# Primordial Gravitational Waves and Reheating in a New Class of Plateau-Like Inflationary Potentials

## Abstract

**:**

## 1. Introduction

## 2. The $\mathcal{H}\left(\mathit{\phi}\right)$ Parametrization

## 3. Reheating and $\mathit{e}$-Folding

## 4. The Generalised Gaussian Model

#### 4.1. The Potential for $n=2$

#### 4.2. The Potential for $n>2$

## 5. Results

#### 5.1. ${n}_{s}$ and r

#### 5.2. Reheating Temperature

#### 5.3. Primordial Gravitational Waves at 1 Hz

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1. | As long as $\dot{H}={H}^{\prime}\left(\phi \right)\dot{\phi}<0$, which is a consequence of the Null-Energy Condition. |

2. | In particular, Reference [16] used the Hamilton-Jacobi approach to construct a family of plateau-like potentials using truncated series with stochastic coefficients drawn from special distributions, whereas we construct similar models using a simple Gaussian function. |

**Figure 1.**The inflaton potential $V\left(\phi \right)/V\left(0\right)$ for the Generalised Gaussian model $\mathcal{H}\propto \mathrm{exp}(-{\phi}^{n})$, with $n=4,6,8$. On each curve, the bottom left corner marks the end of inflation at $\phi =0$. All these potentials are consistent with Planck’s constraints in the $({n}_{s},r)$ plane.

**Figure 2.**Predictions in the ${n}_{s}$-r plane for the Generalised Gaussian Model (17) with $n=2$ (left block of four panels) and $n=4$. Each block contains four panels for reheating temperatures ${T}_{\mathrm{reh}}=1-{10}^{15}$ GeV. In each panel, the lines show predictions for the reheating equation of state $\overline{w}=-1/3,0,0.25$ and 1. On each line, as the model parameter $\alpha $ is varied from small to large, r decreases steadily towards zero. See Section 5.1 for discussion.

**Figure 3.**Predictions in the ${n}_{s}$-${T}_{\mathrm{reh}}$ plane for the Generalised Gaussian Model (17) with $n=4,6,8$, with $\alpha =1$. The shaded region shows Planck’s $2\sigma $ constraint on ${n}_{s}$. The curves intersect where reheating would occur instantaneously.

**Figure 4.**The amplitude of inflationary gravitational waves, ${\mathsf{\Omega}}_{\mathrm{gw}}{h}^{2}$ measured at 1 Hz (Equation (25)) plotted against ${n}_{s}$, for the GG model with $n=4$ and 8, and ${T}_{\mathrm{reh}}=1$ and ${10}^{10}$ GeV. The shaded region is the $2\sigma $ constraint on ${n}_{s}$ from Planck. Various curves in each panel correspond to the values of $\overline{w}$ as before. The thick part of each line corresponds to where $r<0.07$ [41]. In the lower panels, we omit the $\overline{w}=-1/3$ curves as they are already ruled out by the constraint on ${n}_{s}$.

**Figure 5.**${\mathsf{\Omega}}_{\mathrm{gw}}{h}^{2}$ plotted against r for the same models as in Figure 4. The dashed vertical line in each panel shows the current upper bound $r<0.07$.

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**MDPI and ACS Style**

Chongchitnan, S. Primordial Gravitational Waves and Reheating in a New Class of Plateau-Like Inflationary Potentials. *Universe* **2018**, *4*, 77.
https://doi.org/10.3390/universe4070077

**AMA Style**

Chongchitnan S. Primordial Gravitational Waves and Reheating in a New Class of Plateau-Like Inflationary Potentials. *Universe*. 2018; 4(7):77.
https://doi.org/10.3390/universe4070077

**Chicago/Turabian Style**

Chongchitnan, Siri. 2018. "Primordial Gravitational Waves and Reheating in a New Class of Plateau-Like Inflationary Potentials" *Universe* 4, no. 7: 77.
https://doi.org/10.3390/universe4070077