Hilltop Supernatural Inflation and SUSY Unified Models

In this paper, we consider high scale (100 TeV) supersymmetry (SUSY) breaking and realize the idea of hilltop supernatural inflation in concrete particle physics models based on flipped-SU(5) and Pati-Salam models in the framework of supersymmetric grand unified theories (SUSY GUTs). The inflaton can be a flat direction including right-handed sneutrino and the waterfall field is a GUT Higgs. The spectral index is $n_s=0.96$ which fits very well with recent data by PLANCK satellite. There is no both thermal and non-thermal gravitino problems. Non-thermal leptogenesis can be resulted from the decay of right-handed sneutrino which plays (part of) the role of inflaton.

n s = 0.96 with natural scales coming from SUSY breaking. It was later shown in [9] (see also [10]) that this hilltop supernatural inflation can evade both thermal and non-thermal gravitino problems.
In this paper, we materialize this model in some solid SUSY GUT models including flipped-SU (5) and Pati-Salam model 1 .
This paper is organized as follows: In section II, we briefly review the model of hilltop supernatural inflation. In section III, we present the potential. In section IV, flipped SU(5) model is considered.
In section V, Pati-Salam model is considered. In section VI, we describe the topological defects after the inflation in our models. We discuss issues after inflation including gravitino problem and leptogenesis in section VII and section VIII is our conclusion. In Appendix A, we present some calculation detail for hilltop inflation models and in Appendix B, we review flipped SU(5) and Pati-Salam models.

II. HILLTOP SUPERNATURAL INFLATION
The potential for a hybrid inflation is given by where ψ is the inflaton field and φ is the waterfall field. The effective mass of the waterfall field (at φ = 0) is During inflation, the field value of ψ gives a large positive mass to φ therefore it is trapped to φ = 0 and the potential during inflation is of the form where V 0 = κ 2 Λ 4 . The end of inflation is determined by m 2 φ = 0 when the waterfall field starts to become tachyonic which implies For original supernatural inflation, because the potential in Eq. (3) is concave upward, a blue spectral index n s > 1 is obtained in the simplest form of this model. We can get a red spectral index if we extend the model into a hilltop supernatural inflation by considering where η 0 ≡ m 2 ψ M 2 P /V 0 and M P = 2.4 × 10 18 GeV is the reduced Planck mass. Given the potential, we can solve for the spectrum, spectral index, and the field value. We put the detailed calculation in the Appendix A. The potential becomes concave downward (for λ > 0) when cosmological scales leave the horizon at N = 60 and as can be seen in Fig 1 a spectral index n s = 0.96 can be obtained by η 0 = 0.02 (for λ = 4.4 × 10 −14 ), 0.03 (for λ = 2.2 × 10 −13 ).
The quartic term in the scalar potential with a tiny coupling constant can be obtained by considering a non-renormalizable term in the superpotential 2 : where a is a dimensionless coupling constant. This makes the quartic term in the scalar potential during inflation 3  In this paper, we address to construct the hilltop supernatural hybrid inflation in unified models.
In the models, V 0 ≡ κ 2 Λ 4 is a parameter in the GUT potential, which is not necessarily related to the SUSY breaking. Let us describe model-independent constraints and features of the type-III hilltop inflation.
1. As can be seen in Fig. 3 and 4, in this model ψ(N = 60) ∼ ψ end ∼ 10 −7 M P is obtained. This also implies a small tensor-to-scalar ratio and primordial gravity waves is unobservable.
3. In order to make inflation ends promptly once the waterfall field becomes tachyonic, we require |m 2 φ | ≫ H 2 when ψ approaches to the origin. By using Eqs. (1) and (2), we have |m 2 φ | ∼ κ 2 Λ 2 2 We consider ψ field as a flat-direction, therefore no renormalizable term are relevant here. 3 There is also a positive F -term ∼ ψ 6 but we can easily check that it is negligible due to the smallness of the field value ψ.
and H 2 ∼ V 0 /M 2 P = κ 2 Λ 4 /M 2 P . Therefore the condition |m 2 φ | ≫ H 2 implies Λ 2 ≪ M 2 P . As we will see in the following sections, this condition is automatically satisfied.

