Natural Inflation from 5D SUGRA and Low Reheat Temperature

Motivated by recent cosmological observations of a possibly unsuppressed primordial tensor component $r$ of inflationary perturbations, we reanalyse in detail the 5D conformal SUGRA originated natural inflation model of Ref. [1]. The model is a supersymmetric variant of 5D extra natural inflation, also based on a shift symmetry, and leads to the potential of natural inflation. Coupling the bulk fields generating the inflaton potential via a gauge coupling to the inflaton with brane SM states we necessarily obtain a very slow gauge inflaton decay rate and a very low reheating temperature $T_r\stackrel{<}{_\sim }{\cal O}(100)$~GeV. Analysis of the required number of e-foldings (from the CMB observations) leads to values of $n_s$ in the lower range of present Planck 2015 results. Some related theoretical issues of the construction, along with phenomenological and cosmological implications, are also discussed.


Introduction
Inflation solves the problems of early cosmology in a natural way [2] and besides that produces a primordial fluctuation spectrum [3] which allows to discuss structure formation successfully. In detailed models (i) a sufficient number of e-folds for the inflationary phase has to be produced, (ii) guided by bounds presented by the Planck Collaboration [4,5], the cosmic background radiation and a spectral index n s = 0.9603 ± 0.0073 should be generated. And (iii), the normalization of fluctuations has to be reproduced. Rather flat potentials for the inflaton field lead to the "slow roll" needed for (i). Such potentials appear naturally in (tree level) global supersymmetric models; higher loop corrections can be controlled, but the inclusion of supergravity easily produces an inflaton mass of the order of the Hubble scale.
In models with an extra dimension the fifth component of a U(1) gauge field entering in a Wilson loop operator can act as an inflaton field of pseudo Nambu-Goldstone type which is protected against gravity corrections and avoids a transplanckian scale [6], [7], present in the original model of "natural inflation" [8]. We have presented such a model [1] based on 5D conformal SUGRA on an orbifold S 1 /Z 2 with a predecessor based on global supersymmetry with a chiral "radion" multiplet on a circle in the fifth dimension [9]. We also made the interesting observation that a spectral index n s ∼ 0.96 as observed recently [different from a value very close to (1) usually obtained in straightforward SUSY hybrid inflation [11]], is obtained rather generically in gauge inflation. Actually, in the supersymmetric formulation we have a complex scalar field which besides the gauge inflaton A 1 5 contains a further "modulus" field M 1 which also might allow for successful inflation [1]. The main difference between the two inflation types is that gauge inflation leads to a large tensor to scalar ratio r(∼ 0.12 in [1]) whereas modulus inflation leads to very small r( < ∼ 10 −4 in [1]). 5 Recently the BICEP2 data [12] gave strong indication of a large ratio r = 0.2 +0.07 −0.05 though there is a still ongoing discussion if this indeed has primordial origin [13]. We here therefore consider the gauge inflation of ref. [1] again with particular emphasis on the required length of inflation. The well known 62 e-folds solving the horizon problem will turn out to require a substantial expansion during the reheating period within the natural inflation scenario.
