Implementing Hilltop F-term Hybrid Inflation in Supergravity

# Implementing Hilltop F-term Hybrid Inflation in Supergravity

R. Armillis and C. Pallis

Institut de Théorie des Phénomenès Physiques,
École Polytechnique Fédérale de Lausanne,

BSP 730 Cubotron, CH-1015 Lausanne, SWITZERLAND
roberta.armillis@epfl.ch
Department of Physics, University of Cyprus,
P.O. Box 20537, CY-1678 Nicosia, CYPRUS
cpallis@ucy.ac.cy
###### Abstract

F-term hybrid inflation (FHI) of the hilltop type can generate a scalar spectral index, , in agreement with the fitting of the seven-year Wilkinson microwave anisotropy probe data by the standard power-law cosmological model with cold dark matter and a cosmological constant, CDM. We investigate the realization of this type of FHI by using quasi-canonical Kähler potentials with or without the inclusion of extra hidden-sector fields. In the first case, acceptable results can be obtained by constraining the coefficients of the quadratic and/or quartic supergravity correction to the inflationary potential and therefore a mild tuning of the relevant term of the Kähler potential is unavoidable. Possible reduction of without generating maxima and minima of the potential on the inflationary path is also possible in a limited region of the available parameter space. The tuning of the terms of the Kähler potential can be avoided with the adoption of a simple class of string-inspired Kähler potentials for the hidden-sector fields which ensures a resolution to the problem of FHI and allows acceptable values for the spectral index, constraining the coefficient of the quartic supergravity correction to the inflationary potential. Performing a four-point test of the analyzed models, we single out the most promising of these.
Keywords: Cosmology, Inflation
PACS codes: 98.80.Cq, 11.30.Pb

## 1 Prologue

Inflation [2] has been incredibly successful in providing solutions to the problems of the Standard Big Bang cosmology (SBB). It can set the initial conditions, which give rise to the high degree of flatness and homogeneity that we observe in the universe today. From particle physics motivated models, it not only yields a mechanism for accelerated expansion but also explains, through quantum fluctuations, the origin of the temperature anisotropies in the Cosmic Microwave Background (CMB) and the seeds for the observed Large Scale Structure – for reviews see e.g. Refs. [3, 4].

We focus on a set of well-motivated, popular and quite natural models of supersymmetric (SUSY) F-term hybrid inflation (FHI[5]. Namely, we consider the standard [6] FHI and some of its specific versions: the shifted [7] and smooth [8] FHI. They are realized [6] at (or close to) the SUSY Grand Unified Theory (GUT) scale and can be easily linked to several extensions [4] of the Minimal Supersymmetric Standard Model (MSSM) which have a rich structure. Namely, the -problem of MSSM is solved via a direct coupling of the inflaton to Higgs superfields [9] or via a Peccei-Quinn symmetry [10], baryon number conservation is an automatic consequence [9] of an R symmetry and the baryon asymmetry of the universe is generated via leptogenesis which takes place [11] through the out-of-equilibrium decays of the inflaton’s decay products.

Although quite successful, these models have at least two shortcomings:

• The problem of the enhanced (scalar) spectral index, . It is well-known that under the assumption that the problems of SBB are resolved exclusively by FHI, these models predict just marginally consistent with the fitting of the seven-year results [12] from the Wilkinson Microwave Anisotropy Probe Satellite (WMAP7) data with the standard power-law cosmological model with cold dark matter and a cosmological constant (CDM).

• The so-called problem. This problem is tied [3, 5, 13] on the expectation that supergravity (SUGRA) corrections generate a mass squared for the inflaton of the order of the Hubble parameter during FHI and so, the criterion is generically violated, ruining thereby FHI. Inclusion of SUGRA corrections with canonical Kähler potential prevents [5, 14, 15] the generation of such a mass term due to a mutual cancellation. However, despite its simplicity, the canonical Kähler potential can be regarded [5] as fine tuning to some extent and, in all cases, increases even more.

