Inflation in supergravity from field redefinitions

Supergravity (SUGRA) theories are specified by a few functions, most notably the real K\"ahler function denoted by $G(T_i, \bar {T}_i) = K + \log |W|^2$, where K is a real K\"ahler potential, and W is a holomorphic superpotential. A field redefinition $T_i \rightarrow f_1(T_i)$ does not change neither the theory, nor the K\"ahler geometry. Similarly, the K\"ahler transformation, $K \rightarrow K + f_2 + \bar f_2, W \rightarrow e^{-f_2} W$ where $f_2$ is holomorphic also leaves G and hence the theory and the geometry invariant. However, if we perform a field redefinition only in $K(T_i,\bar{T}_i) \rightarrow K(f(T_i),f(\bar{T}_i))$, while keeping the same superpotential $W(T_i)$, we get a different theory, as G is not invariant under such a transformation while maintaining the same K\"ahler geometry. This freedom of choosing $f(T_i)$ allows constructing an infinite number of new theories given a fixed K\"ahler geometry and a predetermined superpotential W. Our construction generalizes previous ones that were limited by the holomorphic property of $W$. In particular it allows for novel inflationary SUGRA models and particle phenomenology model building, where the different models correspond to different choices of field redefinitions. We demonstrate this possibility by constructing several prototypes of inflationary models (hilltop, Starobinsky-like, plateau, log-squared and bell-curve) all in flat K\"ahler geometry and an originally renormalizable superpotential $W$. The models are in accord with current observations and predict $r\in[10^{-6},0.06]$ spanning several decades that can be easily obtained. In the bell-curve model, there also exists a built-in gravitational reheating mechanism with $T_R\sim \mathcal{O}( 10^7 GeV)$.


