A scheme of simplified smooth hybrid inflation is realized in the framework of supersymmetric . The smooth model of hybrid inflation provides a natural solution to the monopole problem that appears in the breaking of gauge symmetry. The supergravity corrections with nonminimal Kähler potential are shown to play important role in realizing inflation with a red-tilted scalar spectral index , within Planck’s latest bounds. As compared to shifted model of hybrid inflation, relatively large values of the tensor-to-scalar ratio are achieved here, with nonminimal couplings and and the gauge symmetry-breaking scale GeV.
Simplified Smooth Hybrid Inflation in
Mansoor Ur Rehman111Email: firstname.lastname@example.org and Umer Zubair222Email: email@example.com
Department of Physics,
Quaid-i-Azam University, Islamabad 45320, Pakistan.
Supersymmetric (SUSY) hybrid inflation - offers a natural framework to realize inflation within grand unified theories (GUTs). The GUT symmetry breaking in these models is associated with inflation in accordance with the cosmological observations on the cosmic microwave background (CMB) radiation. In the standard version of SUSY hybrid inflation [1, 2] GUT symmetry is broken at the end of inflation whereas in the shifted  and the smooth  versions of the model it is broken during inflation. The copious formation of topological defects (such as magnetic monopoles) are, therefore, naturally avoided in the shifted and smooth versions of SUSY hybrid inflation.
The supersymmetric is the simplest GUT gauge symmetry where all three gauge couplings of Standard Model (SM) gauge group, , unify at GUT scale GeV. The gauge symmetry is usually broken into SM by the non-vanishing vacuum expectation value (VEV) of 24 adjoint Higgs field. This then leads to the overproduction of catastrophic magnetic monopoles in conflict with the cosmological observations. To avoid this problem shifted and smooth variants of SUSY hybrid inflation can be employed333See ref.  for an alternative scheme to avoid monopole problem in SUSY GUTs.. In shifted hybrid inflation, the addition of leading-order nonrenormalizable term in the superpotential generates an additional classically flat direction. With the necessary slope provided by the one-loop radiative corrections, this shifted track can be used for inflation; see ref.  for details. As gauge symmetry is broken in the shifted track, catastrophic magnetic monopoles are inflated away. In the smooth version of hybrid inflation, this additional direction appears with a slope at classical level which then helps to drive inflation. The radiative corrections are usually assumed to be suppressed in smooth hybrid inflation. Furthermore, inflation ends in the smooth hybrid model due to slow-roll breaking whereas in standard and shifted hybrid models, termination of inflation is abrupt followed by a waterfall.
In this paper we consider the simplified version of smooth hybrid inflation in SUSY . In simplified smooth hybrid inflation , the ultraviolet (UV) cutoff is replaced with the reduced Planck mass, GeV, in contrast to the standard smooth hybrid inflation  where is allowed to vary below . With the minimal Kähler potential, inflation requires trans-Planckian field values which is inconsistent with the supergravity (SUGRA) expansion. However, with the nonminimal Kähler potential, successful inflation is realized with sub-Planckian field values. The predicted values of the scalar spectral index and the tensor-to-scalar ratio are easily obtained within the Planck’s latest bounds . As compared to the shifted case we obtain relatively large values of tensor-to-scalar ratio in smooth hybrid inflation.
Smooth hybrid inflation
In SUSY , the matter superfields are accommodated to the , and dimensional representations, while the Higgs superfields belong to the fundamental representations , and adjoint representation . The superpotential of the model which is consistent with the , and symmetries with the leading-order nonrenormalizable terms is given by 444For SUSY SU(5) inflation without R-symmetry, see ref. .