III. THE POTENTIAL
In this section, we present the potential form which will be used in subsequent sections for concrete particle physics models in the framework of SUSY GUT.
The scalar potential in supergravity is given by where The gravitino mass is Using the condition of elimination of cosmological constant , we usually express the gravitino mass as We consider the following superpotential.
The hilltop potential (for φ = 0) is where the Kähler potential for the matter field is assumed to be canonical, and F X is an F -term of SUSY breaking superfield X. We assume ψ ′ = 0 which is the case if it is heavy during inflation. As we will see, for example, the field N in Eq. (47) can play the role of ψ ′ . The term V 0 is from W (φ) which will be addressed in Eq. (15). If the condition W 0 ≫ aψ 4 /M P is satisfied 4 , the potential can be "hilltop", and we obtain where A = m 3/2 e K/2M 2 P (1 + W X K X /W 0 ) ∼ m 3/2 . One can show that ψ 6 term (and higher order terms) can be negligible around the hilltop of the potential. We note that the potential can be written as where θ is a phase of aAψ 4 and λ = 2|aA|/M P . The hilltop configuration can be obtained if the inflation starts at θ = π.
The term V 0 can be obtained from the superpotential: The key feature of this potential is that there is only linear term of a singlet field S, and the mass term of S and cubic term are forbidden (or suppressed by an approximate symmetry) 5 . This can be achieved by R-symmetry.
Combining Eqs. (11), (13) and (15) and setting S = ψ ′ = 0, the whole potential in our model is given by This can be compared with Eq. (1) (by rotating the phase to the real component of the complex scalar fields) except the second term which is introduced to make the potential into a hilltop form by choosing a negative a. During inflation, the large expectation value of the inflaton field ψ would force φ = 0 through the third term in the potential therefore the fourth term becomes V 0 = κ 2 Λ 4 which provides the vacuum energy to drive inflation.
We note that the linear term of the singlet field S can be always generated due to the SUSY breaking [14]. In fact, the following term in the Kähler potential can be always generate the linear term where X is a SUSY breaking superfield, whose F -term is non-zero. Because the gravitino mass is The scale κΛ 2 can be also obtained by strong SU(N) dynamics (which is different form color SU(3) c ). The condensation of "squark" fields of SU(N), Q Q , can induce the dynamical scale Λ via SQQ interaction term (In this example of model, thus, Λ has non-trivial R-charge), e.g.
where M, B andB are "meson" and "(anti)baryon" condensations andμ is a Lagrange multiplier [15]. In this building of waterfall potential, the condensation Q Q generates V 1/2 0 =Λ = κΛ 2 , which can be much less than the Planck scale, and it is a free parameter in the model. Smallness of the coupling constant κ induces the hierarchy between V 1/4 0 and the VEV of the waterfall field (namely, unification scale). We do not address the detail of the mechanism in this paper, and we mention the feature of this potential.
It is well-known that fine-tuning between the SUSY breaking order parameter F X and superpotential W is needed to eliminate the cosmological constant In order to realize the proper SUSY breaking scale, we need In naive GUT superpotential W GUT , one obtains W 1/3 GUT ≡ M G ∼ 10 16 GeV. We need 3|Ŵ + M 3 G | 2 /M 2 P = |F | 2 (Ŵ is a non-GUT superpotential) for vanishing the cosmological constant. Therefore, to realized the proper SUSY breaking scale and vanishing the cosmological constant, two-step cancellation is needed (among three quantities): unless no-scale supergravity is considered. If the waterfall potential V (φ) is employed to break GUT symmetry, the first cancellation in the superpotential is "automatic" via the F -flatness condition.
