Using $f\left(T\right)$ gravitational theory, we construct modified cosmological models via the first law of thermodynamics by using the non-extensive thermodynamics framework, the effects of which are captured by the parameter $\delta$. The resulting cosmological equations are modified compared to the standard Einstein-Hilbert ones, with the modifications coming from the $f\left(T\right)$ gravitational theory and from the non-extensive parameter which quantifies the non-extensive thermodynamics effects quantified by the parameter $\delta$, which when is set equal to unity, one recovers the field equations of $f\left(T\right)$ gravity. We study in detail the cosmological evolution of the model in the presence of collisionless non-relativistic matter case, and we derive the exact forms of the dark energy density parameter and of the dark energy equation of state parameter, from which we impose constraints on the non-extensive thermodynamics parameter, $\delta$, by using the Planck 2018 data on cosmological parameters. Accordingly, we repeat our calculations after including the relativistic matter along with the non-relativistic one, and we derive the new forms of the dark energy density parameter and of the dark energy equation of state parameter. Our study shows that the inclusion of non-extensive thermodynamic effects, quantified by the parameter $\delta$, for a flat Friedmann-Robertson-Walker Universe, has measurable differences compared with the normal thermodynamics case. We confront our results with Type Ia supernovae observations for $z\ge 0.4$ and we obtain reasonably agreement with the observational data.

中文翻译：

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f（T）引力的宇宙学中的非广义热力学效应

使用 $F（Ť）$ 引力理论，我们使用非扩展的热力学框架，通过热力学第一定律构造了修正的宇宙学模型，其效果被参数捕获 $\delta$。与标准的爱因斯坦-希尔伯特方程相比，对所得的宇宙学方程进行了修改，这些修改来自于$F（Ť）$ 引力理论，并从非扩展参数量化了由该参数量化的非扩展热力学效应 $\delta$，当设为等于1时，可以恢复的场方程 $F（Ť）$重力。我们详细研究了在无碰撞非相对论情况下模型的宇宙学演化，并推导了暗能量密度参数和状态参数暗能量方程的精确形式，从中将约束强加于非广泛的热力学参数，$\delta$，通过使用有关宇宙学参数的Planck 2018数据。因此，我们在将相对论和非相对论都包括在内之后，重复了计算，得出了暗能量密度参数和状态参数暗能量方程的新形式。我们的研究表明，通过参数量化了非广泛的热力学效应$\delta$与平坦的热力学情况相比，对于扁平的Friedmann-Robertson-Walker宇宙，具有可测量的差异。我们用Ia型超新星观测结果来应对我们的结果$ž\ge 0.4$ 并且我们与观测数据合理地达成了一致。