Consistent cosmology with Higgs thermal inflation in a minimal extension of the MSSM

# Consistent cosmology with Higgs thermal inflation in a minimal extension of the MSSM

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###### Abstract

We consider a class of supersymmetric inflation models, in which minimal gauged F-term hybrid inflation is coupled renormalisably to the minimal supersymmetric standard model (MSSM), with no extra ingredients; we call this class the “minimal hybrid inflationary supersymmetric standard model” (MHISSM). The singlet inflaton couples to the Higgs as well as the waterfall fields, supplying the Higgs -term. We show how such models can exit inflation to a vacuum characterised by large Higgs vevs, whose vacuum energy is controlled by supersymmetry-breaking. The true ground state is reached after an intervening period of thermal inflation along the Higgs flat direction, which has important consequences for the cosmology of the F-term inflation scenario. The scalar spectral index is reduced, with a value of approximately 0.976 in the case where the inflaton potential is dominated by the 1-loop radiative corrections. The reheat temperature following thermal inflation is about GeV, which solves the gravitino overclosure problem. A Higgs condensate reduces the cosmic string mass per unit length, rendering it compatible with the Cosmic Microwave Background constraints without tuning the inflaton coupling. With the minimal U(1) gauge symmetry in the inflation sector, where one of the waterfall fields generates a right-handed neutrino mass, we investigate the Higgs thermal inflation scenario in three popular supersymmetry-breaking schemes: AMSB, GMSB and the CMSSM, focusing on the implications for the gravitino bound. In AMSB enough gravitinos can be produced to account for the observed dark matter abundance through decays into neutralinos. In GMSB we find an upper bound on the gravitino mass of about a TeV, while in the CMSSM the thermally generated gravitinos are sub-dominant. When Big Bang Nucleosynthesis constraints are taken into account, the unstable gravitinos of AMSB and the CMSSM must have a mass O(10) TeV or greater, while in GMSB we find an upper bound on the gravitino mass of O(1) TeV.

a,c]Mark Hindmarsh b] D. R. Timothy Jones

HIP-2012-27/TH

LTH 958

Prepared for submission to JCAP

Consistent cosmology with Higgs thermal inflation in a minimal extension of the MSSM

• Dept. of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, U.K.

• Dept. of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, U.K.

• Helsinki Institute of Physics, P.O. Box 64, 00014 Helsinki University, Finland

Keywords: Supersymmetry, Higgs, inflation, cosmic strings

## 1 Introduction

Inflation is the accepted paradigm for the very early universe, thanks to its power to account accurately for cosmological data in one simple framework. However, it raises a number of theoretical problems, principally the identity of the inflaton, the flatness of its potential, and how it is coupled to the Standard Model.

A technically natural way of achieving a flat potential is through supersymmetry (SUSY). However, the flatness is generically spoiled in supergravity [1], which must be taken into account if the inflaton changes by an amount of order the Planck scale or more (“large-field” inflation). Given the large parameter space of supergravity theories, this motivates starting the search for a supersymmetric theory of inflation with small-field inflation, in the context of a renormalisable theory.

At the same time, low energy supersymmetry remains an attractive theoretical framework in which to understand the smallness of the electroweak scale relative to the Planck scale. The Minimal Supersymmetric Standard Model (MSSM) is the most economical possibility to combine low energy SUSY with the phenomenological triumph of the Standard Model (although the high Higgs mass and the absence of positive results from the Tevatron and LHC increases the amount of parameter tuning required).

Indeed, the MSSM itself can realise inflation along one of the many flat directions [2] with the addition of non-renormalisable couplings. Inflation takes place near an inflection point in the potential, where trilinear and soft mass terms are balanced against each other, although the amount of tuning required [3] reduces the attractiveness of the scenario. The tuning can be reduced by extending the MSSM [4, 5].

