Grant-in Aid for Scientific Research c (2)

1)对于爱因斯坦方程的长波展开，我们得到了描述长波涨落的非线性动力学的重整化群方程。重整化群方程的解表明，重整化度规很好地定性地描述了膨胀宇宙中引力坍缩的行为。在现代宇宙学中，广义相对论在解释我们的宇宙的起源和演化方面起着至关重要的作用。然而，它的基本方程--爱因斯坦方程是非线性偏微分方程组，一般很难求其解。我们必须用某种近似方法或数值方法来研究广义相对论的问题。本研究的目的是构造基于重整化群方法的爱因斯坦方程的非线性逼近方法。2)利用重整化群方法研究了宇宙微扰对宇宙演化的反作用。从二阶微扰爱因斯坦方程出发，对宇宙的标度因子进行了重整化，导出了包含由于宇宙不均匀引起的反反应效应的有效标度因子的演化方程。

1 ) For long wavelength expansion of Einstein's equation, we obtained the renormalization group equation that describes non-linear dynamics of the long wave fluctuation. The solution of the renormalization group equation shows that the renormalized metric describes the behavior of the gravitational collapse in the expanding universe qualitatively well.In modern cosmology, general relativity plays an essential role in explaining the origin and the evolution of our universe. However its basic equation, Einstein's equation, is non-liner partial differential equation and it is difficult to obtain its solution in general. We must use some approximation method or numeral method to investigate problems in general relativity. The purpose of this research is to construct non-linear approximation method for Einstein's equation based on the renormalization group method. The results of this research are2)We investigate the back reaction of cosmological perturbation on the evolution of the Universe using the renormalization group method. Staring from the second order perturbed Einstein's equation, we renormalize a scale factor of the Universe and derive the evolution equation for the effective scale factor that includes back reaction effect due to inhomogeneities of the Universe.