Let us present the organization of the paper and summarize some of the results. In Sec. 2 we perform a detailed analysis of natural inflation with cos-type potential. For the calculation of the spectral index n s and the tensor to scalar ratio r, we use a second order approximation with respect to slow roll (SR) parameters. Since these quantities (n s , r) are determined at the point where the SR parameters are tiny, this approximation is sufficient for all practical purposes. However, near the end of the inflation, when SR breaks down, we perform an accurate numerical determination of the point via the condition ǫ H = 1 on the Hubble slow roll parameter (see [14]- [16] for definitions). This is needed to compute, with desired precision, the number of e-foldings (N inf e ) before the end of the inflation. Within 1σ deviation of the r (provided by BICEP2), we find that N inf e < 54, indicating that nearly ∼ 10 e-foldings of the Universe expansion should occur during the reheating process. For this calculation (together with n s and r) we use the value of the amplitude of curvature perturbations measured by the Planck collaboration [5]. All this allows to calculate the reheat temperature, which turns out to be low ( < ∼ O(10) TeV). We have also obtained results for 3σ and 2σ deviations of r and n s respectively. 6 In Sec. 3 and Appendix A we shortly review our model of ref. [1] in a more self-contained way and discuss how natural inflation emerges from 5D SUGRA. Using a superfield formulation, we do not need to go in the details of the component expressions in conformal 5D SUGRA of Fujita, Kugo and Ohashi (FKO) [17]. Indeed this emerged from our discussion [18,19] (see also Ref. [21]) bringing the 5D conformal SUGRA formulation closer to the 4D global SUSY language [19]. We concentrate here on gauge inflation, i.e. on the case M 1 = 0 (stabilized moduli in the origin or a choice of initial conditions 7 ). In section 3, discussing the realization of natural inflation within 5D SUGRA, we present the mechanism for inflaton decay. We show that the inflaton's slow decay is a natural consequence of the 5D construction, being realized by couplings of the heavy bulk supermultiplets (generating the inflaton potential through their gauge coupling) with brane SM states. Because the inflaton decay proceeds by 4-body decay and the decay width is additionally suppressed by the 2-nd power of the tiny U(1) gauge coupling constant (of the gauge inflaton-charged fields) and a relatively small inflaton mass, the proper suppression of the reheat temperature T r also comes out naturally. Our 5D SUGRA construction allows us to make an estimate T r ∼ 0.34ρ Appendix A discusses the Kaluza-Klein spectrum of the fields involved, as well as the SUSY breaking effects for brane fields. We also perform a derivation of higher dimensional operators involving the inflaton φ Θ and light (MSSM) states relevant for the inflaton decay. As it turns out, the dominant decay channel is φ Θ → llhh (with l and h denoting SM lepton and Higgs doublets respectively). Sec. 4 includes a discussion and concluding remarks about some related issues.

Natural inflation
In this section we analyse inflation with the potential of natural inflation [8] given by: where φ Θ is a canonically normalized real scalar field of inflation. In the concrete scenario of Ref. [1], we focus later on, the inflaton originates from a 5D gauge superfield, while the parameters/variables of (1) are derived through the underlying 5D SUGRA. See Eqs. (24), (25), (A.17) and also the comment underneath Eq. (A.17). The slow roll parameters ("VSR") are given by 6 Awaiting for more accurate measurements and a refined analysis, for this moment we still keep in mind possible changes corresponding to such deviations, and present appropriate numerical results. 7 For a discussion of moduli stabilization in the superfield formalism within 5D SUGRA see [22]. For a choice of initial conditions leading approximately to M 1 = 0 see Ref. [10].
where M P l = 2.4 · 10 18 GeV is the reduced Planck mass. The number of e-foldings during inflation, i.e. during exponential expansion, denoted further by N inf e , is calculated as In this exact expression the HSR parameter ǫ H (defined below), participates. The point φ e Θ , at which inflation ends, is determined by the condition ǫ H = 1. The point φ i Θ corresponds to the begin of the inflation.