The simplest class of renormalisable supersymmetric inflation models is minimal F-term hybrid inflation, by which we mean the first supersymmetric model of Ref. [1], characterised by the superpotential

 WI=λ1Φ¯¯¯¯ΦS−M2S. (1.1)

General theoretical considerations of small-field inflation drive one towards this model [6], which works without a Planck-scale inflaton field, non-renormalisable operators, or supersymmetry-breaking terms. It invokes an inflaton sector of (at least) 3 chiral superfields, consisting of the inflaton itself, , and two waterfall [7] fields, , with an optional gauge superfield.111The number of chiral superfields can be reduced to 2 without a gauge field, or if they are in a real representation. Because the the inflaton field appears linearly in the superpotential, it does not suffer from the generic supergravity problem of Hubble-scale mass terms during inflation [1].

In its standard form, however, F-term hybrid inflation suffers from a number of problems which reduce its power to fit cosmological data. First and foremost is the gravitino problem, which limits the reheat temperature to be unnaturally small compared with the inflation scale. Of less severity is the spectral index problem. If the inflaton potential is dominated by the 1-loop radiative corrections, F-term hybrid inflation predicts that the spectral index of cosmological perturbations e-foldings before the end of inflation is . For the canonical 60 e-foldings, this is more than 1 above the WMAP7 value . Finally, many models generate cosmic strings, and the CMB constraints on their mass per unit length forces one to very weak inflaton couplings, where [8].

There also remains the question of how the inflaton sector is coupled to the MSSM. If we restrict ourselves to renormalisable theories combining minimal U(1)-gauged F-term hybrid inflation with the MSSM, with no other fields, and preserving all the symmetries, the choices are limited. The singlet inflaton can couple in the superpotential only to the product of the Higgs fields or the square of the right-handed neutrino fields (which we take to be included the MSSM). If the MSSM fields have non-trivial charge assignments under the U(1) of F-term inflation, the coupling of to the neutrinos is forbidden, and its place taken by one of the waterfall fields. This has the nice feature of generating a see-saw mechanism, with the neutrino masses also controlled by the vev of the waterfall fields. Neutrino masses are also allowed if the waterfall fields are U(2) triplets, with SU(2) as a subgroup.

We will refer to the minimal case where the symmetry of the waterfall fields is U(1) as the Minimal Hybrid Inflationary Supersymmetric Standard Model (MHISSM). In the model, it is very natural that the gauge singlet inflaton should be coupled both to the waterfall fields and to the Higgs fields, which mixes the standard MSSM Higgs flat direction with the hybrid inflation waterfall direction. If the coupling of the inflaton to the Higgs is smaller than to the waterfall fields, inflation ends with the development of vevs for the Higgs multiplets, , breaking the electroweak symmetry. Soft terms lift the flat direction, and if certain constraints are satisfied, the Higgs fields will finally reach the standard vacuum after a period of thermal inflation, with a reheat temperature of about GeV. This solves the gravitino overclosure problem, and Big Bang Nucleosynthesis constraints can be satisfied with massive (O(10) TeV or more) or stable gravitinos [9, 10, 11, 12].

We call this second period of accelerated expansion Higgs thermal inflation. It is a natural consequence of the coupling of the F-term hybrid inflaton to the Higgs fields, and offers a generic solution to the gravitino problem. At the same time, a TeV-scale vacuum expectation value for the inflaton generates an effective -term. The model was first introduced in Ref. [13] in the context of Anomaly-Mediated Supersymmetry Breaking (AMSB). We termed the version of AMSB there deployed strictly anomaly mediated supersymmetry breaking (sAMSB), because D-terms associated with the U(1) symmetry resolve the AMSB tachyonic slepton problem, without requiring an additional explicit source of supersymmetry breaking.