In this topical review we reconsider one set of possible resolutions (for other proposals, see Ref. [16, 17, 18, 19]) of the tension between FHI and the data. This is relied on the utilization of three types of quasi-canonical [20] Kähler potential with or without the inclusion of extra fields, . The term “extra fields” refers to hidden-sector [21] fields or fields which do not participate [22] in the inflationary superpotential but may only affect the Kähler potential. The consideration of extra fields assists us in solving the problem of FHI as well. In particular, we review the following embeddings of FHI in SUGRA:

• FHI in next-to-minimal SUGRA (nmSUGRA) – see Sec. 5. A convenient choice of the next-to-minimal term [23, 24, 25] of the Kähler potential leads to a negative mass (quadratic) term for the inflaton and therefore can be diminished sizeably.

• FHI in next-to-next-to-minimal SUGRA (nnmSUGRA) – see Sec. 6. A convenient choice of the next-to-minimal and the next-to-next-to-minimal term generates [26, 27] a positive mass (quadratic) term for the inflaton and a sizeable negative quartic term which yield acceptable enhancing somehow the running of , .

• FHI with extra fields, , obeying a string-inspired Kähler potential (hSUGRA) – see Sec. 7. In the presence of ’s, we can establish [28] a type of FHI which avoids the tuning – required in the cases (i) and (ii) above – of the quadratic SUGRA correction and is largely dominated by the quartic SUGRA correction. Namely, the coefficients of the Kähler potential are constrained to natural values (of order unity) so as the mass term of the inflaton field is identically zero.

In all the cases above and in the largest part of the parameter space the inflationary potential acquires a local maximum and minimum. Then, FHI of the hilltop [29, 30] type can occur as the inflaton rolls from this maximum down to smaller values. However, the value of the inflaton field at the maximum is to be sufficiently close to the value that this field acquires when the pivot scale crosses outside the inflationary horizon. Therefore, can become consistent with data, but only at the cost of an extra indispensable mild tuning [23] of the initial conditions. Another possible complication is that the system may get trapped near the minimum of the inflationary potential, thereby jeopardizing the attainment of FHI. On the other hand, we can show [24] that acceptable ’s can be obtained even maintaining the monotonicity of the inflationary potential, i.e. without this minimum-maximum problem in the case of nmSUGRA.

In this presentation we reexamine the above ideas for the reduction of within FHI, updating our results in Ref. [25, 28] and incorporating recent related developments in Ref. [26]. In particular, the text is organized as follows: In Sec. 2, we review the basic FHI models and in Sec. 4 we recall the results holding for FHI in minimal SUGRA (mSUGRA). In the following we demonstrate how we can obtain hilltop FHI using various types of Kähler potentials – see Secs 5, 6 and 7. Our conclusions are summarized in Sec. 8. Throughout the text, charge conjugation is denoted by a star and brackets are, also, used by applying disjunctive correspondence.

## 2 FHI within SUGRA

We outline the salient features of the basic types of FHI. Namely we present the relevant superpotentials in Sec 2.1 and the SUSY potentials in Sec. 2.2. We then (in Sec. 2.3) describe the embedding of these models in SUGRA and extract the relevant inflationary potential in Sec. 2.4.

### 2.1 The Relevant Superpotential

The F-term hybrid inflation can be realized adopting one of the superpotentials below:

 (1)

where we allow for the presence of a part, , which depends exclusively on the hidden sector superfields, . Here (and hereafter) we use the hat to denote such quantities. To keep our analysis as general as possible, we do not adopt any particular form for – for some proposals see Ref. [31, 32]. Note that our construction remains intact even if we set  as it was supposed in Ref. [22]. This is due to the fact that is expected to be much smaller than the inflationary energy density – see Sec. 2.3. For , though, we need to assume that ’s are stabilized before the onset of FHI by some mechanism not consistently taken into account here. As a consequence, we neglect the dependence of , and on and so, these quantities are treated as constants. We further assume that the D-terms due to ’s vanish – contrary to the strategy adopted in Ref. [22].

