# Constraints on Non-Flat Starobinsky f(R) Dark Energy Model

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## Abstract

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## 1. Introduction

## 2. Starobinsky f(R) Gravity in the Non-Flat Universe

## 3. Numerical Results

#### 3.1. Cosmological Evolutions

#### 3.2. Global Fitting Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Evolutions of ${\rho}_{DE}/{\rho}_{DE}^{0}$ for the Starobinsky $f\left(R\right)$ model with ${\lambda}^{-1}=0.4$ in the flat and non-flat universe, where ${\rho}_{DE}^{0}$ represents the energy density of dark energy at the present time, while the initial values are given by ${\mathsf{\Omega}}_{m}^{\left(0\right)}\simeq 0.3144$, and ${\mathsf{\Omega}}_{\mathsf{\Lambda}}^{\left(0\right)}\simeq (0.6742,0.6842,0.6942)$ for ${\mathsf{\Omega}}_{K}^{0}=(0.01,0,-0.01)$.

**Figure 2.**Evolutions of ${w}_{DE}$ for the Starobinsky $f\left(R\right)$ model with ${\lambda}^{-1}=0.4$ in the flat and non-flat universe, where the initial values are the same as Figure 1.

**Figure 3.**One and two-dimensional distributions of ${\mathsf{\Omega}}_{b}^{0}{h}^{2}$, ${\mathsf{\Omega}}_{c}^{0}{h}^{2}$, $\tau $, ${\mathsf{\Omega}}_{K}$, $\sum {m}_{\nu}$, ${\lambda}^{-1}$ and ${H}_{0}$ for the Starobinsky $f\left(R\right)$ and $\mathsf{\Lambda}$CDM models without the flatness assumption with the combined data of CMB, BAO and Pantheon data sets, where the contour lines represent 68% and 95% C.L., respectively.

**Table 1.**Priors of cosmological parameters for Starobinsky $f\left(R\right)$ and $\mathsf{\Lambda}$CDM models in the non-flat universe.

Parameter | Prior |
---|---|

$f\left(R\right)$ model parameter ${\lambda}^{-1}$ | ${10}^{-4}\le {\lambda}^{-1}\le 1$ |

Curvature parameter ${\mathsf{\Omega}}_{K}$ | $-0.1\le {\mathsf{\Omega}}_{K}\le 0.1$ |

Baryon density | $0.5\le 100{\mathsf{\Omega}}_{b}{h}^{2}\le 10$ |

CDM density | $0.1\le 100{\mathsf{\Omega}}_{c}{h}^{2}\le 99$ |

Optical depth | $0.01\le \tau \le 0.8$ |

Neutrino mass sum | $0\le \Sigma {m}_{\nu}\le 2$ eV |

$\frac{\mathrm{Sound}\phantom{\rule{4pt}{0ex}}\mathrm{horizon}}{\mathrm{Angular}\phantom{\rule{4pt}{0ex}}\mathrm{diameter}\phantom{\rule{4pt}{0ex}}\mathrm{distance}}$ | $0.5\le 100{\theta}_{MC}\le 10$ |

Scalar power spectrum amplitude | $1.61\le ln\left({10}^{10}{A}_{s}\right)\le 3.91$ |

Spectral index | $0.8\le {n}_{s}\le 1.2$ |

**Table 2.**Fitting results in Starobinsky $f\left(R\right)$ and $\mathsf{\Lambda}$CDM models without the flatness assumption with the CMB, BAO and Pantheon data sets, where the cosmological parameters and the model parameter ${\lambda}^{-1}$ are constrained at 95% C.L. and 68% C.L., respectively.

