The thermodynamics-gravity conjecture reveals that one can derive the gravitational field equations by using the first law of thermodynamics and vice versa. Considering the entropy associated with the horizon in the form of non-extensive Tsallis entropy, S ∼ Aβ here, we first derive the corresponding gravitational field equations by applying the Clausius relation δQ = TδS to the horizon. We then construct the Friedmann equations of Friedmann-Lemaître-Robertson-Walker Universe based on Tsallis modified gravity (TMG). Moreover, in order to constrain the cosmological parameters of TMG model, we use observational data, including Planck cosmic microwave background, weak lensing, supernovae, baryon acoustic oscillations, and redshift-space distortions data. Numerical results indicate that TMG model with a quintessential dark energy is more compatible with the low redshift measurements of large scale structures by predicting a lower value for the structure growth parameter σ8 with respect to ΛCDM model. This implies that TMG model would slightly alleviate the σ8 tension.

中文翻译：

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对 Tsallis 修正重力的观测约束

热力学-引力猜想表明，利用热力学第一定律可以推导出引力场方程，反之亦然。考虑以非广延 Tsallis 熵 S ∼ Aβ 形式与视界相关的熵，我们首先通过将克劳修斯关系 δQ = TδS 应用于视界推导出相应的引力场方程。然后，我们基于 Tsallis 修正重力 (TMG) 构建弗里德曼-勒梅特-罗伯逊-沃克宇宙的弗里德曼方程。此外，为了约束 TMG 模型的宇宙学参数，我们使用观测数据，包括普朗克宇宙微波背景、弱透镜、超新星、重子声振荡和红移空间畸变数据。数值结果表明，通过预测结构生长参数 σ8 相对于 ΛCDM 模型的较低值，具有典型暗能量的 TMG 模型与大尺度结构的低红移测量更兼容。这意味着 TMG 模型将略微缓解 σ8 张力。