The remaining symbols in the right hand side (r.h.s) of Eq. (1) are identified as follows:

• is a left handed superfield, singlet under a GUT gauge group ;

• , is a pair of left handed superfields belonging to non-trivial conjugate representations of , and reducing its rank by their vacuum expectation values (v.e.vs);

• is an effective cutoff scale comparable with the string scale;

• and are parameters which can be made positive by field redefinitions.

The superpotential in Eq. (1) for standard FHI is the most general renormalizable superpotential consistent with a continuous R-symmetry [6] under which

 S → eirS, ¯ΦΦ → ¯ΦΦ, W→ eirW. (2)

After including in this superpotential the leading non-renormalizable term, one obtains the superpotential of shifted [7] FHI in Eq. (1). Finally, the superpotential of smooth [8] FHI can be obtained if we impose an extra symmetry under which the combination has unit charge.

### 2.2 The SUSY Potential

The SUSY potential, , extracted (see e.g. ref. [3]) from in Eq. (1) includes F and D-term contributions. Namely,

 VSUSY=VF+VD, (3)

where

##### (i)

The F-term contribution can be written as:

 VF=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩κ2M4((\footnotesizeΦ2−1)2+2\footnotesizeS2\footnotesizeΦ2)for standard FHI,κ2M4((\footnotesizeΦ2−1−ξ\footnotesizeΦ4)2+2\footnotesizeS2\footnotesizeΦ2(1−2ξ\footnotesizeΦ2)2)for shifted FHI,μ4S((1−\footnotesizeΦ4)2+8\footnotesizeS2\footnotesizeΦ6)for smooth FHI, (4)

where the scalar components of the superfields are denoted by the same symbols as the corresponding superfields and with [7]. In order to recover the properly normalized energy density during FHI, we absorb in the constants of Eq. (4) some normalization pre-factors emerging from the SUGRA potential – see below – so that their definition is

 κ=eˆK/2m2PˆZ−1/2ˆκ,  μS=eˆK/4m2PˆZ−1/4ˆμS  and  MS=e−ˆK/4m2PˆZ1/4ˆMS (5)

where and are the -dependent parts of the Kähler potential, , considered in Sec. 2.3. The last relation is introduced so as and . Also, we use [7, 8] the following dimensionless quantities

 ⎧⎨⎩\footnotesizeΦ=|Φ|/M  and  \footnotesizeS=ˆZ1/2|S|/Mfor standard or shifted % FHI,\footnotesizeΦ=|Φ|/√μSMS  and  \footnotesizeS=ˆZ1/2|S|/√μSMSfor smooth FHI. (6)

In Fig. 1-(a), Fig. 1-(b) and 2 we present the three dimensional plot of versus and for standard, shifted and smooth FHI, respectively.

##### (ii)

The D-term contribution vanishes for since has the form:

where is the (unified) gauge coupling constant, are the generators of and the notation used is explained below Eq. (11) – recall that and belong to the conjugate representation of .

From the form of in Eq. (4), we can understand that in Eq. (1) plays a twofold crucial role:

##### (i)

It leads to the spontaneous breaking of . Indeed, the vanishing of gives the v.e.vs of the fields in the SUSY vacuum. Namely,

 ⟨S⟩=0  and  |⟨¯Φ⟩|=|⟨Φ⟩|=vG=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩Mfor standard FHI,M√1−√1−4ξ√2ξfor shifted FHI,√μSMSfor smooth FHI (8)

(in the case where , are not Standard Model (SM) singlets, , stand for the v.e.vs of their SM singlet directions). The non-zero value of the v.e.v signalizes the spontaneous breaking of .

