# Non-Extensive Thermodynamics Effects in the Cosmology of f(T) Gravity

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## Abstract

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## 1. Introduction

#### Historical Review and Introductory Remarks

## 2. Essential Features of $\mathit{f}\left(\mathit{T}\right)$ Gravity

**Torsion tensor**$${T}^{\alpha}{\phantom{\rule{1.42271pt}{0ex}}}_{\mu \nu}={h}_{a}{\phantom{\rule{1.42271pt}{0ex}}}^{\alpha}\phantom{\rule{2.84544pt}{0ex}}[{\partial}_{\mu}{h}^{a}{\phantom{\rule{1.42271pt}{0ex}}}_{\nu}-{\partial}_{\nu}{h}^{a}{\phantom{\rule{1.42271pt}{0ex}}}_{\mu}]\phantom{\rule{0.166667em}{0ex}},$$$${h}^{a}{\phantom{\rule{1.42271pt}{0ex}}}_{\mu}{h}_{a}{\phantom{\rule{1.42271pt}{0ex}}}^{\nu}={\delta}_{\mu}{}^{\nu}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}{h}^{a}{\phantom{\rule{1.42271pt}{0ex}}}_{\mu}{h}_{b}{\phantom{\rule{1.42271pt}{0ex}}}^{\mu}={\delta}_{b}^{a}\phantom{\rule{0.166667em}{0ex}}.$$**Contortion Tensor**$${K}^{\mu \nu}{\phantom{\rule{1.42271pt}{0ex}}}_{\alpha}=-\frac{1}{2}\left({T}^{\mu \nu}{\phantom{\rule{1.42271pt}{0ex}}}_{\alpha}-{T}^{\nu \mu}{\phantom{\rule{1.42271pt}{0ex}}}_{\alpha}-{T}_{\alpha}{\phantom{\rule{1.42271pt}{0ex}}}^{\mu \nu}\right)\phantom{\rule{0.166667em}{0ex}}.$$The contortion tensor is skew symmetric in its first pair of indices as it is clear from Equation (10).**Superpotential**$${S}_{\alpha}{\phantom{\rule{1.42271pt}{0ex}}}^{\mu \nu}=\frac{1}{2}\left({K}^{\mu \nu}{\phantom{\rule{1.42271pt}{0ex}}}_{\alpha}+{\delta}_{\alpha}^{\mu}\phantom{\rule{2.84544pt}{0ex}}{T}^{\beta \nu}{\phantom{\rule{1.42271pt}{0ex}}}_{\beta}-{\delta}_{\alpha}^{\nu}\phantom{\rule{2.84544pt}{0ex}}{T}^{\beta \mu}{\phantom{\rule{1.42271pt}{0ex}}}_{\beta}\right)\phantom{\rule{0.166667em}{0ex}},$$

## 3. Including Non-Extensive Thermodynamics Effects in $\mathit{f}\left(\mathit{T}\right)$ Gravity

**The Black Hole entropy**The black hole entropy, namely the Bekenstein−Hawking entropy, is defined as follows [81],$$S=\frac{A}{4G}\phantom{\rule{0.166667em}{0ex}},$$$${r}_{a}=\frac{1}{\sqrt{{H}^{2}+\frac{k}{{a}^{2}}}}\phantom{\rule{0.166667em}{0ex}}.$$**Entropy and $\mathit{f}\left(\mathit{T}\right)$ Linkage**

#### 3.1. Late-Time Evolution of the Universe

#### 3.2. Evolution Including Relativistic Matter

#### 3.3. The Distance Modulus within Non-Extensive $f\left(T\right)$-Gravity

## 4. Results of Our Numerical Analysis and Confrontation with the Observations

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The relation between the non-extensive parameter, $\delta $, versus the integration constant C given in Equation (42) and also a comparison between these two parameters presented in the study in Reference [91] (red curve) (

**b**) The relation between the non-extensive thermodynamics $\delta $ and the dimensional parameter $\beta $.

**Figure 2.**The left-panel is the matter density ${\Omega}_{m}$ (solid-red) and the dark energy density ${\Omega}_{DE}$ (dot-dashed black), as a function of the redshift z, for which $\beta =-0.027$, ${H}_{0}=1$, $\alpha =1$ and $\delta $ = 1. The right panel is the same densities at $\delta $ = 1.1, and $\beta =-0.103$.

**Figure 3.**Different behaviors of the EoS of dark energy, ${\omega}_{DE}$, (right-panel) verses the red-shift z, for different values of the non-extensive $\delta =1,1.1,1.2,1.3$. The deceleration parameter $q\left(z\right)$ at different values of the non-extensive $\delta =1,1.1,1.2$ is draw in (right-panel).

**Figure 4.**The behavior of the dark-energy and matter content densities given in the left and the right panel calculated for each value of $\beta $, and for ${H}_{0}=1$, $\alpha =1$.

**Figure 5.**The dark energy EoS parameter for different values of $\delta $ including the relativistic matter perfect fluid effects.

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Shalaby, A.G.; Oikonomou, V.K.; Nashed, G.G.L.
Non-Extensive Thermodynamics Effects in the Cosmology of *f*(*T*) Gravity. *Symmetry* **2021**, *13*, 75.
https://doi.org/10.3390/sym13010075

**AMA Style**

Shalaby AG, Oikonomou VK, Nashed GGL.
Non-Extensive Thermodynamics Effects in the Cosmology of *f*(*T*) Gravity. *Symmetry*. 2021; 13(1):75.
https://doi.org/10.3390/sym13010075

**Chicago/Turabian Style**

Shalaby, Asmaa G., Vasilis K. Oikonomou, and Gamal G. L. Nashed.
2021. "Non-Extensive Thermodynamics Effects in the Cosmology of *f*(*T*) Gravity" *Symmetry* 13, no. 1: 75.
https://doi.org/10.3390/sym13010075