模拟光度距离的宇宙学常数的光滑LTB模型是否也能满足年龄约束？
Can smooth LTB models mimicking the cosmological constant for the luminosity distance also satisfy the age constraint?
摘要：
定义LTB解的函数的中心光滑度在它们模仿宇宙学常数的影响的能力中起着至关重要的作用。到目前为止，人们的注意力都集中在$C^{1}$模型，而在本文中，我们接近它一个更一般的方式，调查的影响，高阶中心光滑条件的LTB模型再现亮度距离的$ΛCDM$宇宙。我们的分析是基于一个低的红移展开，并扩展了以前的调查，包括来自宇宙年龄的约束，并重新表达方程的反演问题的解决方案，在一个明显的无量纲形式，这使得明显的自由，以适应任何值的$H_0$，以及，纠正了一些错误的说法，即$H_0$的观测值足以排除LTB模型。高阶光滑性条件相对于一阶条件强烈地限制了可能解的数量。无论是$C^{1}$还是$C^{i}$ LTB模型都不能同时满足年龄约束和模拟光度距离的宇宙学常数。一个区别是在不包括年龄约束并且bang函数为零的情况下，其中对于$C^1$模型存在唯一解，但是对于$C^{i}$情况没有解。另一个区别是在不包括年龄约束和bang函数不为零的情况下，其中$C^1$和$C^{i}$模型的解都是不确定的，但后者的剩余参数自由度要小得多。我们的研究结果意味着，任何LTB模型能够适应光度距离数据，并满足年龄的约束，要么不完全模仿的$ΛCDM$红移空间观测值的理论预测，或者它不是$C^{\infty}$光滑。
英文摘要：
The central smoothness of the functions defining a LTB solution plays a crucial role in their ability to mimick the effects of the cosmological constant. So far attention has been focused on $C^{1}$ models while in this paper we approach it a more general way, investigating the implications of higher order central smoothness conditions for LTB models reproducing the luminosity distance of a $ΛCDM$ Universe. Our analysis is based on a low red-shift expansion, and extends previous investigations by including also the constraint coming from the age of the Universe and re-expressing the equations for the solution of the inversion problem in a manifestly dimensionless form which makes evident the freedom to accommodate any value of $H_0$ as well, correcting some wrong claims that the observed value of $H_0$ would be enough to rule out LTB models. Higher order smoothness conditions strongly limit the number of possible solutions respect to the first order condition. Neither a $C^{1}$ or a $C^{i}$ LTB model can both satisfy the age constraint and mimick the cosmological constant for the luminosity distance. One difference is in the case in which the age constraint is not included and the bang function is zero, in which there is a unique solution for $C^1$ models but no solution for the $C^{i}$ case. Another difference is in the case in which the age constraint is not included and the bang function is not zero, in which the solution is undetermined for both $C^1$ and $C^{i}$ models, but the latter ones have much less residual parametric freedom. Our results imply that any LTB model able to fit luminosity distance data and satisfy the age constraint is either not mimicking exactly the $ΛCDM$ red-shift space observables theoretical predictions or it is not $C^{\infty}$ smooth.
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