永恒暴胀中的数学问题
Mathematical Issues in Eternal Inflation
摘要：
在本文中，我们考虑的问题的存在性和唯一性的爱因斯坦场方程的解的空间平坦FLRW宇宙的背景下，随机永恒的通货膨胀的随机机制是通过添加一个随机强迫项表示高斯白噪声的Klein-Gordon方程。我们表明，在这些考虑下，Klein-Gordon方程实际上成为一个随机微分方程。因此，爱因斯坦方程解的存在唯一性取决于该随机微分方程的系数是否满足Lipschitz连续性条件。我们表明，对于任何选择的$V(ϕ)$，爱因斯坦场方程不是全球适定的，因此，任何解决方案，这些方程是不保证是唯一的。相反，这些系数在动力学变量的物理状态空间中至多是局部Lipschitz连续的，而动力学变量只存在于有限的爆炸时间内。我们进一步执行Feller的爆炸测试的任意幂律暴胀子潜力，并证明所有的解决方案，EFE爆炸在有限时间内的概率为1。这意味着，这样考虑的随机暴胀机制不能被描述为永恒的，因为永恒暴胀的概念本身就意味着这个过程无限地继续下去。因此，我们认为，基于随机强迫项的随机暴胀不会在某些多重宇宙系综中产生无限多个宇宙。总的来说，由于爱因斯坦场方程在这两种情况下都不是适定的，我们进一步得出结论，通过本文考虑的随机永恒膨胀机制的多重宇宙的存在仍然是一个悬而未决的问题，需要更深入的研究。
英文摘要：
In this paper, we consider the problem of existence and uniqueness of solutions to the Einstein field equations for a spatially flat FLRW universe in the context of stochastic eternal inflation where the stochastic mechanism is modelled by adding a stochastic forcing term representing Gaussian white noise to the Klein-Gordon equation. We show that under these considerations, the Klein-Gordon equation actually becomes a stochastic differential equation. Therefore, the existence and uniqueness of solutions to Einstein's equations depend on whether the coefficients of this stochastic differential equation obey Lipschitz continuity conditions. We show that for any choice of $V(ϕ)$, the Einstein field equations are not globally well-posed, hence, any solution found to these equations is not guaranteed to be unique. Instead, the coefficients are at best locally Lipschitz continuous in the physical state space of the dynamical variables, which only exist up to a finite explosion time. We further perform Feller's explosion test for an arbitrary power-law inflaton potential and prove that all solutions to the EFE explode in a finite time with probability one. This implies that the mechanism of stochastic inflation thus considered cannot be described to be eternal, since the very concept of eternal inflation implies that the process continues indefinitely. We therefore argue that stochastic inflation based on a stochastic forcing term would not produce an infinite number of universes in some multiverse ensemble. In general, since the Einstein field equations in both situations are not well-posed, we further conclude that the existence of a multiverse via the stochastic eternal inflation mechanism considered in this paper is still very much an open question that will require much deeper investigation.
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