Influence of spatial curvature in cosmological particle production

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

💡 Research Summary

The paper investigates how spatial curvature influences cosmological particle production during the early‑universe inflationary epoch. The authors consider a conformally coupled scalar field (ξ = 1/6) evolving on Friedmann‑Lemaître‑Robertson‑Walker (FLRW) backgrounds with three possible curvature signs: closed (κ > 0), flat (κ = 0) and open (κ < 0). They model inflation as an exact de Sitter phase with scale factor a(η)=−1/(Hη) and constant Ricci scalar R = 12 H², and they assume the field starts in the Bunch‑Davies vacuum.

The analysis proceeds by expanding the rescaled field χ = a φ in eigenfunctions of the spatial Laplace‑Beltrami operator. The eigenvalues λ²(k) differ for each curvature: λ² = \bar{k}(\bar{k}+2)|κ| for closed, λ² = k² for flat, and λ² = (\bar{k}²+1)|κ| for open geometries, where \bar{k}=k/√|κ|. Notably, the open case possesses a non‑zero minimum eigenvalue, acting as an effective mass gap for low‑k modes.

Time‑dependent mode functions v_k(η) satisfy a harmonic‑oscillator equation v_k’’+ω_k²(η)v_k=0 with ω_k²(η)=λ²(k)+a²(η)m²+(ξ−1/6)R(η). The authors treat two distinct ways of ending inflation:

1. Instantaneous exit – at a conformal time η₁ the scale factor abruptly becomes constant, mimicking a sudden transition to a static universe.
2. Smooth (adiabatic) exit – the scale factor is interpolated so that the adiabaticity parameter C_k(η)=|ω’_k/ω_k²| remains ≪ 1 throughout the transition, and the adiabatic vacuum is used as the out‑state.

For each scenario they match the “in” Bunch‑Davies modes to the “out” modes, compute the Bogoliubov coefficients α_k and β_k (with |α_k|²−|β_k|²=1), and obtain the particle number density per mode n_k=|β_k|²/(2π²). The total density is integrated over the curvature‑dependent momentum measure dμ_k, with appropriate regularisation for the non‑compact (κ≤0) cases (introducing a finite radial cutoff R and taking R→∞).

The results reveal a strong curvature dependence:

• Closed universe (κ>0) – low‑k modes experience a reduced effective frequency, leading to a large enhancement of |β_k|². For light fields (m≪H) the particle density can be amplified by two to four orders of magnitude relative to the flat case.
• Flat universe (κ=0) – reproduces the standard de Sitter particle‑production spectrum found in earlier literature.
• Open universe (κ<0) – the presence of a minimum λ² suppresses low‑k excitations, yielding a particle density significantly lower than the flat case; the suppression can also reach several orders of magnitude for light fields.

Comparing the two exit prescriptions, the smooth adiabatic transition reduces |β_k|² modestly compared with the instantaneous jump, reflecting the reduced non‑adiabaticity. Nevertheless, the qualitative curvature trends persist in both cases.

The authors discuss the implications for scenarios where such scalar particles constitute dark matter (e.g., “spectator” or “freeze‑in” production). Since curvature can alter the relic abundance by orders of magnitude, even the current observational bound on spatial curvature (|Ω_k| ≲ 10⁻³) may have non‑negligible impact on model predictions. Consequently, analyses that assume strict spatial flatness could misestimate the viable parameter space for ultra‑light dark‑matter candidates.

In conclusion, the paper provides a fully analytic treatment of curvature‑induced modifications to cosmological particle production, clarifies the role of the Laplace‑Beltrami spectrum, and highlights the necessity of incorporating spatial curvature when evaluating early‑universe particle‑creation mechanisms.