Induced-Gravity Inflation in Supergravity Confronted with Planck 2015 & Bicep2/Keck Array
Supersymmetric versions of induced-gravity inflation are
formulated within Supergravity (SUGRA) employing two gauge singlet
chiral superfields. The proposed superpotential is uniquely
determined by applying a continuous and a discrete
symmetry. We also employ a logarithmic Kähler potential respecting the symmetries above and including all the allowed
terms up to fourth order in powers of the various fields. When the
Kähler manifold exhibits a no-scale-type symmetry, the model predicts
spectral index and tensor-to-scalar
. Beyond no-scale SUGRA, and depend
crucially on the coefficient involved in the fourth order
term, which mixes the inflaton with the accompanying
non-inflaton superfield in the Kähler potential, and the prefactor
encountered in it. Increasing slightly the latter above , an
efficient enhancement of the resulting can be achieved putting
it in the observable range favored by the Planck and Bicep2/Keck Array results.
In all cases, imposing a lower bound on the parameter ,
involved in the coupling between the inflaton and the Ricci scalar
curvature, inflation can be attained for subplanckian values of the
inflaton while the corresponding effective theory respects the
Published in PoS CORFU 2014, 156 (2015).
Induced-Gravity Inflation in Supergravity Confronted with Planck 2015 & Bicep2/Keck Array
Departament de Física Teòrica and IFIC,
Universitat de València-CSIC,
E-46100 Burjassot, SPAIN
Induced-gravity inflation (IGI)  is a subclass of non-minimal inflationary models in which inflation is driven in the presence of a non-minimal coupling function between the inflaton field and the Ricci scalar curvature and the Planck mass is determined by the vacuum expectation value (v.e.v) of the inflaton at the end of the slow roll. As a consequence, IGI not only is attained even for subplanckian values of the inflaton – thanks to the strong enough aforementioned coupling – but also the corresponding effective theory remains valid up to the Planck scale [2, 3]. In this talk we focus on the implementation of IGI within Supergravity (SUGRA) [4, 5] revising and updating the findings of Ref.  in the light of the recent joint analysis [6, 7] of Planck and Bicep2/Keck Array results.
Below, in Sec. 2, we describe the generic formulation of IGI in SUGRA. The established in Sec. 3 inflationary models are investigated in Sec. 4. The ultraviolet (UV) behavior of these models is analyzed in Sec. 5. Our conclusions are summarized in Sec. 6. Throughout the text, the subscript denotes derivation with respect to (w.r.t) the field ; charge conjugation is denoted by a star, and we use units where the reduced Planck scale is set equal to unity.
2 Embedding IGI in SUGRA
According to the scheme proposed in Ref. , the implementation of IGI in SUGRA requires at least two singlet superfields, i.e., , with () and ( being the inflaton and a stabilized field respectively. The superpotential of the model has the form
which is (i) invariant under the action of a global discrete symmetry, i.e.,
and (ii) consistent with a continuous symmetry under which
Confining ourselves to and assuming relatively low ’s we hereafter neglect the second term in the definition of in Eq. (2). The Supersummetric (SUSY) F-term scalar potential obtained from in Eq. (2) is
where the complex scalar components of and are denoted by the same symbol. From Eq. (2), we find that the SUSY vacuum lies at the direction
where we take into account that the phase of , , is stabilized to zero during and after IGI. If is the holomorphic part of the frame function and dominates it, Eq. (2) assures a transition to the conventional Einstein gravity realizing, thereby, the idea of induced gravity .