(Surely, we still need a fine-tune between SUSY breaking and total W for vanishing cosmological constant). The waterfall superpotential requires only one cancellation even if a field acquires a VEV ∼ 10 16 GeV. This is one of the important conceptual merits of this scenario in the view of GUT model building. The size of W is related to the R-symmetry breaking scale, which should related to the SUSY breaking order parameters. In the supernatural inflation models, the size of V 1/4 0 can be also related to the scale naturally.
Another implication of the waterfall potential comes from a symmetry of the potential. The symmetry can be utilized to suppress dangerous proton decay operators. Due to the symmetry, some of the fields remains light (∼ κΛ) in the multiplet to break GUT symmetry. Contrary to the usual GUT superpotential, cubic terms of the field whose VEV breaks GUT symmetry is absent. As a consequence, there can be an accidental global symmetry (which may be softly broken), and the fields can be light to be TeV scale (or SUSY breaking scale), which can have an phenomenological implication.

IV. UNIFIED MODEL BUILDING
In the potential of the hybrid inflation, a large VEV of the waterfall field is suggested. The large VEV is available to break a unified symmetry. As we have mentioned in the previous section, the waterfall potential should have a symmetry which can be adopted to explain a phenomenological issue in the unified models. To explain these features, we consider the unified model, in which the unified gauge symmetry is directly broken down to SM by a VEV of the waterfall field, (1). flipped-SU(5) model [16,17], whose gauge symmetry is SU(5) × U(1) X , (2). Pati-Salam model [18], whose gauge symmetry is SU(4) c × SU(2) L × SU(2) R . The brief introductions of the models are given in Appendix B.

A. Waterfall potential
In the flipped-SU(5) model, the gauge symmetry can be broken by under the SU(5) × U(1) X symmetry. The 10 representation under SU (5) is two-rank anti-symmetric tensor. The VEVs of T 45 andT 45 (those are singlet components under SM gauge group) can directly break SU(5)×U(1) X down to SM. The symmetry breaking can occur via the waterfall superpotential: As many people concern, GUT models have doublet-triplet splitting problem. In the flipped-SU(5) model, the doublet-triplet splitting can be realized simply. The Higgs multiplets which include MSSM Higgs doublets (H u , H d ) are: The multiplets also include colored-triplets H C ,H C . Then, the following superpotential is allowed.
The VEV of T 45 andT 45 (= T ) to break the GUT symmetry makes the colored triplet heavy because T multiplet has a colored-triplet Higgs component T C (so-called missing partner). The Higgs doublet, on the other hand, does not acquire masses from T . The Higgs mass term is where M T and M H are mass parameters of T and H multiplets. The doublet Higgs mass (usually called Higgsino mass µ) should be small, and therefore M H should be small, which can originate from the accidental discrete symmetry (by assuming R-symmetry 6 ) to construct the waterfall potential in GUT.
The interesting feature of this mechanism is that the proton decay via the exchange of colored Higgs fields (dimension 5 operator) can be suppressed. Indeed, we assume that the matter fields does not couple to T , but couple to H via Yukawa interaction (neglecting to show coupling constants and generation indices explicitly) by a symmetry to construct the waterfall potential: Then, integrating out the heavy colored Higgs field, we obtain and the operator is suppressed by a factor M T / T : It is interesting to note that M T has to be much smaller than the GUT scale ∼ T = Λ in our inflation framework. In fact, even if we consider the correction from SUSY breaking sector, the mass parameter M T is up to the SUSY breaking order parameter, (m SUSY M P ) 1/2 . This is responsible for a symmetry to obtain the waterfall potential for the hybrid inflation. This means our inflation model is compatible with proton decay suppressing. The inflation model in the flipped-SU(5) model with the waterfall potential is also constructed in Ref. [11]. In their model, the singlet field is the inflaton. In this paper, we construct the quartic hilltop potential term in terms of the matter fields.