##### (ii)

It gives rise to FHI. This is due to the fact that, for large enough values of , there exist valleys of local minima of the classical potential with constant (or almost constant in the case of smooth FHI) values of . In particular, we can observe that takes the following constant value

 VHI0=⎧⎪ ⎪⎨⎪ ⎪⎩κ2M4κ2M4ξμ4Salong the % direction(s):\footnotesizeΦ=⎧⎪ ⎪⎨⎪ ⎪⎩0% for standard FHI,0or1/√2ξfor % shifted FHI,0or1/2√3\footnotesizeSfor smooth FHI, (9)

with . From Figs. 1 and 2 we deduce that the flat direction corresponds to a minimum of , for , in the cases of standard and shifted FHI and to a maximum of in the case of smooth FHI. The inflationary trajectories are depicted by bold points, whereas the critical points by red/light points. Note that critical points exist only in the case of standard – for – and shifted – for – FHI but not for smooth FHI. In the case of Fig. 1-(b), the implementation of shifted FHI is ensured by restricting in the range [7]. Under this assumption, the shifted track lies lower that the trivial one and so, it is energetically more favorable to drive FHI.

Since FHI can be attained along a minimum of , we infer that, during standard FHI, the GUT gauge group is necessarily restored. As a consequence, topological defects such as strings [33, 23], monopoles, or domain walls may be produced [8] via the Kibble mechanism [34] during the spontaneous breaking of at the end of FHI. This can be avoided in the other two cases, since the form of allows for non-trivial inflationary valleys along which is spontaneously broken due to non-zero values that and acquire during FHI. Therefore, no topological defects are produced in these cases.

### 2.3 SUGRA Corrections

The consequences that SUGRA has on the models of FHI can be investigated by restricting ourselves to the inflationary trajectory (possible corrections due to the non-vanishing and in the cases of shifted and smooth FHI are expected to be negligible). Therefore, in Eq. (1) takes the form

 W=ˆW+I ,whereI=−ˆV1/2HI0SwithˆVHI0=e−ˆK/m2PˆZVHI0. (10)

The SUGRA scalar potential (without the D-terms) is given (see, e.g., Ref. [3]) by

 VSUGRA=eKm2P(KM¯NFMF∗¯N−3|W|2m2P)whereFN=WN+KNWm2P (11)

are the SUGRA-generalized F-terms, the subscript denotes derivation with respect to (w.r.t) the complex scalar field which corresponds to the chiral superfield with and the matrix is the inverse of the Kähler metric . In this paper we consider a quite generic form of Kähler potentials, which do not deviate much from the canonical one and respect the R symmetry of Eq. (2). Namely we take

 K = ˆK+ˆZ|S|2+14k4SˆZ2|S|4m2P+16k6SˆZ3|S|6m4P+18k8SˆZ4|S|8m6P (12) +110k10SˆZ5|S|10m8P+112k12SˆZ6|S|12m10P+|Φ|2+|¯Φ|2+⋯,

where and are positive or negative constants of order unity and the ellipsis represents higher order terms involving the waterfall fields ( and ) and . We can neglect these terms since they are irrelevant along the inflationary path.

Substituting Eq. (12) into Eq. (11) and expanding in powers of , we end up with an expansion of the form:

 VSUGRA≃VHI0⎛⎜⎝1−2c1Ke−ˆK2m2P|ˆW|ˆZ1/2√VHI0|S|cosθ+5∑ν=1(−1)νc2νKˆZν(|S|mP)2ν⎞⎟⎠ (13)

where the phase reads . In the r.h.s of the expression above, we neglect terms proportional to which are certainly subdominant compared with those which are proportional to . From the terms proportional to we present the second term of the r.h.s of Eq. (13) which expresses the most important contribution [15, 31] to the inflationary potential from the soft SUSY breaking terms. For natural values of and this term starts [15] playing an important role in the case of standard FHI in mSUGRA – see Sec. 4.2 – for whereas it has [15] no significant effect in the cases of shifted and smooth FHI.