In this paper we demonstrate that the interesting cosmological consequences, in particular Higgs thermal inflation, are a result of the structure of the model at the inflation scale, and not of the particular supersymmetry-breaking scenario. We derive the effective potential for the combination of fields driving thermal inflation, and the constraints on the soft breaking parameters for a phenomenologically acceptable ground state, in three popular supersymmetry-breaking scenarios: anomaly-mediated (AMSB), gauge-mediated (GMSB) and the constrained minimal supersymmetric standard model (CMSSM). We find that the lower reheat temperature following thermal inflation solves the gravitino problem in the CMSSM, while in AMSB enough gravitinos can be produced to account for the observed dark matter abundance through decays into neutralinos. In GMSB we find an upper bound on the gravitino mass of about a TeV, derived from constraints on NLSP decays during and after Big Bang Nucleosynthesis (BBN).

F-term models with Higgs thermal inflation have other important features. The spectral index of scalar Cosmic Microwave Background fluctuations is reduced, as fewer e-foldings of F-term inflation are required. In the range of couplings for which the 1-loop radiative corrections dominate the inflaton potential, we find , where the uncertainty comes from the spread of reheat temperatures in that range. The cosmic string mass per unit length is greatly reduced by the presence of a Higgs condensate at the string core, and is rendered independent of the inflaton coupling. Finally, thermal inflation sweeps away the gravitinos generated at the first stage of inflation, and any GUT-scale relics such as magnetic monopoles.

There are other models which renormalisably couple F-term hybrid inflation to the MSSM. hybrid inflation [14, 15] has the same field content as ours, but the MSSM has no U(1) charges; and it requires a Fayet Iliopoulos term. Also potentially in the class is the BL model of Refs. [16, 17, 18], although there is no explicit discussion of the coupling of the inflaton to the Higgs fields. In the model of Ref. [19] the waterfall fields are SU(2) triplets. The authors identified a flat direction involving the Higgs, without pursuing its consequences. The original F-term inflation model [1] had a spontaneously broken global U(1) symmetry, and models based on coupling it to the MSSM have recently been explored in [20], again without the possibility of Higgs thermal inflation being noticed. The same field content can also produce a promising superconformal D-term inflation model [21].

Further afield, it is also possible to construct renormalisable models of inflation in the Next-to-Minimal Supersymmetric Standard Model using soft terms to generate the vacuum energy [22]. Inflation along a flat direction which mixes a singlet with an MSSM flat direction has also been investigated recently in Ref. [23]. In that work, a single stage of inflation was envisaged, and in order to supply a satisfactory spectral index, the coupling to the inflaton has to be non-renormalisable.

The spectral index problem can also be solved with a non-minimal Kähler potential [24], or tuning the inflaton coupling to be small enough that the linear soft term dominates its potential [25]. In this paper we will restrict ourselves to the case where radiative corrections dominate the inflaton potential, and the Kähler potential is canonical.

## 2 Coupling F-term inflation and the Mssm

Our guiding principle is to couple minimal F-term hybrid inflation and the MSSM (which we take to include 3 families of right-handed neutrinos) in a renormalisable way, preserving all symmetries including supersymmetry (while allowing soft breaking terms in both sectors). Hence the superpotential will take the form

 W=WI+WA+WX (2.1)

where is the standard linear F-term hybrid inflation superpotential of Eq. (1.1), is the MSSM Yukawa superpotential

 WA=H2QYUU+H1QYDD+H1LYEE+H2LYNN, (2.2)

and is the coupling between the inflaton sector and the MSSM superpotential, containing renormalisable terms only. We will assume that the U(1) symmetry of the waterfall fields

 Φ→Φ′=eiqΦθΦ,¯¯¯¯Φ→¯¯¯¯Φ′=eiq¯¯¯Φθ¯¯¯¯Φ (2.3)

is gauged. The inflaton must be a gauge singlet, and so . The mass scale sets the inflation scale and the vevs of and . Given that the inflation scale is of order GeV, the waterfall fields must be SU(3)SU(2)U(1) singlets. Note that has a global U(1) R-symmetry, which forbids the terms , and . In order to preserve the flat potential for the inflaton, we must preserve this symmetry; we will discuss more of its implications in a moment.