B. Hilltop potential
The hilltop potential can be constructed in terms of the matter superfields, and the scalar partners of the quarks and leptons can cause the inflation. Let us describe the hilltop potential in the flipped-SU(5) model.
The matter content of the flipped-SU(5) model (See Appendix B for more detail) is 6 The terms (Mass of HH, TT , HHTT , etc) which can spoil the DT splitting can be forbidden by R-symmetry. The concrete R symmetry is written in [11].
where i is a generation index. The Yukawa couplings are: The Dirac neutrino Yukawa coupling is unified with the up-type quark Yukawa coupling Y u . Because the representation of the fields T to break the SU(5) × U(1) X symmetry down to SM are same as the matter filed 10 i , one has to adopt a discrete symmetry (or R-symmetry) to distinguish them. The symmetry can be also useful to obtain the waterfall potential. Cubic terms of the matter representations are not invariant under the gauge symmetry. In the usual SU(5) GUT, λ ijk 10 i5j5k term is allowed. This term is not singlet under U(1) X in the flipped-SU(5) model.
In usual SU (5) GUT, since5 andH have same quantum charges, and they can be replace. However, in the flipped-SU(5) model, since they have different U(1) X charges, the replacement is forbidden. We note that the quartic Higgs term HHHH is allowed. It can however destabilize the electroweak symmetry breaking (which is the same circumstances as in MSSM) due to a diagram with quadratic divergence, and both HH and HHHH have to be suppressed by a symmetry The auxiliary field in the vector multiplet (so-called D-term) generates the D-term potential. The Kähler potential of the matter superfields are given as Then, the D-term for SU(5) and U(1) X are obtained as One can find that D-flatness (vanishing D-term potential) condition is satisfied, for example, for where subscripts denote the flavor indices and the italic superscripts denote the gauge group indices.
We stress that the F -term potential is lifted-up if the generation indices are given as their mass eigenstates: The up-quark Yukawa coupling is about 10 −5 , and the coupling of quartic term is about 10 −10 , which is not negligible. The F -term potential always provides a concave upward piece, which is not suitable to obtain n s < 1. Without loss of generality, using the unitary rotation in the generation space, one can take a basis where where x stands for any non-zero values. Under this generation configuration, both F -and D-flatness can be satisfied. This is because there are three generations. The components correspond to the SM component fields, We are using the conventional flavor notation for simplicity, but these are not given as mass eigenstates. Considering a non-renormalizable superpotential term we obtain the hilltop potential term (up to a phase) One can also find that the D-flatness condition is satisfied for which corresponds to and ψ 4 = ν c d c s c u c . We note that the D-term vanishes if (D 5 ) b a ∝ δ b a . This is because the group generator is traceless in simple groups. Through this superpotential term, baryon number B and lepton number L can be generated by Affleck-Dine (AD) mechanism, when the inflaton oscillates.
However, B − L is not generated.
Other example of the D-flat direction is which are in the SM component fields, and ψ 4 = ee c ν µ ν c e . In this direction, both baryon and lepton numbers are not generated by AD mechanism.
The g coupling (φψψ ′ term in the superpotential) is needed to obtain the hybrid potential. One can consider a variety of models for this coupling. We will present examples to construct a model 7 .
In the flipped-SU(5) model, the right-handed neutrino is not a gauge singlet. To acquire the righthanded neutrino Majorana mass, the GUT symmetry breaking VEV is available. For example, one can consider the following superpotential to obtain the right-handed neutrino Majorana mass: where N is a gauge singlet. Then, the right-handed neutrino Majorana mass is obtained as More precisely, the seesaw mechanism to obtain the sub eV active neutrino mass becomes so-called double seesaw. Naively, M N is of the order of GUT scale (or Planck scale), and the g coupling should be much less than O(1), depending on the generation of 10 we choose. For example, if we choose M N to be the GUT scale (VEV of T ), suitable size of g coupling is about 0.01 to obtain the right-handed Majorana mass to be 10 12 GeV. Choosing the D-flat direction with the right-handed neutrino components (in 10): e.g., or the superpotential term 10 2 10 1 10 25 or 10 25152 1 1 can induce the hilltop potential. From the Dflatness, generations of 10 12 and 10 23 have to be different. This is same for the generations of5 4 and5 5 .