Taking in Eq. (12) and (as in Secs. 46) the coefficients are found to be

 c(0)1K = −2, (\theparentequationa) c(0)2K = k4S, (\theparentequationb) c(0)4K = 12−7k4S4+k24S−3k6S2, (\theparentequationc) c(0)6K = −23+3k4S2−7k24S4+k34S+10k6S3−3k4Sk6S+2k8S, (\theparentequationd) c(0)8K = 38−5k10S2−13k4S24+41k24S32−7k34S4+k44S−13k6S4 (\theparentequatione) +143k4Sk6S24−9k24Sk6S2+9k26S4−39k8S8+4k4Sk8S, c(0)10K = −215+32k10S5+3k12S+k4S24−5k10Sk4S−13k24S24+41k34S32 (\theparentequationf) −7k44S4+k54S+5k6S3−29k4Sk6S6+103k24Sk6S12−6k34Sk6S−5k26S +27k4Sk26S4+5k8S−67k4Sk8S8+6k24Sk8S−6k6Sk8S.

We observe that terms of order in the expansion of Eq. (12) give rise to contributions of order equal or greater than in the expansion of Eq. (13).

### 2.4 The Inflationary Potential

The general form of the potential which can drive the various versions of FHI reads

 VHI≃VHI0⎛⎝1+cHI−aSσ√2VHI0+5∑ν=1(−1)νc2νK(σ√2mP)2ν⎞⎠, (1)

where is the canonically (up to the order ) normalized inflaton field and we take which minimizes for given . To facilitate our numerical analysis, we introduce the real tadpole parameter defined in terms of the parameter, by the relation

 aS=2c1Ke−ˆK/2m2P|ˆW|. (2)

In Eq. (1), besides the contributions originating from in Eq. (13), we include the term which represents a correction to resulting from the SUSY breaking on the inflationary valley, in the cases of standard [6] and shifted [7] FHI, or from the structure of the classical potential in the case of smooth [8] FHI. Indeed, breaks SUSY and gives rise to logarithmic radiative corrections to the potential originating from a mass splitting in the supermultiplets. On the other hand, in the case of smooth [8] FHI, the inflationary valleys are not classically flat and, thus, the radiative corrections are expected to be subdominant. The term can be written as follows:

 cHI=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩κ2\footnotesizeN[2ln(κ2xM2/Q2)+frc(x)]/32π2%forstandardFHI,κ2[2ln(κ2xξM2ξ/Q2)+frc(xξ)]/16π2for shifted FHI,−2μ2sM2S/27σ4% for smooth FHI, (3)

with and

 frc(x)=(x+1)2ln(1+1/x)+(x−1)2ln(1−1/x)⇒frc(x)≃3forx≫1. (4)

Also is the dimensionality of the representations to which and belong and is a renormalization scale. For the values of encountered in our work renormalization group effects [35] remain negligible.

In our applications in Secs. 4.2, 5.3, 6.3 and 7.3 we take . This choice corresponds to the left-right symmetric GUT gauge group with and belonging to doublets with and 1 respectively. No cosmic strings are produced during the GUT phase transition and, consequently, no extra restrictions on the parameters (as e.g. in Refs. [33]) have to be imposed. As regards the case of shifted [7] FHI we identify with the Pati-Salam gauge group . Needless to say that the case of smooth FHI is independent on the adopted GUT since the inclination of the inflationary path is generated at the classical level and the addition of any radiative correction is expected to be subdominant. Negligible is also the third term in the r.h.s of Eq. (1) for , besides the case of standard FHI in mSUGRA – see Sec. 4 – where it may be important for . For simplicity, we neglect it, in the analysis of the remaining cases – see Secs. 5, 6 and 7.

## 3 Constraining FHI

The parameters of FHI models can be restricted imposing a number of observational constraints described in Secs. 3.1 and 3.2. Additional theoretical considerations presented in Sec. 3.3 can impose further limitations.

