No-scale hybrid inflation with R-symmetry breaking

In this paper we provide a no-scale supergravity scenario of hybrid inflation with R-symmetry being broken maximally. We investigate the inflation dynamics in details in both cases of pure F-term hybrid inflation and when adding constant Fayet-Iliopoulos D-terms. The effective inflation potential is asymptotically flat in a region of the parameter space in both cases. We explore all regions in the parameter space when discussing the constraints from the observables. We point out a connection between inflation, R-symmetry breaking and GUT scales. The moduli backreaction and SUSY breaking effects are investigated in a specific stabilization mechanism. We emphasis that a successful reheating is not affected by R-symmetry breaking, but it has interesting consequences. We study the reheating in flipped GUT model. We argue in favor of Z2 symmetry associated with flipped GUT models to avoid phenomenologically dangerous operators and allow for decay channels for the inflaton to right-handed neutrinos (sneutrinos).


Introduction
Since its discovery, the accumulation of the data from the Cosmic Microwave Background (CMB) over the past years supports the cosmological inflation paradigm. The most recent data by Planck collaboration [1], confirmed that the spectral index of the scalar fluctuations is n s = 0.955 − 0.974, up to 2 sigma exclusion limits, while the upper bound on the tensor to scalar ratio is r < 0.08. This may hint at a connection between the ideas of cosmological inflation and supersymmetric grand unification. It turns out that the inflation energy scale is estimated as One of the key issues in cosmology, is building a cosmological inflation model that accommodates the current observational constraints and connects to particle physics via reheating phase. Supergravity offers a promising framework for constructing inflationary models. At such large scale of inflation, supergravity effects should be taken into account. Building supergravity models of inflation is not an easy task. Basically, three problems arise:

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• The η-problem: it appears due to the supergravity contributions to the inflaton mass which spoils the slow-roll conditions, 1, η 1. More specifically, the expansion of the inflaton scalar potential yields V = V 0 1 + |φ| 2 M 2 P + · · · , where V 0 is the inflation scale. Now the Hubble constant squared during inflation is given by H 2 = V 0 /3M 2 P , therefore the inflaton acquires a mass of order Hubble constant which is a generic feature of supergravity models. Hence the slow roll parameter • Effective single field inflation: on top of its simplicity, it is a sufficient condition for avoiding unacceptably large isocurvature fluctuations. In supergravity models of inflation, the inflaton is not the only scalar field. Even in simple models containing only the inflaton superfield with a complex scalar component, we need only one real degree of freedom to play the role of inflaton and the other being integrated out from the inflation dynamics. Furthermore, in many supergravity models other scalar degrees of freedom appear such as the moduli fields and SUSY breaking superfields in hidden sectors, as well as fields of observable sector containing the Standard model. Effective single field inflation can be guaranteed if other fields acquire large masses of order Hubble scale and hence frozen during the inflation without affecting the inflation dynamics.
• Supersymmetry breaking in an approximate flat space: after the end of inflation, the inflaton goes to its true minimum and supersymmetry should be broken in an approximate flat space with infinitesimal vacuum energy density V 10 −120 M 4 P according to recent observations that supports a very tiny cosmological constant. This scale is very small compared to the other scales of particle physics. On the other hand connection to low-energy physics imposes a lower bound on the SUSY breaking soft masses m 10 −15 M P and on the other hand they are bounded from above by the SUSY breaking scale M S = F I 1/2 . This implies that the supergravity scalar potential V 10 −60 M 4 P , which contradicts the above tiny value of the cosmological constant. One solution of such problem is to consider SUSY breaking with Minkowski vacuum as in models of no-scale supergravity [27,28]. Moreover the SUSY breaking sector has a non trivial backreaction on the inflation potential and may spoil the inflation.
The η-problem in models of supergravity can be solved by defining a shift symmetry on a singlet inflaton [2][3][4], and moreover it can be defined on charged inflaton [6,7]. The Kähler potential K(|X| 2 , S +S) is invariant under the shift symmetry S → S + ic, with c is a real constant, while the superpotential takes the form where X is a stabilizer field that is introduced to avoid negative quartic terms for large values of the inflaton S, and f (S) is a holomorphic function in S. The class of models where f (S) is chosen to be a monomial [2][3][4][5][6][7] has a common imprint of having unacceptably large value of tensor to scalar ratio r 0.1 which is excluded by Planck recent observations [1].