We note that the O(1) size of the coupling a (in the Planck mass unit) is favored for the quartic hilltop term to obtain n s = 0.96. The 10 ·5 ·5 · 1 and 10 · 10 · 10 ·5 terms contain d c u c c c e c and qqqℓ terms, respectively. The O(1) size of this term can be problematic to cause a rapid proton decay depending on the flavor configuration. To satisfy the F -flatness, the flavor configuration has to be chosen (as described), and the required hilltop term may contain the 1st and 2nd generation much. Then, it is dangerous even if the sfermion masses are 100 TeV, especially for the operator 10 · 10 · 10 ·5, which contains left-handed proton decay operator qqqℓ. The gaugino dressing for the proton decay operators can suppress the proton decay amplitude by a factor m gaugino /m 2 q . The light gaugino scenario is favored to suppress the proton decay via the hilltop term. We note that for the case of 10 ·5 ·5 · 1 ⊃ ν c ℓℓe c + d c u c u c e c , one can choose 1 i = e c i to be the 3rd generation (right-handed tau lepton in the mass eigenstate), but 10 and5 are 1st and 2nd generations. (In order not to lift the F -term potential from the Yukawa coupling, 10 and5 are not given in the mass eigenstates for up-type quarks. One can choose the F -flat flavor configuration for 10 and5, without loss of generality). Then, the right-handed operator d c u c u c e c is not dangerous because tau lepton is heavier than proton unless a large flavor changing current is affected by SUSY breaking slepton masses.
One can adopt 50 + 50 representations as a waterfall φ field, whose VEV gives a right-handed neutrino Majorana mass directly by a 10 · 10 · 50 coupling. For example, there is a Majorana mass term g10 2 10 3 50, which can be identified to the gφψψ ′ coupling to obtain the hybrid potential, for the right-handed neutrino component in 10 3 is a (part of) inflaton, and 10 2 is identified as ψ ′ which does not contribute to the inflation. The smallness of g coupling can be related to the Majorana mass scale, which should be smaller than the GUT/Planck scale. In this case, the VEV of 50 + 50 can also break the SU(5) × U(1) X down to SM. Contrary to the 10 + 10 waterfall fields, they are not utilized to realize the doublet-triplet splitting. Therefore, as a building block for this structure, one can consider Pati-Salam model in a simple manner. Before moving to describe the inflation in the Pati-Salam model, we comment on the decays of inflaton and waterfall fields. Sometimes, it is said that the decay of the lightest particle of ψ, φ and ψ ′ is kinematically blocked due to the symmetry to obtain the hybrid inflation potential. Actually, the superpotential term has a parity symmetry (assigning − to ψ ′ , φ andφ, and + to the inflaton ψ). Inflaton ψ can decay to the SM particle, via the Yukawa coupling, One may think that one of ψ ′ and φ cannot decay due to the kinematical block. However, the parity symmetry is violated via the non-zero values of the waterfall fields. Let us explain it in a concrete example: where the right-handed neutrino is the (part of) inflaton and the waterfall field isφ =T 45 . Because there has to be multiple N fields to obtain the Majorana masses of three right-handed neutrinos, we write the index of N explicitly. The inflaton field is the right-handed Majorana neutrino and its mass is a little less than the unification scale, g 2 φ 2 /M N after falling down the waterfall potential.
