Sneutrino driven GUT Inflation in Supergravity

In this paper, we embed the model of flipped GUT sneutrino inflation -in a flipped SU(5) or SO(10) set up- developed by Ellis et al. in a supergravity framework. The GUT symmetry is broken by a waterfall which could happen at early or late stage of the inflationary period. The full field dynamics is thus studied in detail and these two main inflationary configurations are exposed, whose cosmological predictions are both in agreement with recent astrophysical measurements. The model has an interesting feature where the inflaton has natural decay channels to the MSSM particles allowed by the GUT gauge symmetry. Hence it can account for the reheating after the inflationary epoch.


Introduction
In the last years, the results published by the Planck collaboration about the Cosmic Microwave Background (CMB) [1] have lead to the speculation of a connection between the ideas of cosmological inflation and supersymmetric grand unification. Using the measured value for the amplitude of scalar perturbations in the CMB, A s = (2.19 ± 0.11) × 10 −9 , it is possible to estimate the energy density during the inflationary epoch as where r is the ratio of tensor to scalar perturbations. Therefore, for a value of r ∼ 0.1, perfectly compatible with Planck's observations, Eq. (1) shows the remarkable coincidence of the energy density during inflation V 1 4 and the unification scale predicted by Supersymmetric GUTs, M GU T ∼ 2 × 10 16 GeV. The issue of inflationary Grand Unification has been studied extensively in the past [2][3][4][5]. These models typically require an epoch of hybrid inflation [6][7][8], during which the inflaton field(s), as it rolls down the slope of its potential, destabilizes the GUT preserving minimum and thus breaks the symmetry. After dropping down to its new vacuum, the GUT Higgs field backlashes into the inflationay potential, causes the end of the inflationary phase.
Many of the SUSY GUTs in the literature, built to accommodate low-energy phenomena [9][10][11] cannot be consistently constructed within a hybrid inflationary framework. This is due to the known GUT monopole problem [12,13], which refers to the overproduction of magnetic monopoles during the GUT epoch, in tension with experimental searches, and having the unfortunate side effect of overclosing the universe [14]. This issue is then avoided if the GUT symmetry is broken way before the inflationary era so that these monopoles, along with other topological defects such as domain walls or cosmic strings, are diluted away by the rapid expansion of the universe.
However, hybrid inflation expects the unification symmetry to be broken towards the end of inflation, with potentially not enough e-folds left over to wash away the topological defects. Fortunately, as pointed out by 't Hooft [12], monopoles do not arise in systems with nonsemisimple symmetry groups, i.e. Lie-groups of the form G × U (1) X , where the charge of the abelian factor Q X enters in the linear combination of the electromagnetic charge where the charge Q i and parameters α i and β (i = 1, . . . , n) depend on the structure of G and the pattern of symmetry breaking. Therefore, we will take non-semisimple groups as the unification symmetry, in particular SU (5) × U (1) and SO(10) × U (1), similar to previous works [2][3][4], where the role of the inflaton is played by the right-handed sneutrino [4,[15][16][17][18]. We will also embed the model into a supergravity framework, with the addition of a shift symmetry which avoids the intrinsic η-problem [5,[19][20][21]. Although it is not easy to build a consistent hybrid inflation model within this framework, due mostly to the chaotic nature of the inflationary potential, there are a couple of scenarios in which it can successfully be constructed, and where the model predictions for the cosmological observables lie within 2σ of the experimental measurements.
This paper is structured as follows. In section 2 we describe in detail the model chosen, giving the details of the relevant field content in two flipped GUT scenarios, and part of the superpotential relevant for inflation. In section 3 we perform an analysis of the inflationary trajectories and the conditions required by the model to satisfy successful symmetry breaking, with a few comments on the predictions for the cosmological observables. In section 4 we will discuss reheating after the period of inflation and in the last section 5 we will conclude with some remarks and summarize the results.