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However, models with small tensor to scalar ratio such as the Starobinsky potential of inflation, can be accommodated in the above setup [30]. Hybrid inflation models [11] can connect the inflation physics and particle physics via introducing a GUT gauge symmetry, where the inflaton is coupled to the GUT higgs fields. A supersymmetric model of hybrid inflation, with an exact U(1) R symmetry (R-symmetry), was introduced in [12]. With the following superpotential and minimal Kähler potential inflation can be realized along the flat direction in which the GUT higgs fields φ 1 , φ 2 are frozen at the origin. The universe is dominated by a constant energy density V = κ 2 M 4 as long as |S| > |S c | = M . The Coleman-Weinberg 1-loop correction to the potential provides a slope for the inflaton to slowly roll resulting in small field inflation. The previous superpotential is the most general renormalizable one which is consistent with U(1) R symmetry. The R-charge assignments are as follows: In that context, R-symmetry has important advantages. First, it prevents higher degree terms such as S 2 and S 3 which spoil the small field inflation. Furthermore, the term µ φ φ 1 φ 2 is not allowed which spoils the inflation trajectory, φ 1 = φ 2 = 0, and breaks SUSY. Second it avoids naturally the η-problem when supergravity corrections are included, since the calculated mass squared of the inflaton from the supergravity potential cancels at the tree level [13]. Third, U(1) R symmetry has many phenomenological advantages in the low energy effective theory [14,15,60]. It forbids higher dimensional operators that contribute to proton decay as it gives rise to an accidental U(1) B . Moreover its unbroken Z 2 subgroup acts as matter parity which prevents couplings that lead to the LSP decay. In addition, R-symmetry may contribute to a solution to the µ-problem as it forbids the MSSM Higgs mixing term µ H H u H d . The latter term can be generated via Giudice-Masiero (GM) mechanism [17]. Nevertheless, the standard hybrid inflation models with R-symmetry predicts a large spectral index n s ∼ 0.98 far from the observation limits, and small tensor to scalar ratio r ∼ 10 −5 .
Interestingly, no-scale supergravity offers a natural solution to the η-problem and can yield an inflation potential with a plateau adequate for slow rolling. Indeed the supergravity scalar potential resembles that in a globally supersymmetric version, where a cancellation occurs between |W | 2 and terms of Kähler derivatives products in |DW | 2 . The latter happens due to the noncompact SU(N, 1)/SU(N )×U(1) no-scale symmetry [27,28]. Therefore the scalar potential goes like In some cases [29][30][31], the resulting inflationary potential is the Starobinsky potential of inflation [8][9][10], with the predicted inflation observables are in the core of the allowed regions of Planck data. The Cecotti model [29] and its modification [30] depend on the no scale symmetry SU(2, 1)/SU(2) × U(1), with two superfields employed as a stabilizer superfield and inflaton superfield. The model of Ellis-Nanopoulos-Olive (ENO) [31] relies on JHEP02(2021)230 the no-scale symmetry SU(2, 1)/SU(2) × U(1) with two superfields as well. One superfield corresponds to the inflaton S and the other is identified as a modulus superfield T . The superpotential was chosen as the Wess-Zumino model, hence the superpotential and the Kähler potential are given by The modulus field is stabilized at high scale by string theory mechanisms such as the KKLT [24], LVS [25,26] or other mechanisms such as [38,39]. Therefore, one ends up with a single field inflation.
In this class of models R-symmetry needn't be exact to have a successful inflation. As a matter of fact embedding the hybrid inflation, with a superpotential respecting the R-symetry (1.3), in an ultraviolet theory containing moduli fields, such as no-scale supergravity, has some difficulties. The stabilized moduli backreact nontrivially on the inflation trajectory and in many cases spoil the inflation [41,46]. Furthermore, including the inflaton in the no-scale Käher potential requires redefining the fields to have canonical kinetic terms. This implies an effective inflaton potential ∼ cosh 4 x which is too steep for inflation.
R-symmetry breaking in connection to inflation was studied in the literature [19][20][21][22][23]. In refs. [19,20], R-symmetry was allowed to be broken softly by adding a Planck suppressed dimension four operator to the superpotential, while R-symmetry was exact on the tree level. In ref. [19], non-canonical kähler potential was considered as well as the superpotential (1.3) corrected by a dimension four operator, which results in large field inflation. While in ref. [20] Starobinsky like inflation results due to considering no-scale supergravity with a toy superpotential containing dimension two and dimension four operators.
It is worth mentioning that an exact R-symmetry is a necessary condition for supersymmetry breaking according to Nelson-Seiberg theorem [18]. On the other hand breaking SUSY spontaneously in a hidden supergravity sector implies a non-vanishing vev of the superotential W = 0, since our universe is associated with an infinitesimally small vacuum energy. Therefore R-symmetry is broken as W has a non-trivial R-charge.
Our prime aim in this paper is to establish model independent hybrid inflation scenario in no-scale supergravity and waive the R-symmetry constraint applied to the standard hybrid inflation models. We like to stress that both the inflation and the low energy consequences will be consistent with observation when R-symmetry is broken maximally.
R-symmetry can be broken explicitly by adding the terms µS 2 and λS 3 to the superpotential and choose µ φ to vanish. Alternatively, R-symmetry may be broken spontaneously. The latter may be stemming from a hidden sector containing a superfield Ψ, with R-charge R[Ψ] = −1, that acquires a non-zero vev. In that case the undesirable term µ φ φ 1 φ 2 can be avoided since the term Ψ φ 1 φ 2 is not allowed as well as any higher order opera- to an effective superpotential containing the terms µS 2 and λS 3 with the identifications µ ≡ ν 1 Ψ and λ ≡ ν 2 M P Ψ 2 . The values of µ, λ depend on the R-symmetry breaking scale Ψ 10 16 GeV, ν 1 ∼ 10 −1 − 10 −4 and ν 2 ∼ 10 −1 − 10 −3 . At such large scale of breaking R-symmetry, the associated R-axion problem does not exist [18]. 1 This paper is organized as follows. In section 2 we investigate the F-term hybrid inflation in no-scale supergravity. We present a complete analysis of the effective inflation potential and explore all allowed regions of the parameter space versus the Planck limits on the inflation observables. We consider also hybrid inflation with Fayet-Iliopoulos D-term in section 3 and analyse the effective inflation potential. In section 4 we discuss the SUSY breaking and moduli backreaction on the inflation in both models. Section 5 is devoted for discussing the reheating. We emphasize on the specific choice of Z 2 discrete symmetry as well as a gauge group such as flipped SU(5) to study the reheating phase and some other phenomenological consequences when R-symmetry is not exact. Finally we conclude in section 6.

