从Noether对称性看$Λ$ CDM和$f(T)$引力的区别
The distinctions between $Λ$CDM and $f(T)$ gravity according Noether symmetry
摘要：
Noether理论为我们研究各种修正引力理论的守恒量和对称性提供了一个有用的工具，其中$f(T)$理论是一种广义修正的遥平行引力理论，被用来解释暗能量现象。利用Noether对称性方法，我们研究了$f(T)$理论的幂律形式、指数形式和多项式形式。所有形式的$f(T)$都具有时间平移对称性，这与能量条件或Hamilton约束有关。此外，我们发现幂律和指数形式的性能并不令人满意。在$T^n$中增加一个线性项$T$是合理的，它是在小尺度上模拟平行引力或广义相对论的最有效修正，即，太阳系的规模。相应的Noether对称性表明只有时间平移对称性保留。通过数值计算和观测数据约束，得到了最佳形式的$αT + βT^{-1}$，它的宇宙学解与标准$Λ$ CDM最接近，但宇宙年龄略有减小，这可以通过引入另一项$T^m$来缓解.更重要的是，我们发现了$Λ$ CDM和$f(T)$重力之间的显著差异。$Λ$ CDM模型除了有两个负守恒量外，还有两个额外的对称性和相应的正守恒量.
英文摘要：
Noether's theory offers us a useful tool to research the conserved quantities and symmetries of the modified gravity theories, among which the $f(T)$ theory, a generally modified teleparallel gravity, has been proposed to account for the dark energy phenomena. By the Noether symmetry approach, we investigate the power-law, exponential and polynomial forms of $f(T)$ theories. All forms of $f(T)$ concerned in this work possess the time translational symmetry, which is related with energy condition or Hamilton constraint. In addition, we find out that the performances of the power-law and exponential forms are not pleasing. It is rational adding a linear term $T$ to $T^n$ as the most efficient amendment to resemble the teleparallel gravity or General Relativity on small scales, ie., the scale of the solar system. The corresponding Noether symmetry indicates that only time translational symmetry remains. Through numerically calculations and observational data-sets constraining, the optimal form $αT + βT^{-1}$ is obtained, whose cosmological solution resembles the standard $Λ$CDM best with lightly reduced cosmic age which can be alleviated by introducing another $T^m$ term. More important is that we find the significant differences between $Λ$CDM and $f(T)$ gravity. The $Λ$CDM model has also two additional symmetries and corresponding positive conserved quantities, except the two negative conserved quantities.
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