# Improved Model of Primordial Black Hole Formation after Starobinsky Inflation

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## Abstract

**:**

## 1. Introduction

## 2. The New Model

## 3. Inflaton Potential and Slow-Roll

## 4. Power Spectrum of Scalar Perturbations and PBH Masses

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Note

1 | Accordingly, we changed the notation for the AB-parameter denoted by ${\u03f5}_{AB}$ in Ref. [20] to ${E}_{AB}$. |

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**Figure 1.**The inflaton potential for selected values of ${R}_{0}$ and $\delta $ at fixed $g=2.25$ and $b=2.89$ with ${V}_{0}=\frac{3}{4}{M}_{\mathrm{Pl}}^{2}{M}^{2}$ (on the

**left**). The potential for low values of $\varphi /{M}_{\mathrm{Pl}}$ are shown on the

**right**.

**Figure 3.**The SR parameter ${\u03f5}_{\mathrm{sr}}\left(\varphi \right)$ (on the

**left**) and the SR parameter ${\eta}_{\mathrm{sr}}\left(\varphi \right)$ (on the

**right**) for selected values of ${R}_{0}$ and $\delta $ at fixed $g=2.25$ and $b=2.89$. The end of Starobinsky inflation is reached at ${\varphi}_{end}\approx 2.98{M}_{\mathrm{Pl}}$ when ${\u03f5}_{\mathrm{sr}}\left({\varphi}_{end}\right)\approx 1$.

**Figure 4.**The evolution of inflaton field $\varphi \left(t\right)$ and Hubble function $H\left(t\right)$ with the initial conditions ${\varphi}_{in}=7.01\phantom{\rule{3.33333pt}{0ex}}{M}_{\mathrm{Pl}}$ and ${\dot{\varphi}}_{in}=0$, and the parameters $\delta =2.7\xb7{10}^{-8}$ and ${R}_{0}=3.0\phantom{\rule{3.33333pt}{0ex}}{M}^{2}$.

**Figure 5.**The evolution of the Hubble flow parameters ${\u03f5}_{H}\left(t\right)$ and ${\eta}_{H}\left(t\right)$ with the initial conditions ${\varphi}_{in}=7.01\phantom{\rule{3.33333pt}{0ex}}{M}_{\mathrm{Pl}}$ and ${\dot{\varphi}}_{in}=0$, and the parameters $\delta =2.7\xb7{10}^{-8}$ and ${R}_{0}=3.0\phantom{\rule{3.33333pt}{0ex}}{M}^{2}$.

**Figure 6.**The primordial power spectrum ${P}_{\zeta}\left(t\right)$ of scalar (curvature) perturbations, derived from Equation (21).

${\mathit{\varphi}}_{\mathbf{in}}/{\mathit{M}}_{\mathbf{Pl}}$ | $\mathit{\delta}$ | ${\mathit{n}}_{\mathit{s}}$ | r | ${\mathit{M}}_{\mathbf{PBH}}$, g | ${\mathit{N}}_{\mathbf{peak}}$ | ${\mathit{N}}_{\mathbf{total}}$ | $\mathbf{\Delta}\mathit{N}$ |
---|---|---|---|---|---|---|---|

6.36 | $2.55\times {10}^{-7}$ | 0.964959 | 0.0359 | $5.0\times {10}^{19}$ | 26 | 47 | 21 |

6.70 | $8.74\times {10}^{-8}$ | 0.964905 | 0.0182 | $2.0\times {10}^{19}$ | 34 | 54 | 20 |

7.01 | $2.70\times {10}^{-8}$ | 0.964944 | 0.0095 | $1.0\times {10}^{20}$ | 43 | 65 | 22 |

7.07 | $2.05\times {10}^{-8}$ | 0.964917 | 0.0083 | $2.6\times {10}^{18}$ | 45 | 64 | 19 |

7.12 | $1.60\times {10}^{-8}$ | 0.964925 | 0.0074 | $1.0\times {10}^{17}$ | 47 | 65 | 18 |

7.15 | $1.36\times {10}^{-8}$ | 0.964908 | 0.0070 | $5.0\times {10}^{16}$ | 49 | 66 | 17 |

7.20 | $1.02\times {10}^{-8}$ | 0.964961 | 0.0062 | $1.6\times {10}^{15}$ | 51 | 64 | 13 |

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Saburov, S.; Ketov, S.V.
Improved Model of Primordial Black Hole Formation after Starobinsky Inflation. *Universe* **2023**, *9*, 323.
https://doi.org/10.3390/universe9070323

**AMA Style**

Saburov S, Ketov SV.
Improved Model of Primordial Black Hole Formation after Starobinsky Inflation. *Universe*. 2023; 9(7):323.
https://doi.org/10.3390/universe9070323

**Chicago/Turabian Style**

Saburov, Sultan, and Sergei V. Ketov.
2023. "Improved Model of Primordial Black Hole Formation after Starobinsky Inflation" *Universe* 9, no. 7: 323.
https://doi.org/10.3390/universe9070323