Introduction
Cosmic inflation [1][2][3] is a hypothetical period of accelerated expansion of the Universe.Inflation solves certain problems of classical cosmology [4] and is responsible for generating the primordial inhomogeneities of the Universe.Predictions of inflation are also highly consistent with experimental data [5].Inflation usually requires flat potentials or flat directions in field space (for single and multi field inflationary models respectively), which appear naturally in Supersymmetry (SUSY) and Supergravity (SUGRA).On the other hand, the well-known eta-problem is most clearly evident in SUGRA, where canonical Kähler potential of the form K = T T immediately generates a very steep potential along the radial direction of the field due to the exponential nature of the scalar potential V = e K (• • • ).In the last decade many SUGRA inflationary models have been developed, with the emphasis on the α-attractors [6][7][8], for which a non-canonical Kähler potential generates a kinetic term with a pole and, in consequence, a scalar potential with an inflationary plateau.
α-attractors are characterized by Kähler potentials with non-zero Kähler curvature.Such a non-canonical Kähler potential generally assumes some knowledge of the UV theory, and restricts the possible UV completions.An alternative to this approach is to investigate Kähler potentials of flat geometry with R i jk l = 0.For instance, [9] Such a Kähler potential gives the scalars canonical kinetic terms and can always be viewed as a low energy approximation of some UV complete model without restrictions.
Obviously, such a Kähler potential has a flat Kähler direction along T R ≡ T (T I ≡ T ) for − (+) in (1.1).One can use this flat direction in order to generate inflation.The crucial ingredient is the shift symmetry of the Kähler potential, T → T ± c for pure imaginary (real) c.Generalization of this form of Kähler potentials has been used throughout the literature.Generalizations beyond the simple shift symmetry of singlets to Higgs doublets and other symmetry groups have been suggested in [10,11].
The existing SUGRA literature connects the possibility and the type of model of inflation with the Kähler geometry.Perhaps the well known examples are necessary conditions on the sectional curvature of the Kähler manifold [12][13][14].It is therefore tempting to consider inflationary models as different classes of Kähler geometries.Obviously this is not the case, since for a fixed Kähler manifold and metric, one can have various superpotentials leading to different models of inflation.A rather generic construction was specified in [15], where the authors considered K = K − and its generalizations with a superpotential of the form W = S F (T ), which give rise to a stable inflation trajectory of the form V = F (φ/ √ 2) 2 (where φ is a real part of the field T ).The parameters of the superpotential W still need to fulfill the slow-roll conditions, and F (T ) is a holomorphic function.We wish to generalize the above construction and remove the limitation of holomorphism of the function F .
The suggestion is the following.In SUGRA, the theory is specified, by a few functions.The true gauge invariant one for our discussion is the real Kähler function1 : where K(T i , Ti ) is the real Kähler potential and W (T i ) is the holomorphic superpotential.G fixes the Kähler geometry and the full action can be written in terms of G and its derivatives without the artificial separation between K and W .Using G one defines a Kähler geometry, defined by The holomorphic sectional curvature defined by R i jk l = G i jk l − g mn G ikn Gjl m plays a crucial role in determining the stability of the model, since the Hessian matrix V IJ (i.e. the matrix of the second derivatives of the scalar potential) is positively define for [14] where Note that the sectional curvature depends only on K.The particular example of (1.1) gives a flat Kähler geometry, since in such a case , where f 2 (T i ) is holomorphic, so such a transformation does not change the physics.Field redefinitions T i → f i (T i ) also do not change the physics.However, for this to hold, this field redefinition has to be applied to the full action, otherwise a new theory is constructed.Field redefinitions are in general not limited by holomorphism, and non holomorphic functions such as logarithm or a square root may also be considered.Our suggestion is that given a theory with a fixed K, W (≡ fixed G), we perform a field redefinition T i → f i (T i ) to K only.As a result we get a new theory, that allows us to construct various new inflationary models in SUGRA.
Similar models in prior work required for instance, the use of non-canonical Kähler/non-flat Kähler geometry, or were limited by the holomorphicity of the superpotential.We show that by judiciously choosing T i → f i (T i ), we can get various models, all with a simple fixed superpotential W , and a flat Kähler geometry.In particular we will show that for a single Kähler K ± = ± 1 2 f (T ) ± f ( T ) 2 + S S and superpotential W = ΛS T we can generate small field, large field and Starobinsky type inflation, the sole difference being the field redefinition that we picked.Hence, classes of inflation models differ neither by their Kähler geometry, nor by their superpotential, but only by the field redefinition of the Kähler potential that we have specified.
Let us stress that this method is applicable for any G and therefore for any SUGRA theory, any Kähler geometry and any superpotential W .However, to demonstrate the method's effectiveness and power we limit ourselves to the basic model of flat Kähler and a simple renormalizable superpotential.Analyzed inflationary scenarios consist of both small and large field models, with 10 −6 r 0.06 and n s consistent with observational data.In the bell-curve model, there also exists a built-in gravitational reheating mechanism with T R ∼ O(10 7 GeV ).

Transformation of the Kähler potential
Consider the following field redefinition for (1.1): where f (T ) has a dimension of mass.Let us stress that T → f (T ) in the full action is a simple field redefinition and does not change the physics.The novelty here is performing the transformation only on the Kähler potential.Hence, the geometry is unchanged and in (1.1) field space is flat.Our purpose here is to use it to construct new SUGRA models that support inflation.For the superpotential of the form of one finds the F-term scalar potential from the equation where The S field is the so-called stabilizer and plays no other role.The dependence on S allows us to integrate it out supersymmetrically, giving a VEV S = S = 0, yielding: In order to obtain a canonical Kähler metric let us introduce the following variable which finally gives where 2 and u R and u I are real fields with canonical kinetic terms.Note that after the field redefinition one finds K U Ū = 1 and the Kähler curvature remains zero.In this work we investigate models with several forms of f (T ) and we show how they can be used to generate inflation.Inflation happens along the flat Kähler direction, which means that during inflation one finds In further part of this work we will assume K = K − , unless explicitly stated otherwise.This comes from the fact that for the considered range of models K − always generates inflation with sufficiently flat potential and graceful exit, which is not the case for K = K + .As mentioned, we will focus on the simplest case of the superpotential W = ΛS T , for which one finds (2.9) As a consequence of the canonical kinetic terms for u R and u I the equation of motion of the inflaton field takes its well know form where H is a Hubble parameter, φ = u R or u I is the inflaton field and Inflation ends when the absolute value of one of the following slow-roll parameters and η becomes of order of 1, where Finally, in this work we will compare the predictions of the considered inflationary models with the data [5], which requires to obtain two quantities, which define main properties of the power spectrum of primordial inhomogeneities, namely tensor-toscalar ratio r and a spectral index n s r = 16 , n s = 1 − 6 + 2η .
(2.13) r and n s should be taken at a moment, when the pivot scale leaves the horizon.This moment corresponds to N e-folds before the end of inflation, where N is usually taken to be around 50 or 60.In all the figures, the green region corresponds to values of observables allowed by current observations.
3 Prototypes of models