### 3.1 Inflationary Observables

Applying standard formulae – see e.g. Refs. [3, 4] – we can estimate the inflationary observables of FHI. Namely, we can find:

##### (i)

The number of e-foldings that the scale suffers during FHI,

 NHI∗=1m2P∫σ∗σfdσVHIV′HI, (5)

where the prime denotes derivation w.r.t , is the value of when the scale crosses outside the horizon of FHI, and is the value of at the end of FHI, which can be found, in the slow roll approximation, from the condition

 \footnotesizemax{ϵ(σf),|η(σf)|}=1,  where  ϵ≃m2P2(V′HIVHI)2  and  η≃m2P V′′HIVHI⋅ (6)

In the cases of standard [6] and shifted [7] FHI and in the parameter space where the terms in Eq. (11) do not play an important role, the end of inflation coincides with the onset of the GUT phase transition, i.e. the slow roll conditions are violated close to the critical point [] for standard [shifted] FHI, where the waterfall regime commences. On the contrary, the end of smooth [8] FHI is not abrupt since the inflationary path is stable w.r.t for all ’s and is found from Eq. (6). An accurate enough estimation of ’s – suitable for our analytical expressions presented below – is

 σf≃⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩√2Mfor standard % FHI,Mξfor shifted FHI,√23√5/33√μSMSmPfor smooth FHI. (7)
##### (ii)

The power spectrum of the curvature perturbations generated by at the pivot scale

 ΔR=12√3πm3PV3/2HI|V′HI|∣∣ ∣∣σ=σ∗⋅ (8)
##### (iii)

The spectral index

 ns=1−6ϵ∗ + 2η∗, (\theparentequationa)

its running

 αs=23(4η2∗−(ns−1)2)−2ξ∗, (\theparentequationb)

with and the scalar-to-tensor ratio

 r=16ϵ∗ (\theparentequationc)

where all the variables with the subscript are evaluated at .

### 3.2 Observational Constraints

Under the assumption that the contribution in Eq. (8) is solely responsible for the observed curvature perturbation – i.e. there are no contributions to from curvatons [3] or topological defects [33] – and (ii) there is a conventional cosmological evolution after FHI – see point (i) below –, the parameters of the FHI models can be restricted imposing the following requirements:

##### (i)

The number of e-foldings computed by means of Eq. (5) has to be set equal to the number of e-foldings elapsed between the horizon crossing of the observationally relevant mode and the end of FHI. can be found as follows [3]:

 k∗H0R0=H∗R∗H0R0 = H∗H0R∗RHfRHfRrhRrhReqReqR0 (1) = √VHI0ρc0e−N∗(VHI0ρrh)−1/3(ρrhρeq)−1/4(ρeqρm0)−1/3 ⇒ N∗≃lnH0R0k∗+24.72+23lnV1/4HI01 GeV+13lnTrh1 GeV,

where is the reheating temperature after the completion of the FHI. Moreover, is the scale factor, is the Hubble rate, is the energy density and the subscripts , , Hf, rh, eq and m denote respectively values at the present (except for the symbol ), at the horizon crossing () of the mode , at the end of FHI, at the end of the reheating period, at the radiation-matter equidensity point and in the matter dominated era. In our calculation we take into account that for decaying-particle domination or matter dominated era and for radiation dominated era. We use the following numerical values:

 ρc0=8.099×10−47h20 GeV4  with  h0=0.71, (\theparentequationa) ρrh=π230gρ∗T4rh  %with  gρ∗=228.75, (\theparentequationb) ρeq=2Ωm0(1−zeq)3ρc0  with  Ωm0=0.26  and  zeq=3135. (\theparentequationc)

Setting and in Eq. (1) we arrive at

 NHI∗≃22.6+23lnV1/4HI01 GeV+13lnTrh1 GeV⋅ (1)

Throughout our investigation we take as in the majority of these models [4, 11, 15] saturating conservatively the gravitino constraint [36]. This choice for does not affect crucially our results, since appears in Eq. (1) under its logarithm raised to the one third power and therefore, its variation over two or three orders of magnitude has a minor influence on the final value of .