To combine this idea with an inflationary setting we have to define a suitable relation between and the Kähler potential so as the scalar potential far away from the SUSY vacuum to admit inflationary solutions. To this end, we focus on Einstein frame (EF) action for ’s within SUGRA  which is written as
where is the F–term SUGRA scalar potential given below, summation is taken over the scalar fields , with , is the determinant of the EF metric . If we perform a conformal transformation defining the Jordan frame (JF) metric through the relation
where is a dimensionless (small in our approach) parameter which quantifies the deviation from the standard set-up , is written in the JF as follows
with being the JF potential in Eq. (2). If we specify the following relation between and ,
and employ the definition  of the purely bosonic part of the on-shell value of the auxiliary field
we arrive at the following action
where in Eq. (2) takes the form
It is clear from Eq. (2) that exhibits non-minimal couplings of the ’s to . However, also enters the kinetic terms of the ’s. To separate the two contributions we split into two parts
where is a dimensionless real function including the kinetic terms for the ’s and takes the form
with coefficients and of order unity. The fourth order term for is included to cure the problem of a tachyonic instability occurring along this direction , and the remaining terms of the same order are considered for consistency – the factors of are added just for convenience. On the other hand, in Eq. (2) is a dimensionless holomorphic function which, for , represents the non-minimal coupling to gravity – note that is independent of since . If is stabilized to zero, then and from Eqs. (2) and (2) we deduce that Eq. (2) recovers the conventional term of the Einstein gravity at the SUSY vacuum implementing thereby the idea of induced gravity. The choice , although not standard, is perfectly consistent with the set-up of non-minimal inflation  since the only difference occurring for is that the ’s do not have canonical kinetic terms in the JF due to the term proportional to in Eq. (2). This fact does not cause any problem since the canonical normalization of keeps its strong dependence on , whereas becomes heavy enough during IGI and so it does not affect the dynamics – see Sec. 3.1.
In conclusion, through Eq. (2) the resulting Kähler potential is
We set throughout, except for the case of no-scale SUGRA which is defined as follows:
This arrangement, inspired by the early models of soft SUSY breaking [9, 2], corresponds to the Kähler manifold with constant curvature equal to . In practice, these choices highly simplify the realization of IGI, rendering it more predictive thanks to a lower number of the remaining free parameters.
3 Inflationary Set-up
In this section we describe – in Sec. 3.1 – the derivation of the inflationary potential of our model and then – in Sec. 3.2 – we exhibit a number of observational and theoretical constraints imposed.
3.1 Inflationary Potential
Along the inflationary track determined by the constraints
if we express and according to the standard parametrization
the only surviving term in Eq. (3.1) is
Here we take into account that
where the functions and are defined along the direction in Eq. (3.1) as follows:
Given that with , in Eq. (3.1) is roughly proportional to . Therefore, an inflationary plateau emerges for and a chaotic-type potential (bounded from below) is generated for . More specifically, and the corresponding EF Hubble parameter, , can be cast in the following form:
where we introduce the functions and .
The stability of the configuration in Eq. (3.1) can be checked verifying the validity of the conditions
where are the eigenvalues of the mass matrix with elements and hat denotes the EF canonically normalized fields defined by the kinetic terms in Eq. (2) as follows
where the dot denotes derivation w.r.t the JF cosmic time and the hatted fields read
Upon diagonalization of , we construct the mass spectrum of the theory along the path of Eq. (3.1). Taking advantage of the fact that and the limits and we find the expressions of the relevant masses squared, arranged in Table 1, which approach rather well the quite lengthy, exact expressions taken into account in our numerical computation. We have numerically verified that the various masses remain greater than during the last e-foldings of inflation, and so any inflationary perturbations of the fields other than the inflaton are safely eliminated. They enter a phase of oscillations about zero with reducing amplitude and so the dependence in their normalization – see Eq. (3.1) – does not affect their dynamics. As usually – cf. Ref. [10, 2] –, the lighter eignestate of is which here can become positive and heavy enough for – see Sec. 4.2.
Inserting, finally, the mass spectrum of the model in the well-known Coleman-Weinberg formula, we calculate the one-loop corrected inflationary potential
where is a renormalization-group mass scale. We determine it by requiring  with the radiative corrections (RCs) to . To reduce the possible dependence of our results on the choice of , we confine ourselves to ’s and ’s which do not enhance the RCs. Under these circumstances, our results can be exclusively reproduced by using .