Monomial Models
Consider U = f (T ) = T p for p ≥ 1 and w(T ) = T .In such case the scalar potential for the canonically normalized field U will be: where inserting the correct powers of M pl to account for the correct dimensionality is obvious.It is also obvious that u I = 0 is an extremum and the global minimum is at u I = u R = 0. Hence inflation takes place along the u I = 0 direction and the potential is simplified to a monomial model: The predictions of such models are well known with As shown in [5] the predictions of the monomial model lies within the 2σ regime the Planck/BICEP data only for N = 50 and p 3. This model is the only case considered in this paper, for which K − and K + gives exactly the same inflationary potential.

Locally flat potentials
The successful inflation requires only ∼ 60 e-folds, one can safely assume, that an inflationary potential may only be locally flat, just like in the case of hilltop models [16][17][18].In this section we present the Kähler potential, for which the local flatness appears around a local maximum of the potential, which is a stationary point of the order of p − 1.Let us assume that After carrying out the procedure specified in the previous section, one finds the scalar potential of the form of Inflation takes place along the flat Kähler direction, i.e. for u I = 0, which gives The potential (3.6) is plotted in Fig 1 .Around the minimum at u R = 0 one finds We want to emphasize that the desired form of the inflationary potential may in this case only be obtained with K = K − .
The potential (3.6) has a local maximum at u R = 0, around which one finds a locally flat area of the potential.The u R = 0 is a stationary point of the theory, thus this model can be understood as a mixture between a hill-top inflation and a higher order saddle point inflation 2 .In principle, as shown in [19], models with stationary point of the order of p tend to give low scale of inflation together with n s 1 − 2p (p−1)N , which gives correct value of n s for p 5. The results for the (3.6) model are plotted in the Fig. 2. Note how increasing M increases r and therefore the scale of inflation and field excursion, which can be estimate as Clearly, the model M = 1 (M = 10) is a good example of a small (large) field model.The (3.4) model can be generalized into ), which is equivalent to the (3.4) for M = µ p.In such a case the position of the local maximum is p-dependent.

Generalization of the Starobinsky inflation
one finds the scalar potential of the form of   The potential obtains its global minimum for u I = u R = 0, while the inflation happens for u I = 0 and u R > M .In such a case the potential (3.9) takes the following approximate form which is a simple generalization of the Starobinsky inflation, which has been already investigated in e.g.[20].In the M 1 limit the results of this model correspond to α-attractors, namely Note that this model is a certain limit of the (3.6) scenario.If in (3.6) one would consider M = µ p, then (3.6) would be equivalent to (3.10) for p → ∞.In such a case the stationary point moves towards u R → ∞.
A similar model has been investigated in [23], where the authors consider K = K − and f (T ) = arctan(T ).In such a case one recovers the scalar potential equivalent to α-attractors, which give predictions very similar to the Starobinsky inflation.
The (3.8) model is a good example on how a choice of K = K + may give radically different results.In such a case one finds which is a well known potential of natural inflation [24], which, unlike the Starobinsky inflation, is inconsistent with the data [5].

3.4
The log 2 φ model for K = K − Following the general model defined in the Sec. 2 let us consider the following theory which from (2.7) gives the following form of the scalar potential The inflation takes place in the valley of u I = 0. Global minima of the potential are u I = 0 and u R = ±M , for which one finds V = 0.The potential along the inflationary trajectory reads The entire evolution of the field may be described as reaching the u I = 0 valley from random initial conditions and then following the valley until u R reach global minimum at u R = ±M .The results for such a model line plotted in Fig. 5 (dashed line).

