圈量子引力作为一种候选的量子引力方案经过三十年的发展已经取得了非 常大的进展。作为它的对称约化模型，圈量子宇宙学由于解决了宇宙学大爆炸 奇点而备受人们关注。为了进一步考察圈量子宇宙学的合理性以及它的预言力， 圈量子宇宙学微扰的研究是非常必要的。在考察圈量子宇宙学的微扰时，人们 发现了约束反常现象，即约束代数不再封闭了。为了解决约束代数的不封闭性， 人们提出了所谓的反常自由的微扰方法。圈量子宇宙学对经典宇宙学的改造主 要体现为和乐修正和逆体积修正，其中由于逆体积修正一方面对于宇宙反弹不 起主要作用，另一方面依赖于宇宙元胞的选择而更加复杂，本文主要考虑和乐 修正。反常自由的和乐修正微扰理论揭示了畸变的时空约束代数，特别是在能 量密度接近反弹点密度时，可以发现时空结构是类欧式的，这意味着一个可能 的时间起点：我们应该将时间起点定义在从欧式结构到闵氏结构的转变点处。 更有趣的的是基于畸变的时空约束代数和微扰动力学方程，在转变点之后可以 自然定义一个不同于经典广义相对论的因果结构。本文主要基于这些认识，考 察圈量子宇宙学在暴胀理论中的应用。 暴胀理论试图解决标准宇宙学模型的三大疑难：视界疑难，平直型疑难以 及磁单极子疑难。暴胀理论的解决方案是设想早期宇宙（辐射主导期之前）曾 经经历过一次近指数膨胀过程，只要膨胀足够多（大于 60 个 e-叠数），再加上 其它自然的过程 (包括退出机制和粒子产生机制)，三大疑难可以得到解决。人 们自然会问到，对于一个待定模型，如此多的膨胀数本身是否有足够多的可能。 本文第四章将基于圈量子宇宙的理论框架，利用圈量子宇宙学微扰论的新认识 分析了视界疑难的解决可能性问题。 暴胀理论另一个精彩的地方是它能自然地解释宇宙大尺度结构。本文第五 章讨论圈量子宇宙学对暴胀功率谱的量子修正。我们发现虽然将初值设在号差 转变点是非常自然的，然而不幸的是真空无法在那一时刻构造，同时我们也发 现在暴胀微扰计算中普遍使用的较强要求的慢滚近似也不再成立。因此我们放 弃在转逆点定义真空的企图，而将初值设在转变点之后的临近点，幸运的是不 仅真空对大 k 模式有良好定义，而且在这一点处强要求的慢滚近似也得到了保 证。基于这种初值选择，我们定义了微扰场的量子真空，并计算了标量功率谱、张量功率谱以及它们的谱指数。我们发现当考虑的 k–模式穿出视界时的能量密 度远低于普朗克能标时，圈量子宇宙学的结论与经典理论一致。 值得注意的是暴胀理论面临了一个基本难题：跨 Planck 问题（trans-Planckian problem），即虽然我们在 CMB 观察到的模式在穿出视界时能标远低于普朗克 能标，但这些模式的物理尺度在暴胀早期却是普朗克长度的。圈量子宇宙学的 微扰论现在还没有考虑到这个问题，这表明了今天的圈量子宇宙学微扰论尽管 保持了约束代数的封闭性，但仍不是最后的微扰论。作为展望我们将在最后一 章讨论这个问题和其它圈量子宇宙学中有趣的问题。 另外，前三章的基本结构如下：首先我们在第一章中对圈量子引力和圈量 子宇宙学作一些基础的介绍，第二章将讨论和乐修正的圈量子宇宙学微扰论，第 三章我们通过对微扰论方程的特征线方程的分析，试图获得圈量子宇宙中的有 效因果，这个有效因果为第四章和第五章的讨论作了必要的准备。

Loop Quantum Gravity as a candidate of quantum gravity has been developed widely in the recent 30 years. Loop Quantum Cosmology(LQC) as its symmetry-reduced model attracts much attention because it solves the Big Bang singularity. To test LQC’s validity and to obtain more predications from loop quantum cosmology, the theory of perturbations in the framework of LQC is required. When considering the perturbations of Loop quantum cosmology, one will find the constraints would be closed. To solve the closure problem of constraint algebra, people proposed the anomaly-free perturbations of loop quantum cosmology. LQC has two main corrections to classical cosmology, i.e. holonomy corrections and inverse volume corrections. Since inverse volume corrections on the one hand do not play the main role in the solving Big Bang singularity, on the other hand depend on the choice of cosmos’s volume cell, we will only consider holonomy corrections in this paper. The perturbations based on holonomy corrections reveal a deformed space-time algebra, especially near the big bounce, the structure of space-time is more like Euclidean, which means a possible starting point time: We should take time starts at the transition point from Euclidean space to Minkowskian space. A more interesting thing is that based on deformed space-time constraint algebra and the equations of perturbations, one can define a causal structure different from general relativity. Our work will based on these understandings to consider some LQC’s applications to inflation theory. Inflation theory tries to solve three puzzles of the standard cosmological model: horizon problem, flatness problem and the monopole problem. The idea to solve these problem of inflation theory is try to insert an almost exponential expansion epoch (before radiation dominated epoch), and if the expansion is enough (more than 60 e-foldings), and associated with some other natural processes, the three puzzles can be solved. It is natural to ask, for inflation model, whether so much expansion is possible. In chapter IV, we will consider this problem in the framework of loop quantum cosmology and use the new results in the perturbations of LQC, especially the problem of horizon. Another promising thing of inflation is it can give a natural explanation of large structure for Universe. In chapter V, we will consider the quantum correction of loop quantum cosmology to the power spectra of inflation. We find although the initial value proposed at the transition point is natural, the vacuum can not be well defined at that time. Another problem is the slow roll approximation also can not be used at the transition point. Thus we give up to give initial problem at the transition point but near after it. Fortunately, one can define the vacuum for large k-modes and slow roll approximations also can be used.. Based on this choice, we define the quantum vacuum of perturbation fields and calculate the power spectra of scalar perturbation and tensor perturbation and their indexes. We find the energy density of Universe when the k-mode considered is crossing horizon is much lower than Planck energy density, loop quantum cosmology gives almost same result as classical theory. One notable thing is the trans-Planckian problem of inflation which says although the energy density when the k-modes we observed cross horizon are much lower than Planck scale, their wave lengths are of the same order of Planck length at the begining time of inflation. The perturbation theory of loop quantum cosmology still can not solve this problem, which means although the closure of constraint algebra is satisfied, the theory is not the final framework of perturbation of loop quantum cosmology. To make a outlook, we will consider this problem and some other interesting problems in loop quantum cosmology. The structure of the first three chapters is the followings: In Chapter-I, we make a introduction to loop quantum gravity and loop quantum cosmology; Then in Chapter-II, we will consider the perturbation of loop quantum cosmology under holonomy corrections; In Chapter-III, we will analyse the characteristics of perturbation equations obtained in chapter two and try to get an effective causal structure which will be used in chapter IV and V.