Abstract
During the inflationary epoch,geometry of the universe may be described by quasi-de Sitter space. On the other hand,maximally extended de Sitter metric in the comoving coordinates accords with a special FLRW model with positive spatial curvature,so in this article we focus on the positively curved inflationary paradigm.For this purpose,first we derive the power spectra of comoving curvature perturbation and primordial gravitational waves in a positively curved FLRW universe according to the slowly rolling inflationary senario. It can be shown that the curvature spectral index in this model automatically has a small negative running parameter which is compatible with observational measurements.Then,by taking into account the curvature factor,we investigate the relative amplitude of the scalar and tensor perturbations.It would be clarified that the tensor-scalar ratio for this model against the spatially flat one,depends on the waelength of the perturbative models directly.
Highlights
Rqo ∝ q − −2 −δ, (1)where Rq is the Fourier component of a comoving curvature perturbation with comoving wave number q
Where Rq is the Fourier component of a comoving curvature perturbation with comoving wave number q
According to the standard slowrolling inflationary theory rq = 16 (q stands for the comoving wave number of perturbations in the spatially flat universe) [13], so if it is supposed that = 0.008 [11], r = 0.128, which is greater than the data result BICEP2/Keck Array and Planck released (r0.05 < 0.12) [25] whereby the question is brought up of whether it is possible to eliminate this flaw by considering a curvature factor
Summary
Where Rq is the Fourier component of a comoving curvature perturbation with comoving wave number q (the superscript “o” stands for “outside the Hubble horizon”). Massó et al [46] investigated the imprint of spatial curvature on the scalar power spectrum in a non-flat inflationary universe with an exact de Sitter space as the background. They evaluated the power spectrum at the horizon exit instant and discussed the effect of curvature on the angular power spectrum of CMB. The present paper investigates the slow-rolling inflationary scenario in a spatially closed background with trivial topology, namely a positively curved FLRW universe, for which the curvature power spectrum has been obtained in an inflationary universe with positive spatial curvature by imposing appropriate initial conditions.
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