Consistent cosmology with Higgs thermal inflation in a minimal extension of the MSSM
Abstract
We consider a class of supersymmetric inflation models, in which minimal gauged Fterm 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 supersymmetrybreaking. 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 Fterm 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 1loop 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 righthanded neutrino mass, we investigate the Higgs thermal inflation scenario in three popular supersymmetrybreaking 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 subdominant. 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
HIP201227/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
Contents
 1 Introduction
 2 Coupling Fterm inflation and the MSSM
 3 The Higgs potential and its extrema
 4 Supersymmetrybreaking and the true minimum
 5 Inflation and reheating
 6 Review of gravitino constraints
 7 Higgs thermal inflation and gravitinos
 8 Conclusions
 A sAMSB soft parameters and inflaton coupling constraints
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 (“largefield” inflation). Given the large parameter space of supergravity theories, this motivates starting the search for a supersymmetric theory of inflation with smallfield 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 nonrenormalisable 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 Fterm hybrid inflation, by which we mean the first supersymmetric model of Ref. [1], characterised by the superpotential
(1.1) 
General theoretical considerations of smallfield inflation drive one towards this model [6], which works without a Planckscale inflaton field, nonrenormalisable operators, or supersymmetrybreaking 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.^{1}^{1}1The 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 Hubblescale mass terms during inflation [1].
In its standard form, however, Fterm 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 1loop radiative corrections, Fterm hybrid inflation predicts that the spectral index of cosmological perturbations efoldings before the end of inflation is . For the canonical 60 efoldings, 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 Fterm 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 righthanded neutrino fields (which we take to be included the MSSM). If the MSSM fields have nontrivial charge assignments under the U(1) of Fterm 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 seesaw 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 Fterm hybrid inflaton to the Higgs fields, and offers a generic solution to the gravitino problem. At the same time, a TeVscale vacuum expectation value for the inflaton generates an effective term. The model was first introduced in Ref. [13] in the context of AnomalyMediated Supersymmetry Breaking (AMSB). We termed the version of AMSB there deployed strictly anomaly mediated supersymmetry breaking (sAMSB), because Dterms 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 supersymmetrybreaking 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 supersymmetrybreaking scenarios: anomalymediated (AMSB), gaugemediated (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).
Fterm models with Higgs thermal inflation have other important features. The spectral index of scalar Cosmic Microwave Background fluctuations is reduced, as fewer efoldings of Fterm inflation are required. In the range of couplings for which the 1loop 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 GUTscale relics such as magnetic monopoles.
There are other models which renormalisably couple Fterm 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 Fterm 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 Dterm inflation model [21].
Further afield, it is also possible to construct renormalisable models of inflation in the NexttoMinimal 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 nonrenormalisable.
The spectral index problem can also be solved with a nonminimal 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 Fterm inflation and the Mssm
Our guiding principle is to couple minimal Fterm hybrid inflation and the MSSM (which we take to include 3 families of righthanded 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
(2.1) 
where is the standard linear Fterm hybrid inflation superpotential of Eq. (1.1), is the MSSM Yukawa superpotential
(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
(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) Rsymmetry, 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 anomalyfree 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 Rsymmetry, with superfield charges
(2.4) 
now uniquely specify the coupling term as
(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 Rsymmetry. Note in particular that the Rsymmetry forbids the Higgs term . Moreover, the Rsymmetry 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 Bviolating operators in the superpotential of arbitrary dimension.
Soft terms break the continuous Rsymmetry to the usual Rparity. 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 :

The field content of minimal Fterm inflation and the MSSM.

The symmetries of minimal Fterm inflation and the MSSM.

Renormalisable couplings only.

