Influence of spatial curvature in cosmological particle production

TL;DR

This work analyzes cosmological particle production in FLRW spacetimes with nonzero spatial curvature for a conformally coupled scalar field, modeling inflation as an exact de Sitter phase followed by a transition to a static era. By constructing Bunch-Davies mode functions adapted to curved patches and employing Bogoliubov transformations, the authors obtain nonperturbative particle spectra under instantaneous and adiabatic exits from inflation. They find that spatial curvature can significantly modify the production spectra, enhancing production in open universes and suppressing it in closed ones, with the largest effects for ultra-light fields. Although the curvature-induced relic density remains subdominant for realistic parameters, the results reveal a strong sensitivity of gravitational particle production to global geometry and motivate further studies on isocurvature signatures and curved-inflation dynamics.

Abstract

We analyze cosmological particle production driven by spacetime expansion in the early universe for homogeneous and isotropic cosmologies with positive, negative, and zero spatial curvature. We prioritize analytical results to gain a deeper understanding of curvature-induced effects. Specifically, for a conformally coupled scalar field, we model the inflationary epoch as an exact de Sitter phase followed by a transition to a static universe. Both instantaneous and smooth exits from inflation are considered, the latter being implemented via the adiabatic vacuum prescription. Starting from an initial Bunch-Davies vacuum, we derive the associated mode functions carefully adapted to each curvature sign. Using the Bogoliubov formalism, we non-perturbatively compute the number density of produced scalar particles. Our results demonstrate that spatial curvature significantly impacts the resulting particle spectra, particularly for light fields, where the deviation from the flat-space scenario is most prominent and can reach several orders of magnitude

Figures (5)

• Figure 1: Time dependence of the scale factor for the three different spatial curvatures, assuming an instantaneous transition to a static geometry at $\eta_{\text{f}}$. The scale factor is normalized so that the value it takes at the end of inflation is $a(\eta_{\text{f}}) = 1$.
• Figure 2: Penrose diagrams for the different coordinate charts in de Sitter: (a) flat, (b) closed, and (c) open coordinates, in terms of cosmological time $t$. The shadowed region corresponds to the covering of the corresponding coordinates, which are denoted with the same symbol in each case.
• Figure 3: Abundance for today-non-relativistic particles by assuming flat quadratic inflation (blue), de Sitter with instantaneous vacuum (orange), and de Sitter with adiabatic vacuum (yellow). The mass of the spectator field is given in GeV. The dashed line denotes the observed dark matter abundance.
• Figure 4: Spectra of produced particles with different spatial curvature in de Sitter inflation using the adiabatic vacuum prescription, for a large value of the curvature abundance $\Omega_\kappa$ (corresponding to $|\kappa| = m_\phi$) and $m = 10^{-2}m_\phi$. Here, $\mathcal{N}_k = (k/m_\phi)^2|\beta_k|^2$ for $\kappa \leq 0$, whereas $\mathcal{N}_k = (k + \sqrt{|\kappa|})^2/m_{\phi}^2|\beta_k|^2$ for $\kappa > 0$ (see also Eq. (2.13)). The abundances $\Omega$ in the legend denote the associated relic abundance obtained by using Eq. (5.1) with $T_{\text{rh}} = 10^{15}$ GeV. Note that the spectrum for positively curved scenarios is discrete; in particular, the blue point at the far left of the spectrum corresponds to the $k=0$ mode. Positive and negative curvature lead to a smaller and larger abundance than in the flat case, respectively. The differences can be of several orders of magnitude.
• Figure 5: Spectra of produced particles with zero and negative spatial curvature in de Sitter inflation using the adiabatic vacuum prescription for $\lvert\kappa\rvert = 10^{-40}m_{\phi}$ and $m=10^{-25}m_{\phi}$. It shows that significant deviations in the spectra are obtained for fields in the ultra-light regime even considering curvature values compatible with current observations. Although the total abundance for such light fields represents a subleading contribution to the dark matter density, these results reveal a strong sensitivity of the production mechanism to the global geometry of the universe. The abundance is obtained using $T_{\text{rh}}=10^{15} \, \text{GeV}$. Note that for such light fields and small spatial curvature, the spectrum of produced particles in the case of positive curvature is negligible compared to the other two cases shown in the Figure.