In this work, we revisit the dynamics of pre-inflationary universe with a family of $\alpha-$attractor potentials, in the framework of loop quantum cosmology, in which the big bang singularity is generically resolved purely with quantum geometric effects, and replaced by a quantum bounce. At the bounce, the background evolution is divided into two distinct classes, the first is dominated by the kinetic energy of the inflaton field and the second by the potential energy. In both classes, we find the physically viable initial conditions numerically that provide not only the slow-roll inflation, but also sufficient e-folds to be compatible with observations. In the entire range of kinetic energy dominated initial conditions (except some subsets of Models 2 and 4), the background evolution prior to reheating is always split into three different phases: bouncing, transition and slow-roll inflation. In the bouncing phase, the numerical evolution of the scale factor is independent not only of the initial data, but also the inflationary potentials, as long as it is dominated by the kinetic energy, and can be well approximated by an analytical solution, whereas in the potential energy dominated case, such approximated results do not exist. Moreover, we study the phase space analysis for a class of $\alpha-$attractor potentials, and discuss the phase space trajectories for physically viable initial conditions of the inflaton field.

中文翻译：

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重温环量子宇宙学中α-吸引子家族的暴胀前宇宙

在这项工作中，我们在环量子宇宙学的框架内，重新审视了暴胀前宇宙的动力学，其中大爆炸奇点一般纯粹用量子几何效应解决，并替换为通过量子反弹。在反弹时，背景演化分为两个不同的类别，第一类由暴胀场的动能主导，第二类由势能主导。在这两类中，我们在数值上找到了物理上可行的初始条件，它不仅提供慢滚膨胀，而且提供足够的 e-folds 以与观测兼容。在整个动能主导的初始条件范围内（模型 2 和模型 4 的某些子集除外），再加热前的背景演化总是分为三个不同的阶段：反弹、过渡和慢速通胀。在弹跳阶段，尺度因子的数值演化不仅与初始数据无关，而且与暴涨势无关，只要它以动能为主导，并且可以很好地被解析解近似，而在在势能占优势的情况下，这种近似结果是不存在的。此外，我们研究了一类$\alpha-$吸引子势的相空间分析，并讨论了暴胀子场物理可行初始条件的相空间轨迹。并且可以通过解析解很好地近似，而在势能主导的情况下，这种近似结果不存在。此外，我们研究了一类$\alpha-$吸引子势的相空间分析，并讨论了暴胀场物理可行初始条件的相空间轨迹。并且可以通过解析解很好地近似，而在势能主导的情况下，这种近似结果不存在。此外，我们研究了一类$\alpha-$吸引子势的相空间分析，并讨论了暴胀子场物理可行初始条件的相空间轨迹。