2 The model 2.1 GUT structure Candidate groups of the form G ×U (1) X must preserve the chiral structure of the SM, i.e. the SM fermions must live in conjugate representations of the group. This condition restricts the landscape of groups to a handful of them, e.g. SU (5) ′ ×U (1) X or SO(10) ′ ×U (1) X [22]. These, so called flipped GUT groups, have in common a non-standard embedding of SM fermions, which follows from the requriment of Q X charge assignments so that the SM hypercharge is obtained as [22,23] where Q Y ′ is the charge associated with the first abelian factor of the broken U (1) Y ′ ×U (1) X , subalgebra of SU (5) ′ × U (1) X and, similarly, Q Z is the charge associated with the subalgebra has been thoroughly studied in the literature [2,11,[24][25][26][27][28], and it is one of the preferred candidates for an pseudo-unified group for it solves or avoids many of the everpresent problems in GUT model building. It predicts the unification of the gauge couplings α 2 and α 3 at M GU T ≳ 10 16 GeV, which allows for long-lived protons [3]; it provides a natural solution to the doublet-triplet splitting via the missing partner mechanism [2,26]; and it can easily be extended, via the addition of sterile neutrinos, to accommodate neutrino masses [28].
The field content of the flipped SU (5) group, following the steps of [4], can be sketched as follows : • The standard model (SM) matter content is contained in representations 10 F ,5 F and 1 F , whose respective U (1) X charges are 1, -3 and 5.
• The Brout-Englert-Higgs bosons acting the electroweak symmetry breaking are contained in 5 Hu and5 H d .
• The model is added a singlet 1 S necessary to provide the mixing 5 Hu5H d required for the electroweak symmetry breaking.
• The heavy scalars Σ andΣ, triggering the breaking of the flipped SU(5) to the standard model gauge group are contained in representations 10 H and 10 H .
• An additional superfield, in the conjugate representation of the 10 dimensional matter multiplet, 10 F with U (1) X charge opposite -1, is added to allow us the introduction of a shift symmetry in the Kähler potential, which will be described below.
With these field representations, and taking canonic hypercharge normalization through eq. (3), the generator Q Y ′ can be written as In addition to this field content, we add Z 2 matter parity so as to forbid undesirable RP-violating couplings [2]. The U (1) X and Z 2 charges of all the involved fields, can be seen in table 1. In this scenario the inflaton can be taken to be the singlet 1 S or the right handed sneutrino, N c (N c ), embedded in the representation 10 F (10 F ), both of which were covered in [4]. As was mentioned in the introduction, in this work we will focus on the latter case, because it presents a more interesting case due to the constraint nature of the problem.
The minimal flipped SO(10) model [22,23], though not as studied as the flipped SU (5) model, still presents a very appealing candidate as an unified model. In addition to the advantages mentioned above for the SU (5) case, which this model shares, an SO(10) ′ ×U (1) X model provides a natural link with ultraviolet completions, such as heterotic string theories, for it does not require high dimensional representations which could not be obtained via a manifold compatification of string theory [29,30].
The field content of this theory can be summarized as follows: • All the SM matter content is embedded in the representations 16 F , 10 F and 1 F , which also contain the two MSSM Higgs bosons, two coloured Higgs fields and an additional SM singlet.
• Typically, one needs two pairs of symmetry breaking fields, Σ,Σ and Σ ′ ,Σ ′ in the representations 16 H , 16 H and 16 ′ H , 16 ′ H , respectively. One pair would break the symmetry to the flipped SU (5) model and the other, containing the fields 10 H , 10 H from above, will continue the symmetry breaking all the way to the low energies.
• Again we find the need to add an additional 16 F representation, to realize the shift symmetry that will be described below.
The U (1) X charges of all this fields, as well as their Z 2 parities can be found in table 2 It is worth noting that the flipped SO(10) model provides the possibility of implementing a missing partner mechanism, analogous to the case of flipped SU (5), via non-renormalizable operators of the type [22,23] W ⊃ 16 As in SU (5) × U (1), inflation can be driven by two different components of the 16 F field, corresponding to the SU (5) singlet and right handed sneutrino cases from above, and again we will focus henceforth on the latter case.