The observables n s and r depend on the value of φ i Θ (the point at which scales cross the horizon). This allows to determine φ i Θ as follows. Via HSR parameters, the expressions for n s and r are given by [14], [16], [15]: where we have limited ourself with second order corrections. The HSR parameters ǫ H , η H , ξ H are given by: with the Hubble parameter H and it's derivative with respect to the inflaton field. The subscript ′ i ′ in (4) indicates that the parameter is defined at the point at which scales cross the horizon. As it turns out, at this scale the slow roll parameters are small and second order corrections in n s and r are small and the approximations made in (4) are pretty accurate. Exact relations between VSR (ǫ, η, ξ, · · · ) and HSR parameters (ǫ H , η H , ξ H , · · · ) are given by [14], [16], [15]: When the slow roll parameters are small, from (6), the HSR parameters to a good approximation can be expressed in terms of VSR parameters as Using these approximations in (4), we can write n s and r in terms of VSR parameters: where we have still restricted the approximations up to the second order. Applying these expressions, for the model (determining ǫ, η and ξ as given in Eq. (2)), we arrive at: and r = 8(M P l α) 2 From Eq. (10) we can express tan αφ i Θ 2 in terms of r and M P l α. As will turn out, the latter's value is small, so to a good approximation we find: Plugging this into Eq. (9) for the spectral index we get: This will be used as orientation for further analysis and various predictions. So far, we have performed calculations in a regime of small slow roll parameters, determining the value of φ i Θ via Eq. (11). As was mentioned, the value of φ e Θ is determined from the condition ǫ H = 1. Near this point both ǫ and η parameters turn out to be large and instead of an expansion we need to perform numerical calculations. This will be relevant upon the calculation of the number of e-foldings N inf e . Since, within our model, via Eq. (2) VSR parameters are related to each other as the three equation in (6) can be rewritten as where σ H has been dropped because of it's smallness. From the system of (15), for a fixed value of M P l α, the parameters ǫ H , η H and ξ H can be found in terms of the single parameter ǫ. The dependance of these parameters on the value of ǫ, for M P l α = 0.04 are shown in Fig. 1 (for different values of M P l α shapes of the curves are similar). We see that ǫ H = 1 is achieved when ǫ = ǫ e ≈ 2 and thus, the expansion with respect to ǫ, η within this stage of inflation is invalid. On the other hand, the values of η H and ξ H remain relatively small. From the relation 2ǫ = (M P l α) 2 tan 2 αφ Θ 2 one derives: Using this, the integral in (3) can be rewritten as Having the numerical dependence ǫ H = ǫ H (ǫ) (depicted in Fig. 1), we can evaluate the integral in (17) and find N inf e for various values of M P l α. The results are given in Fig. 2. Curves in Fig.  2 demonstrate that, within our model, there is an upper bound on N inf e . Namely, within 1σ error bars of r and n s we have This, on the other hand, leads to another striking prediction and constraint.
As discussed in Refs. [23], [5], the N eff e , guaranteeing causality of fluctuations, should satisfy: where for the scale k we take k = 0.002 Mpc −1 corresponding to the Planck's data [4], [5], while the present horizon scale is a 0 H 0 ≈ 0.00033 Mpc −1 . The factor γ accounts for the dynamics of the inflaton's oscillations [24], [25] after inflation, and can be for our model approximated as γ ≃ 1− 1 16 Ve V 0 (will turn out to be a pretty good approximation).  table 3 of Ref. [5], corresponds to the ΛCDM model). Generated by inflation, this parameter is given by: In order to obtain the observed value of A 1/2 Using these values in (19) we see that the sum of the 3 rd and 4 th terms is≈ 3.4. Thus, the last term should be responsible for a proper reduction of N inf e . Namely, during the reheating process, the universe should expand by nearly 10 (or even more) e-foldings. This means that, for this case, the model should have a significant reheat history with ρ 1/4 reh ∼ 400 GeV. 9 Within the scenario of natural inflation, this has not been appreciated before. For lower values of r (deviating from the central value by 2σ or more) the reheating temperature can be larger. For appropriate values of r and M P l α (and n s ) the reheating temperature can be big. The numerical results are given in Table 1, where we considered cases with ρ 1/4 reh not smaller than 10 −3 GeV, and N inf e ≤ 62. The values of the spectral index running dns d ln k = 16ǫ i η i − 24ǫ 2 i − 2ξ i are also presented. The first three row-blocks correspond to the values of (r, n s ) within 2σ ranges of the current experimental data. Central (r = 0.2) and 9 The reheating process can continue even till the epoch of nucleosynthesis. In this case one should have ρ  Below we will show that within our scenario of natural inflation, a low reheat temperature is realized naturally and fits well with the proposed construction.

