Impact of string and monopole-type junctions on domain wall dynamics: implications for dark energy

We investigate the potential role of string and monopole-type junctions in the frustration of domain wall networks using a velocity-dependent one-scale model for the characteristic velocity, $v$, and the characteristic length, $L$, of the network. We show that, except for very special network configurations, $v^2 \lsim (HL)^2 \lsim (\rho_\sigma + \rho_\mu)/\rho_m$ where $H$ is the Hubble parameter and $\rho_\sigma$, $\rho_\mu$ and $\rho_m$ are the average density of domain walls, strings and monopole-type junctions. We further show that if domain walls are to provide a significant contribution to the dark energy without generating exceedingly large CMB temperature fluctuations then, at the present time, the network must have a characteristic length $ L_0 \lsim 10 \Omega_{\sigma 0}^{-2/3} {\rm kpc}$ and a characteristic velocity $v_0 \lsim 10^{-5} \Omega_{\sigma 0}^{-2/3}$ where $\Omega_{\sigma 0}=\rho_{\sigma 0}/\rho_{c 0}$ and $\rho_c$ is the critical density. In order to satisfy these constraints with $\Omega_{\sigma 0} \sim 1$, $\rho_{m 0}$ would have to be at least 10 orders of magnitude larger than $\rho_{\sigma 0}$, which would be in complete disagreement with observations. This result provides very strong additional support for the conjecture that no natural frustration mechanism, which could lead to a significant contribution of domain walls to the dark energy budget, exists.