Superpotential
For either of the flipped scenarios mentioned in section 2.1, one can construct a superpotential containing a piece relevant for inflation given by 1 where we have rewritten the complex fields as This superpotential is consistent with the Z 2 matter parity for both types of models, but can only be generated at the renormalizable level for the SU (5) ′ × U (1) X model. For the SO(10) ′ × U (1) X model one needs to add non-renormalizable operators like that of eq. (5), with 16 H ↔ 16 F , which reduces to (6) onceΣ ′ acquires a v.e.v. in the S direction, thus breaking the symmetry and recovering the flipped SU (5) model.
Using this set up, the inflaton fields will be contained among the field φ 1,2 real components, whereas the field S will play the role of stabilizer during inflation and h 1,2 are the waterfall fields. Here the term µ φ φ 1 φ 2 is important to give the inflaton its mass at the true vacuum.
It is to note that the inflaton fields, contrarily to many supergravity scenarios of large field inflation, carries here quantum numbers since it lives in multiplets of the unification gauge group. This situation has been studied in [5] where it was shown in particular that the shift symmetry (necessary for releasing slow roll polynomial inflation in supergravity) can be incorporated using conjugate representations of the gauge group in the following way 2 It is important to note that the shift symmetry enforces us to define another 10 dimensional fermionic multiplete which has U (1) X charge opposite to 10 F , in order to have a gauge invariant Kähler potential. In such a set up, part of the combinations φ 1 ±φ 2 will constitute the inflaton (multi) fields, while the other component will get Hubble scale masses, being hence stabilized during inflation. We hence write the kinetic part of the lagrangian The quartic term η S 4 is added to stabilize S at a Hubble-scale mass during inflation in order not guarantee that it does not perturb inflation. Such term can be generated for instance through loops of heavy fermions coupling exclusively to the S field [31].

Supergravity vacua
Before studying in detail the inflationary trajectory, let us mention that the true vacuum is in this scenario preserving supersymmetry. The latter is defined by As sketched above, the fields h i will be responsible for the breaking of the SU (5)×U (1) at the end of inflation. The scale of such breaking is expected to happen at the Grand Unification scale, therefore we will for now on impose that which will unambiguously fix the parameter λ h in what follows, the mass term M remaining a priori free. The breaking of the global U (1) symmetry will naturally generate a Goldstone mode which one can identify by performing the following field redefinitions, consistent with kinematic normalization and finally With these definitions, θ is exactly massless and is the Goldstone boson of the theory. This will be true both during and after inflation, and as described in [5] the field θ can be removed from the theory by a simple gauge transformation. Hence, in the vacuum the remaining fields, after elimination of the goldstone mode, get masses

Inflationary trajectory
We will now enter the description of the inflationary trajectory, ensuring that all directions, except the inflaton one are prevented to run away during the hubble scale inflation period. One also has to control that spectator fields acquire hubble scale masses during inflation, such that their dynamics does not interfere with the inflation slow rolling. As described above, the inflationary direction is triggered, due to the shift symmetry introduced, by the combination φ 1 − φ 1 ≡ α 2 + iβ 2 . At very large values of the latter, all the spectator fields S, h, α 1 , β 1 , as well as the heavy scalar ρ (= H) are stabilized closed to the origin, while the inflaton pair of fields (α 2 , β 2 ) rolls down the potential. At some critical value of the inflaton, the field ρ becomes tachyonic at the origin and a waterfall happens, ending in the SUSY vacuum described above, and spontaneous breaking of the unification gauge group occurs.
At early stages of inflation -far before the waterfall happened -the vacuum expectation values (vev) of the spectator fields are given by ⟨σ⟩ = ⟨α 1 ⟩ = ⟨β 1 ⟩ = ⟨h r ⟩ = ⟨h i ⟩ = 0 (15) and where we expanded for large values of the inflaton fields (α 2 , β 2 ). The masses of these fields are then at the Hubble scale, where H 2 is given by the (early)inflationary potential Note that we expanded here for large values of the inflaton pair, and large parameter η 3 . Hence, for reasonable values of η ≳ 1 12, all the spectator fields get masses higher than the Hubble scale, ensuring that the slow rolling of the inflaton is not perturbed by light fields dynamics at early stages.