The form of is now tightly constrained by symmetry and anomaly cancellation. Possible anomaly-free charge assignments for the MSSM fields are shown in Table 1.

The SM gauged is . is ; in the absence of this would have and U(1)-gravitational anomalies. The diagonal subgroup of SU(2) is . Note that quite generally , so we will write . We will assume that the MSSM fields couple to a U(1) distinct from , i.e. that , and moreover that in the AMSB case the values of and result in a solution to the AMSB tachyonic slepton problem [26]. For the resulting sparticle spectra in this case, see Ref. [13]. (Note that if the U(1) does not couple to MSSM fields, we are driven to inflation [14, 15]). Three SU(3)SU(2)U(1) singlets quadratic in the MSSM fields are available for , namely , and [27]. The U(1) charge assignments, combined with the global R-symmetry, with superfield charges

 S=2,L=E=N=U=D=Q=1,H1=H2=Φ=¯¯¯¯Φ=0, (2.4)

now uniquely specify the coupling term as

 WX = 12λ2NNΦ−λ3SH1H2, (2.5)

where we have set to permit the first term. All renormalisable B, L violating interactions and the and mass terms are forbidden by the U(1) gauge invariance, and the superpotential Eq. (2.1) contains all renormalisable terms consistent with U(1) and the R-symmetry. Note in particular that the R-symmetry forbids the Higgs -term . Moreover, the R-symmetry forbids the quartic superpotential terms and , which are allowed by the symmetry, and give rise to dimension 5 operators capable of causing proton decay [28, 29]. In fact the charges in Eq. (2.4) disallow B-violating operators in the superpotential of arbitrary dimension.

Soft terms break the continuous R-symmetry to the usual R-parity. The lightest supersymmetric particle (LSP) is therefore stable. (From Eq. (2.4), the LSP is a scalar quark or lepton, or a gaugino, or a fermionic Higgs, , or .)

To summarise the assumptions which force us to this unique class of theories, we require a theory with :

1. The field content of minimal F-term inflation and the MSSM.

2. The symmetries of minimal F-term inflation and the MSSM.

3. Renormalisable couplings only.

4. An inflaton-sector U(1) gauge symmetry which is coupled to the MSSM.

Note that if and are gauged under a larger symmetry group, the coupling is not allowed, unless they are triplets of SU(2) and are doublets [19].

The parameters are real and positive and is a symmetric matrix which we will take to be real and diagonal. The sign of the term above is chosen because with our conventions, in the electroweak vacuum

 H1=(v1√2,0)TandH2=(0,v2√2)T (2.6)

we have .

In the following we will denote the SU(3)SU(2)U(1) gauge couplings by , and , and the U(1) gauge coupling by . The normalisation of the U(1) gauge coupling corresponds to the usual SM convention, not that appropriate for SU(5) unification. We will denote the soft parameters for the gaugino masses , for a cubic interaction with Yukawa coupling , and for a mass term (where denotes a scalar field), . For the one mass term of the form in the MSSM () we will use .

## 3 The Higgs potential and its extrema

In this section we explore the important extrema of the Higgs potential, and demonstrate that there is a 1-parameter family of supersymmetric ground states with non-zero vevs for and before supersymmetry-breaking is taken into account. We will assume that , the scale of inflation and symmetry-breaking, is much larger than the scale of supersymmetry-breaking.