The inflaton field can decay into MSSM field directly through the Dirac neutrino Yukawa coupling y. The field N is mixed with ν c in 10 via the non-zero value of φ, it can decay into the MSSM field directly (without decaying into φ). As a consequence, both φ and ψ ′ can decay into MSSM fields after all. In the case where we adopt 50 + 50 as waterfall fields, the situation is the same as above.

V. PATI-SALAM MODEL
In the Pati-Salam model, the gauge symmetry is SU(4) c × SU(2) L × SU(2) R , and the matter content of the model (See Appendix B for more detail) is Matter The Pati-Salam gauge symmetry is directly broken down to SM gauge symmetry by (4, 1, 2) + (4, 1, 2) or (10, 1, 3)+ (10, 1, 3). The non-zero values of them can generate the right-handed neutrino Majorana masses as same as the flipped-SU(5) model. Therefore, the hybrid inflation model with hilltop potential can be easily constructed. Because the dimension of the representation is less than the one in the flipped-SU(5) model, it is easier to understand the field contents in it as a building block. In the model using (10, 1, 3) + (10, 1, 3) to obtain the waterfall potential, the remnants in the representations can remain light due to the accidental discrete symmetry for the waterfall potential. The 10 dimensional representation in SU(4) c is a symmetric tensor, and it can be decomposed under where subscript denotes the U(1) B−L charge. The B − L charge can be obtained by the B − L generator, The SU(3) c sextet can remain light till TeV scale, and it can have an interesting phenomenological implication for the baryon number violation [19]. The implication in the inflation model will be studied in an accompanied paper.
The quartic invariants in terms of the matter representation to obtain the hilltop potential are LLLL,RRRR, LLRR.
Due to the SU(4) c symmetry, matter cubic terms are not allowed. For LLLL, one can write the group indices explicitly, where ǫ is a total anti-symmetric tensor, the indices a, b, c, d are for SU(4) c , and α, β, γ, δ are for SU(2) L . We do not write the group index explicitly later. We note that all the generation of L cannot be same in the LLLL operator due to anti-symmetricity. If the D-flat direction to lift LLLL andRRRR potential, the F -term potential from the Yukawa coupling is not lifted. To use LLRR potential, one has to care about the generation configuration in order not to lift the F -term potential as we have explained in the flipped-SU(5) model. If (4, 1, 2) + (4, 1, 2) is employed to break the gauge symmetry, R-parity violating terms can be generated since one of the fields has the same quantum number as the right-handed fieldR. If In the Pati-Salam model, one can make the right-handed neutrino to be inflaton, and to become heavy after the waterfall fields falls down, similarly to the flipped-SU(5) model. Here we describe a different configuration as an example. The Higgs representation to break the elecroweak symmetry is bi-doublet representation: H : (1, 2, 2). (58) The Yukawa interaction to generate masses of quarks and leptons is LRH.
If there is only one bi-doublet field, all the up-type quark, down-type quark, and charged-lepton and Dirac neutrino Yukawa couplings are same. Therefore, to break the wrong prediction of the fermion mass, one has to extend the model as either (case 1) there are multiple bi-doublets, or (case 2) quarks and lepton fields are mixed with the other representation. The case 2 is compatible to the hybrid inflation scenario.
We exhibit the scenario to mix the right-handed strange quark field with the inflaton by employing a field S D = (6, 1, 1). (60) The 6-dimensional representation is a anti-symmetric tensor of SU(4) c . The representation can be The representation is equivalent to the right-handed down-type quark's one, and they can mix.