The K = K + scenario
This model has this interesting feature, that both V + and V − may generate successful inflation.The scalar potential V + as a function of u R and u I is presented in the Fig. 4. In the case of K = K + the evolution of fields may look way more complicated.First, the field reaches the u R = 0 valley, for which the inflationary potential may be approximated as Inflation occurs while |u R | M and u I M .In addition, in order to enable the inflaton to reach the global minimum at (u I , u R ) = (0, ±M ) one requires M < M c ≡ √ 2e 2.33.Otherwise the inflaton finds itself in a de Sitter (DS) minimum at the end of the logarithmic slope and graceful exit becomes possible only via quantum tunneling.In the close vicinity of M = M c one obtains another possibility of inflation along the u R direction.In order to see that let us investigate the evolution of the field for u I M , when inflation along the u I direction is over.In such a case, one finds As shown in the Fig. 6, the potential develops a local minimum for M > M c , which prevents the field to reach its global minimum.For M ≤ M c the only existing minima are in u R = ±M .For M = M c one obtains a saddle point at u R = 1, which means that inflation have two phases.The first one happens at the logarithmic slope of the log 2 (u I ) inflation, while the second one happens at the plateau in the vicinity of u R √ 2. The inflation along the u R direction requires certain fine tunning.In order to obtain at least 60 e-folds one requires M c − M O(10 −4 ).In addition, as shown in the Fig. 6, the consistency with the data requires M c − M 10 −5 M c .The reason for the fine-tuning is the following -for the saddle point inflation one obtains n s 0.92, which is highly inconsistent with Planck/BICEP data.Thus, one must deviate from the saddle point scenario and assume an inflection-point potential.On the other hand, the plateau obtained in this model is quite short, so even small deviations from the saddle-point inflation lead to insufficient number of e-folds.

A simple plateau model
One may easily obtain a flat inflationary potential using the following field redefinition and superpotential In such a case the potential is equal to Inflation happens in the u I = 0 valley, which leads to simple inflationary potential of the form of Both, spectral index and tensor-to-scalar ratio can be founded analytically.Using the N 1 approximation one finds u R = u R (N ) (2n(n + 2)N ) This form of f (T ) can be motivated by the modular transformation, for which ad−bc = 1.For the modular transformation one requires a, b, c and d to be integers.One can also consider more generic case of the P SL(2, R) group, for which one finds a, b, c, d ∈ R.
For K = K − and u I = 0 one finds One can simplify this potential using ad − bc = 1 and a simple field transformation u R → u R + √ 2a/c, which gives (3.24) The (3.24) model is simply a generalization of the (3.20).On the other hand, for K = K + one finds This potential obtains a minimum for u I = 0.In order to obtain a Minkowski vacuum one requires b = 0, which together with ad − bc = 1 gives The a4 term is a part of the normalization of the potential, which could be absorbed into Λ and does not affect r and n s .Thus, from the point of view of the predictions on the (n s , r) plane, the model has only one parameter, which is c/a.The (3.26) potential has two limits.For φ 2 a 2 /c 2 one finds V ∝ φ 2 , while for φ 2 a 2 /c 2 the potential obtains inflationary plateau.The potential for the (3.26) model is shown in the Fig. 8. Assuming the slow-roll approximation one finds r 16 In the small/big 4N c 2 /a 2 limit r and n s can be further simplified to In the 4N c 2 /a 2 1 one finds r ∼ 10 −1 , which is inconsistent with PLANCK/Bicep data.The comparison to the Planck/BICEP data is presented in 9.  where M = (2p − 1) √ 0 (2(1 − 2p) 0 ) −p p −1 and n = 2(p − 1)/(2p − 1).0 < p < 1/2 corresponds to n > 2, while p < 0 to 1 < n < 2. Note that the p → 0 limit corresponds models, small field, Starobinsky, large field and even a squared logarithmic potential as suggested in [28].We have also constructed the plateau model form a modular or PSL(2,R) transformation, that covers most of the allowed parameter space.The predictions of investigated span r ∈ [10 −6 , 0.06], and a valid n s for reasonable number of e-folds.Of particular interest are the bell curve models since their predictions cover all allowed values of r and n s .In addition, due to the kination of the inflaton after inflation, the model has a build-in mechanism of gravitational reheating, which does not require additional couplings of T to matter fields.The specific details of analyzed models are given in the Sec. 3.