where is a gauge singlet superfield, is a superheavy mass and the UV cutoff (reduced Planck mass) has replaced the cutoff  usually employed in smooth hybrid inflation models . In the superpotential above, , , are the Yukawa couplings for quarks and leptons, and is the neutrino mass matrix. The first term in the first line of Eq. (1) is relevant for inflation, while the last two terms are required for the solution of the doublet-triplet splitting problem, for which a fine-tuning on the parameters is required. The terms in the second line generate fermion masses after the electroweak symmetry breaking. The global symmetry ensures the linearity of the superpotential in -omitting terms such as which could spoil inflation by generating an inflaton mass of Hubble size, . Also, respects a symmetry under which and, hence, only cubic powers of are allowed. All other fields are neutral under the symmetry. The -charges of the superfields are assigned as follows :
In component form, the above superpotential takes the following form,
where we have employed the adjoint basis for with Tr and Tr. Here the indices , , vary from 1 to 24 whereas the indice , , , vary from 1 to 5. The -term scalar potential obtained from the above superpotential is given by
where the scalar components of the superfields are denoted by the same symbols as the corresponding superfields. The vacuum expectation values (VEVs) of the fields at the global SUSY minimum of the above potential are given by,
where , and the superscript ‘0’ denotes the field value at its global minimum. Using symmetry transformation the VEV matrix can be aligned in the -direction. This implies that and , where and have been assumed. Thus the gauge symmetry is broken down to SM gauge group by the non-vanishing VEV of which is a singlet under . The -term scalar potential,
also vanishes for this choice of the VEV (since ) and for .
The scalar potential in Eq. (4) can be written in terms of the dimensionless variables
This potential is displayed in Fig. (1) which shows a valley of minimum given in the large limit by
This valley of local minimum is not flat and possess a slope to drive inflaton towards SUSY vacuum. Here we assume special initial conditions for inflation to occur in the valley. However, see the relevant references in  for detailed discussion of the fine-tuning of initial conditions in various models of SUSY hybrid inflation. During inflation (), the global SUSY potential is given by,
The inflationary slow roll parameters are given by,
Here, the derivatives are with respect to whereas the canonically normalized field . In the slow-roll (leading order) approximation, the tensor-to-scalar ratio , the scalar spectral index , and the running of the scalar spectral index are given by
The last number of e-folds before the end of inflation is,
where is the field value at the pivot scale , and is the field value at the end of inflation, defined by . The amplitude of the curvature perturbation is given by
where is the Planck normalization at . In the large limit we obtain following results for various inflationary parameters,
For , we obtain , and , with , and GeV.
Supergravity corrections and non-minimal Kähler potential
Above analysis is incomplete unless we include SUGRA corrections which have important effect on the global SUSY results. The -term SUGRA scalar potential is given by
with being the bosonic components of the superfields and where we have defined
and The Kähler potential may be expanded as
As is an adjoint superfield, many other terms of the form,
can appear in the Kähler potential. The effective contribution of all these terms is either suppressed or can be absorbed into other terms already present in the Kähler potential. Therefore, the supergravity (SUGRA) scalar potential during inflation becomes
where . As expected, the dominant contribution in the potential comes only from the terms with higher powers of as all other fields ( and ) are suppressed as compared to . The one-loop radiative corrections and the soft SUSY breaking terms are expected to have a negligible effect on the inflationary predictions, therefore, in numerical calculations we can safely ignore these contributions .
Including SUGRA corrections usually generates a large inflaton mass of the order of the Hubble parameter which makes the inflationary slow-roll parameter and spoils inflation. This is known as the problem . In SUSY hybrid inflation with minimal Kähler potential, this problem is naturally resolved as a result of a cancellation of the mass squared term from the exponential factor and the other part of the potential in Eq. (22). This is a consequence of -symmetry 555See ref. , for a SUSY hybrid inflation scenario (with minimal Kähler potential) in which the symmetry is explicitly broken by Planck scale suppressed operators in the superpotential. which ensures this cancellation to all orders [2, 3]. With non-minimal Kähler potential, however, a mass squared term of the form,
appears which requires some tuning of the parameter (), so that the scalar potential is flat enough to realize successful inflation.
It can readily be checked that, for the minimal Kähler potential (with ), the SUGRA corrections dominate the global SUSY potential for the values and GeV obtained earlier. This, in turn, alter the values of and significantly, making their predictions lie outside the Planck data bounds. Stating the same fact in a different way, the SUGRA corrections require trans-Planckian field values to obtain and within Planck’s bounds. But this invalidates the SUGRA expansion itself. The minimal case () is, therefore, inconsistent with the Planck’s data.