No scale F-term Hybrid Inflation (FHI)
We consider the following superpotential which is renormalizable and breaks R-symmetry Here S is the singlet inflaton superfield and φ 1 , φ 2 represent conjugate representations of the Higgs supermultipletes that transform non-trivially under GUT gauge group. The scalar components φ 1 , φ 2 acquire vevs in the SM neutral direction. The parameter M is the GUT symmetry breaking scale and κ is a dimensionless coupling. The parameter µ determines the scale of the inflation and λ is a dimensionless coupling and they are responsible for the R-symmetry breaking in the superpotential. The gauge invariant Kähler potential has the no-scale structure which has the no-scale symmetry SU(3, 1). The modulus T can be stabilized at high scale [24-26, 38, 39] with Re(T ) = τ 0 , Im(T ) = 0. The total scalar potential is the sum of the F-term and D-term potentials V = V F +V D . The F-term scalar potential is given by and D I is the Kähler derivative defined by D I = ∂ ∂Z I + ∂K ∂Z I . We use the lower case letters i, j to run over the inflation sector fields S, φ 1 , φ 2 . Here and in the rest of the paper we JHEP02(2021)230 work in the units where the reduced Planck mass M P is unity. The D-term potential is given by where f AB is the gauge Kinetic function and the indices A, B are corresponding to a representation of the gauge group under which Z i are charged. The D-term D A is given by with T A are generators of the GUT gauge group in the appropriate representation. Working in the D-flat direction, the total potential will be given by F-term scalar potential It is clear that, the above scalar potential is positive semidefinite. Therefore it has a global minimum which is supersymmetric and Minkowskian, located at In fact D i W = W = 0 at the minimum. Looking at the superotential (2.1) which contains three complex degrees of freedom, one notices that it depends on the combination φ 1 φ 2 .
Taking into account that R-symmetry is broken as well as the D-flat direction, |φ 1 | = |φ 2 | = ρ, hence one real degree of freedom cancel. It is convenient to parametrize the complex scalar fields in terms of their real components as follows It is clear that the scalar potential (2.6) depends only on four real degrees of freedom, namely s, σ, ρ, θ, while the fifth degree of freedom Σ will correspond to the massless goldstone boson which is unphysical and will be eaten by the massless gauge boson to render it massive, hence it will not contribute to the dynamics of inflation. In that representation, the minimum of the potential is located at However we rewrite the scalar potential (2.6) in terms of the Cartesian variables s, σ, α, β, , when discussing the simulation and the mass matrices.