The existence of the one-parameter family (before thermal effects and soft terms are taken into account), is demonstrated as follows. The minimum of the scalar potential is determined by the requirement that both the F- and D-terms vanish. The vanishing of the D-terms ensures that , and , while the vanishing of the F-term is assured by . The minimum can therefore be parametrised by an SU(2) gauge transformation and angles defined by

 ⟨h1⟩ ≃ iσ2⟨h2⟩∗≃(M√λ3cosχ,0), ⟨ϕ⟩ ≃ ⟨¯¯¯ϕ∗⟩≃M√λ1sinχeiφ. (3.1)

The angle can always be removed by a U(1) gauge transformation (where the residual symmetry unbroken by the Higgs vevs alone is ), so the physical flat direction just maps out the interval . At the special point the U(1) symmetry is restored, and at the is restored. Away from these special points only U(1) is unbroken.

The degenerate minima have been noted before [19] in a model with gauge group . However, the important cosmological consequences which follow was first explored in Ref. [13].

Let us first consider the limiting cases where either or vanish.

### 3.1 The ϕ,¯¯¯ϕ,s extremum (ϕ-vacuum)

In the subspace (lower case fields denote the scalar component of the superfields) the scalar potential (including soft supersymmetry-breaking terms) is:

 V = λ21(|ϕs|2+|¯¯¯ϕs|2)+|λ1ϕ¯¯¯ϕ−M2|2+12q2Φg′2(|ϕ|2−|¯¯¯ϕ|2)2 (3.2) + m2ϕ|ϕ|2+m2¯¯¯ϕ|¯¯¯ϕ|2+m2s|s|2+ρM2m32(s+s∗) + hλ1ϕ¯¯¯ϕs+c.c..

We will assume that the term linear in is small enough not to be important for inflation (and quantify this smallness in Section 5). In AMSB there are arguments [30] to show that, without a quadratic term in the superpotential, the only RG invariant solution for is .

Let us establish the minimum in this subspace, under the assumption that . We shall call this the -vacuum. With the notation , and , we find

 vϕ[m2ϕ+12λ21v2s+12g2q2Φ(v2ϕ−v2¯¯¯ϕ)]+v¯¯¯ϕ[λ1(12λ1vϕv¯¯¯ϕ−M2)+hλ1√2vs] = 0, (3.3) v¯¯¯ϕ[m2¯¯¯ϕ+12λ21v2s−12g2q2Φ(v2ϕ−v2¯¯¯ϕ)]+vϕ[λ1(12λ1vϕv¯¯¯ϕ−M2)+hλ1√2vs] = 0, (3.4) vs[m2s+12λ21(v2ϕ+v2¯¯¯ϕ)]+hλ1√2vϕv¯¯¯ϕ+√2ρM2m32 = 0. (3.5)

From Eqs. (3.3), (3.4) we find

 λ1(12λ1vϕv¯¯¯ϕ−M2) = −vϕv¯¯¯ϕv2ϕ+v2¯¯¯ϕ[m2ϕ+m2¯¯¯ϕ+λ21v2s]−hλ1√2vs, (3.6) 12g′2q2Φ(v2ϕ−v2¯¯¯ϕ) = v2¯¯¯ϕm2¯¯¯ϕ−v2ϕm2ϕ+(v2¯¯¯ϕ−v2ϕ)12λ21v2sv2ϕ+v2¯¯¯ϕ. (3.7)

Then from Eqs. (3.6), (3.7), to leading order in an expansion in we have

 v2ϕ≃v2¯¯¯ϕ≃2λ1M2, (3.8)

and from Eq. (3.5) that is . It follows from Eq. (3.7) that

 v2ϕ−v2¯¯¯ϕ=m2¯¯¯ϕ−m2ϕg′2q2Φ+O(m432/M2), (3.9)

and from Eq. (3.5) that

 vs=−hλ1√2λ21−m32ρ√2λ1+O(m232/M). (3.10)

From now on we neglect , assuming that

 |ρ|≲∣∣ ∣∣hλ1λ1m32∣∣ ∣∣. (3.11)

Substituting back from Eqs. (3.8), (3.10) into Eq. (3.2), we obtain to leading order

 Vϕ=1λ1M2⎛⎝m2ϕ+m2¯¯¯ϕ−h2λ12λ21⎞⎠ (3.12)

and from Eq. (3.10) a Higgs -term

 μh=λ3hλ12λ21, (3.13)

naturally of the same order as the supersymmetry-breaking scale.