Natural inflation from 5D SUGRA
In order to address the details of inflaton decay, related to the reheat temperature, we need to specify the underlying theory natural inflation emerged from. A very good candidate is a higher dimensional construction [6]. Here we present a 5D conformal SUGRA realization [1] of this idea, using the off-shell superfield formulation developed in Refs. [18], [19]. 10 Lagrangian couplings, for the bulk H = (H, H c ) hypermultiplets', components are: where the odd fields V i are set to zero. 11 Σ 1 is the Z 2 even 5 th component of the 5D U(1) vector supermultiplet. With the parity assignments the KK decomposition for H and H c superfields is given by With these decompositions, and steps given in Appendix A, we can calculate the mass spectrum of KK states, their couplings to the inflaton and with these, the one loop order inflation potential (dropping higher winding modes) having the form of (1) with The 4D inflaton field φ Θ is related to the 5D U(1) gauge field A 1 5 as: Since the model is well defined, we also can write down the inflaton coupling with the components of H. The latter, having a coupling with the SM fields, would insure the inflaton decay and the reheating of the Universe. In our setup, we assume that all MSSM matter and scalar superfields are introduced at the y = 0 brane. Since H is even under orbifold parity and a singlet under all SM gauge symmetries, it can couple to the MSSM states through the following brane superpotential couplings where l and h u are 4D N = 1 SUSY superfields corresponding to lepton doublets and up type higgs doublet superfields respectively. In Eq. (26), without loss of generality, only one lepton doublet (out of three lepton families) is taken to couple with the H, where l now denotes the fermionic lepton doublet and h u an up-type higgs doublet. Statesl andh u stand for their superpartners respectively. H (n) and ψ (n) H in Eq. (27) indicate scalar and fermionic components of the superfield H. 12 11 The bulk hypermultiplet action of Eq. (21), derived from 5D off shell SUGRA construction [18], including coupling with a radion superfield T , in a rigid SUSY limit coincides with the one given in Ref. [27]. 12 In Eq. (27) we have omitted HF l h u and HlF hu type terms, which because of the smallness of the µ term(∼few×TeV) and suppressed lepton Yukawa couplings( < ∼ 10 −2 ) can be safely ignored in the inflaton decay proccess. Upon eliminating all F -terms and heavy fermionic and scalar states (in the H and H c superfields), we can derive effective operators containing the inflaton linearly. As it will turn out within the model considered (see discussion in Appendix A.1), thel states are heavier than the inflaton and operators containingl are irrelevant for the inflaton decay. Thus, the effective operators, needed to be considered, are These terms should be responsible for the inflaton decay. Derivation and form of the C-coefficients are given in Appendix A.

Inflaton Decay and Reheating
As was mentioned above and shown in Appendix A.1, the slepton statesl have masses 1 2 |F T | ∼ 1/(2R) and thus are heavier than the inflaton. Indeed, the latter's mass, obtained from the potential, is: (g 4 ≪ 1 for successful inflation). Thus the inflaton decay in channels containingl is kinematically forbidden. Therefore, among operators generated via exchange of heavy fermionic χ (n) i and scalar S (n) i states, only those given in Eq. (28) are relevant. For calculating the decay widths (in a pretty good approximation) it is enough to have the form of the C i coefficients.
