Krein approach is used to study the particle creation during quasi-de Sitter inflation in different background space-times. In the conventional method for calculating the created particles spectrum, the background space-time is automatically considered flat. Selecting a flat background poses two fundamental problems: First, the method of calculating is not covariant relative to the curved space-time. Second, the number of created particles becomes negative. Krein approach can be considered as a covariant method to calculate two-point functions in curved space-time. So we extend this method for particle creation during early cosmic inflation. As a new proposal, we choose the background vacuum based on the smallest number of created particles in that space-time. The calculations will show that in order to solve the above mentioned problems, the background space-time must be non-flat and the number of particles in the background must be minimum.

中文翻译：

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非平坦背景下粒子创建的协变方法

Kerin 方法用于研究不同背景时空下准德西特暴胀过程中的粒子产生。在计算所产生粒子谱的传统方法中，背景时空被自动视为平坦的。选择平坦的背景会带来两个基本问题：首先，计算方法相对于弯曲的时空不是协变的。其次，创建的粒子数变为负数。Kerin 方法可以被认为是一种协变方法来计算弯曲时空的两点函数。因此，我们将这种方法扩展到早期宇宙暴胀期间的粒子创建。作为一项新提议，我们根据该时空内产生的最小粒子数来选择背景真空。计算表明，为了解决上述问题，