The theory is approximately supersymmetric at the scale , so the gauge boson, the Higgs boson, the gaugino and one combination of form a massive supermultiplet with mass , while the remaining combination of and and the other combination of form a massive chiral supermultiplet, with mass .222A detailed explanation of the symmetry-breaking is contained in Ref. [18].

The large vev for generates inflation-scale masses for the triplet, thus naturally implementing the see-saw mechanism.

### 3.2 The h1,2,s extremum (h-vacuum)

. In the subspace, the scalar potential is

 V = λ23(|h1s|2+|h2s|2)+|λ3h1h2−M2|2+12g′2q2H(|h1|2−|h2|2)2 (3.14) + 18g21(h†1h1−h†2h2)2+18g22∑a(h†1σah1+h†2σah2)2 + m2h1|h1|2+m2h2|h2|2+m2s|s|2+ρM2m32(s+s∗) + hλ3h1h2s+c.c..

Note that we assume there is no mass term; its absence follows from the absence of the corresponding term in the superpotential (which is forbidden by the R-symmetry) when the source of supersymmetry breaking can be represented by a non-zero vev for a spurion (or conformal compensator) field.

The structure is similar to Eq. (3.2), with the addition of SU(2) and D-terms. Without loss of generality the SU(2) D-term vanishes with the choice and , and . The values of the fields at the minimum (which we term the -vacuum) and the value of the potential at this extremum can then be recovered from the result of the previous section with the replacement ), leading to a potential energy density

 Vh=M2λ3⎛⎝m2h1+m2h2−h2λ32λ23⎞⎠. (3.15)

### 3.3 Potential along the ϕ, ¯¯¯ϕ, h1, h2 flat direction

As we outlined at the beginning of the section, the supersymmetric minima are parametrised by an angle , defined in (3.1). Soft terms lift this degeneracy, and the leading terms in the effective potential for can be found in an expansion in . After solving for , it is found that

 V(χ)≃−M22(~hλ1sin2χ+~hλ3cos2χ)2λ1sin2χ+λ3cos2χ+M2(¯m2ϕλ1sin2χ+¯m2hλ3cos2χ), (3.16)

where we have defined

 ~hλ1=hλ1λ1,~hλ3=hλ3λ3,¯m2ϕ=m2ϕ+m2¯ϕ,¯m2h=m2h1+m2h2. (3.17)

## 4 Supersymmetry-breaking and the true minimum

In this section we investigate under which conditions the phenomenologically acceptable large- solution is the true minimum, in three popular supersymmetry-breaking scenarios. Hence we are looking for constraints on the soft supersymmetry-breaking parameters such that

 Vh−Vϕ = M2⎛⎜⎝~h2λ12λ1−~h2λ32λ3−¯m2ϕλ1+¯m2hλ3⎞⎟⎠>0, (4.1) V′′(π/2) = 2M2λ1⎡⎢⎣−~h2λ12(2~hλ3~hλ1−λ3λ1−1)+¯m2hλ1λ3−¯m2ϕ⎤⎥⎦>0. (4.2)

We will also check that the false vacuum at is a local maximum, from the sign of , which can be recovered from by the replacements and . A metastable false vacuum, as we will demonstrate in Section 7, would lead to the universe remaining trapped in an inflating phase.

We assume that the U(1) symmetry is broken by a vev of order , and evaluate the soft terms at this scale, rather than running down to the electroweak scale. This is the appropriate renormalisation scale to investigate a potential with vevs of order , whose important radiative corrections are from particles of mass of order and . Note that in inflation models, with inflaton couplings and are generally small, and so the U(1) gauge boson mass is much greater than , unless is also small.