As shown in Appendix A, within our model C 2 = 0 and the corresponding operator does not play any role. Moreover, according to Eqs. (A.26) and (A.30) we have C 0 ∼ R 2 and C 1 ∼ R 3 (with |F T | ∼ 1/R, dictated from the inflation). Thus, we get an estimate for the following branching ratio This means that the inflaton decay is mainly due to the C 0 operator, via the channel φ Θ → llh u h u (the diagram in Fig. 3 The factor 9 in the numerator accounts for the multiplicity of final states. (The final llh u h u channel includes three combinations e − e − h + h + , ννh 0 h 0 , e − νh + h 0 and for each pair of identical final states a factor 2 should be included.) The denominator factors in (31) come from the phase space integration. Using the form of C 0 , given by Eq. (A.26), in expression (31), we get: Expressing ρ reh = π 2 30 g * T 4 r through the reheat temperature [28] T r = 90 (g * is the number of relativistic degrees at temperature T r ) and using expressions (32) and (29), we get From this, with RM P l ∼ 10, R|F T | ∼ 1 and g 4 ∼ 1.5 · 10 −3 we obtain ρ 1/4 reh ∼ |λ| 2 × 100 GeV. Our 5D SUGRA construction allows more accurate estimates, because some of the parameters are related to each other. For instance, from (24) we have From (35) we see that in order to have RM P l > ∼ 10 we need V 1/4 0 < ∼ 3.4 · 10 16 GeV. 14 The latter value suites well with most of the values of V 1/4 0 given in Table 1 (calculated from the inflation potential). At the same time, we see from (35) that |F T | can not be suppressed and should be |F T | ∼ 1/R. Using Eqs.
This expression is useful to find the maximal value of ρ 1/4 reh . Using the pairs of (αM P l , V 0 ) given in Table 1  and M P l α were taken from Table 1, which correspond to successful inflation. Also, we have selected the values of R|F T | in such a way as to get RM P l > ∼ 10. We see that within 2σ deviations of r and n s we have ρ 1/4 reh < ∼ |λ| 2 × 386 GeV, while a reduced value r = 0.05 gives ρ 1/4 reh < ∼ |λ| 2 × 619 GeV. These correspond to reheat temperatures T r < ∼ |λ| 2 × 130 GeV and T r < ∼ |λ| 2 × 210 GeV respectively. These values can be easily reconciled with those low values of ρ 1/4 reh , given in Table 1, by natural selection of the brane Yukawa coupling λ in a range 1/300 < ∼ λ < ∼ 1.

Discussion and Concluding Remarks
In the effective action of our 5D conformal SUGRA model the 5-th component (Σ 1 ) of a U(1) vector supermultiplet couples to a charged hypermultiplet (H, H c ). This, due to a fixed compactification radius R leads to the potential of natural inflation for the CP odd part of Σ 1 , neglecting the suppressed higher winding modes. We analysed this potential like in [8] putting emphasis on the potential of inflation and the number of e-folds of perturbations leaving the horizon. This we compared with the number of e-folds required by a causal connection between the observed universe background fluctuations and by the size of observed curvature perturbations. For a large tensor component r, a small reheating temperature is needed for agreement. We inspected the decay of the gauge inflaton to the light MSSM fields living on a brane. These decays are mediated by the bulk hypermultiplet H. The very same H hypermultiplet, together with it's SU(2) R partner H c , generates the inflation potential. The H is assumed to have superpotential Yukawa couplings to brane fields with a Yukawa strength λ < ∼ 1. This naturally led to a suppressed decay width and reheat temperature T r ∼ |λ| 2 × 100 GeV. Within the considered scenario the dominant 4body decays of the inflaton are mediated by fermionic components ψ H (of H) with llhh final states (two lepton and two higgs doublets' components). Other channels are either kinematically forbidden due to heavy sleptonsl gaining large masses through the large F T term, a case of split SUSY, or are suppressed (due to the small inflaton mass M φ Θ ≪ 1/R) by an additional small factor (RM φ Θ ) 2 . Therefore, a similar mechanism can be realized also for extranatural inflation [6] without supersymmetry with a bulk fermionic ψ H generating the inflation potential and brane Yukawa coupling λlhψ H . Within our model (as shown in Appendix A), due to specific bulk couplings and degeneracy, the lepton number is conserved and neutrinos remain massless. The situation can be changed by introducing a brane Majorana mass term 1 2 M br HH and it is inviting to exploit such a possibility. Since this is not directly related to inflation, on one side, and trying to keep the calculus simple on the other side, we have not pursued this possibility in this paper and reserved it for future studies.