Inflation trajectory
Along the inflationary trajectory the potential (2.6) is minimized along the D-flat direction φ 1 = φ 2 = 0, and the higgs fields are fixed at the origin during the inflation and we have Fterm hybrid inflation (FHI). Accordingly the effective inflationary potential will be given by

and the Hubble scale during inflation is
It is worth mentioning that the effective infationary potential is Starobinsky-like and it is similar to the one obtained in [32,33]. 2 We turn to discuss the stability of the inflation trajectory resulting in effectively a single field inflation. In order to have a canonical kinetic terms for the inflaton S we should have the following field redefinition where x is the slow rolling inflaton. The target space metric during inflation is diagonal in the basis (x, α, y, β) and is given by The fields α, β, y are fixed at the origin during the inflation, since the scalar potential is minimized for α = β = y = 0 and their inflaton field dependent masses are larger than the Hubble scale during inflation as follows The above equations have been extracted for large values of the inflaton field x. After inflation ends, the fields β, y are fixed at zero value. On the other hand, α will be fixed at α = 0 during inflation as its field dependent mass is positive. As the inflaton rolls down, its value decreases to smaller values until it reaches a critical value x c at which the field dependent mass m 2 α changes to negative and α = 0 becomes a local maximum as indicated in figure 1. This triggers the waterfall phase and α goes to its true minimum α = √ 2M .
2 In [32,33] they assumed the no-scale supergravity realization of Starobinsky like inflation, with adding a Polonyi term to break SUSY along and after inflation. In our scenario, we have Starobinsky like potential due to stabilizing the higgs at the origin with broken SUSY during inflation. After the inflation ends, SUSY is exact at the global minimum. Therefore the phenomenology is different. Moreover, we will give a complete analysis of the potential from the point of view of the phenomenology of our model and will investigate all regimes of the parameter space that are not discussed in [32,33]. In particular, for small x, to leading order

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Therefore, for small values of x, m 2 α < 0 whenever M 2 < 3τ 0 . The critical value of inflaton x c which triggers the waterfall, can be computed from the inflaton dependent mass squared of α which is given for small x by Accordingly, the critical value of the inflaton x c at which the sign of m 2 α flips to a negative sign, is given to leading order in µ and λ by In figure 2, we show the potential of x, α. For large values of x the potential is minimum α direction at α = 0 and the x direction is flat, whereas for small x inflation ends and waterfall happens.
We will simulate the time evolution of the scalar fields by solving the supergravity equations of motion:Ψ

Inflaton effective potential
The potential (2.10) is positive semi-definite and its global minimum is V inf = 0 when

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Here, it is clear that for M = 0, we return to the original ENO model of [31]. Using the field redefinition (2.11), the resulting potential will have the form are dimensionless quantities. Apparently the simple Starobinsky case (ENO) [31], is recovered for f = 1, b = 0. Expanding the hyperbolic functions, considering that y is frozen at the origin, the effective inflationary potential will be given by Clearly, the location of the minimum of the potential at x 0 is shifted from the origin The lower sign is chosen such that the minimum is shifted to the left and hence x * > x 0 is guaranteed. In that respect, we have the important constrain x c > x 0 is satisfied also. The latter allows for the waterfall and hence the fields α, x stabilize to their true minima at 0, √ 2M , respectively. We study two regimes in the parameter space: • Case I An interesting case for the inflation potential is the limit when b → 1−f , the potential becomes flat for large values of x with constant hight a 4 (1 + b) 2 . 3 In this limit the potential will have the following form

• Case II
If f = 1 − b + ε, the effective potential (2.21) can be written as x .
where the supertrace is taken over all superfields with inflaton dependent masses m(S). As advocated above (2.1), the stabilized fields during the inflation have m(S) ∼ H. Since H 2 ∼ a M 2 p ∼ 10 −10 M 2 p , the 1-loop correction V 1-loop ∼ H 4 64π 2 10 −22 which is negligible compared to the tree level potential.

