Topological defects and scalar field modes in cosmological backgrounds

TL;DR

This work develops a unified, higher-dimensional framework for quantum scalar fields in cosmological backgrounds that host angular deficit topological defects. By separating variables, it constructs a complete set of mode functions phi_sigma = T(t) W(r) Y(theta) for the Klein-Gordon equation with curvature coupling, with angular components built from associated Legendre functions and angular deficits entering the eigenvalues gamma. The approach covers global monopole and cosmic string geometries, and provides explicit radial solutions for maximally symmetric spaces and de Sitter/Milne expansions, including vacuum-state choices that determine the time-evolution and spectra. Overall, the results facilitate the computation of vacuum polarization and two-point functions in complex geometries, and offer concrete de Sitter and Milne examples to illustrate the impact of defects on quantum fields in expanding spacetimes.

Abstract

We study topological defects with a general structure in higher-dimensional cosmological backgrounds described by a set of angle deficit parameters. As special cases, they include higher-dimensional generalizations of cosmic strings and global monopoles. The corresponding complete set of mode functions is presented for a massive scalar field with a general curvature coupling parameter. For general scale factors and radial functions in the line element, the angular parts of the scalar modes are expressed in terms of associated Legendre functions. De Sitter and Milne universes are considered as examples of cosmological expansion. For the de Sitter bulk, we present the time-dependent parts of the mode functions in inflationary, hyperbolic, and global coordinates.