Parameter | Starobinsky $\mathit{f}\left(\mathit{R}\right)$ | $\mathbf{\Lambda}$CDM |
---|---|---|

${\mathsf{\Omega}}_{\mathit{b}}{\mathit{h}}^{\mathbf{2}}$ | $0.{02241}_{-0.00030}^{+0.00030}$ | $0.{02242}_{-0.00031}^{+0.00031}$ |

${\mathsf{\Omega}}_{\mathit{c}}{\mathit{h}}^{\mathbf{2}}$ | $0.{1195}_{-0.0027}^{+0.0028}$ | $0.{1195}_{-0.0026}^{+0.0027}$ |

$\mathbf{100}{\mathit{\theta}}_{\mathit{MC}}$ | $1.{04097}_{-0.00063}^{+0.00062}$ | $1.{04100}_{-0.00062}^{+0.00059}$ |

$\mathit{\tau}$ | $0.{056}_{-0.014}^{+0.016}$ | $0.{056}_{-0.014}^{+0.016}$ |

${\mathsf{\Omega}}_{\mathit{K}}$ | $0.{00099}_{-0.0042}^{+0.0044}$ | $0.{0005}_{-0.0040}^{+0.0040}$ |

$\mathbf{\Sigma}{\mathit{m}}_{\mathit{\nu}}$ | $<0.137$ eV | $<0.132$ eV |

$\mathrm{ln}\left({\mathbf{10}}^{\mathbf{10}}{\mathit{A}}_{\mathit{s}}\right)$ | $3.{047}_{-0.028}^{+0.030}$ | $3.{046}_{-0.027}^{+0.031}$ |

${\mathit{n}}_{\mathit{s}}$ | $0.{9664}_{-0.0089}^{+0.0087}$ | $0.{9666}_{-0.0084}^{+0.0082}$ |

${\mathit{\lambda}}^{-\mathbf{1}}$ | $<0.283$ | − |

${H}_{0}$ | $67.{6}_{-1.5}^{+1.5}$ | $68.{0}_{-1.2}^{+1.2}$ |

${\sigma}_{8}$ | $0.{811}_{-0.021}^{+0.020}$ | $0.{814}_{-0.019}^{+0.018}$ |

$\mathrm{Age}/\mathrm{Gyr}$ | $13.{74}_{-0.17}^{+0.16}$ | $13.{76}_{-0.15}^{+0.15}$ |

${\mathit{\chi}}_{\mathit{best}-\mathit{fit}}^{\mathbf{2}}$ | $3821.72$ | $3821.84$ |

**Table 3.**The results of AIC, BIC and DIC computed from the sample we used for both $\mathsf{\Lambda}$CDM and exponential $f\left(R\right)$ models, where $\Delta AIC=AI{C}_{f\left(R\right)}-AI{C}_{\mathsf{\Lambda}CDM}$, $\Delta BIC=BI{C}_{f\left(R\right)}-BI{C}_{\mathsf{\Lambda}CDM}$, and $\Delta DIC=DI{C}_{f\left(R\right)}-DI{C}_{\mathsf{\Lambda}CDM}$.

Model | ${\mathit{\chi}}_{\mathit{min}}^{2}$ | AIC | $\mathbf{\Delta}$AIC | BIC | $\mathbf{\Delta}$BIC | DIC | $\mathbf{\Delta}$DIC |
---|---|---|---|---|---|---|---|

$\mathsf{\Lambda}$CDM | $3821.84$ | $3837.84$ | 0 | $3887.35$ | 0 | $3850.38$ | 0 |

Starobinsky $f\left(R\right)$ | $3821.72$ | $3839.72$ | $1.88$ | $3895.42$ | $8.07$ | $3852.41$ | $2.03$ |

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**MDPI and ACS Style**

Geng, C.-Q.; Hsu, Y.-T.; Lu, J.-R.
Constraints on Non-Flat Starobinsky *f*(*R*) Dark Energy Model. *Entropy* **2021**, *23*, 1320.
https://doi.org/10.3390/e23101320

**AMA Style**

Geng C-Q, Hsu Y-T, Lu J-R.
Constraints on Non-Flat Starobinsky *f*(*R*) Dark Energy Model. *Entropy*. 2021; 23(10):1320.
https://doi.org/10.3390/e23101320

**Chicago/Turabian Style**

Geng, Chao-Qiang, Yan-Ting Hsu, and Jhih-Rong Lu.
2021. "Constraints on Non-Flat Starobinsky *f*(*R*) Dark Energy Model" *Entropy* 23, no. 10: 1320.
https://doi.org/10.3390/e23101320