3.2 Inflationary Requirements
Based on in Eq. (3.1) we can proceed to the analysis of IGI in the EF , employing the standard slow-roll approximation. We have just to convert the derivations and integrations w.r.t to the corresponding ones w.r.t keeping in mind the dependence of on , Eq. (3.1). In our analysis we take into account the following observational and theoretical requirements:
where is the value of when crosses outside the inflationary horizon and is the value of at the end of IGI, which can be found from the condition
are the well-known slow-roll parameters and is the reheat temperature after IGI, which is taken throughout. We also assume canonical reheating  with an effective equation-of-state parameter and the effective number of relativistic degrees of freedom at temperature is taken corresponding to the MSSM spectrum.
The amplitude of the power spectrum of the curvature perturbation generated by at has to be consistent with data 
where the variables with subscript are evaluated at .
The remaining inflationary observables (the spectral index , its running , and the tensor-to-scalar ratio ) – estimated through the relations:
with – have to be consistent with the data , i.e.,
at 95 confidence level (c.l.) – pertaining to the CDM framework with . Although compatible with Eq. (3.2b) the present combined Planck and Bicep2/Keck Array results  seem to favor ’s of order since at 68 c.l. has been reported.
Since SUGRA is an effective theory below the existence of higher-order terms in and , Eqs. (2) and (2), appears to be unavoidable. Therefore, the stability of our inflationary solutions can be assured if we entail
where the UV cutoff scale of the effective theory for the present models is , as shown in Sec. 5.
The structure of as a function of for various ’s is displayed in Fig. 1, where we depict versus imposing . The selected values of and , shown in Fig. 1, yield and for increasing ’s – gray, light gray and black line. The corresponding values are . We remark that a gap of about one order of magnitude emerges between – and – for of order and due to the larger and values employed for ; actually, in the former case, – and – approaches the SUSY grand-unification scale, – cf. Ref. . This fact together with the steeper slope that acquires close to for is expected to have an imprint in elevating in Eq. (3.2) and, via Eq. (3.2c), on .
Confronting our inflationary scenario with the requirements above we can find its allowed parameter space. We here present our results for the two radically different cases: taking in Sec. 4.1 and in Sec. 4.2.
We focus first on the form of Kähler potential induced by Eq. (2) with . Our analysis in Sec. 4.1.1 presents some approximate expressions which assist us to interpret the numerical results exhibited in Sec. 4.1.2.
4.1.1 Analytic Results
Upon substitution of Eqs. (3.1) and (3.1) into Eq. (3.2), we can extract the slow-roll parameters which determine the strength of the inflationary stage. Performing expansions about , we can achieve approximate expressions which assist us to interpret the numerical results presented below. Namely, we find
As it may be numerically verified, the termination of IGI is triggered by the violation of the criterion at , which does not decline a lot from its value for . Namely we get
In the same approximation and given that , can be calculated via Eq. (3.2) with result
Obviously, IGI with subplanckian ’s can be achieved if
for . Therefore we need relatively large ’s.
Therefore, a clear dependence of and on arises, with the first one being much more efficient. This depedence does not exist within no-scale SUGRA since vanishes by definition – see Eq. (2).
4.1.2 Numerical Results
Our results are independent of , provided that – see in Table 1. The same is also valid for – see Eq. (3.1). We therefore set . Besides these values, in our numerical code, we use as input parameters and . For every chosen , we restrict and so that the conditions Eqs. (3.2), (3.2) and (3.2) are satisfied. By adjusting we can achieve ’s in the range of Eq. (3.2). Our results are displayed in Fig. 2-(a) and (b) where we delineate the hatched regions allowed by the restrictions of Sec. 3.2 in the  plane. The conventions adopted for the various lines are also shown. In particular, the dashed [dot-dashed] lines correspond to , whereas the solid (thick) lines are obtained by fixing – see Eq. (3.2). Along the thin line, which provides the lower bound for the regions presented in Fig. 2, the constraint of Eq. (3.2b) is saturated. At the other end, the allowed regions terminate along the dotted line where , since we expect values of order unity to be natural. From Fig. 2-(a) we see that remains almost proportional to and for constant , increases as decreases. From Fig. 2-(b) we remark that is confined close to zero for and or – see Eq. (4.1.1). Therefore, a degree of tuning (of the order of