They have components which can mix with q, u c , ℓ, e c respectively. In the Pati-Salam model and left-right gauge model (SU(3) c × SU(2) L × SU(2) R × U(1) B−L ), the gDd cφ term is motivated. The non-zero value ofφ breaks the gauge symmetry down to SM one. It is interesting that it is compatible with the hybrid inflation model. For example, we can consider the following D-flat direction: One can easily check that this satisfies the D-flatness condition for SU(4) c × SU(2) R . The s c field is mixed with the D c field in S D by gs c Dφ term, which generates the hybrid potential. The hilltop term is obtained by where the subscripts denotes the generation index. We note that this operator contains Since the inflaton is mixed with the matter by the non-zero value of the waterfall field φ, it can decay to MSSM field via the Yukawa interaction term, qs c H = cos θqψH − sin θqŝ c H. The ψ ′ field corresponds to D. The Lagrangian includes a term Therefore, afterφ acquires a non-zero value, ψ ′ can directly decay into MSSM fields. In this way, the decays are not kinematically blocked after all.

VI. TOPOLOGICAL DEFECTS
When a gauge symmetry G breaks down to H, monopoles are produced if the second homotopy group π 2 (G/H) is non-trivial [20]. In fact, the standard model gauge group contains U(1) Y , and therefore, it is possible that the monopoles are produced if the unified gauge group is semi-simple (e.g. SU(5), SO (10), and SU(4) c × SU(2) L × SU(2) R ). As it is well-known, the inflation (after the GUT symmetry breaking phase transition) can be motivated to dilute the monopole density.
In the current setup, the phase transition by waterfall potential occurs after the inflation ends, and the vacuum manifold has to be considered to avoid the monopole problem. Because the ψ fields have quantum numbers of the unified gauge group, the unified gauge symmetries (i.e. SU(5) × U(1) X and SU(4) c × SU(2) L × SU(2) R in our models) are not fully maintained during inflation. Due to the situation, the monopoles are not necessarily generated at the waterfall phase transition even in the case of Pati-Salam semi-simple gauge group, depending on the D-flat configuration of ψ fields.
Let us investigate how the configuration ofR : (4, 1, 2) along the D-flat direction breaks the SU(4) c × SU(2) R gauge group in the Pati-Salam model. As given in Appendix B, the representation contains the the right-handed matter fields as where sub(super)script ofR stands for the SU(4) c (SU(2) R ) index. As the first example, let us consider the D-flat configuration in terms of 2nd and 3rd generations: and the vacuum configuration is In this case, the remained SU(2) symmetry is broken down to U(1). As a result, at the phase transition (after the inflation ends), the monopoles are generated.
The vacuum configuration of the waterfall fields can be chosen to be Eq.(77) without loss of generality. On the other hand, the D-flat configuration of ψ (which causes the inflation) is not necessarily aligned to the Φ direction. For example, one can consider the configuration, where U is a 2 × 2 unitary matrix. The D-flat conditions can be satisfied similarly if |a| = |b| = |c| = |d|, and SU(2) symmetry is remained along the D-flat direction. At the phase transition by the waterfall field, the SU(2) symmetry is completely broken unless U is a diagonal matrix. Therefore, the monopoles are not necessarily generated at the phase transition. Only when the ψ and φ configuration are aligned, the monopoles can be generated. Next, let us consider the D-flat configuration in terms of three generations. For example, The D-flat conditions are satisfied if |a| = |b| = |c| = |d|, again. In this case, one can find that the remained symmetry along the flat direction is U(1), and the remained U(1) symmetry is not broken at the waterfall phase transition. Therefore, no topological defects are generated in this example.
Similar to the two generation case, it is not necessary to align the ψ and φ configuration, and a unitary matrix can be multiplied. In general, therefore, the remained U(1) symmetry is broken at the phase transition, and a cosmic string can be generated. However, the broken U(1) symmetry is recovered after the ψ field settles on the vacua, and the cosmic string disappear. We stress that the monopole production can be avoided at the waterfall phase transition in the current setup of the Pati-Salam model. We also note that in the flipped-SU(5) model, no serious topological defects are generated at the waterfall phase transition. For example, in the case of D-flat configuration in Eq.(37), the remained symmetry is SU(3) ⊂ SU(5) and it breaks down to SU(2) by the waterfall fields.

