# Inflation in Supergravity from Field Redefinitions

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## Abstract

**:**

## 1. Introduction

## 2. Transformation of the Kähler Potential

## 3. Prototypes of Models

#### 3.1. Monomial Models

#### 3.2. Locally Flat Potentials

#### 3.3. Generalization of the Starobinsky Inflation

#### 3.4. The Log${}^{2}\varphi $ Model for $K={K}_{-}$

#### The $K={K}_{+}$ Scenario

#### 3.5. A Simple Plateau Model

#### 3.6. Plateau from the Modular Transformation

#### 3.7. Bell-Curve Potentials

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Both panels show the potential of the (22) model for different values of p for $M=1$ and $M=p\phantom{\rule{0.166667em}{0ex}}\mu $ (left and right panels respectively), where M and $\mu $ 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.**Left Panel: results for the model (22) model for $M=1$, $p\in (7,39)$ and ${N}_{\U0001f7c9}=50$ and ${N}_{\U0001f7c9}=60$ (orange and blue dots respectively). Right panel: The same results for $M=10$. Note how the predicted value of r rise up with M.

**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 $M\gg 1$ the inflationary potential resembles the $V\propto {\varphi}^{2}$ model. Right panel: $r\left({n}_{s}\right)$ for ${N}_{\U0001f7c9}=50$ and ${N}_{\U0001f7c9}=60$ (dashed and solid lines respectively). Red, pink, orange and green points corresponds to $M=\sqrt{3/2}$, $M=5$, $M=10$ and $M=50$ respectively. The black points correspond to $r=4(1-{n}_{s})$, which is the result of the $V\propto {\varphi}^{2}$ model. For ${N}_{\U0001f7c9}=50$ and ${N}_{\U0001f7c9}=60$ one obtains the consistency with the data for $M<11$ and $M<17.5$ respectively.

**Figure 4.**Results for the (33) model for $M=1$. Solid (dashed) lines correspond to $K={K}_{+}$ ($K={K}_{-}$), while red and blue dots represent ${N}_{\U0001f7c9}=60$ and ${N}_{\U0001f7c9}=50$ respectively. The consistency with the data requires ${N}_{\U0001f7c9}\simeq 50$.

**Figure 5.**Both panels present scalar potential from Equation (31) with $K={K}_{+}$, as a function of ${u}_{R}$ and ${u}_{I}$. During inflation $|{u}_{R}|\ll M\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}|{u}_{I}|\gg M$ and after inflation the field rolls towards one of the minima at ${u}_{I}=0$ and ${u}_{R}=\pm M$ (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). Blue lines present the evolution of fields ${u}_{R}$ and ${u}_{I}$ from the inflationary plateau to their minima.

**Figure 6.**Left panel: The potential (35) for $M\simeq {M}_{c}$. 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 (35) on the $({n}_{s},r)$ plane. Dots correspond to $M=(1-k\times 5\times {10}^{-6}){M}_{c}$, where $k\in \{12,13,\dots ,20\}$ and bigger k corresponds to bigger ${n}_{s}$. Inflation can be only consistent with the data for $(1-9.5\times {10}^{-5}){M}_{c}\lesssim M\lesssim (1-7\times {10}^{-5}){M}_{c}$, which means that the model requires significant fine-tuning in order to be within the $2\sigma $ regime of Planck/BICEP results.

**Figure 7.**Left panel: The inflationary potential of model (36) and its generalization $V={\Lambda}^{2}{(1-{u}_{R}^{-1})}^{2n}$. Right panel: The tensor-to-scalar ratio and the spectral index for ${N}_{\U0001f7c9}=50$ and ${N}_{\U0001f7c9}=60$ (solid line, blue and red dots respectively). Dashed and dotted lines represent results for the generalization of the model, for which one has taken $f\left(T\right)={T}^{-n}/\sqrt{2}$. Solid, dashed and dotted lines correspond to $n=1$, $n=2$ and $n=4$ respectively.

**Figure 8.**Potential (45) for different values of $c/a$. Please note that for $c\gg a$, inflation takes place predominantly at the plateau, while for $c\ll a$ the last 60 e-folds of inflation happen around the ${\varphi}^{2}$ slope.

**Figure 9.**Results for the (45) model for both $c/a>1$ and $c/a<1$. Blue and red dots represent $N=50$ and $N=60$ respectively. Left panel: We consider $k=c/a$ and $k\in \{1,2,\dots ,10\}$. Right panel: $k=a/c$ and $k\in \{1,2,\dots ,20\}$. Note how in the $c\ll a$ limit one moves towards the results of ${\varphi}^{2}$ inflation.

**Figure 10.**Potential (51) for $p>0$ and $p<0$ (left and right panel respectively). For $p>0$ inflation happens around the local maximum and the potential does not have a minimum. For $p<0$ inflation happens on one of the infinite plateaus separated by the minimum.

**Figure 11.**Results for the (51) model for ${N}_{\U0001f7c9}=60$ and $p<0$ or $p>0$ (solid and dashed lines respectively). Red, orange, gray, blue and pink lines represent $M=1$, $M=10$, $M=20$, $M=30$ and $M=100$ and $p\in (-0.44,0.16)$, $p\in (-1.3,0.152)$, $p\in (-1.7,0.132)$, $p\in (-0.28,0.106)$ and $p\in (-0.053,0.042)$ respectively. Note how results for different M reproduce the result of the Starobinsky inflation in the $p\to 0$ limit. In addition, the results of the (51) model may cover the whole $2\sigma $ regime of the Planck/BICEP data.

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Artymowski, M.; Ben-Dayan, I. Inflation in Supergravity from Field Redefinitions. *Symmetry* **2020**, *12*, 806.
https://doi.org/10.3390/sym12050806

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Artymowski M, Ben-Dayan I. Inflation in Supergravity from Field Redefinitions. *Symmetry*. 2020; 12(5):806.
https://doi.org/10.3390/sym12050806

**Chicago/Turabian Style**

Artymowski, Michał, and Ido Ben-Dayan. 2020. "Inflation in Supergravity from Field Redefinitions" *Symmetry* 12, no. 5: 806.
https://doi.org/10.3390/sym12050806