Closed Universes from Cosmological Instantons

TL;DR

This work addresses the viability of closed inflationary universes by formulating the Euclidean path integral about constrained singular instantons and computing the resulting scalar and tensor perturbation spectra. It develops suitable gauge-invariant variables for closed geometries, demonstrates how non-singular instantons harbor multiple negative modes, and imposes Kirklin-type boundary conditions to project them out for singular instantons. The authors compute the full power spectra, perform analytic continuation to predict CMB anisotropies using a curved-space extension of CAMB, and show that the resulting spectra closely resemble flat slow-roll predictions on small scales with controlled deviations on the largest scales due to curvature. This provides a practical framework for testing closed-universe scenarios against CMB data, especially in regimes where curvature effects are most pronounced.

Abstract

Current observational data is consistent with the universe being slightly closed. We investigate families of singular and non-singular closed instantons that could describe the beginning of a closed inflationary universe. We calculate the scalar and tensor perturbations that would be generated from singular instantons and compute the corresponding CMB power spectrum in a universe with cosmological parameters like our own. We investigate spatially homogeneous modes of the instantons, finding unstable modes which render the instantons sub-dominant contributions in the path integral. We show that a suitable condition may be imposed on singular closed instantons, constraining their instabilities. With this constraint these instantons can provide a suitable model of the early universe, and predict CMB power spectra in close agreement with the predictions of slow-roll inflation.

Figures (6)

• Figure 1: This figure illustrates the continuation from singular (left) and non-singular (right) instantons into Lorentzian universes. Analytic continuation at the equator yields a closed universe. In the non-singular instanton continuation at the poles gives open universes.
• Figure 2: The scale factor (solid line) and $z^2$ (dashed line) versus Euclidean conformal time for a closed singular instanton with an ${\frac{1}{2}} m^2\phi^2$ potential. The closed universe is obtained by analytic continuation on the central hypersurface, and the singularities are at the two end points.
• Figure 3: The evolution of the $n=3$ mode, ${\cal X}(t)/\sqrt{2 p_0 {\cal X}'_0 {\cal X}_0}$. The left hand side shows the Euclidean conformal time evolution of the classical solution. The RHS shows the real time evolution of the real (dotted line) and imaginary (dashed line) parts until the modes are driven to a constant when they are well outside the horizon.
• Figure 4: Evolution of the $n=10$ mode, axes as for Fig. \ref{['fig:ant2']}
• Figure 5: The scalar power spectrum (solid line) ${\cal P}_{\cal X}(n)$ and tensor spectrum (dashed line) ${\cal P}_h(n)$ for an ${\frac{1}{2}} m^2\phi^2$ potential.