VII. NON-THERMAL LEPTOGENESIS AND GRAVITINO PROBLEM
The inflaton in our model is a flat direction containing right-handed sneutrino. It has a soft mass roughly 10 TeV during inflation. However, after inflation, the GUT Higgs develops a VEV and the right-handed sneutrino becomes massive by the double seesaw mechanism 8 . It decays mainly via Yukawa coupling y into slepton and Higgs or into lepton and Higgsino with a decay width given by Γ N = M N y 2 /(4π). The decay of the sneutrino after inflation reheats the universe to a temperature T R ∼ √ Γ N M P if the waterfall field decays earlier enough which we will assume to be the case for a simple estimation. In this case, non-thermal leptogenesis may happen. The baryon asymmetry is given by where ε ∼ 3 8π ∆m 2 31 M N δ/( H u 2 ), ∆m 2 31 ∼ 2.6×10 −3 eV 2 is the atmospheric neutrino mass squared difference and H u ∼ 174 GeV, and δ is the effective CP violating phase [22,23]. Therefore successful non-thermal leptogenesis can happen if the temperature at right-handed sneutrino decay is 10 6 GeV [24][25][26].
Since the gravitino mass we consider is 100 TeV, the gravitino does not affect big bang nucleosynthesis (BBN). However, an upper bound of reheating temperature ∼ 10 10 GeV(100 GeV/m LSP ), with m LSP LSP mass is given by the production of LSP cold dark matter. Interestingly we may have both leptogenesis and dark matter if this upper bound is saturated.

VIII. CONCLUSION AND DISCUSSION
In this paper, we present two concrete models of hilltop supernatural inflation based on flipped SU(5) and Pati-Salam models. The phenomenology in our model is very rich. Our inflation model is closely connected to particle physics. For example, our parameter space is constrained by both proton decay and CMB. Hilltop inflation fits very well in recent PLANCK data concerning spectral index, non-Gaussianity, tensor to scalar ratio and basically all the observables. As a hybrid inflation model, we consider a flat direction containing right-handed sneutrino to be the inflaton field and the waterfall field is a GUT Higgs. Non-thermal leptogenesis can happen after inflation. It is also possible to generate LSP dark matter. We have also shown that topological defects are not produced in the current setups. We consider a potential of the hilltop form, where V 0 /M 4 P = 10 −24 in our model. The number of e-folds is given by This model can be solved analytically and we have The spectrum and the spectral index are given respectively by From the above equations we can obtain Here ψ means the field value of inflaton at number of e-folds N and ψ end means the field value of inflaton at N = 0. By imposing the CMB normalization P (A8) This is ploted in Fig. 1. For example, for η 0 = 0.04, we have λ = 5.9 × 10 −13 . By using those numbers, we show the typical potential form in Fig. 2.
Having the relation of λ as a function of η 0 , we can express ψ (at N = 60) as a function of a single parameter η 0 by using Eq. (A3) and obtain ψ M P = 2.4 × 10 −9 η 0 − 0.01 .
This is ploted in Fig. 3. In order to know ψ end , we can use Eqs.
This is ploted in Fig. 4.

p = 6
By the same calculation procedure as p = 4 case, the relation between λ and η 0 is given by This is plotted in Fig. 5.
The field value at N = 60 is given by This is plotted in Fig. 6. In order to know ψ end we can obtain This is plotted in Fig. 7. In the usual SU(5) GUT model, the hypercharge Y is same as Y ′ (T Y = T Y ′ ). Therefore, the matter contents in the usual SU(5) are It is easy to check that the gauge anomaly is absent. This is obvious because the SU(5) × U(1) X is a subgroup of SO(10), and the above matter multiplets can be embedded into the 16 representation under SO (10). In the flipped-SU(5) model, the hypercharge in the standard model is assigned as It is easy to find that the places of the right-handed quarks (u c and d c ) and leptons (ν c and e c ) are flipped, respectively, compared to the usual SU(5) assignment.