Inflation observables
Here we will investigate the inflation observables and see the constraints on the different scales µ and M . We investigate the inflation observables such as the tensor-to-scalar ratio r, scalar tilt n s and the scalar amplitude A s (sensitive to the scale of the inflation), and they can be expressed in terms of the slow-roll parameters and η as follows r = 16 where the above observables are computed at the crossing horizon value of the inflaton field x * . The number of efolding is given by where x e is the value of the inflaton at the end of inflation. The value of a is fixed by observed value of the scalar amplitude A s 1.95896 ± 0.10576 × 10 −9 at 68% CL [1]. Now we analyze different regimes in the parameter space that leads to successful inflation.
In the limit f = 1 − b, we analyze the inflation described by effective potential (2.22). It is clear that the slow roll parameters depend only on b. In that case x e is given by This imposes the constrain 0 < b < 0.25, such that x e is positive. The observables n s and r are independent of b, and depend only on N (see appendix A). They have the following form In that case b has only upper bound is not fixed by n s , r and A s . If the flat regime is perturbed as f = 1 − b + ε, then corresponding potential (2.23) is not asymptotically flat. Hence, the inflation may not succeed.
In figure 6 we show a logarithmic plot for n s and r prediction of the inflationary potential eq.

No scale Hybrid Inflation with constant Fayet-Iliopoulos D-terms (FDHI)
In this section we add Fayet-Iliopoulos D-term and study the hybrid inflation by considering the same Kähler potential (2.2) and the following renormalizable superpotential that breaks R-symmetry Again S is the singlet inflaton superfield while φ + , φ − have opposite charges under U(1) gauge group which is anomalous or non-anomalous [35]. The total scalar potential is the

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sum of the F-term and D-term potentials. Here the D-term D A is given by with ξ A are Fayet-Iliopoulos D-terms exist for the U(1) gauge groups. We consider U(1) gauge group and the gauge kinetic function as the Kronecker delta, hence the total potential will be given by Again the scalar potential is positive semidefinite. The global minimum is supersymmetric and Minkowskian and is corresponding to D i W = W = D A = 0. It is located at We will parametrize the complex scalar fields in terms of their real components as follows 2ρ 0 with ρ 0 being the vev, hence θ will correspond to the massless goldstone boson and the dynamics will depend on five real degrees of freedom [6]. However will work on the basis s, σ, α 1 , α 2 , β 1 , β 2 and β 2 is mainly the goldstone boson, hence the minimum of the potential is located at (3.6)

Inflation trajectory and effective potential
The scalar potential is minimized in the direction φ + = φ − = 0 and the effective inflation potential is given by Clearly, the above potential is the same as the ENO model [31] but shifted by the energy density g 2 ξ 2 2 . We use the same field redefinition (2.11), then the target space metric during inflation is found to be diagonal in the basis (x, α 1 , y, β 1 , α 2 , β 2 ) and is given by Similarly the fields α 1 , β 1 , α 2 , β 2 , y are fixed at the origin during the inflation, since the scalar potential is minimized for α 1 = β 1 = α 2 = β 2 = y = 0. As a matter of fact, minimizing the potential in the direction of α 2 gives two solutions, namely α 2 = 0 and the other solution for x 1 is given by The latter gives complex

