我們利用 Tsallis 熵與包含指數修正項的熵，並且將熱力學第一定律應用在宇宙
的視界，藉此建構出新的暗能量模型，並且利用 CosmoMC 軟體進行數值運算
與擬和。我們發現利用此方法建構的暗能量模型，在微中子總質量的限制範圍
會被放寬。
我們也針對一個包含三階向量場交互作用的廣義 Proca 理論來建構暗能量模
型。我們利用 CMB 與 CMB+HST 觀測數據對此模型進行參數範圍的限制，我
們發現了在此模型中，哈伯常數在 68% 信賴區間為 H0 = 71.80+1.07
−0.60 (72.48+0.72−0.60) kms−1Mpc−1 基於 CMB(CMB+HST) 數據，這有助於解決哈伯常數不一致的問題。此外，此模型的約化卡平方接近 1，也代表著此模型是可以描述宇宙演化的候選模型。
另外，我們研究了其中一個可行 f(R) 暗能量模型在非平坦的 FLRW 時空背
景中的宇宙學特性。我們利用 CosmoMC 將此模型與觀測數據比較，並且給了
此模型的參數限制。我們也計算了此模型與 ΛCDM 模型的 AIC、BIC 與 DIC
數值。

We apply the first law of thermodynamics to the apparent horizon of the universe with the power-law corrected and non-extensive Tsallis entropies rather than the Bekenstein-Hawking one. We examine the cosmological properties in the two entropy models by using the CosmoMC package. In particular, the first numerical study for the cosmological observables with the power-law corrected entropy is performed. We also show that the neutrino mass sum has a non-zero central value
with a relaxed upper bound in the Tsallis entropy model comparing with that in the ΛCDM one.

We consider a specific dark energy model, which only includes the Lagrangian up to the cubic order in terms of the vector field self-interactions in the generalized Proca theory. We examine the cosmological parameters in the model by using the data sets of CMB and CMB+HST, respectively. In particular, the Hubble constant is found to be H0 = 71.80+1.07−0.72 (72.48+0.72−0.60) kms−1Mpc−1 at 68% C.L. with CMB (CMB+HST), which would alleviate the Hubble constant tension. We also obtain that the reduced χ^2 values in our model are close to unity when fitting with CMB and CMB+HST, illustrating that our model is a good candidate to describe the cosmological evolutions of the universe.
We study the viable Starobinsky f(R) dark energy model in spatially non-flat FLRW backgrounds, where f(R) = R − λ*Rch[1 − (1 + R2/R2
ch)−1] with Rch and λ representing the characteristic curvature scale and model parameter, respectively.
We modify CAMB and CosmoMC packages with the recent observational data
to constrain Starobinsky f(R) gravity and the density parameter of curvature ΩK.
In particular, we find that the model and density parameters to be λ
−1 < 0.283 at 68% C.L and ΩK = −0.00099+0.0044
−0.0042 at 95% C.L, respectively. The best χ^2 fitting result shows that χ^2_{f(R)} < χ^2_{ΛCDM} , indicating that the viable f(R) gravity model is consistent with ΛCDM when ΩK is set as a free parameter. We also evaluate the values of AIC, BIC and DIC for the best fitting results of f(R) and ΛCDM models in the non-flat universe.

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