Waterfall
Looking closer at the mass of ρ, assuming for simplicity that µ φ ≪ λ φ , M , and noticing and it appears that, while the inflaton fields decrease, the latter can reach negative values at a critical value Reaching this point, a waterfall is expected to take place. Asking that the waterfall happens at fields values of order α 2 2 + β 2 2 c ∼ 1 would hence imply that

Inflationary scenarios
In most of Hybrid inflation models, it is usually assumed that reaching the critical point is equivalent to the ending of the inflationary period. Though it was shown in [8,32] that this assumption is far from being exact. Indeed, the transverse momentum of the inflaton field is highly diluted during the inflationary phase and it can reach the critical point with almost zero transverse impulsion. Moreover, quantum excitations of the waterfall fields can make it backreact and contribute highly to the energy density, releasing most of the required inflationary e-folds in late stages of inflation. In a nutshell, inflation can actually continue (and even be entirely released in some cases) during the waterfall period itself. In our setup the critical point can be tuned to be at very low values of the inflaton or very large ones. In principle, for an intermediate case, the inflaton could begin falling from hightransplanckian -values, far beyond the critical point, starting inflating the universe. Inflation could then be continued during or after the waterfall phase, the inflaton finally running down along the φ-dependant vev of the ρ field.
Since the waterfall phase is rather technical to describe in an analytic way, we better consider two extreme scenarios in this paper: In this case there is not enough excursion between the critical point and the true vacuum in order to release a significant number of e-folds during the waterfall phase. Spontaneous breaking of the grand unification scenario takes place at the end of inflation. In this case the effective potential during the inflation is given by It turns out that the potential has a rotational symmetry in the plane of α 2 and β 2 and in the polar representation it depends only on the radial part, which we denote by Φ = α 2 2 + β 2 2 . Hence the potential reduces to the form 2. High critical value: φ c ≳ 20M p In this case the horizon crossing happens far after the waterfall. The field ρ stabilizes to its field dependant vev, and the scalar potential is hence modified, but still almost quadratic. Inflation thus happens in the valley of non vanishing ρ. Breaking of the grand unification symmetry group is then very high at the beginning of inflation and fall to the GUT scale when inflation ends. In this case there is back-reaction to the above inflationary potential and hence the potential is given by Both scenarios are depicted in Fig. 1 where the inflationary trajectories are drawn for φ c = 1M p and 30M p .

Observables
In both of the cases described above the quartic contribution to the potential is triggered by a term scaling like Φ 4 (λ 2 φ − 3µ 2 φ ). The only way inflation observables can fit with experimental constraints is to kill such a steep contribution to the scalar potential, by requiring that In both scenarios, we perform a scan over the parameters µ φ and < 1. Moreover we fix η to be of order 4 10M p , while other parameters are fixed by the choice of φ c (Eq. (21)) and requiring a GUT scale symmetry breaking (say M GU T = 10 −2 M p in Eq. (10)). In the case of low critical point we fix φ c = φ end = 1M p whereas in the case of large critical point we impose φ c = 30M p . Such choices are made so that one can safely sit either in one or the other of the two extreme cases described above. Observables are depicted in Fig. 2. The value of is indicated in color, in order to quantify the fine-tuning required to produce acceptable observables. It turns out that the scenario where the GUT symmetry is spontaneously broken before the horizon crossing is the less fine tuned one 5 . From this point of view, the choice of the flipped SU (5) case as a unification gauge group becomes less necessary since magnetic monopoles generated by such phase transition will naturally be diluted by the 60 e-folds of inflation coming after all. However, this choice of gauge group has interesting consequences for reheating, since it provides natural couplings of the inflation to the MSSM particles, as described below.

