Modelling the dynamics of global monopoles

Abstract

A thin wall approximation is exploited to describe a global monopole coupled to gravity. The core is modelled by de Sitter space; its boundary by a thin wall with a constant energy density; its exterior by the asymptotic Schwarzschild solution with negative gravitational mass $M$ and solid angle deficit, $ΔΩ/4π= 8πGη^2$, where $η$ is the symmetry breaking scale. The deficit angle equals $4π$ when $η=1/\sqrt{8πG} \equiv M_p$. We find that: (1) if $η<M_p$, there exists a unique globally static non-singular solution with a well defined mass, $M_0<0$. $M_0$ provides a lower bound on $M$. If $M_0<M<0$, the solution oscillates. There are no inflating solutions in this symmetry breaking regime. (2) if $η\ge M_p$, non-singular solutions with an inflating core and an asymptotically cosmological exterior will exist for all $M<0$. (3) if $η$ is not too large, there exists a finite range of values of $M$ where a non-inflating monopole will also exist. These solutions appear to be metastable towards inflation. If $M$ is positive all solutions are singular. We provide a detailed description of the configuration space of the model for each point in the space of parameters, $(η, M)$ and trace the wall trajectories on both the interior and the exterior spacetimes. Our results support the proposal that topological defects can undergo inflation.