For the first time, we choose non-flat vacuum mode for background spacetime based on the minimum number of created particles during early non-de Sitter inflation. In conventional methods for calculating the number of created particles, the flat background is selected automatically and causes a negative number problem for created particles during asymptotic-de Sitter inflation. In a covariant approach to curved spacetime, both real and background spacetimes should be selected, curved and consequently the relation for particle creation should be modify. As an interesting finding from this research, flat space does not include minimum number of particles and there are some asymptotic de Sitter spacetimes with fewer number. Therefore, in the generalized formula for particle creation, we choose a non-flat background containing the minimum number of created particles.

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 in covariant approach will show that in order to solve the above second mentioned problem, the background space-time must be non-flat and the number of particles in the background must be minimum.

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