##### (ii)

The power spectrum of the curvature perturbations given by Eq. (8) is to be confronted with the WMAP7 data [12]:

 ΔR≃4.93⋅10−5  at  k∗=0.002/Mpc. (2)
##### (iii)

According to the fitting of the WMAP7 results by the cosmological model CDM, at the pivot scale has to fall within the following range of values [12]:

 ns=0.968±0.024 ⇒ 0.944≲ns≲0.992  at 95% c.l. (3)
##### (iv)

Limiting ourselves to ’s consistent with the assumptions of the power-law CDM cosmological model, we have to ensure that remains negligible. Since, within the cosmological models with running spectral index, ’s of order 0.01 are encountered [12], we impose the following upper bound:

 |as|≪0.01. (4)

### 3.3 Theoretical Considerations

From a more theoretical point of view, the models of (hilltop) FHI can be better refined using the following criteria:

##### (i)

Gauge coupling unification. When contains non-abelian factors (beyond the SM one), the mass, , of the lightest gauge boson at the SUSY vacuum, Eq. (8) is to take the value dictated by the unification of the gauge coupling constants within MSSM, i.e.,

 gvG≃2⋅1016 \rm GeV⇒vG≃2.86⋅1016 \rm GeV  with  g≃0.7, (5)

being the value of the unified gauge coupling constant. However, we display in the following results for standard FHI which do not fulfill Eq. (5). This is allowed since the relevant restriction can be evaded if includes only abelian factors (beyond the SM one) which do not disturb the gauge coupling unification. Otherwise, threshold corrections may be taken into account in order to restore the unification.

##### (ii)

Boundness of . The inflationary potential is expected to be bounded from below. This requirement lets open the possibility that the inflaton may give rise to an inflationary expansion under generic initial conditions set at .

##### (iii)

Convergence of . The expression of in Eq. (1) is expected to converge at least for . This fact can be ensured if, for , each successive term in the expansion of (and ) Eq. (13) (and Eq. (12)) is smaller than the previous one. In practice, this objective can be easily accomplished if the ’s in Eq. (12) – or Eq. (1) – are sufficiently low.

##### (iv)

Monotonicity of . Depending on the values of the coefficients ’s in Eq. (1), is a monotonic function of or develops a local minimum and maximum. The latter case leads to the possible complication in which the system gets trapped near the minimum of the inflationary potential and, consequently, no FHI takes place. It is, therefore, crucial to check if we can avoid the minimum-maximum structure of . In such a case the system can start its slow rolling from any point on the inflationary path without the danger of getting trapped. This can be achieved, if we require that is a monotonically increasing function of , i.e. for any or, equivalently,

 V′HI(¯σmin)>0  with  V′′HI(¯σmin)=0  and  V′′′HI(¯σmin)>0 (6)

where is the value of at which the minimum of lies.

##### (v)

Tuning of the initial conditions. When hilltop FHI occurs with rolling from the region of the maximum down to smaller values, a mild tuning of the initial conditions is required [23] in order to obtain acceptable ’s. In particular, the lower we want to obtain, the closer we must set to , where is the value of at which the maximum of lies. To quantify somehow the amount of this tuning in the initial conditions, we define [23] the quantity:

 Δm∗=(σmax−σ∗)/σmax. (7)

The naturalness of the attainment of FHI increases with .

## 4 FHI in mSugra

The simplest choice of Kähler potential emerging from the expression of Eq. (12) is the one which assures canonical kinetic terms for the inflaton field, , with the minimal number of terms. This choice is specified in Sec. 4.1 and our results are discussed in Sec. 4.2.

### 4.1 The Relevant Set-up

The used Kähler potential in this case can be derived from Eq. (12) by setting:

 ˆK=0, ˆZ=1  and  k4S=k6S=k8S=k10S=k12S=0. (8)

Upon substituting Eqs. (\theparentequationa) – (\theparentequationf) into Eq. (1) we infer that the resulting takes the form

 VHI≃VHI0(1+cHI−aSσ√2VHI0+σ48mP), (9)

since in this case and . It is worth mentioning that mSUGRA is, in principle, beneficial for the implementation of FHI, since it does not generate any new contribution in the parameter, Eq. (6), due to a miraculous cancellation emerging in the computation of . Despite this fact, fixing all the remaining terms in Eq. (12) beyond the quadratic term equal to zero can be regarded as an ugly tuning.