Let us consider
The idea of the field redefinition bypasses the limit of holomorphicity used for generating general potentials by having an arbitrary holomorphic function f (T ) in W .This is because at the fundamental level, with the T field, we have the same holomorphic W , and what one really has are different real Kahler potentials, but all with the same Kahler geometry.As such, the construction abides the standard rules of SUGRA models.We have focused on the F-term scalar potential in SUGRA.It would be interesting to apply this method for D-term inflationary model building as well.

M = μ p , μ = 1 Figure 1 .
Figure 1.Both panels show the potential of the (3.6) model for different values of p for M = 1 and M = p µ (left and right panels respectively), where M and µ are p-independent constants.Inflation takes place around the local maximum of the potential.Right panel: note how the local maximum is moving towards bigger u R for bigger p.

Figure 2 .
Figure 2. Left Panel: results for the model (3.6) model for M = 1, p ∈ (7, 39) and N = 50 and N = 60 (orange and blue dots respectively).Right panel: The same results for M = 10.Note how the predicted value of r rise up with M .

. 11 ) 2 Figure 3 .
Figure 3. Left panel: inflationary potential for different values of M .For small M the last 60 e-folds of inflation happens on the plateau, while for M 1 the inflationary potential resembles the V ∝ φ 2 model.Right panel: r(n s ) for N = 50 and N = 60 (dashed and solid lines respectively).Red, pink, orange and green points corresponds to M = 3/2, M = 5, M = 10 and M = 50 respectively.The black points correspond to r = 4(1 − n s ), which is the result of the V ∝ φ 2 model.For N = 50 and N = 60 one obtains the consistency with the data for M < 11 and M < 17.5 respectively.

Figure 4 .
Figure 4.Both panels present scalar potential from the Eq.(3.13) with K = K + , as a function of u R and u I .During inflation |u R | M , |u I M | and after inflation the field rolls towards one of the minima at u I = 0 and u R = ±M (left panel).For M > M c , the field is stuck in the DS vacuum separated from the true vacuum by a local maximum (right panel).

Figure 5 .
Figure 5. Results for the (3.15) model for M = 1.Solid (dashed) lines correspond to K = K + (K = K − ), while red and blue dots represent N = 60 and N = 50 respectively.The consistency with the data requires N 50.

Figure 6 .
Figure 6.Left panel: The potential (3.17) for M M c .For M = M c one obtains a saddle point, around which one obtains a second phase of inflation.For M > M c one obtains a local maximum, which separates the inflaton from the Minkowski vacuum.Right panel: results for the (3.17) on the (n s , r) plane.Dots correspond to M = (1 − k × 5 × 10 −6 )M c , where k ∈ {12, 13, . . ., 20} and bigger k corresponds to bigger n s .Inflation can be only consistent with the data for (1 − 9.5 × 10 −5 )M c M (1 − 7 × 10 −5 )M c , which means that the model requires significant fine tuning in order to be within the 2σ regime of Planck/BICEP results.

. 20 ) 1 √ 2 T
In such a case one finds the global minimum of the potential at u R = ±1 and u I = 0.The scalar potential (3.20) is presented in Fig.7.The model can be easily generalized into f (T ) = −n , which gives V = Λ 2 (1 − u −1 R ) 2n .Increasing n decreases r and moves n s inside the 2σ of the Planck/BICEP data.The results for n = 1, n = 2 and n = 4 have been presented in the Fig.7.

.21) 3 . 6 Λ 2 Figure 7 .
Figure 7. Left panel: the inflationary potential for the (3.18) model and its generalization V = Λ 2 (1 − u −1 R ) 2n .Right panel: tensor-to-scalar ratio and spectral index for N = 50 and N = 60 (solid line, blue and red dots respectively).Dashed and dotted lines represent results for the generalization of the model, for which one has taken f (T ) = T −n / √ 2. Solid, dashed and dotted lines correspond to n = 1, n = 2 and n = 4 respectively.

Figure 8 .
Figure 8. Potential (3.26) for different values of c/a.Note that for c a the inflation takes place predominantly at the plateau, while for c a the last 60 e-folds of inflation happen around the φ 2 slope.