Reheating
The reheating after inflation may be one of the most interesting features of the above scenario. In our model the right handed sneutrino is the inflaton and hence it can decay to MSSM particles via natural decay channels allowed by the gauge symmetry. Here the situation is different from [5], where it was necessary to add extra components to allow the inflaton to decay.
In this context, successful inflation via R-sneutrino would imply that its mass is of order O(10 12−13 ) GeV which is the same as the right handed neutrino, since SUSY is preserved at the end of inflation. This is consistent with the seesaw mechanism, which hints at a relation between the neutrino mass scale to the inflaton mass scale. Now the renormalizable superpotential that results in interactions between the inflaton (Ñ c ) and the MSSM particles is given by where α, β, ... = 1, 2, 3, 4, 5 and sum over flavour indices is implied. Here it is clear that the up-quark sector has the same Yukawa coupling as the neutrino and this can be understood in the seesaw framework. The right handed neutrino superfield exists in the components (10 F ) ab , where a, b = 4, 5. Accordingly, the inflaton can decay to the MSSM lighter fermions only via the following interaction Lagrangian which can be extracted from the first term in eq. 27. The reheating temperature T R is defined by [34,35] T R ≈ (8π) 1 4 7 where Γ is the total decay width of the inflaton field and M p is the reduced Planck mass. In this respect, the decay of the sneutrino to massless fermions is given by The cosmological constraints on the reheating temperature [36][37][38][39][40], imply that T R ≲ 10 9 GeV. Figure 3 shows the relation between the reheating temperature and the Yukawa coupling, for different values of the inflaton masses. For typical sneutrino masses of order 10 12−13 GeV, we require that the Yukawa couplings Y u ∼ 10 −5 which is typically of order the up quark Yukawa 6 . From this, we can conclude that the inflaton should correspond to sneutrino from the first generation in order to be consistent with the constraints on the reheating temperature. Other generations of sneutrinos, with larger Yukawa couplings, would potentially contribute significantly to the reheating temperature, driving it to very large values. In order to avoid this, we could assume the existence of a flavour violating sector at high energies that would stabilize the fields of the heavy generations prior to inflation, thus not affecting the reheating temperature. Specific details about this fall beyond the scope of this work.

Conclusions
In this paper we have proposed a way to embed in supergravity a full set up of chaotic inflation within the framework of a GUT flipped group of the sort G × U (1) X . The η-problem is evaded using the conjugate shift symmetry developed in [5] while the inflationary material is released in a similar fashion than in [4]. Inflation is driven by the sneutrino whereas the spontaneous breaking of the GUT symmetry is triggered by a U (1) X charged heavy Higgs. The dynamics of all the fields is treated in full detail, including that of spectator fields, as well as the backreaction of the waterfall field on the inflationary potential.
Two scenarios of inflation are detailed: either (i) the critical point when the waterfall takes place is located at very low values of the inflaton and inflation is mostly released in the region where the waterfall field is stuck at the origin, or (ii) the waterfall happens at early stages of inflation, and the last 60 e-folds of inflation take place after the waterfall has happened. In the latter case, the inflaton direction goes along the minimum of the waterfall field.
Cosmological observables are computed in both scenarios and can both be accommodated to lay in the 2-σ region of the last Planck measurements. However, we note that some fine tuning is necessary on the parameters in Eq. (26) in order to have consistency with the observables, particularly in scenario (i) above. In the second case (ii) the breaking of the GUT symmetry takes place much earlier than inflation itself. Although this seems to suggest that the flipped structure of the unification is not needed anymore, we have seen that it remains actually crucial for releasing a satisfactory reheating period.
The reheating temperature for these scenarios is computed as well and it turns out to be rather constraining. Indeed Yukawa couplings of order unity would lead to a unacceptably high reheating temperature and overclose the universe. The required value for the Yukawa coupling, Y u ∼ 10 −5 , corresponds to that of the up quark, leading to the conclusion that the inflaton belongs to the first generation and, assumming the decoupling of the other two generations, could satisfy the constraints of reheating. This further motivates the choice of a flipped SU (5)×U (1) unification group, where the right-handed sneutrino has direct couplings to the rest of the MSSM particles.