Inflationary stochastic anomalies
Inflationary stochastic anomalies
复制标题
DOI：
10.1088/1361-6382/ab097f
复制
发表时间：
2018-06
影响因子：
3.5
通讯作者：
L. Pinol;S. Renaux-Petel;Y. Tada
中科院分区：
文献类型：
--
作者：
L. Pinol;S. Renaux-Petel;Y. Tada
随机方法旨在通过经典随机理论描述暴胀期间量子场的长波部分。它通常用朗之万方程来表述，从而得出场（可能还有它们的动量）的概率分布函数的福克 - 普朗克方程。一般来说，这两种描述之间的联系是模糊的，因为它取决于一种隐含的离散化过程，其中两种主要的是伊藤和斯特拉托诺维奇规定。在这里我们表明，只有在后一种情况下场重新定义下的广义协变性要求才能得到验证，然而这是以引入虚假的“框架”依赖性为代价的。当只有一个随机性来源时，比如在慢滚单场暴胀中，这种随机反常现象会消失，但在考虑整个相空间或存在多个场时就会显现出来。尽管存在这些困难，我们利用物理论据写下了一个协变的福克 - 普朗克方程，它描述了在过阻尼极限下非线性西格玛模型中轻标量场的扩散。我们将其应用于德西特空间中的测试标量场，并表明一类具有导数相互作用的双场模型的一些统计性质可以通过与具有有效势的单场模型的对应来重现。我们还在单场理论到双曲场空间几何的简单扩展中给出了明确的结果。我们所描述的困难似乎是在量子理论中维持广义协变性这一众所周知的难题以及相关的算符排序和路径积分构造选择的随机对应物。因此，我们的工作在宇宙学、统计物理学和量子场论的交叉领域开辟了新的研究途径。
The stochastic approach aims at describing the long-wavelength part of quantum fields during inflation by a classical stochastic theory. It is usually formulated in terms of Langevin equations, giving rise to a Fokker–Planck equation for the probability distribution function of the fields, and possibly their momenta. The link between these two descriptions is ambiguous in general, as it depends on an implicit discretisation procedure, the two prominent ones being the Itô and Stratonovich prescriptions. Here we show that the requirement of general covariance under field redefinitions is verified only in the latter case, however at the expense of introducing spurious ‘frame’ dependences. This stochastic anomaly disappears when there is only one source of stochasticity, like in slow-roll single-field inflation, but manifests itself when taking into account the full phase space, or in the presence of multiple fields. Despite these difficulties, we use physical arguments to write down a covariant Fokker–Planck equation that describes the diffusion of light scalar fields in non-linear sigma models in the overdamped limit. We apply it to test scalar fields in de Sitter space and show that some statistical properties of a class of two-field models with derivative interactions can be reproduced by using a correspondence with a single-field model endowed with an effective potential. We also present explicit results in a simple extension of the single-field theory to a hyperbolic field space geometry. The difficulties we describe seem to be the stochastic counterparts of the notoriously difficult problem of maintaining general covariance in quantum theories, and the related choices of operator ordering and path-integral constructions. Our work thus opens new avenues of research at the crossroad between cosmology, statistical physics, and quantum field theory.