Evolution and simulation of spacetime structure in quantum inflationary universe

作为将量子效应引入时空结构的准备，我们研究了实现非均匀宇宙模型的梯度展开方法。该方法利用关于空间导数阶数的展开来构造爱因斯坦方程的摄动解。我们将重整化群方法应用于Friedmann-Robertson-Walker宇宙的宇宙学微扰，成功地重构了梯度展开的最低阶解。作为这一结果的直接应用，我们发现只有非绝热涨落模才能反向反应Friedmann-Robertson-Walker宇宙并修正其膨胀定律。然后，我们利用最低阶梯度展开式，数值研究了膨胀后预热阶段的动力学。在这一阶段，度规涨落被膨胀子场的相干振荡引起的参数共振放大。对于具有两个标量场的模型，我们得到了涨落功率谱的演化。由于所考虑的模型本质上是混沌系统，随着系统的演化，场的空间相关性丢失，空间模式演化为白噪声的空间模式。我们发现度规微扰功率谱的演化依赖于起伏的初始功率指数，非线性效应对小尺度结构的影响很大，而波长大于视界尺度的大尺度结构不受参数共振引起的反反应效应的影响。

As a preparation to incorporate the quantum effect into spacetime structure, we investigated the gradient expansion approach which is a method to realize a model of inhomogeneous universes. This approach constructs the perturbative solution of the Einstein equation using expansion about the order of the spatial derivative. We succeeded to re-construct the lowest order solution of the gradient expansion by applying the renormalization group method to the cosmological perturbation about a Friedmann-Robertson-Walker universe. As a direct application of this result, we found that only non-adiabatic mode of fluctuation can back react a Friedmann-Robertson-Walker universe and modify its expansion law.Then using the lowest order gradient expansion, we investigated the dynamics of preheating stage after inflation numerically. In this stage, the metric fluctuation is amplified by the parametric resonance caused by the coherent oscillation of the inflaton field. For a model with two scalar fields, we obtained the evolution of the power spectrum of fluctuations. As the considering model is essentially chaotic system, the spatial correlation of the fields is lost as the system evolves and the spatial pattern evolves to that of the White noise. We found that the evolution of the power spectrum of metric perturbation depends on the initial power index of fluctuation and non-linear effect strongly affects the small scale structure while the large scale structure of which wave length is larger that the horizon scale is not affected by the back reaction effect caused by the parametric resonance.