### 4.2 Results

The investigation of this model of FHI depends on the parameters:

 σ∗,vG,aS  and{κfor standard and shifted FHI,MSfor% smooth FHI,

where we fix in the case of shifted FHI. In our computation, we use as input parameters and or . We then restrict and so as Eqs. (1) and (2) are fulfilled. Using Eqs. (\theparentequationa) and (\theparentequationb) we can extract and respectively. Our findings for standard [shifted and smooth] FHI are displayed in Sec. 4.2.1 [Sec. 4.2.2].

We can obtain a rather accurate estimation of the expected ’s if we omit the third term in the r.h.s of Eq. (9), calculate analytically the integral in Eq. (5), replace the ’s by their values in Eq. (7) and solve the resulting equation w.r.t . Taking into account that we can extract from Eq. (\theparentequationa) and find

 ns≃⎧⎪ ⎪⎨⎪ ⎪⎩1−1/NHI∗+3κ2\footnotesizeNNHI∗/4π2for standard FHI,1−1/NHI∗+3κ2NHI∗/2π2for shifted FHI,1−5/3NHI∗+2(6μ2SM2SNHI∗/m4P)1/3for smooth FHI. (10)

Observing that the last term in the r.h.s of the expressions above arise from the last term in the r.h.s of Eq. (9), we can easily infer that mSUGRA increases significantly for relatively large ’s or ’s.

On the other hand, the third term in the r.h.s of Eq. (9) can be important for and – cf. Ref. [15]– since it becomes comparable with the second term there. In this regime, the required by Eq. (1) becomes comparable to and the approximation of in Eq. (4) is no longer valid. Instead, we here have

 frc(x)=3−x−26−x−430−x−684−x−8180−x−10330−x−12546−x−14840−x−161224−x−181710−x−202310−⋯ (11)

As a consequence, decreases sharply (enhancing ) whereas (or ) increases adequately, lowering thereby to an acceptable level.

#### 4.2.1 Standard FHI

In the case of standard FHI (with ), we display in Fig. 3-(a) the allowed by Eqs. (1) and (2) values of versus . The corresponding variation of versus is depicted in Fig. 3-(b) where the observationally compatible region of Eq. (3) is also delimited by thin lines. Solid, dashed and dot-dashed lines stand for the results obtained for and respectively. We observe that the various lines coincide for . For the sake of clarity we do not show in Fig. 3 solutions with or – cf. Ref. [15] – which are totally excluded by Eq. (3). The third [last] term in the r.h.s of Eq. (9) become important for [] whereas for , the second term in the r.h.s of Eq. (9) becomes prominent. As a consequence, the last term in the r.h.s of Eq. (9) drives to values close to or larger than unity whereas the third one succeed in reconciling it with Eq. (3) for discriminated ’s related to the chosen ’s. Namely, from Fig. 3 we deduce that there is a marginally allowed area for

 0.0015≲κ≲0.032,  5.5≲vG/(1015 GeV)≲7.5, (\theparentequationa) 0.983≲ns≲0.99  and  1.2≲|αs|/10−4≲3.5. (\theparentequationb)

In addition, from Fig. 3 we find isolated corridors consistent with Eq. (3), e.g.

 κ≃3⋅10−4, 5⋅10−4, 6⋅10−4  % with  vG≲3⋅1015 \rm GeV  for%   aS=1,5  and  10 \rm TeV, (\theparentequationc)

respectively. We remark that the ’s allowed here lie well below the ones required by Eq. (5). In conclusion, although standard FHI in mSUGRA can not be excluded, it can be considered as rather disfavored since the allowed region is extremely limited.