An inflatonsector 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
(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 1parameter family of supersymmetric ground states with nonzero vevs for and before supersymmetrybreaking is taken into account. We will assume that , the scale of inflation and symmetrybreaking, is much larger than the scale of supersymmetrybreaking.
The existence of the oneparameter 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 Dterms vanish. The vanishing of the Dterms ensures that , and , while the vanishing of the Fterm is assured by . The minimum can therefore be parametrised by an SU(2) gauge transformation and angles defined by
(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 extremum (vacuum)
In the subspace (lower case fields denote the scalar component of the superfields) the scalar potential (including soft supersymmetrybreaking terms) is:
(3.2)  
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
(3.3)  
(3.4)  
(3.5) 
From Eqs. (3.3), (3.4) we find
(3.6)  
(3.7) 
Then from Eqs. (3.6), (3.7), to leading order in an expansion in we have
(3.8) 
and from Eq. (3.5) that is . It follows from Eq. (3.7) that
(3.9) 
and from Eq. (3.5) that
(3.10) 
From now on we neglect , assuming that
(3.11) 
Substituting back from Eqs. (3.8), (3.10) into Eq. (3.2), we obtain to leading order
(3.12) 
and from Eq. (3.10) a Higgs term
(3.13) 
naturally of the same order as the supersymmetrybreaking 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 .^{2}^{2}2A detailed explanation of the symmetrybreaking is contained in Ref. [18].
The large vev for generates inflationscale masses for the triplet, thus naturally implementing the seesaw mechanism.
3.2 The extremum (vacuum)
. In the subspace, the scalar potential is
(3.14)  
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 Rsymmetry) when the source of supersymmetry breaking can be represented by a nonzero vev for a spurion (or conformal compensator) field.
The structure is similar to Eq. (3.2), with the addition of SU(2) and Dterms. Without loss of generality the SU(2) Dterm 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
(3.15) 
3.3 Potential along the , , , 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
(3.16) 
where we have defined
(3.17) 
4 Supersymmetrybreaking and the true minimum
In this section we investigate under which conditions the phenomenologically acceptable large solution is the true minimum, in three popular supersymmetrybreaking scenarios. Hence we are looking for constraints on the soft supersymmetrybreaking parameters such that
(4.1)  
(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 Anomalymediated supersymmetrybreaking
With anomaly mediation, the soft breaking parameters take the generic renormalisation group invariant form
(4.3)  
(4.4)  
(4.5)  
(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 anomalymediated supersymmetrybreaking (sAMSB) originates.
The Higgs term, , is generated by the the vev of the inflation , which in turn is triggered by the U(1) symmetrybreaking. Hence the parameter , and the equation for , are relevant only below the symmetrybreaking 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:
(4.7)  
(4.8) 
while the mass soft terms are
(4.9)  
(4.10)  
(4.11)  
(4.12) 
The one loop function is
(4.13) 
where
(4.14)  
for . Hence
(4.15)  
(4.16) 
Thus the difference in the energy densities between the two vacua is, in this approximation,
(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
(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 nonzero vevs, whereas if it is which become nonzero; this statement is independent of the nature of the soft breaking terms. Now is easy to show that unless
(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 electroweak vacuum [13], the condition becomes (from Eq. (4.17))
(4.20) 
or from the approximation Eq. (4.18).
We see, therefore, that there will generally be a domain
(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 Gaugemediated supersymmetrybreaking
In the GMSB framework (see e.g. [31]), supersymmetrybreaking is communicated by a set of messenger fields which have SM gauge charges in a vectorlike 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 nonzero Fterm , 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
(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,
(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 2loop corrections of
(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 supersymmetrybreaking.
We thus have
(4.25) 
Thus the difference between the vacuum energies is
(4.26)  
so that, if we assume dominance of the terms, the condition that becomes
(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
(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,
(4.29) 
and hence
(4.30) 
Hence if (so that inflation ends in the vacuum) then for we require
(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
(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
(4.33) 
5 Inflation and reheating
5.1 Fterm inflation
We assume that the vevs of MSSM fields apart from the Higgs are negligible, in which case the relevant tree potential is