For non-minimal Kähler potential, we obtain the following approximate results for and in the large limit,
Now with the addition of two extra parameters, and we expect to find the red-tilted () solutions consistent with the latest Planck bounds on . For example, with , and we obtain and from above expressions. This rough estimate guides us to the region of parameters where we can possibly find the large solutions.
In our numerical calculations we take (, ) 1 and . Employing (next-to-leading order) slow-roll approximations [17, 18], the predicted values of various parameters are displayed in Figs. (2-4). The results obtained here are quite similar to those obtained in the simplified smooth model of hybrid inflation , where instead of adjoint superfield a conjugate pair of chiral superfields () is employed. For the predictions of standard, shifted and smooth hybrid inflation with a conjugate pair of chiral Higgs superfields and non-minimal Kähler potential see refs. -. It can be seen from the Figs. (2-4) that by employing non-minimal Kähler potential, there is a significant increase in the tensor-to-scalar ratio . Moreover, both and play the crucial role to bring the scalar spectral index to the central value of Planck data bounds i.e. , with a large value of tensor-to-scalar ratio i.e. . For and , we obtain scalar spectral index within Planck - bounds.
The behavior of and with respect to the scalar spectral index is presented in Fig. (2). The relation between these parameters looks very similar which can be understood by noting that and are proportional to each other. From Eq. (16), we obtain the following approximate relation between the tensor-to-scalar ratio and ,
which gives an adequate estimate of the numerical results displayed in Fig. (2). The upper boundary curve in Fig. (2) represents the constraint, whereas the lower boundary curve corresponds the constraint. The impact of and on the behavior of is of particular interest. The variation of with respect to is shown in the left panel of Fig. (3), while the right panel shows the relationship between , and . As depicted in Fig. (3), a red tilted () scalar spectral index and a large tensor-to-scalar ratio require (, ) (, ). Therefore, the large values of are obtained with the potential of the form,
In the right panel of Fig. (3), one can see that in the large limit both and are tuned to make very small.
For the curve depicted in Fig. 4 (right panel), varies in the range GeV. This shows that small values of particularly favors ( GeV), whereas large tensor-to-scalar ratio requires greater than . The range of running scalar spectral index is found to be (see left panel of Fig. 4), which is compatible with the Planck’s data assumptions on the power-law expansion of .
It is worth comparing our results with the model of shifted hybrid inflation , where we find the scalar spectral index with a much smaller tensor-to-scalar ratio , taking on values . The reduction in the value of is mainly due to the dominant contribution of the radiative corrections which keeps the value of small. This is in contrast to the above model of smooth hybrid inflation where the radiative corrections are suppressed enough to have any influence on the inflationary predictions. However, in shifted model is easily obtained whereas in above model GeV.
After the symmetry breaking, the fields , , acquire super heavy masses, while the fields and acquire masses of order . The octet , , and the triplet , , , , remain massless as shown in . The presence of these massless particles (or flat directions) in simple groups like  and with a symmetry is a generic feature as pointed out in . For a relevant discussion also see . These massless octets and triplets spoil the gauge coupling unification. To preserve unification we can add vector-like fermions as discussed in . However, these vector-like fermions do not form a complete multiplet of and thus their presence does not respect symmetry. In short we have to give up gauge coupling unification in all SUSY models of inflation with a symmetry.
To summarize, we have analyzed the simplified smooth hybrid inflation in supersymmetric model. As gauge symmetry is broken during inflation monopole density is diluted and remains under the observable limits. With minimal Kähler potential, SUGRA corrections over-dominate all other terms to have any inflation. However, with non-minmal terms in the Kähler potential successful inflation is realized. We obtain tensor-to-scalar ratio with the non-minimal couplings and consistent with the Planck’s 2- bounds on the scalar spectral index and tensor-to-scalar ratio . If the detection of gravitational waves is confirmed by Planck’s B-mode polarization data then these models will be ruled out.
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