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value for α 2 and the only allowed minimum during inflation is α 2 = 0. The field dependent masses are larger than the Hubble scale during inflation as follows The above equations have been extracted for large values of the inflaton field x. In fact the field dependent squared mass matrices of (α 1 , α 2 ) and (β 1 , β 2 ) have mixing terms which are very small in the large limit of the inflaton field. The fields α 1 , β 1 , β 2 , y will be fixed at zero during and after inflation. On the other hand α 2 will be fixed at α 2 = 0 during inflation with positive field dependent mass squared. As the inflaton rolling down, its value decreases to smaller values until it reaches a critical value x c at which the field dependent mass m 2 α 2 changes to negative and α 2 = 0 becomes a local maximum. This triggers the waterfall phase and α 2 goes to its true minimum. In particular, for small x, to leading order m 2 To find the critical value of the inflaton x c which triggers the waterfall, we expand the masses for small x, hence Accordingly, the critical value x c at which the sign of m 2 α flips to negative sign, is given to leading order in ξ by (3.10) The effective inflationary potential has a plateau for λ =μ and is given by which is the Starobinsky potential shifted byξ 2 = g 2 ξ 2 2 . On the other hand, perturbing the plateau with λ =μ + ε, the potential is given by x sinh 3 2 3 x + 9 ε 2 e 2 2 3 x sinh 4 2 3 x .
(3.12) Therefore the plateau is spoiled by the last two terms which are very steep. Here we stress that the infation potentials (3.11), (3.12) are only valid for x > x c , otherwise the potentials are minimized at x = 0 with cosmological constant of orderξ 2 which is inconsistent with the observation of infinitesimally small cosmological constant. Now we turn to discuss the observables. The value of x at the end of inflation is In order to have x e > 0, the constraintξ < √ 3μ should be satisfied. The shift byξ 2 doesn't alter the predictions of the ENO model [31], wheneverξ < √ 3μ. The scale of inflation is JHEP02(2021)230 determined by observed value of the scalar amplitude A s 1.95896 ± 0.10576 × 10 −9 [1], and is given by Since the predicted r ∼ 10 −3 , M inf is of order GUT scale. Figure 8 displays the allowed region by the observed value of scalar amplitude, in the ε −μ plane with fixingξ = 10 −6 . The range of values ofμ ∼ 3.2 × 10 −5 − 9.2 × 10 −5 .

Moduli backreaction and SUSY breaking
An essential component of the no-scale inflationary models is the modulus field T . It turns out that the stabilization mechanism of the modulus field can affect the inflation trajectory [41][42][43][44][45][46][47][48][49][50][50][51][52]. In this section we study the effect of the modulus stabilization and the expected backreaction on the inflationary trajectory. We will focus on the mechanism proposed in [38,39] which provides a strong stabilizing terms in the Kähler potential as follows where the scale Λ 1 such that the modulus acquires large mass and stabilizes during the inflation. The above Kähler potential preserves the no-scale structure with stabilizing the JHEP02(2021)230 modulus at τ 0 , where τ 0 represents the minimum of the modulus in absence of inflation sector. Including the inflation sector, the large positive energy density during inflation shifts the modulus minimum of T by δT . The effective scalar potential is then given, to leading order in δT , δT , in terms of the total supergravity scalar potential V by The displacement δT is obtained by imposing the minimization condition: It turns out that the exact no-scale symmetry will preserve the inflation potential from dangerous terms such as the soft mass term and the term proportional to −3|W | 2 [49,52]. However, the stabilizing term in the Kähler (4.1) doesn't have an origin from UV theory such as string theory. If instead we used mechanisms of moduli stabilization in string theory such as KKLT or LVS models, the no-scale structure is broken by the non-perturbative terms in the moduli superpotential and the backreaction of the moduli results in dangerous terms arising from |W | 2 in the scalar potential, which spoils the plateau [52].

Backreaction on no-scale FHI
We add a constant W 0 to the superpotential (2.1) and use the Kähler potential (4.1). At the global minimum, SUSY is broken via the F-term only in the directions of T and φ 1 , φ 2 with zero cosmological constant. The gravitino mass and the modulus mass are given by On the other hand during the inflation SUSY is broken via D-term and via F-term in the direction of T and S. The waterfall fields and y are still fixed at the origin during the inflation. Hence, the effective potential is given by As expected, the modulus backreaction on the inflation potential results in corrections suppressed by powers of the large modulus mass which doesn't affect the plateau.

Backreaction on no-scale FDHI
Similarly we add a constant W 0 to the superpotential (3.1) and use the Kähler potential (4.1). At the global minimum, SUSY is broken via the F-term only in the direction of T and φ − with zero cosmological constant. The gravitino mass and the modulus mass are given by

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On the other hand during the inflation SUSY is broken via D-term and via F-term in the direction of T . The effective inflation potential is given by where the waterfall fields and y are still fixed at the origin during the inflation. Again, the modulus backreaction on the inflaton potential is negligible and the plateau is not affected.