The model of [1], reanalyzed here in more detail, is by no means complete. A concrete mechanism for radion stabilization like in Ref. [22] has to be presented and the breaking of 4D SUSY has to be worked out in more detail. Here and in [1] we concentrated on the aspects that our model originates in a very straightforward way from 5D conformal SUGRA -which can be also interpreted as a result of M-theory [19] -and that the inflaton is related to a gauge field. If the new BICEP2 data, advertising large tensorial fluctuations, will turn out not to be mainly dust effects, then our gauge inflation is indeed a suitable and attractive candidate for inflationary model building. If further findings will discriminate the primordial origin, reveal dust effects and indicate a suppressed value of r, then as an alternative, the 'modulus' inflation of [1] should be pursued. This would mean that the inflaton is the real part of the Σ 1 chiral supermultiplet scalar component. Also, a more general two field inflation [10] from complex Σ 1 could get into focus again.

A KK spectrum and the inflaton effective couplings
First let us discuss the emergence of the non zero F T term of the T radion superfield. This can be easily understood by the effective 4D SUGRA description developed in [19]. The 4D supergravity action is given by [31] where K and W (Φ) are the Kähler potential and the superpotential respectively, while f IJ (Φ) is the gauge kinetic function. φ is the 4D compensator chiral superfield. Being a 4D effective theory, (A.1) would include zero modes of the 5D supermultiplets and the brane fields as well. Therefore, for the bulk states the form of (A.1) will be dictated by the 5D construction [19]. For instance, the 4D compensator φ is related with the 5D compensator as φ = √ 2πRκ  . From (A.1) we find the expressions for the F-terms: where I runs over all scalars. By plugging Eq. (A.2) back in to (A.1), one derives the F -term scalar potential (by setting φ = M P and going to the 4D Einstein-frame, rescaling the metric g µν → e K/3 g µν ): For the T modulus (the radion) the Kähler potential is K = −3 ln(T + T † ). For the time being we take W =const. for the superpotential. 15 With these, it is easy to check that we get a flat potential V F = 0 with F φ = 0 and F T = M P W * . Thus, we have fixed a non zero F T which plays a crucial role for the generation of the inflaton potential. This is enough for performing a calculation of the KK spectrum and the 1-loop inflaton potential. We will come back to the SUSY breaking at the end, upon discussion of the superpartners' spectrum from the MSSM brane fields. Any bulk state transforming non trivially under SU(2) R feels F T SUSY breaking. This happens of course with the bulk hypers described by the terms in (21). With the parametrization setting the scalar component of T to one, and making a phase redefinition of the scalar components H, H c : the couplings in (21) give the potential: With the decomposition of Eq. (23) and integrating along the fifth dimension 2πR 0 for n = 1, · · · , +∞ , are real scalars) and the potential mass terms will get diagonal and canonical forms: for n = 1, · · · , +∞ : (m As for the spectrum of the fermionic components of the H, H c superfields, with the phase redefinition from Eq. (21) we get the couplings With the mode expansion of Eq. (23) and integration over the fifth dimension for n = 1, · · · , +∞ : ψ from Eq. (A.13) we will get diagonal and canonically normalized mass terms: With this spectrum, integrating out the corresponding KK states (including zero modes) leads to the 1-loop effective potential [1], [9]: written in terms of canonically normalized 4D scalar fields φ Θ = √ 2πRA 1 5 , φ M = √ 2πRM 1 and dimensionless 4D gauge coupling g 4 = g 1 / √ 2πR. In (A.17) summation is performed with k winding modes. The dominant contribution comes from k = 1 [29]. With this leading term, the minimum of the potential is achieved for g 4 R φ Θ = g 1 R A 1 5 = 1 and φ M = 0. Further, we assume that the modulus φ M (i.e. M 1 ) is sitting in its minimum and study only the motion of φ Θ 's quantum part as the inflaton. We add to the potential (A.17) a constant term in such a way as to set the ground state vacuum energy to be zero (usual fine tuning of 4D cosmological constant). With these, the inflaton potential (part with k = 1) gets the form of Eq. (1) with the parametrization given in Eq. (24).
