Title:CMB Constraints on Quantized Spatial Curvature $Ω_K$ in globally CPT-symmetric universes

Abstract:The periodic solution of the Friedmann equation in conformal time, implies that only cosmological perturbations exhibiting corresponding symmetries are physically permissible, leading to a discrete spectrum of allowed wave vectors. Furthermore, in a spatially closed universe, these wave vectors are independently constrained to be integers. Matching these two distinct quantization conditions provides a novel theoretical constraint on the possible values of spatial curvature. In this work, we numerically solve the cosmological perturbation equations, incorporating radiation anisotropy and higher-order Boltzmann terms, to calculate these discrete wave vectors with improved precision. Subsequently, we generate Cosmic Microwave Background (CMB) power spectra for different characteristic spacings of these quantized wave vectors. Finally, we apply the constraint to Planck 2018 observational data to determine the cosmological parameters. This analysis yields a discrete set of allowed values for the spatial curvature, $\Omega_K$, including $[-0.076,-0.039, -0.024, -0.016, -0.012, \dots]$.