Reheating
In this section we study the reheating after inflation which is one of the interesting consequences of the FHI model. The mass matrix of the inflaton and the higgs (B.2) is not diagonal. The mixing between the inflaton and the higgs fields is proportional to µ. The later is stemming from R-symmetry breaking term in the superpotential (2.1). This will have important impacts on the reheating scenario. Indeed that provides additional motivation for the inflation scenario 2 where a natural coupling of the inflaton (via the mixing with the GUT higgs) to the MSSM sector can arise and may contribute to the reheating stage. The mixing angle is given, in terms of the mass matrix (B.2) entries, by The physical states (we use the canonical inflaton x) and the physical masses are given as follows The reheating is dependent on the choice of the gauge symmetry group. We will consider the flipped GUT (FGUT) gauge group SU(5) × U(1) X (or flipped SU(5)) which has many appealing features as well as the advantage of being free from the monopoles [53]. Inflation and reheating in the context of FGUT scenarios was considered in [7,[60][61][62][63]. In [7] the inflaton was right-handed sneutrino that is charged under FGUT gauge group which allows for natural decay channels to the MSSM particles, while in [61][62][63] the inflaton was a singlet. In the latter scenario, it was pointed out a connection between inflation, neutrino masses and reheating via the couplings between the inflaton, the right handed neutrino and the higgs. On the other hand, FGUT hybrid inflation with singlet inflaton was explored in [60].
The field representations of the flipped SU(5) group are listed in table 1 as well as the respective U(1) X charges. The Q X charges are assigned such that the SM hypercharge is obtained as [7] 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 . Y is the diagonal generator of SU(5) and Q Y = 1 6 diag (−2, −2, −2, 3, 3).
In this regard the particle content is accommodated in the flipped SU(5) representations as follows [7,54,60] • The standard model (SM) matter content is contained in the representations 10 F , 5 F and 1 F as follows • The SM Brout-Englert-Higgs bosons responsible for the electroweak symmetry breaking, are contained in5 Hu and 5 H d . the tiny neutrino masses via type I seesaw mechanism. In that respect, the interaction Lagrangian responsible for the inflaton decay is given by After diagonalizing the mass matrix (B.2), we rewrite the Lagrangian (5.4) in terms of the physical states x , α The reheating temperature is given by [68,69] where Γ x is the total decay width of the inflaton field which is given by An upper bound on the reheating temperature arises from cosmological constraints such as the gravitino overproduction problem [70][71][72][73][74], and it is given by [74] T R < 10 7 − 10 10 GeV. Therefore the inflaton should decay only to the first or second generations in the neutrino sector, as contributions from the third generation would drive the reheating temperature to values exceeding the above constrain. We may assume the existence of a flavour violating sector at high energies, without specific details, that prevents the decay to the third generation.

Conclusions
In this paper, we have proposed a scenario for hybrid inflation with a maximal breaking of R-symmetry, in no-scale supergravity context. In that respect, we have studied the FHI model and found a region in the parameter space in which the effective potential is asymptotically flat, which is not studied in [32,33]. We have treated the dynamics of all fields in full detail and realized the waterfall phase. Moreover, we have discussed the case of adding FI D-term which results in an effective potential similar to the ENO model but shifted.
The question of moduli stabilization and their backreaction as well as SUSY breaking has been discussed. It has been emphasised that inflation trajectory will not be affected in specific type of strong moduli stabilization proposed by Ellis et al.
Finally, the reheating phase has been studied in the context of flipped GUT scenario. We have stressed on the role of the associated Z 2 symmetry in low energy phenomenology and in allowing for decay channels for the inflaton in connection to the neutrino masses. We emphasised that the calculated reheating temperature prefers GUT symmetry breaking scale M ∼ 10 15 GeV, in order to be consistent with the cosmological constraints. Moreover the latter constrain implies that the inflaton should decay only to the first or second generations in the neutrino sector, while the third generation should decouple.

A No-scale FHI observables
The crossing horizon value of the inflaton x * as a function in N, b is given by

B Mass matrices
For FHI and in the basis (x, α, y, β), the target space metric g ij is diagonal at the minimum, and is given by For FDHI, in the basis (x, α 1 , y, β 1 , α 2 , β 2 ), the target space metric g ij is diagonal at the minimum, and is given by Hence the masses are given by