Further, we work out the effective couplings of the inflaton with the MSSM states. For this purpose, in couplings (A.9), (A.16) (and in any relevant term) we make the substitution and put M 1 = 0. With this, we obtain the linear couplings of the inflaton with the heavy S i states: At the same time, with (A.18) from (A.14) we have ω n = −ω n = π/4, and Eq. (A.15) gives inflaton couplings with heavy χ i states: Furthermore, we derive couplings of S i and χ i states with the corresponding components of the brane superfields l, h u . As shown in Appendix A.1, thel states are heavy. Because of this, they will not be relevant for the inflaton decay and we will omit any term containing thel. From the part of Eq. (27) involving ψ H states we obtain On the other hand, making (A.5) phase redefinitions, the part of Eq. (27) involving H gives: From Eq. (A.22) we get S i lh u type couplings: Now, we integrate out the heavy χ i and S i states, in order to obtain effective operators. Starting with the integration of the fermionic modes, at relatively low energies, we can ignore kinetic terms for the χ i states. With this, via equations of motions δL δχ 1 (n) = δL δχ 2 (n) = 0, we can solve χ 1,2 (n) and plug them back into the Lagrangian. Doing so [using couplings of Eqs. (A.15), (A.20) and (A.21)] and keeping terms up to the first power of φ Θ , we obtain: = (2n + 1)/(2R). Using this in (A.24), we see that the first sum-term (coefficient in front of d = 5 operator) cancels out, i.e. no d = 5 lepton number violating operator emerges. This is understandable, because the whole theory has U(1) gauge symmetry and the lepton number is a residual global symmetry (with M 1 = 0) at d = 5 level. 16 Thus, from Eq. (A.25) we obtain where subscript (χ) indicates that this d = 6 operator is obtained through the integration of the heavy χ i states. The sum in (A.25) is well convergent because +∞ n=0 1 (2n+1) 2 = π 2 8 . It turns out that a φ Θ (lh u ) 2 type operator emerges only via integration of the χ i states. Taking into account these, comparing Eq. (A.25) and (28) we have Next, by integrating out heavy S i states, the φ Θ (lh u ) 2 and φ Θ (lh u )(lh u ) type dimension 7 operators will be C 1 φ Θ (lh u ) 2 + h.c. + C 2 φ Θ (lh u )(lh u ) , While the C 2 is precisely zero, the C 1 vanishes in the F T → 0 limit. However, with |F T | ∼ 1/R, we have C 1 ∼ R 3 . Remaining operators, as discussed in Sec. 3.1, will not have any relevance for the inflaton decay and we will not present them here.

A.1 SUSY breaking on a brane
We assume that all MSSM states, that are matter {f }, gauge {V } and higgs h u , h d superfields, live on a 4D brane. Matter superfields can be included in the Kähler potential as follows where K(h u,d ) account for part of the higgs superfields and will be specified below. With (A.31), from Eq. (A.3), for squark and slepton masses we get Due to the brane superpotential coupling of l, h u with H state, there will be also a loop induced contribution to the soft mass 2 , which we do not display here. Thus, with the large F T -term all squark and sleptons are heavier than the inflaton field and they play no role for the inflaton decay.
On the other hand we need to keep at least one Higgs doublet to be light. Since the SUSY breaking scale is very high, this can be achieved only by price of fine tuning: assuming for instance that the light Higgs mainly resides in h u , and selecting its Kähler potential as ( ), we obtain M hu ∼ O(100 GeV) -the needed value. As far as the gaugino masses are concerned, since the MSSM gauge supermultiplets are introduced on a brane they will not have direct couplings neither with the T modulus nor with the compensator. By selecting, in Eq. (A.1), the gauge kinetic function f IJ = δ IJ , the corresponding gauginos will remain light. By the same token, the higgsino mass -the µ parameter, can be around the TeV scale. Therefore, the lightest neutralino can be a dark matter candidate. This is the split SUSY scenario, which, as was shown [32], can have various remarkable phenomenological features and interesting implications.