 vacuum expectation value (VEV) also provides large masses ( 10 14 GeV), via higher dimensional operators, to the right-handed neutrinos. In addition, the adjoint 45-plet, in conjunction either with a 54-plet or using higher dimensional operators, is employed to complete the breaking of SO(10) to the MSSM gauge symmetry. Finally, following [13], we can employ two Higgs 10-plets to implement electroweak symmetry breaking and accommodate the charged fermion masses and mixings as well as neutrino oscillation data. This summarizes the basic structure of a realistic supersymmetric SO(10) model.
Recall that a non-minimal λφ 4 inflation scenario is defined by the following action in the Jordan frame: where we have set to unity the reduced Planck mass, M P = 2.44 × 10 18 GeV. In the limit ξ → 0 + the non-minimal gravitational coupling term ξϕ 2 R vanishes and we approach minimal λϕ 4 chaotic inflation. In the Einstein frame with a canonical gravity sector, we can describe the action with a new inflaton field (σ) which has a canonical kinetic term. The relation between σ and ϕ is given by The action in the Einstein frame is then given by with The slow-roll parameters in terms of the original scalar field (ϕ) are expressed as where a prime denotes a derivative with respect to ϕ. The amplitude of the curvature perturbation ∆ R is given by with ∆ 2 R = 2.195 × 10 −9 from the Planck measurement [10] with the pivot scale chosen at k 0 = 0.002 Mpc −1 . The number of e-folds is given by where ϕ 0 is the inflaton value at horizon exit of the scale corresponding to k 0 , and ϕ e is the inflaton value at the end of inflation, which is defined by max[ (ϕ e ), |η(ϕ e )|] = 1. The value of N 0 depends logarithmically on the energy scale during inflation as well as on the reheating temperature, and is typically taken to be N 0 = 50 − 60.
The slow-roll approximation is valid as long as the conditions 1, |η| 1 and ζ 1 hold. In this case, the scalar spectral index n s , the tensor-to-scalar ratio r, and the running of the spectral index α = dns d ln k , are given by Here the inflationary predictions are evaluated at ϕ = ϕ 0 . With the constraint ∆ 2 R = 2.215 × 10 −9 , once N 0 is fixed, the inflationary predictions as well as the quartic coupling λ are determined as a function of ξ. In Table 1 we list the numerical results for selected values of ξ. The inflationary predictions are consistent with the Planck results (n s = 0.9655±0.0062, r 0.07 and α = 0.0057±0.0071 at 68% C.L.) for ξ 0.01. As ξ increases, the inflationary predictions approach n s 0.968, r 0.00296 and α −5.23 × 10 −4 , while the quartic coupling is monotonically increasing.
Next we discuss how this scenario is implemented in a realistic supersymmetric SO(10) model. The relevant superpotential terms for inflation are given by where z,z denote the 16 − 16 fields, A represents the 45-plet, and the last term has been included so that W = 0 at the desired supersymmetric minimum with SO(10) broken to the SM gauge group. To implement non-minimal λφ 4 inflation an appropriate Kähler potential, following [14], is given by where the parameter coefficients γ and γ A are assumed to be real and positive constants.
The inflaton trajectory is parametrized by the D-flat direction where the field VEVs ϕ and a break SO(10) → SU (5) and SO(10) → SU (3) × SU (2) L × SU (2) R × U (1) B−L , respectively. Thus, the initial theory reduces to a model with two real scalars ϕ and a, and the Jordan frame action is given by Here, the Kähler potential is expressed as where ξ = (γ − 1)/6 and ξ A = (γ A − 1)/6. The scalar potential V J in the Jordan frame is calculated as [15] where M −1 is the inverse of the matrix To compute the potential (14) we write W in terms of ϕ and a Then we have The potential minimum where SO(10) is broken to the SM gauge group lies at The dynamics of inflation is encoded in the scalar potential V E = V J /Φ 2 in the Einstein frame. In Fig. 1 (17)). In this case, the scalar potential along the trajectory is greatly simplified as Here we have used ξ > ξ A and ϕ 2 M 2 G for ϕ 0.1. This potential is exactly the same as Eq. (4) with the identification λ = y 2 . Since the inflaton value at the end of inflation is found to be ϕ e = 0.677 for ξ = 252 (see Table 1), the displacement of a during inflation is small and hence our inflation scenario in the context of supergravity is well approximated by λφ 4 inflation with non-minimal gravitational coupling. Table 1 shows the inflationary predictions as n s 0.968, r 0.00297 and α −5.23 × 10 −4 , which are consistent with the Planck results. Along the inflaton trajectory SO(10) is broken to the SM, and hence the primordial monopoles are inflated away.
In our analysis, we have set y = 0.01. We find that the shape of the inflaton trajectory shown in the right panel of Fig. 1 is almost unchanged for a variety of choices of the model parameters, y, ξ and ξ A . 4 In order to identify ϕ with the inflaton, the condition ϕ e 0.1 is crucial. According to the results listed in Table 1, this means ξ 100, or equivalently λ = y 2 10 −4 . Following the SO(10) symmetry breaking to the SM, the components 10 + 10 of SU (5) from 16, 16 and 45 fields, have masses of O(yM G ) 10 14 GeV. There is some mass splitting of the same order within these multiplets but gauge coupling unification is essentially preserved. With the intermediate scale yM G of order 10 11 − 10 14 GeV, the tensor to scalar ratio r varies between 0.01 to 0.003 which should be testable in the foreseeable future.
The VEV of the 16 Higgs field not only breaks the SO(10) symmetry but also generates Majorana masses for the right-handed neutrinos through higher dimensional operators of the form, where 16 i (i = 1, 2, 3) denotes the matter field, and the coefficient c i is taken to be flavordiagonal. Associated with the SO(10) symmetry breaking, the right-handed neutrinos acquire masses Another important role of the higher dimensional operators is that after inflation the inflaton ϕ decays into right-handed neutrinos through them to reheat the Universe. We estimate the reheating temperature as where M 3 is the heaviest right-handed neutrino mass, compatible with kinematics, and Γ ϕ 1 16π is the total decay width of the inflaton (assuming c 3 < 1/2). Since T RH < M 3 , we expect that the reheating occurs after scatterings among the produced heavy neutrinos and their decays, and hence the actual reheating temperature is lower than the value estimated above. In order to avoid the cosmological gravitino problem [16], we consider the upper bound on the reheating temperature of T RH < 10 6 − 10 9 GeV with the gravitino mass in the range of 100 GeV mG 10 TeV [17], and take c 3 small enough to satisfy this upper bound. Depending on the value of reheating temperature and the right-handed neutrino mass spectrum we can consider either thermal [18] or non-thermal [3] letogenesis scenarios.
In summary, we have shown that λφ 4 inflation with non-minimal coupling to gravity can be realized in the framework of realistic supersymmetric SO(10) models. An attractive feature is the utilization as inflaton of a field already present for particle physics reasons. In the example provided inflation is driven by the field that breaks SO(10) to SU (5) and provides masses to the right-handed neutrinos. Depending on additional details, thermal or non-thermal leptogenesis is a natural outcome. The field associated with monopole production is non-zero during inflation and so these topological defects are inflated away. With a scalar spectral index in the vicinity of 0.96 − 0.97 the tensor to scalar ratio r is estimated to be of order 10 −3 − 10 −2 . Significantly larger values of r require appreciably smaller values of the quartic coupling.