### 4.1 Anomaly-mediated supersymmetry-breaking

With anomaly mediation, the soft breaking parameters take the generic renormalisation group invariant form

 Ma = m32βga/ga, (4.3) hU,D,E,N = −m32βYU,D,E,N, (4.4) (m2)ij = 12m232μddμγij+kY′iδij, (4.5) m23 = κm32μh−m32βμh. (4.6)

Here is the renormalisation scale, and is the gravitino mass; are the gauge -functions and is the chiral supermultiplet anomalous dimension matrix. are the Yukawa matrices, is the superpotential Higgs -term, and are constants, and are charges corresponding to the symmetry.

In the MSSM, is an arbitrary parameter, which in practice is fixed by minimising the Higgs potential at the electroweak scale. The parameter is generated by the breaking of the symmetry at a large scale, and forms the basis of the solution to the tachyonic slepton problem within the framework of AMSB, as explained in [13], whence the name strictly anomaly-mediated supersymmetry-breaking (sAMSB) originates.

The Higgs -term, , is generated by the the vev of the inflation , which in turn is triggered by the U(1) symmetry-breaking. Hence the parameter , and the equation for , are relevant only below the symmetry-breaking scale .

As a first approximation, we will assume that the terms dominate throughout, as and are generally large, in which case the and trilinear soft terms are given from Eq. (4.4) as:

 hλ1 ≃ m32λ116π2(4q2Φg′2), (4.7) hλ3 ≃ m32λ316π2(4q2Hg′2), (4.8)

while the mass soft terms are

 m2ϕ ≃ −m232132π2μddμ(2g′2q2Φ), (4.9) m2¯¯¯ϕ ≃ −m232132π2μddμ(2g′2q2Φ), (4.10) m2h1 ≃ −m232132π2μddμ(2g′2q2H), (4.11) m2h2 ≃ −m232132π2μddμ(2g′2q2H). (4.12)

The one loop -function is

 βg′=Qg′316π2 (4.13)

where

 Q = nG(403q2L+8q2E+16qEqL)+36q2L+40qEqL+12q2E (4.14) = 76q2L+36q2E+88qEqL

for . Hence

 m2ϕ≃m2¯¯¯ϕ ≃ −2m232(g′216π2)2q2ΦQ, (4.15) m2h1≃m2h2 ≃ −2m232(g′216π2)2q2HQ. (4.16)

Thus the difference in the energy densities between the two vacua is, in this approximation,

 Vh−Vϕ≃M2⎛⎜ ⎜⎝m32g′216π2⎞⎟ ⎟⎠2[4Qq2Φ+8q4Φλ1−4Qq2H+8q4Hλ3]. (4.17)

The coefficient is in general large, and larger than both and , so the condition for to be the true minimum may be written

 λ3λ1≳(qHqΦ)2. (4.18)

It is not hard to check from Eq. (4.2)) that under the same assumptions, the -vacuum is a minimum and the -vacuum is a maximum. Hence no further constraints on the parameters are generated.

In the next section we will see that if , then inflation ends with developing non-zero vevs, whereas if it is which become non-zero; this statement is independent of the nature of the soft breaking terms. Now is easy to show that unless

 −35≤qLqE≤−13. (4.19)

However, the domain defined by Eq. (4.19) does not permit a satisfactory electroweak vacuum in the AMSB case [26]. For example, for the specific choice , which can lead to an acceptable electro-weak vacuum [13], the condition becomes (from Eq. (4.17))

 λ3λ1≳1988. (4.20)

or from the approximation Eq. (4.18).

We see, therefore, that there will generally be a domain

 λ1(qHqΦ)2≲λ3<λ1 (4.21)

such that the universe exits to the false high Higgs vev -vacuum, evolving subsequently to the true vacuum as we shall describe later.

In the Appendix we include a more accurate computation of the vacuum energy difference, taking into account the SM gauge couplings and the top Yukawa coupling.

### 4.2 Gauge-mediated supersymmetry-breaking

In the GMSB framework (see e.g. [31]), supersymmetry-breaking is communicated by a set of messenger fields which have SM gauge charges in a vector-like representation, which should be complete GUT multiplets if gauge unification is to be preserved. The messenger fields are supposed to have a large mass, given by the vev of the scalar component of a chiral superfield , which also has a non-zero F-term , the source of the supersymmetry breaking. Although there are many possibly choices for the field representations of the messenger fields, we can adapt the simple model described in [31] to study our model.

We introduce the following superpotential for the extra fields

 Wgm=λ4SC¯C+λ5XC¯C, (4.22)

assuming that some extra dynamics at a higher scale gives both the scalar component of and a vev. We will assume that . Radiative corrections from the messenger particles then induce masses for the gauginos at one loop,

 Ma=g2a16π2Λg, (4.23)

where , , and is the messenger index, equal to twice the sum of the Dynkin indices of the messenger fields. Scalars acquire masses from 2-loop corrections of

 m2i=2Λ2s∑a(g2a16π2)2Ca(i), (4.24)

where , is the quadratic Casimir associated with the th gauge group for the th scalar, and the sum over includes the four gauge couplings .

Trilinear terms are also induced at 2 loops, and so are of order . They are small compared with the gaugino masses, and it is a reasonable approximation to take them to vanish at the messenger scale . We assume that are of the correct order of magnitude for supersymmetry-breaking.

We thus have

 m2ϕ = m2¯¯¯ϕ=2Λ2s(g′216π2)2q2Φ, m2h1 = m2h2=2Λ2s⎡⎣34(g2216π2)2+14(g2116π2)2+(g′216π2)2q2H⎤⎦, hλ1λ1 = hλ3λ3=0. (4.25)

Thus the difference between the vacuum energies is

 Vh−Vϕ = M2⎛⎜⎝m2h1+m2h2λ3−m2ϕ+m2¯¯¯ϕλ1⎞⎟⎠ (4.26) = 2Λ2sM2(16π2)2[(32g42+12g41+2q2Hg′4)1λ3−2q2Φg′4λ1],

so that, if we assume dominance of the terms, the condition that becomes

 λ3λ1≲(qHqΦ)2. (4.27)

This is precisely the opposite condition to that in AMSB, Eq. (4.18). As in AMSB, the condition that is sufficient to ensure that is a minimum and a maximum.

Now in GMSB, we do not have the constraint on the domain that we described in the AMSB case. Inflation will end in the Higgs phase unless

 (qHqΦ)2>1and1<λ3λ1<(qHqΦ)2, (4.28)

in which case it ends directly in the true -vacuum.

### 4.3 Constrained minimal supersymmetric standard model

At the high scale we will have the CMSSM pattern of soft breaking parameters,

 m2ϕ = m2¯¯¯ϕ=m2h1=m2h2=m20, hλ1λ1 = hλ3λ3=A (4.29)

and hence

 Vh−Vϕ=M2(2m20−A2/2)[1λ3−1λ1]. (4.30)

Hence if (so that inflation ends in the -vacuum) then for we require

 2m20>A2/2. (4.31)

It is easy to check from Eq. (4.2) that this is again a sufficient condition that be positive. On the other hand, there is then a range

 A22<2m20<λ1λ3A22 (4.32)

for which the -vacuum is also a local minimum. We will see that this scenario is not consistent with a graceful exit from Higgs thermal inflation, and hence for a cosmologically acceptable potential, we must demand

 2m20>λ1λ3A22. (4.33)

## 5 Inflation and reheating

### 5.1 F-term inflation

We assume that the vevs of MSSM fields apart from the Higgs are negligible, in which case the relevant tree potential is

 Vtree = |λ1ϕ¯¯¯ϕ−λ3h1h2−M2|2+[λ21(|ϕ|2+|¯¯¯ϕ|2)+λ23(|h1|2+|h2|2)]|s